Controlling a contactless planar actuator with manipulatoroutperform the old one. Magnetically levitated planar actuators with moving magnets [62, 51, 114, 108, 26, 95, 56, 76, 13]

Controlling a contactless planar actuator with manipulator

Citation for published version (APA):Gajdusek, M. (2010). Controlling a contactless planar actuator with manipulator. Technische UniversiteitEindhoven. https://doi.org/10.6100/IR673132

DOI:10.6100/IR673132

Document status and date:Published: 01/01/2010

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https://doi.org/10.6100/IR673132

https://doi.org/10.6100/IR673132

https://research.tue.nl/en/publications/controlling-a-contactless-planar-actuator-with-manipulator(a87c4819-4d89-4ed7-bbbc-fcac8076da41).html

Controlling a Contactless Planar Actuator

with Manipulator

Controlling a Contactless Planar Actuator

with Manipulator

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor Promoties in het openbaar te verdedigen

op dinsdag 11 mei 2010 om 16.00 uur

door

Michal Gajdušek

geboren te Frýdek-Místek, Tsjechië

Dit proefschrift is goedgekeurd door de promotoren: prof.dr.ir. P.P.J. van den Bosch en prof.dr. E.A. Lomonova MSc Copromotor: dr.ir. A.A.H. Damen

This work is part of the IOP-EMVT program (Innovatiegerichte onderzoeksprogramma’s - Elektromagnetische vermogenstechniek). This program is funded by SenterNovem, an agency of the Dutch Ministry of Economic Affairs.

This dissertation has been completed in fulfillment of the requirements of the Dutch Institute of Systems and Control DISC.

Copyright © 2010 by Michal Gajdušek A catalogue record is available from the Eindhoven University of Technology Library. Controlling a contactless planar actuator with manipulator / by Michal Gajdušek. – Eindhoven: Eindhoven University of Technology, 2010. Proefschrift. - ISBN: 978-90-386-2228-6

NUR 959 Printed by Ipskamp Drukkers B.V., Enschede, The Netherlands

to my parents

vii

Contents

Chapter 1

Introduction 1

1.1 Background 2

1.2 Contactless Planar Actuator with Manipulator 3

1.2.1 Project Description 3

1.2.2 Initial Experimental Setup 3

1.3 Thesis Goal 6

1.4 Organization of the Thesis 7

Chapter 2

Commutation 9

2.1 Introduction 9

2.2 Norm-Based Commutation Algorithms 13

2.2.1 l2-norm Based Commutation (L2C) 13

2.2.2 l∞-norm Based Commutation (LiC) 14

2.2.3 Clipped l2-norm Based Commutation (CL2C) 16

2.2.4 Example on a Model of the COPAM 20

2.2.5 Conclusions 26

2.3 Full Yaw Rotation of Planar Actuator 27

2.3.1 Full Yaw Rotation Investigation 28

viii Contents

2.3.2 Concept Design for Full Rotation 31

2.3.3 Conclusions 37

2.4 Commutation Techniques and Full Rotation on the Model of COPAM 37

2.4.1 Maximum Current 39

2.4.2 Maximum Voltage 44

2.4.3 Dissipated Power 47

2.4.4 Error Sensitivity 51

2.4.5 Lower Bound of the Maximum Acceleration and Velocity 55

2.5 Summary and Conclusions 60

Chapter 3

Errors in Commutation 63

3.1 Introduction 63

3.2 Software-related Errors 64

3.2.1 Numerical Precision of the Modeled Coupling 64

3.2.2 Look-Up-Table-Based Description of the Coupling 64

3.3 Hardware-related Errors 67

3.3.1 Coil Position 67

3.3.2 Coil Dimensions 70

3.3.3 Magnet Array Bending 70

3.3.4 Magnets 72

3.3.5 Measurement System 73

3.3.6 Current amplifiers 74

3.4 Comparison and Summary 74

3.5 Auto-Alignment of the Measurement System 75

3.5.1 Consequence of Offsets in the Measurement System 76

3.5.2 Auto-Alignment Procedure 78

3.5.3 Investigation of Stability of the Auto-Alignment Procedure 79

3.5.4 Measurements on the Experimental Setup 82

3.5.5 Discussion and Extension 83

3.6 Conclusions 85

Chapter 4

Control 87

4.1 Introduction 87

4.2 Model of the Planar Actuator 88

ix

4.3 Trajectory Generation and Feedforward Design 96

4.4 Control of the Planar Actuator 100

4.4.1 PID Control 100

4.4.2 Sliding-Mode Control 105

4.4.3 Iterative Learning Control 113

4.5 Control of Planar Actuator with Manipulator 119

4.5.1 Multi-body Model 120

4.5.2 Simulations 122


Chapter 5

Wireless Communication 129

5.1 Introduction 129

5.2 Desired Specifications of the Wireless Link 129

5.3 Overview of Standardized Wireless Solutions 131

5.4 Infrared Wireless Link for Real-Time Motion Control 133

5.4.1 Data Processing 133

5.4.2 Custom Communication Protocol 134

5.5 IR Wireless Link Performance 135

5.6 Conclusions 141

Chapter 6

Experiments 143

6.1 Introduction 143

6.2 Improvements of the Experimental Setup 144

6.2.1 Measurement System 144

6.2.2 Contactless Energy Transfer 147

6.2.3 Wireless Control 150

6.3 Identification 151

6.4 Control 157

6.4.1 Control of the Planar Actuator 157

6.4.2 Control of the Manipulator 165

6.4.3 Control of the Planar Actuator with Manipulator 165

6.4.4 Clipped l2-norm based commutation 172

6.5 Conclusions 173

x Contents

Chapter 7

Conclusions and Recommendations 177

7.1 Commutation 177

7.2 Errors in Commutation 179

7.3 Control 179

7.4 Wireless Communication 180

7.5 Experimental Verification 181

7.6 Thesis Contributions 183

7.7 Recommendations for Future Development 184

Appendix A

Dimensions of the Experimental Setup 187

Appendix B

List of Symbols and Abbreviations 191

Appendix C

Additional Simulation Data 195

Appendix D

(3+2)-DOF Experimental Setup 199

D.1 Introduction 199

D.2 Experimental setup 200

D.3 Model 203

D.4 Identification 206

D.5 Control 211

D.6 Summary and Conclusions 211

Bibliography 213

Summary 223

Samenvatting 225

Acknowledgements 227

Curriculum Vitae 229

1

Chapter 1

Introduction

If, in a hundred years’ time, somebody looks into a book about technological

development of the end of the 20th and the beginning of the 21st century, he or she will definitely see adjectives like smaller, cheaper, faster or more powerful. In order to manufacture devices with these, ever increasing, requirements, the processing machines must have higher precision and performance. Sometimes, the improvement of the machine performance hits its physical or economical limits and a new approach must be found. The new-principle machine is useless without proper control, which helps the new device to outperform the old one. Magnetically levitated planar actuators with moving magnets [62, 51, 114, 108, 26, 95, 56, 76, 13] are an example of such a technological improvement. These machines use a new approach to achieve high-precision planar movement: magnetic levitation without cable connection to the translator assures movement without friction, which otherwise degrades the device precision. Moreover, the planar movement provided by a single 6-degree-of-freedom (DOF) actuator reduces the weight of the moving mass of the machine. However, without proper control these planar actuators are unstable, as was proven by Samuel Earnshaw in 1842 [35]. A novel control strategy for the decoupling of the forces and torques acting on the planar actuator was developed recently [95, 76] and extended in this thesis.

This thesis describes the application of this novel control strategy for a novel planar actuator that combines three contactless technologies: 6-DOF electromagnetic suspension and propulsion, contactless energy transfer and wireless motion control.

2 Chapter 1. Introduction

1.1 Background

High-precision positioning machines with emphasis on planar movement are required in many industrial devices, e.g. for wafer scanners in semiconductor lithography, for pick-and-place machines in assembly lines, or for inspection systems.

These machines usually consist of stacked long- and short-stroke linear and rotary actuators, which are supported by ball or air bearings. Stacking of the actuators leads to high moving mass or reduced stiffness. In extreme situations, the moving mass of the machine can be even a thousand times heavier than the mass of the moved object. On the other hand, ball bearing causes friction and neither ball nor air bearing can be used in high vacuum. Moreover, a cable slab for power and communication cables, and possibly cooling liquid connected to the moving part causes disturbances such as vibrations and friction, and acts as an additional load.

The idea of combining several 1-DOF actuators into one magnetically levitated multi-DOF planar actuator has been studied for some time [62, 51, 24, 10, 114, 25, 65, 108, 26, 95, 56, 76]. Such a solution reduces the moving mass of the machine while the effective force can remain the same. Moreover, the magnetic bearing is frictionless and can be used in vacuum. Although mostly only a 2-DOF planar motion is desired, these actuators need to be controlled in all six DOF’s to obtain a stable active magnetic bearing (stabilizing the three rotations and the levitation). Two main principles for long-stroke horizontal movement have been studied: 1) moving coils with stationary magnets [25] and 2) moving magnets with stationary coils [51, 26, 56, 76].

The first principle needs many more magnets than coils for long-stroke planar movement. This has benefits in a lower number of amplifiers or switches necessary for powering the coils and a simpler, analytical commutation algorithm (dq0-transformation [31-33, 91, 92] can be applied [25]). However, these planar actuators require power cables and cooling tubes to the moving coils.

On the other hand, moving-magnet planar actuators are really contactless actuators with no cable connection to the translator. Since these actuators are more difficult to control [95, 76], because dq0-transformation cannot be applied [76], only a few prototypes have been developed in recent years [26, 56, 76].

It often happens that for an application some sensors, electronics or an additional manipulator (e.g. micromanipulator) need to be placed on the translator. Then the problem with the cables between the stator and translator appears again. The solution is a magnetically levitated planar actuator with built-in contactless energy transfer and wireless communication. Although a combination of contactless energy transfer with wireless communication has been studied in the past, a combination with contactless actuator has not been studied yet [13].


1.2 Contactless Planar Actuator with Manipulator

1.2.1 Project Description

The project concerning the Contactless Planar Actuator with Manipulator (COPAM) consists of two parts. In the first part, described in [13], the experimental setup was designed and built, and simple control was applied. The setup combines three contactless technologies:

6-DOF electromagnetic suspension and propulsion Contactless energy transfer Wireless motion control

The electromagnetic suspension and propulsion were realized in a planar actuator with moving magnets. The contactless energy transfer is for powering a manipulator on top of the translator, which is wirelessly controlled to keep the whole system without any cable connection between the stator and translator.

As the control designed in the first part of this project was only used for verification of the principle of the system, the second part of the project, this thesis, focuses on the control of the prototype.

1.2.2 Initial Experimental Setup

The planar actuator uses an array of permanent magnets in the translator and an array of coils in the stator. The dimensions of the magnets and coils have been optimized to utilize the same coils also for the contactless energy transfer. The coils have two functions. When the coil is in the magnetic field of the magnet array, it is connected to the low-frequency current power amplifier and the force between the coil and the translator is generated. On the other hand, when the coil is under the moving secondary coil (out of the magnet-array magnetic field) it is driven by a high-frequency amplifier and the coil is used as the primary coil for the contactless energy transfer. Due to high frequencies, the coils have to be made of litz wire. Litz-wire coil suffers from a lower filling factor compared to the standard solid-wire coil. The lower number of turns leads to higher current needed to produce the same force and, consequently, more expensive amplifiers to be used. In order to optimize the costs of the setup, only the coils that are really used for the contactless energy transfer are made of litz wire (see Appendix A, Figure A.1). The other (solid wire) coils differ only in the number of the turns (and consequently the impedance).

The relation between the current applied to one particular coil and the produced force and torque vector acting on the translator is called coupling (also known as force constant). This coupling is dependent on the actual position of the coil with respect to the magnet array. The calculation of the coupling developed in [13] together with utilization of the rounded coils allow for the control of the translator in any orientation about the vertical z-axis. Hence, apart from the long-stroke translational xy-movement, the setup was


theoretically capable of full yaw rotation. Therefore, the measurement system for the planar actuator was designed with the goal of exhibiting the rotational capability of the setup. Three eddy-current sensors and three laser triangulation sensors were used. The eddy-current sensors have limited range of 2 mm and they were used for measuring the vertical z-position and two titling angles ( ,q y) of the planar actuator. The laser triangulation sensors were used for measuring the x- and y-position and (yaw) f-orientation (see Figure 1.1). The laser triangulation sensors were chosen for their ability to measure position on a surface tilted by several degrees. With a tilt angle of 5º, the error is less than 0.12 % of the measuring range of the laser triangulation sensors.

Figure 1.1: Drawing of the experimental setup

The manipulator on top of the translator has two three-phase ironless linear actuators connected with a beam (like a small H-bridge). In the center of the beam, a three-phase rotary actuator was assembled with an arm attached to it. The tip of this arm can be positioned anywhere in the xy-plane between the two horizontal linear legs by combining the translation of the beam and the rotary movement of the arm. Because of its shape, the manipulator is sometimes called an H-drive. Next to the manipulator, there are additional electronics on the translator: a contactless energy transfer unit, three power amplifiers for the actuators, a wireless communication module and a module with a Field-Programmable Gate Array (FPGA) (see Figure 1.2). The FPGA communicates with the wireless link and computes the commutation for the manipulator’s actuators based on the position readings from the encoders. Via the wireless link, the current setpoints are received from the controller and actual position and orientation of the manipulator is transmitted back. With the controllers for the planar actuator and the manipulator in one place (in the fixed world), feedforward disturbance compensation can be easily realized. The wireless communication link, developed for the setup, is based on 2.4 GHz radio-frequency (RF) modules with custom packet and data processing [13].

x,θ y,ψ

z,f

Eddy-current sensors

Laser-triangulation sensors

Secondary coil

Manipulator

FPGA and wireless communication unit

Movingmagnet array

Stationary coil array


Figure 1.2: Photo of the experimental setup

The manipulator was not designed to achieve the highest accuracy, but to show the principle of the wirelessly powered and controlled machine on top of the magnetically levitated platform. Therefore, the resolution of the linear and rotary actuator is several times worse than the resolution of the PA measurement system based on the eddy-current sensors. The dimensions and properties of experimental setup are listed in Appendix A.

Since the prototype is a complex system, which for the first time integrates three contactless technologies into one multi-level system, a few problems arose during the building and testing of the setup.

The first item to be solved is in the custom-made RF wireless communication, which has quite high packet-loss average ratio of about 10-2. The ratio is increased even more when the contactless energy transfer is active.

The second issue is with the electromagnetic interference produced by the contactless energy transfer unit. The interference is mostly visible in the increased packet-loss ratio of the wireless link and in the increased noise levels of the eddy-current sensors.

The third problem is caused by the laser-triangulation sensors in the measurement system. The problem is the sensitivity of the sensors to the perpendicular movement of the measured targets. The sensors measure distance by capturing a laser beam reflected on the measured surface. From the principle of these sensors, not the directly reflected but diffused light is captured by the sensor. Therefore, the sensor is sensitive to the roughness of the surface. On one hand, the surface must be rough (otherwise no light is diffused and captured by the sensor and hence no mirror surface can be used). On the other hand, variation in roughness causes uncertainty of the measurement. The problem is not in the actual position error, which is within the linearity specifications of those sensors, but in the fast variation of the error. Even a 10 μm step in the perpendicular direction could cause a measurement error of about 20 μm in the measured direction. Therefore, these sensors are causing vibrations of the levitated and multi-DOF controlled planar actuator. To remove

Secondary coil

CET electronics

Motor amplifiers

Manipulator

FPGA and wireless communication unit


the vibrations, two laser triangulation sensors were temporarily replaced by two short-stroke eddy-current sensors. As a result, a long-stroke translational movement of the PA is only possible in one direction [13].

1.3 Thesis Goal

The goal of this thesis is to systematically explore the opportunities for improving the performance of the existing contactless planar actuator with the manipulator (COPAM) by developing new control algorithms, a new wireless communication strategy, and by proposing some new electromechanical hardware alternatives.

The control of the system can be split into two parts. In the first part, the multi-DOF system will be decoupled. For the planar actuator consisting of many active coils this means the development of a proper commutation method, which decouples the applied forces and torques. In the second part, once the system has been decoupled, the actual controllers can be designed for the separate DOF’s.

As several problems were discovered during building and operation of the final setup, which could not be solved within the time given for the first part of the project, improvement of the setup is also considered in this thesis. In particular, the measurement system, which caused vibrations in the setup, has to be improved. The wireless RF communication link with the high packet-loss ratio has also to be improved or replaced to guarantee reliable control of the manipulator.

The thesis objectives are summarized in the following recommendations:

Commutation: Research new commutation techniques for the decoupling of the forces and torques applied on the planar actuator that can help to achieve higher performance of the system. Since this planar actuator is theoretically capable of full yaw rotation, research on this feature can be performed.

Errors in commutation: Research the bottlenecks that limit the accuracy of the forces and torques decoupling, as the precision of the decoupling is dependent on the accuracy of the commutation.

Control: Research and develop high-performance control strategies suitable for the multi-DOF planar actuators with moving magnets, which exceeds the performance of present solutions.

Wireless communication: Design and implement a reliable, high-performance wireless communication strategy for the final prototype.

Experimental verification: Design and build a suitable pre-prototype of the COPAM on which the control strategies for the final prototype can be tested. Implement the derived control software, wireless communication and hardware improvements on the final prototype to validate the obtained performance improvements.

1.4 Organization of the Thesis 7

1.4 Organization of the Thesis

Chapter 2 is devoted to commutation for ironless, magnetically levitated planar actuators. Here, development of novel commutation methods and their comparison with the up-to-date commutation method [76] is presented. Since full rotation of the planar actuator is, from the control point of view, related to the commutation, it is also included in this chapter.

Comparison and quantification of different kinds of causes that degrade the performance of the commutation is presented in Chapter 3. In [13], an interesting procedure for estimating and removing the offsets in the measurement system of the COPAM was presented. Proof of the correctness of the procedure is given in this chapter. Moreover, an extension to a larger class of planar actuators with moving magnets is shown.

Control of the multi-DOF system is discussed in Chapter 4. At first, a 6-DOF model of the planar actuator with the position-dependent disturbances and cross-coupling based on the available data is derived. Then, several control techniques are described and applied on the model of the planar actuator. The results from the simulations are compared. Later on, the model is extended by the manipulator and its disturbing movement. The simulation results with the different controllers are again compared.

Wireless communication for real-time control of a fast motion system is discussed in Chapter 5. Commercially available wireless solutions are compared and a novel infrared-light based communication link with excellent packet-loss ratio and radically reduced delay is presented.

Chapter 6 is devoted to the experimental verification. As the experiments on the pre-prototype are described in Appendix D, this chapter focuses on the final prototype. Here, the improvements to the hardware are presented as well as the results from identification. The controllers presented in Chapter 4 are applied and the results are shown and compared.

Conclusions and recommendations for the future work are given in Chapter 7.


9

Chapter 2

Commutation

2.1 Introduction

This chapter focuses on commutation algorithms for a class of ironless planar actuators (PA) with moving magnets that can be described by a linear relation between a

current j

i applied to the jth coil in the stator and a wrench vector j

w produced by that coil and acting on the center of mass of a moving magnet array (e.g. [86, 95, 56, 76, 13]):

( )j j j

w K q i= (2.1)

where the coupling K is an 1m´ vector mapping the current to the wrench vector and dim( )m w= . The coupling vector K is dependent on location and orientation of the coil

with respect to the magnet array, T[ , , , , , ]j j j j j j j

q x y z= q y f . The wrench vector w is the composition of forces and torques: T[ , , , , , ]

x y z x y zw F F F T T T= acting on the center of mass of

the magnet array. Consequently, the coupling vector consists of six components ( 6m = ): T[ , , , , , ]

x y z x y zF F F T T TK K K K K K K= . The relation (2.1) can be applied on ironless actuators

assuming that they do not suffer from cogging force and reluctance effects due to the permanent magnets can be neglected [56]. Therefore, relation (2.1) is linear and it is dictated only by the Lorentz force acting on a piece of wire carrying an electrical current in a magnetic field [34]. Moreover, the linear relation is valid also during transient operation assuming that [56]: 1) the magnetic fields are quasi-static 2) the force caused by eddy currents is negligible 3) the wire diameter of the coil is smaller than skin-depth.

The coupling vector K is highly nonlinear with the position of the coil j

q [95, 56, 76, 13] (see Figure 2.1 and Figure 2.2). Accurate computing is difficult if closed-loop sample rates of several kHz are required [76, 56, 13]. One approach is to use a simplified model of the real system as in [56], where the magnetic flux density of the planar actuator

10 Chapter 2. Commutation

Figure 2.1: Position dependent coupling ( )Fz j

K q .

Figure 2.2: Position dependent coupling ( )Tx j

K q .

is represented as a sum of spatial harmonics [60, 112]. Moreover, the coils are modeled as a number of filament surfaces for which it is possible to find an analytical solution for the Lorentz force and the corresponding torque [60]. Although the analytical expressions provide a good insight into the mechanisms that determine the forces and torques on the planar actuator, the simplifying assumptions compromise the accuracy of the model [13]. To allow real-time wrench calculation, only the first spatial harmonic of the magnet array has been included [56] neglecting the end-effects. Therefore, the coils on the boundary of the magnet array cannot be used and the current in these coils must be forced to zero [77, 79]. An addition of multiplicative edges to the model that slightly increases the range in which the coils can be used effectively is discussed in [3]. However, the coils in the model are still reduced to current sheets [56].

Another approach, which uses a look-up table (LUT), was developed in [18]. First, the magnetic flux density distribution is calculated using the surface charge model of the

2.1 Introduction 11

magnet array [56, 16]. Magnetic flux density distribution of a single cuboidal permanent magnet is calculated as:

( ) ( )

( ) ( )

( )

1 1 1

0 0 01 1 1

0 0 01 1 1

0 0 0

1 log ,4

1 log ,4

1 atan2 ,4

i j kr

xi j k

i j kr

yi j k

i j kr

zi j k

BB R T

BB R S

B STB

RU

+ +

= = =

+ +

= = =

+ +

= = =

= - -p

= - -p

æ ö÷ç ÷= - ç ÷ç ÷çp è ø

ååå

ååå

ååå

(2.2)

where atan2 is the four quadrant arctangent function, r

B is the remanent magnetization of the permanent magnet and:

( ) ( )( ) ( )( ) ( )

2 2 2 ,

1 ,

1 ,

1 ,

i

mj

mk

m

R S T U

S x x a

T y y b

U z z c

= + +

= - - -

= - - -

= - - -

(2.3)

where T[ , , ]m m m

x y z is center position of the magnet with sizes 2a, 2b and 2c in the x-, y- and z-direction, respectively. The total magnetic flux density generated by the magnet array is calculated by superposition [120].

The coil is modeled as a bundle of filamentary wires, where for a wire filament dl with current

ci in the magnetic field of the permanent magnet the produced force vector is

d dc c m

F i l B= ´ , (2.4)

where T[ , , ]m x y z

B B B B= is the magnetic flux density vector. The same force in opposite direction is applied on the magnet array (Newton’s third law): d d

m cF F=- . The torque

exerted on the magnet array by the wire filament equals:

d = d dm c m c c m

T r F r i l B´ =- ´ ´ , (2.5)

where c

r is the vector from the center of the mass of the magnet array to the wire filament.

The position-dependent coupling K is pre-computed using this analytical-numerical Lorentz force model [56] for a high-density grid of positions of one coil in the space of the magnetic field of the magnet array [16]. In the last step, the look-up tables are generated from the calculated coupling [18]. The main benefit of a LUT is that the end-effects and higher harmonics of magnetic flux density can easily be included and therefore the LUT can be very accurate as the time-consuming calculations are done offline whereas accessing a LUT is quite fast. Using the coupling calculated for relevant space of the magnetic field with high resolution would require extreme memory requirements. Therefore, symmetries and periodicity of the magnetic field are used to reduce the look-up table size to just a few hundreds of kilobytes [18]. For that reason, the LUT approach is preferred.


When assuming rigid body behavior, superposition of the coil currents can be applied to obtain the total wrench on the center of mass of the translator [76]:

( )j j

j

w K q i=å (2.6)

or in matrix form:

1 1

( ), , ( ), ( ) ( )n n

w K q K q K q i q i-é ù= =ê úë û K , (2.7)

where i is an 1n´ vector of all coil currents and ( )qK is an m n´ coupling matrix varying with the position and orientation q of the translator with the respect to the stator with coil array.

Magnetically levitated planar actuators are usually over-actuated actuators. That means that the number of active coils is always greater than the number of mechanical degrees of freedom (DOF’s). In a mathematical sense, the set of equations (2.7) is under-determined or dim( ) dim( )i w> . The rank of the matrix K must be equal to the amount of DOF’s for

admq S" Î , where

admS is set of admissible coordinates, for the system of equations

to be consistent. To linearize and decouple the system, an inverse mapping of (2.7) is

necessary*:

des

( , )i q w-= K , (2.8)

where des

w is the desired wrench vector and -K is a mapping from the desired wrench vector to the current vector i at the actual position q . For brushed DC machines, the commutator/brush system provides commutation, which can be seen as a transformation that transforms armature voltages to the field-winding reference frame [39]. The process of the coordinate transformation provided by the commutator is analogous to the inverse mapping (2.8) when considering the class of synchronous brushless AC actuators that can be described by (2.1) [76]. Therefore, the transformation provided by the inverse mapping is defined as commutation in this thesis.

The inverse mapping -K is not necessarily linear in w, because over-actuation brings freedom in the choice of the current vector. If -K is linear in w, the inverse mapping, being a vector function of w, can be written as a position-dependent matrix ( )q-K independent of w:

( , ) ( )q w q w- -K » K . (2.9)

This chapter is organized as follows. Novel commutation methods and their comparison with the up-to-date commutation method is presented in Section 2.2. Full rotation of the planar actuator, from the control point of view related to the commutation, is also presented in this chapter, in Section 0. The findings from the both sections are applied on a model of the contactless planar actuator with the manipulator (COPAM) and

* Please note that there is a distinction between the symbols for the coupling vector K defined in (2.1), the

coupling matrix K defined in (2.6), the inverse coupling vector mapping -K - (Greek capital kappa) and the

inverse coupling matrix K-.


several aspects (e.g. maximum current and velocity, dissipated power, error sensitivity and worst-case acceleration and velocity) are studied in Section 2.4.

2.2 Norm-Based Commutation Algorithms

In this section, three norm-based commutation methods will be described of which two novel methods are presented as alternatives for over-actuated actuators that can be described by the linear relation between the applied currents and the produced wrench (2.7).

For most rotational (and linear) actuators, this inverse mapping is achieved by using dq0- or Park’s transformation [31-33, 91, 92]. For linear and planar actuators, this transformation can be used directly to derive a commutation that decouples only the force components. In moving-coil planar actuator, such as [24, 22], it is possible to use design symmetries, which reduce the complexity of the torque equations. Then an additional transformation can be derived, which allows for decoupling of the torques [25]. Moving-magnet planar actuators with integrated magnetic bearing [51, 26, 59, 18] have complex torque equations [77]. As a result, the dq0-transformation is not convenient for them. In literature, attempts can be found to decouple torque using additional transformation after applying dq0-transformation [108, 114]. Nevertheless, the resultant disturbance torque is still significant. The algorithm was further improved by Binnard et al. [10, 11] resulting in commutation for 6-DOF planar actuator, but the full torque equations are still not included.

2.2.1 l2-norm Based Commutation (L2C)

To overcome the problem of torque decoupling, a method for direct wrench-current decoupling has been developed independently and in parallel by [78] and [95]. The under-determined set of equations (2.7) offers the possibility to impose extra constraints while creating an inverse mapping. An interesting additional constraint is to minimize the sum of the Ohmic losses in the coils (or dissipated power). This can be done by minimizing

the l2-norm of the current vector:

des

2 des2 2( )min ( )q i w

i q w-

==

NKK . (2.10)

The matrix 2

-K , which minimizes the l2-norm, is a reflexive generalized inverse of K, also

known as a pseudo-inverse of K [96], and can be calculated explicitly:

( )1

1 T 1 T2( ) ( ) ( ) ( )q q q q

-- - -=K N K K N K , (2.11)

where N is a positive-definite weighting matrix. The main advantage of this method is the

minimum dissipated power if ºN R, where R is a diagonal weighting matrix of which the elements correspond to the resistances of each active coil [78]. If all the coils have the same resistance, the weighting matrix can be omitted. The second benefit is that the commutation is obtained in a single step (no iterations are necessary). The third benefit is the short calculation time due to the single step solution and the simplicity of calculating


the pseudo-inverse. The inverse in (2.11) is calculated only from a m m´ matrix, where dim( )m w= and therefore for any over-actuated actuator 6m £ . The last, but very

important property is the continuous current characteristic with the position q or the wrench

desw variation. For commutation on a real over-actuated actuator, continuous

current change is important, because too high di/dt causes peaks in the terminal voltage j

u

of the jth-coil [40]:

1

d d d

d d d

nj k

j j kkk j

i i xu i R L M

t t x t=¹

¶L= + + +

¶å , (2.12)

where R (W), L (H) and M (H) are resistance, inductance and mutual inductance, respectively, of the coil, x¶L ¶ is a change of the flux linkage of the permanent magnets with the coil.

The drawback is that the l2-norm based commutation cannot imply any constraints on the current maximum. The currents calculated by using the pseudo-inversion (2.11) can easily be over the physical limits of the amplifiers or coils. In such a situation amplifiers saturate the current or, even worse, they turn themselves off due to overload/overheat. In any case, forces and torques of the actuator are no longer linearized and decoupled. For that reason, it might be better to use different criteria for minimization.

2.2.2 l∞-norm Based Commutation (LiC)

The infinity-norm is the only norm that directly puts constraints on the maximum current. The criterion for minimization is given as:

des

des( )min ( , )q i w

i q w-¥¥ ¥=

= KK

. (2.13)

In mathematical literature, this problem is known under term: Chebyshev solution of an underdetermined system of linear equations [1, 6]:

min |x

x x y¥

=A , (2.14)

where ( )1 2sup , , ,

nx x x x

¥= is called l∞- or Chebyshev norm. From the nature of the

problem, the inversion mapping -¥K is not affine in w anymore, and, consequently, -

¥K

cannot be written as a matrix as in the l2-norm situation. The main difficulty of solving this mathematical problem is that it is not possible to obtain the solution explicitly. All the known methods reach the solution of (2.14) iteratively. The methods differ in memory requirements and in the necessary calculation time, which is mostly dependent on the number of iterations. An extensive summary of the methods for calculation of underdetermined system of linear equations can be found in Abdelmalek [1]. However, one additional, fast and effective algorithm is discussed by Ascher [6]. The Ascher’s algorithm employs a linear programming algorithm for the solution of a set of over-determined linear

equations in the l1-norm to obtain a minimum l∞-norm solution to the set of consistent linear equations. To minimize the number of extraneous variables, the vector x is


normalized first. Then the solution is iteratively obtained via the dual l1 problem. The main benefit is a lower number of necessary iterations, which depends on the number of equations and less on the number of variables. Thus, the algorithm is supposed to lead to a shorter calculation time. This is in contrast to a similar principle for the calculation that can be found in [1] where the linear programming is used to obtain only the initial solution. After this step, the iteration process employs a slightly modified simplex method.

The effectiveness of both algorithms has been verified by the following test. Test sets of equations with different dimensions were filled with random numbers. The test was evaluated ten thousand times with each set of equations always filled with new random values. Mean, maximum number of iterations and time were captured as shown in Table 2.1. The calculation time in Table 2.1 is, for obvious reasons, scaled. The results in Table 2.1 clearly confirm that the number of iterations in the Ascher’s algorithm [6] depends more on the number of equations (m) whereas the Abdelmalek’s algorithm [1] is dependent on the number of variables (n). Both algorithms obtain the same solution whereas the Ascher’s is always about twice as fast regardless the dimensions of the matrix A. This is

not what one would expect when comparing only the number of iterations. For a higher dimension of x, the Abdelmalek’s algorithm needs several times more iterations than the first one to obtain the solution, but it needs less calculation time per iteration. Still, the Ascher’s algorithm is more time-effective in obtaining the solution. The probability distribution of the number of necessary iterations for different number of variables and equations is plotted in Figure 2.3. The probability Pr (-) of obtaining the solution in it iterations is defined as ( ) /

itPr it p k= , where

itp the number of solutions in it iterations and

k is the total number of experiments. Hence, ( ) 1itPr itS = .

The advantage of the l∞-norm minimization is a direct minimization of the maximum currents. This goes so far that 1

an m- + variables have the same minimized

absolute value [6], where a

n is number of variables that have nonzero coefficient in at least one of the equations (number of the active coils). From a physical point of view, it means

Table 2.1: Comparison of two approaches for LiC calculation

Iterations (-) Calculation time (scaled)

Ascher Abdelmalek Ascher Abdelmalek A(m,n)

Avg Max Avg Max Avg Max Avg Max

4 6 4 6 7 12 1 2 4 8

4 10 5 9 8 15 2 3 6 10

5 10 6 10 10 15 3 5 7 10

4 50 7 14 23 33 10 14 20 32

4 100 8 14 46 72 27 35 58 75

5 100 11 19 50 81 30 45 70 100

6 100 14 23 57 92 42 55 83 110


0 10 20 30 40 50 60 70 800

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Iterations (-)

Pro

ba

bilit

y (-

)

Ascher 4x10Ascher 6x10Ascher 4x100Ascher 6x100Abdelmalek 4x10Abdelmalek 6x10Abdelmalek 4x100Abdelmalek 6x100

Figure 2.3: Probability distribution of the number of necessary iterations calculated for both

methods and different matrix A sizes.

that as many coils ( 1a

n m- + ) will be energized to the same level even if their contribution to the final wrench is very small. In consequence, although the maximum current is reduced, the actuator needs much more power (so also more heat production)

than the l2-norm solution. In addition, the energized coils will produce more force, which will cancel each other (the final wrench must be the same); hence, the sensitivity to coupling matrix errors will be higher. Another drawback, from an application point of view, is a discontinuous current-variation with a change of position q .

2.2.3 Clipped l2-norm Based Commutation (CL2C)

So far, two limit solutions for commutation of an over-actuated actuator have

been discussed in this section. The first solution minimizes the l2-norm of the current vector and, therefore, the dissipated power with no constraints on the maximum current. The second one minimizes the l∞-norm of the current vector or the maximum currents whereas the power usage increases rapidly. Obviously, neither of the solutions is perfect. The compromise is to minimize the used power whereas the maximum current can be limited if necessary:

des clip

2( ) ,min

q i w i ii

¥= £K

, (2.15)

where clipi is the limit on the current. Minimization of l2-norm of constrained system of

linear equations can be solved by quadratic programming (e.g. [87]) where a cost function ( )f x is minimized subject to inequality and equality constraints:

T T1

( ) ,2

, .

f x x x c x

x b x d

= +

< =

Q

A E (2.16)


By using the notation of (2.15), the quadratic programming problem (2.16) can be defined as:

T

clip clip des

1( ) ,

2, ,

f i i i

i i i i w

=

- < < =K (2.17)

where clip

i is the 1n´ vector of the current limit values. The large calculation time, for

solving this quadratic programming problem, prohibits real-time commutation. A fast analytical solution for one equality constraint was presented in [21]. However, since K has m rows for m-DOF system, the method is not applicable on this problem. For that reason a novel, fast commutation method has been developed that, as well, puts constraints on the maximum current whereas the current vector is still minimized in l2-norm. The calculation process itself is also iterative, but the number of iterations is low and each iteration step is calculated fast.

The algorithm is based on the fact that if the currents are calculated with L2C and are exceeding the limit (

clip| |

ji i> ), they will the most probably have the limit value

clip: sgn( )

j ji i i= when solved according to (2.15). The algorithm has the following steps:

1. Calculate the pseudo-inversion 2-K (2.11) of the coupling matrix K, which

minimizes the current vector in l2-norm (2.10).

2. If any of the values exceeds the clipping limit clip

, 1, ,j

i i j n> = , continue,

otherwise go to end. 3. All exceeding currents are saturated, a vector of clipped (saturated) currents c i is

generated :

clip clip

clip

sgn( ) if [ ] , 1,...,

0 if j jc

j

j

i i i ii j n

i i

ìï >ïï= =íï £ïïî . (2.18)

4. A new matrix 2l K is created from K, where the columns corresponding to nonzero

elements of vector c i are set to zero. A new desired wrench vector 2des

l w is calculated:

2des des

l cw w i= -K (2.19)

5. By using the pseudo-inversion (2.11) a new current vector 2l i is calculated from the reduced system:

2 2 2des

l l li w=K . (2.20)

6. The final current vector 2cl i is the summation of the clipped values c i and the

vector of l2-norm minimized values 2l i (because for each row j either 0cj

i = or 2 0l

ji = ):

2 2cl l ci i i= + . (2.21)


7. The process loops back to the step 2 (with 2: cli i= ) while any of the current values exceeds the clipping limit.

It is important to mention that the solution of this heuristic algorithm is not necessarily optimal. It might happen that by saturating one of the currents some other, which was also saturated, could be lowered to obtain the optimal solution, but it is not lowered. The optimal solution can be obtained for example by quadratic programming. The main advantage of this approach in comparison to the quadratic programming is in the

calculation time, which is just k-times the calculation time of l2-norm solution, where k is number of iterations needed.

The number of iterations mostly depends on the clipping value clipi for the current

limitation. If the clipping value is higher than the actual maximum current obtained with

l2-norm, only one iteration step is needed. If the value is lower than the maximum current, at least one additional iteration step is required. The number of iterations increases as the clipping value decreases down to its lowest limit, which is equal to the value of the minimized l∞-norm of the current vector min i

¥. The maximum theoretical number of

iterations is thus 2a

n m- + , where a

n is number of the active coils. This situation occurs only when in each iteration step one (extra) current is saturated.

Although the algorithm is heuristic, the obtained solution shows an exact match with the solution obtained with quadratic programming in the majority of tests (see the next section). From simulations, the same solution as from quadratic programming was obtained with higher probability if the clipping value was further from the minimum l∞-norm solution (fewer currents had to be saturated). Even if the obtained solution is suboptimal, it still satisfies given constraints and the non-saturated currents are minimized

in l2-norm. Therefore, the dissipated power will always be lower than those calculated by LiC, so:

2 2

2 2 2

l cl li i i

¥£ £ . (2.22)

The proof of (2.22) is trivial. Since L2C minimizes 2-norm of the current vector, 2 2

2 2

l cli i£ and

2

2 2

l li i

¥£ is automatically true. The third part,

2

2 2

cl li i

¥£ , is

true because CL2C also minimizes 2-norm of the current vector with the inequality constraint on the maximum current (see (2.15)). Therefore, if there exists any

2min i

solution with clip

i i¥< , it will be calculated with CL2C.

Moreover, the current steps caused by the CL2C are not as severe as in the LiC, because only the currents close to

clipi value can make the step to/from the saturation limit.

Other currents (further form the boundary) will also make small steps in consequence, to satisfy the desired wrench production. With known parameters of an ironless actuator and by using (2.12), one can calculate how large a current step the current amplifiers can handle. For example, with a coil inductance of 10 mH, sampling frequency of 1 kHz, clipping limit

clip2 Ai = and a current step of 10 % of

clipi (= 0.2 A), the peak in the voltage

will have amplitude of 2 V. Current amplifiers can usually handle much higher voltage peaks.


Continuous current-change is guaranteed if the consecutive solutions are the same as those obtained by quadratic programming and hence optimal. Continuity of solution obtained via quadratic programming is shown e.g. in [98, 49, 28].

In a real application, current amplifiers should not be the limiting factor during the whole trajectory of the actuator. On the contrary, current limitation via commutation should be considered as a safety layer for unexpected circumstances (increased load, disturbing forces etc.) or just for several spots in the trajectory of the actuator. If a significant number of coils need to be limited for most of the time of the motion, the actuator will have increased dissipated power. Obviously, this is not a good design. If only several currents are limited, the obtained solution is with high probability optimal and hence continuous. Therefore, applicability of CL2C on a real actuator should be possible.

For a predefined trajectory of the actuator, the current continuity and the peak voltages in the case of the discontinuity can be tested in simulation in advance.

Reduction of the Number of Iterations

It might happen that several currents are very close to the clipping limit, but just one or a few of them are actually exceeding the limit. Then the algorithm saturates the exceeding currents, which will cause an increase of the rest of the currents including the ones close to the border. Consequently, one or more of the currents under the limit might cross the limit and needs to be saturated again. This causes another iteration step. Clearly, the situation can repeat for several times. To prevent such a situation, we changed the third step of the proposed algorithm:

3. Set the clipping limit in the condition to the a-times the original value with (0,1]a Î . All currents exceeding the lowered boundary value

clip.ia are saturated - a

vector of clipped currents c i is generated :

clip clip

clip

sgn( ) if .[ ] , 1,...,

0 if .j jc

j

j

i i i ii j n

i i

ìï > aïï= =íï £ aïïî. (2.23)

In this way if any of the currents in step 2 exceeds clipi , the algorithm preventively saturates

all the values over clip

.ia to the clipi value. The proposed value of gain a is between 0.8 and

0.95, but it can be tuned as it is presented in the next section. In this way, all currents close to the clipping limit will be preventively saturated. As a result, the number of iteration steps will be reduced. The price for the decreased calculation time is a possibly suboptimal solution and hence increased dissipated power and current discontinuity. This trade-off can be tuned by the gain a according to the application. Tuning of a can lead to

a maximum of two iteration steps (calculations of the l2-norm) if the clipping value is not

too close to the min i¥ for all q. This condition becomes clear if one imagines that if

clipi is

close to the min i¥ limit, then a large number of currents must be limited at once. Hence

prediction of which currents need to be limited if the boundary is even lowered (clip

.ia ), leads to a wrong choice.

In Table 2.2, all three norm-based commutation methods are compared. The calculation time is based on a model of an actuator with n = 100 coils and m = 6 degrees


Table 2.2: Comparison of the norm-based commutation methods

Commutation method l2-norm

(L2C)

l∞-norm

(LiC)

clipped l2-norm

(CL2C)

Current limitation – ++ +

Current continuity ++ – +/–

Power losses ++ – +

Calculation timea 7b

45 iter 7c

Overall Performance + 0 ++

a Scaled by the same factor as in Table 2.1. b Real calculation time in dSpace system D1005 with n = 99 and m = 6 is about 60 ms.

c Calculation time is approximately equal to the calculation time of the L2C multiplied by the number of iterations.

of freedom. From the comparison, clipped l2-norm based commutation comes out as the best trade-off between the algorithm performance and the constraint satisfaction.

Commutation methods based on minimization of other ln-norms with 2 n< <¥ have no practical sense, since they cannot limit the peak current, nor they minimize the dissipated power.

2.2.4 Example on a Model of the COPAM

The presented norm-based commutation algorithms have been tested on a model of the COPAM. Since the number of active coils is always greater than number of DOF’s, the system satisfies the definition of over-actuated actuator. Because the commutation is a static mapping between the desired wrench vector w and the calculated current vector i, simulation of the commutation gives exactly the same results as we would obtain from the real setup. The dimensions of the coupling matrix K from (2.7) are n = 99 and m = 6. The dimensions and properties of the PA are in Appendix A.

Comparison of L2C, CL2C and LiC

Figure 2.4 shows the energized coils example for one position of the PA when the commutation is calculated with the L2C and LiC for

desw = T[ , , , , , ]

x y z x y zF F F T T T =

T[200 N,200 N,200 N, 0 Nm,0 Nm,0 Nm] . The difference is visible immediately. With the L2C, only a few coils are energized to the significant level (those that have the highest effect on the commutation). On the other hand, with the LiC all the coils with nonzero effect on commutation are energized. Moreover, most of them are energized to the same absolute value, which is less than the maximum value reached by the L2C.

The current values are also plotted in Figure 2.5 where all three types of commutation are used: L2C, CL2C and LiC with 2l i , 2cl i , and l i¥ current vectors, respectively. The clipping of the maximum current shifts the CL2C towards the LiC.


i1

i9

i91

i99

i1

i9

i91

i99

3.5 A

-3.5 A

Figure 2.4: Distribution of currents in the coils calculated with the L2C and LiC; the solid square

represents actual position of the PA; the dashed rectangle is working area of the PA.

0 10 20 30 40 50 60 70 80 90 100-4

-3

-2

-1

0

1

2

3

Coil number (-)

Cu

rre

nt (

A)

l2icl2i, i

clip= 2 A

li

Figure 2.5: Distribution of currents in the coils calculated with the L2C, CL2C and LiC for the

position of the PA in Figure 2.4.

1.5 2 2.5 3 3.5 4550

600

650

700

750

800

850

900

950

Clipping current iclip

(A)

Dis

sip

ate

d p

ow

er

(W)

L2CCL2CLiC

Figure 2.6: Dissipated power depending on clipping current clipi .


The dissipated power, calculated as TP i i= R , where R is a diagonal matrix of the coil resistances, for different clipping currents of the CL2C is presented in Figure 2.6. From the figure, one can conclude that reducing the maximum current moderately will not cause significant power increase, however trying to get close to the minimum l∞-norm current limit, where most of the coils are limited, will increase the power demands substantially. In this example, it is more than 50 %. For

clipi > 2

maxl i , the dissipated power

is the same as in the l2-norm solution.

Investigation of Optimality of the Solution

The solution obtained by the CL2C is not always optimal (on the other hand, the solution always satisfies the constraints and the non-clipped currents are minimized in l2-norm). Therefore, a test has been performed on the model.

First, let’s define a relative clipping value [0,1]cr Î . Then the actual clipping

current is calculated as

( )2clip max max max

l l lc

i i r i i¥ ¥= + - . (2.24)

If 0cr = , the actual clipping limit is equal to min i

¥ solution and for 1

cr = there is no

clipping because clipi is equal to the maximum current calculated with l2-norm. The test was

performed for a dense grid of positions q of the PA. The probability of achieving the optimal solution (the same current vector as from commutation calculation with quadratic programming: 2cl qpi iº ) depending on the relative clipping value

cr is presented in

Figure 2.7 left. Already for 0.2cr ³ the probability of an optimal solution was > 97 % and

for 0.8cr ³ the optimal solution was found in 100 % of the cases. The question is: how far

from the optimal solution can the suboptimal one be? To get a notion about that, the

maximum relative l2-norm difference over all positions q between the suboptimal and the optimal solution:

( )2

maxadm

cl qp

qpq S

i i

iÎ

- (2.25)

for different cr is plotted in Figure 2.7 right. The maximal relative difference is less than

0.01 already for 0.1cr = (very close to l∞ limit value) which means that the CL2C will

require less than 1 % more power than necessary. At the LiC limit value the difference is at most 10 %. For the increasing

cr the maximum power difference decreases exponentially.

If both findings are combined, it is possible to get an impression of how accurate the solution is. It is also important to emphasize that the maximal difference occurs only for a limited amount of cases.

Current Discontinuity

If the continuous current-change is required, the optimal solution must be obtained always. Unfortunately, we cannot guarantee this. On the other hand, with increasing

cr value, the probability of the optimal solution increases as well. In the

presented example, for 0.8cr ³ the optimal solution was found in 100 % of cases.


min Linf 0.2 0.4 0.6 0.8 min L270

75

80

85

90

95

100P

rob

ab

ility

of o

ptim

al s

olu

tion

(%

)

Relative clipping value rc (-)

min Linf 0.2 0.4 0.6 0.8 min L210

-16

10-12

10-8

10-4

100

Ma

x. r

ela

tive

diff

ern

ce (

-)

Relative clipping value rc (-)

Figure 2.7: Left: Probability of obtaining the optimal solution with CL2C, the same solution as

from quadratic programming; Right: Worst-case relative l2-norm difference.

The current (dis)continuity was investigated for the change of the wrench des

w and change of the position q of the PA. Figure 2.8 top shows an example of the current in the coils when the desired vertical force

zF increases for one position of the PA. The current

values are clipped at 2 A. For the force z

F around 500 N, most of the currents are already clipped. Steps in the current caused by the increase of the wrench are shown in Figure 2.8 bottom. The force step

zFD was chosen very small (0.03 N, while max(d d

zi F ) < 0.2 A/N)

so only the current steps due to discontinuity of the solution are clearly visible in the figure. The biggest discontinuity is about 0.2 A. With the sampling frequency of 3 kHz and the coil inductance of 8.1 mH, the voltage peak has amplitude of less than 5 V, while the used current amplifiers can operate with up to 50 V.

An example of variation of the currents in the coils as the PA moves at the

maximum velocity of 0.5 m.s-1 is shown in Figure 2.9 top. The number of the limited currents varies from many (at 0.05 s) to almost none (at 0.12 s). Change of the currents in one time step (0.33 ms) is shown in Figure 2.9 bottom. The discontinuity of the solution is visible as the steps or pulses. From the figure, one can see that the discontinuity of the solution does not cause bigger steps in the current than the steps that are required for the continuous change in heavily saturated situation.

Reduction of Iteration Steps

Several tests were performed to verify the effectiveness of the proposed reduction of iteration steps. In the first test, the influence of gain a on the number of iteration steps was investigated for different clipping values, for the same position q and wrench vector w as in Figure 2.4 and Figure 2.5. The results that are presented in Figure 2.10 left show that it is possible to reduce the number of iteration steps. For example already for gain a = 0.9 with clipping current

clipi = 2.2 A, the commutation will still be calculated in two steps. Of

course, this works only if the clipping current is not too close to the min i¥ limit value.

The plot with a = 1.0 is equal to the original CL2C.


200 250 300 350 400 450 500

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Desired vertical force (N)

Cu

rre

nt i

n th

e c

oils

(A)

200 250 300 350 400 450 500-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Desired vertical force (N)

Cu

rre

nt s

tep

i (A

)

Figure 2.8: Influence of the wrench variation on the current continuity calculated with CL2C;

calculated for one position of the PA; Top: Current in the coils; Bottom: Discontinuity iD in the current.

Another aspect is increased dissipated power. The situation for different gains a and clipping currents is shown in Figure 2.10 right. As expected, gain a results in the coise of a wrong set of currents and, therefore, a suboptimal solution is obtained. On the other hand, in the presented example for a = 0.9 the dissipated power does not increase much whereas two iteration steps are sufficient for calculation, which facilitates real-time implementation.

The last performed test checked the maximum number of iterations on the whole workspace of the planar actuator again for the different a and

clipi (see Figure 2.11).

Although the clipping value, for which two iterations are sufficient, is higher, it is still possible to guarantee just two iteration steps for a significantly reduced maximum current.


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time (s)

Cu

rren

t in

the

coils

(A

)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

-0.1

-0.05

0

0.05

0.1

Time (s)

Cu

rre

nt s

tep

i (

A)

Figure 2.9: Influence of the PA position variation on the current continuity calculated with

CL2C; calculated for a constant wrench and v = 0.5 m.s-1; Top: Change of the current in the

coils; Bottom: Steps in the current.

1.5 2 2.5 3 3.5 40

1

2

3

4

5

6


(A)

Itera

tion

s (-

)

= 1.00 = 0.90 = 0.80 = 0.70

1.5 2 2.5 3 3.5 4

550

600

650

700

750

800


(A)

Dis

sip

ate

d p

ow

er

(W)

= 1.00 = 0.90 = 0.80 = 0.70

Figure 2.10: Left: Number of iteration steps; Right: Dissipated power calculated with the CL2C

depending on clipping current clipi and gain a; for the position of the PA in Figure 2.4.


2.2 2.4 2.6 2.8 3 3.2 3.40

2

4

6

8

10


(A)

Ma

x. It

era

tion

s (-

)

= 1.00 = 0.90 = 0.80 = 0.70

Figure 2.11: Worst-case number of iteration steps of the CL2C depending on clipping current clipi

and gain a; over all positions of the PA.

2.2.5 Conclusions

In this section, three norm-based commutation methods have been compared. The first one, l2-norm based commutation (L2C), was developed and successfully used in the past for control of magnetically levitated planar actuators [78, 95]. Its main benefit is in the minimization of dissipated power. Moreover, the commutation is calculated very fast by using pseudo-inversion of the original coupling matrix. The drawback is its inability to constrain the maximum current, which can cause saturation of the amplifiers and, consequently, the difference between the desired and the real forces and torques. This problem can be solved by two other norm-based commutation techniques. Although the minimal l∞-norm solution for underdetermined system of linear equations has existed in the mathematical literature for some time [1, 6], application on commutation for an over-actuated actuator has never been proposed before. The benefit of this kind of commutation is that the maximal current is always minimized. The negative aspects are power losses, which are not optimized. Because almost all active coils are set to the same maximum current level, although they can have minimal influence on the final force/torque, the power usage increases dramatically. Therefore, a novel, third kind of norm-based commutation is proposed which puts constraints on the maximal current whereas the non-saturated coils are minimized in the l2 norm. By varying the clipping (saturation) value,

the obtained solution is closer to either the l∞- or the l2-norm based commutation. The

applied algorithm for the clipped l2-norm based commutation (CL2C) does not necessarily lead to the optimal solution, but the performed tests indicate that the optimal solution in l2-norm sense can be obtained in the majority of cases whereas the suboptimal solution is still very close to the optimal one and practically indistinguishable in the real system. The benefit of this algorithm is in calculation time, which is reduced considerably in comparison to the LiC or optimal CL2C obtained via quadratic programming. Continuous current-variation with position and wrench variation can be guaranteed in the L2C and can be achieved in the CL2C if the solution is optimal. Nevertheless, as was shown in the example, the current discontinuity caused by the CL2C will not introduce high voltage peaks in the current amplifiers.


2.3 Full Yaw Rotation of Planar Actuator

Development of moving-magnet planar actuators (PA) with magnetic bearing and propulsion has been going for some time [51, 26, 59, 13]. Until now, only long-stroke translational movement of the moving magnet planar actuators has been considered and discussed in the literature [10, 63, 78, 95]. In this section, we are focusing on the possibility of full rotation about the vertical axis (yaw) of these planar actuators. Enhancing the planar actuator by another large-range degree of freedom (DOF) would increase its utility value and open new application areas.

For real-time commutation calculation, a simplified model of the real system has been developed in [56], where the magnetic flux density of the planar actuator is represented as a sum of spatial harmonics. The dimensions of the coils were optimized to achieve physical decoupling of the wrench production. The resultant coils are rectangular and produce forces only in two directions perpendicular to the longer side of the coils. Rotational degrees of freedom are included in the model only as the first-order Taylor expansion and hence rotation only about small angles is possible. The possibility of full rotation about the vertical axis would increase the complexity of the design and the commutation algorithm heavily as the forces and torques would be a function of the orientation. Moreover, rectangular coils are mainly advantageous with a 45-degree rotated magnet array [56]. To overcome problems of increased commutation complexity with full rotation, circular coils can be used as in [51] or in the COPAM [13]. With circular coils, the coupling between the wrench vector w and the current

ji in the coil j is independent of the

yaw-orientation of the single coil. In both cases, authors suggest a possibility to rotate their prototypes fully, but no further investigation of this feature has been done.

As was described in the introduction Section 2.1 and in [18], the position-dependent coupling vector K is pre-computed and stored in the form of Look-Up-Tables (LUT) for each relative position

jq of one coil with respect to the magnet array. By using

symmetries and periodicity, the size of the LUT is significantly reduced. Since yaw-orientation of the single coil has no effect on the coupling, it automatically includes full rotation about vertical axis without increased computational complexity. Then, the commutation can be calculated by using one of the commutation methods presented in Section 2.2. In this section, l2-norm based commutation has been chosen for linearization and decoupling of the wrench vector for its optimality in power utilization.

As the model of the COPAM [13, 17] with LUT-based commutation [13, 18] is capable of full rotation, it has been taken and a first investigation of this capability is performed in this section.

Since the focus of this thesis is on the control of a PA, investigation of this and other PA topologies is also carried out mainly from the control point of view based on the condition number, maximum current and dissipative power. However, many other relevant design criteria are neglected.


2.3.1 Full Yaw Rotation Investigation

Since the magnet array (MA) of the COPAM exceeds the coil array in the corners when it is theoretically rotated by 45°, the coil array was extended by additional coils in the model so the results are not distorted due to the limited array of the coils.

The magnet array of the COPAM has the Halbach structure as in some other PAs [26, 59]. The Halbach magnet array has an increased and concentrated magnetic flux density of sinusoidal shape near the coils [16]. The PA in the COPAM was optimized for low coil surface temperature, controllability, low resistive power losses and other criteria [17, 13]. Therefore, the coil spacing

sc is 4/3 of the magnet-pole pitch t (see Appendix A,

Figure A.2), which is equal to 240 electrical degrees (t represents 180 electrical degrees) [17]. However, this spacing is at least approximately optimal only when the PA is not rotated by a large angle, because the optimization criteria did not consider this orientation variation. Hence, several tests have been performed with varying orientation for three different coil spacings: 225, 240 and 270 electrical degrees. With variation of the coil spacing, the coil width needs to vary as well, to maximize wire density per unit area. This results in change of the turns and resistances of the coils. The parameters are summarized in Table 2.3. The middle values are those from the COPAM prototype.

Table 2.3: Comparison of the norm-based commutation methods

Coil topology 5/4 4/3a 3/2

Electrical degrees 225 240 270

Coil spacing (pitch) s

c 5/4t 4/3t 3/2t

Coil width w

c 48 mm 51.0 mm 57.6 mm

Coil bundle width bwc 19.6 mm 21.1 mm 24.4 mm

Winding 610 turns 684 turns 771 turns

Resistance 3.6 W 4.4 W 5.4 W

a The parameters of the COPAM

Maximum Current

The results from the investigation of the maximum current over all positions of the MA over the coils array depending on orientation are shown in Figure 2.12. The plot of the current maximum is an indication for the necessary current amplifier specifications. The plot in Figure 2.12 left is for the desired w vector equal to the lifting force in steady-state situation (required continuous current): w = [0, 0, mg, 0, 0, 0]T, where m is the mass of the moving part and g is the gravity constant. The plot in Figure 2.12 right is for levitation and 1g acceleration (with force F = mg) in horizontal x- and y-direction at the

same time (required peak current): w = [mg, mg, mg, 0, 0, 0]T. From the figures, one can see that the coil spacing of 4/3τ and 3/2τ is not beneficial any more in the situation where the magnet array is rotated by 45° (with 90° period), because the amplifiers and coils must be capable of operation at more than 5 A of the continuous and 20 A of the peak current


(in case of 4/3τ). This is caused by the coil spacing with respect to the orientation of the magnet array, which varies a lot during rotation. Therefore, for some position and orientation of the magnet array the coupling matrix K has very low values for all the coils. Similar results as for three-force w vector have been obtained for three-torque vector w =

[0, 0, 0, x

J e, y

J e, z

J e]T, where x

J , y

J , and z

J are inertias about x-, y-, and z-axis, respectively,

and e is angular acceleration, with e = 10 rad.s-2.

Maximum Dissipative Power

Comparable conclusions can be made from Figure 2.13, where the maximum necessary power over all positions is calculated. The dissipated power P (W) is calculated from the ohmic losses as TP i i= R , where i is the column vector of the currents in the coils and R is the diagonal matrix of the coil resistances.

Condition number

Condition number k is a classical measure for the investigation of the error sensitivity. If there is a system with errors

ˆ( )x b e+ = +A E , (2.26)

where E and e are errors of the matrix A and vector b , respectively, and x x x= +D , where the error-free equation is

x b=A . (2.27)

Then

12( , )

x e

x b-

æ öD ÷ç ÷ç ÷k +ç ÷ç ÷ç ÷è ø

EA A

Aƒ (2.28)

if 12( , ) 1-k

E

AA A , where

1 12( , )- -k =A A A A . (2.29)

The subscript in 12( , )-k A A represents that matrix 2-norm is used. If the condition

number is close to one, the system is well conditioned. If 12( , )-k A A is infinitely large, the

matrix A is singular. If 12( , ) 10k-k A A , approximately k digits of accuracy in the answer

will be lost (compared to the error in the inputs). An extension to non-square A and its

pseudo-inverse -A has been presented in [76]. Then (2.27) is equivalent to (2.7). The

pseudo-inverse matrix 2-K is obtained from K when it is calculated with the l2-norm based

commutation (L2C) (see (2.11)).

Since the coupling matrix K combines different quantities of the wrench vector w (forces and torques), it would not be useful to calculate the condition number directly from K and

2-K while the dimensions of forces and torques are freely chosen. Normalization of K

and 2-K with the mass and effective arm matrices is proposed in [76]:


* 1

* 12 2

,

,

-

- - -

=

=

K NM K

K K MN (2.30)

where M is the diagonal mass/inertia matrix, and the effective arm matrix

diag(1,1,1, , , )yx zJJ J

m m m=N (2.31)

with , , and x y z

J J J being the moments of inertia about the mass m center point of the translator. In this way, the mechanical power of each degree of freedom is equally penalized [76].

Figure 2.14 shows the worst-case normalized condition number depending on orientation. Again, for orientation around 45° the maximum condition number over all positions can be very high for coil spacing 4/3t and 3/2t. This indicates worse controllability of the planar actuator, and causes the high peak current and power (compare to Figure 2.12 and Figure 2.13).

-80 -60 -40 -20 0 20 40 60 801

2

3

4

5

6

7

8

9

10

Ma

x. c

urr

en

t (A

)

Orientation (deg)

5/4 4/3 3/2

-80 -60 -40 -20 0 20 40 60 800

5

10

15

20

25

30

Ma

x. c

urr

en

t (A

)

Orientation (deg)

5/4 4/3 3/2

Figure 2.12: Angle-dependent worst-case maximum current; Left: levitation force T[0, 0, , 0, 0, 0]w mg= ; Right: T[ , , , 0, 0, 0]w mg mg mg= .

-80 -60 -40 -20 0 20 40 60 80

102

103

Ma

x. p

ow

er

(W)

Orientation (deg)

5/4 4/3 3/2

-80 -60 -40 -20 0 20 40 60 80

103

104

Ma

x. p

ow

er

(W)

Orientation (deg)

5/4 4/3 3/2

Figure 2.13: Angle-dependent worst-case power; Left: levitation force T[0, 0, , 0, 0, 0]w mg= ;

Right: T[ , , , 0, 0, 0]w mg mg mg= .


-80 -60 -40 -20 0 20 40 60 800

5

10

15

20

25

Ma

x. n

orm

. co

nd

. nu

mb

er

(-)

Orientation (deg)

5/4 4/3 3/2 1

Figure 2.14: Angle-dependent worst-case condition number.

2.3.2 Concept Design for Full Rotation

In the previous section, it was observed that a large normalized condition number for some orientation could cause increased maximum current and dissipative power, leading to severe performance problems. In this section, new coil and magnet arrays are proposed to reduce this number. Moreover, a sensor topology is presented for designing a full-rotational planar actuator.

Triangular coil array

There is also another possibility for arranging the array of the coils. The use of a triangular (or isometric) array of the coils is proposed instead of a square array (see Figure 2.15). Such an array has higher wire density per unit area (a factor 2 3 of the filling of the square coil array if the coil spacing is the same) and therefore theoretically a lower current for a single coil is required.

The same tests performed for a triangular array of the coils are shown in Figure 2.16 - Figure 2.18. The same coils are used as for the square array. Comparing these figures with the same plots for the square grid (Figure 2.12 - Figure 2.14), one can immediately observe that the peaks in the maximum current, power losses or condition

Figure 2.15: Square and triangular coil array.


-80 -60 -40 -20 0 20 40 60 80

1

1.5

2

2.5

3

3.5M

ax.

cu

rre

nt (

A)

Orientation (deg)

5/4 4/3 3/2

-80 -60 -40 -20 0 20 40 60 802

4

6

8

10

12

14

16

Ma

x. c

urr

en

t (A

)

Orientation (deg)

5/4 4/3 3/2

Figure 2.16: Angle-dependent worst-case maximum current, triangular array; Left: levitation

force T[0, 0, , 0, 0, 0]w mg= ; Right: T[ , , , 0, 0, 0]w mg mg mg= .

-80 -60 -40 -20 0 20 40 60 80

102

Ma

x. p

ow

er

(W)

Orientation (deg)

5/4 4/3 3/2

-80 -60 -40 -20 0 20 40 60 80

103

Ma

x. p

ow

er

(W)

Orientation (deg)

5/4 4/3 3/2

Figure 2.17: Angle-dependent worst-case power, triangular array; Left: levitation force T[0, 0, , 0, 0, 0]w mg= ; Right: T[ , , , 0, 0, 0]w mg mg mg= .

-80 -60 -40 -20 0 20 40 60 800

2

4

6

8

10

12

14

16

18

Ma

x. n

orm

. co

nd

. nu

mb

er

(-)

Orientation (deg)

5/4 4/3 3/2 1

Figure 2.18: Angle-dependent worst-case condition number, triangular array.


number appear with 30° period, instead of 60° period as could be expected. Such behavior originates from additional periodicity at 30° with 60° period. At 30°, a line of the coils is aligned with the MA in

my -direction of the MA local coordinate system in the same was as

a different line of coils is aligned in x-direction for zero rotation (see Figure 2.19). As the Halbach magnet array for planar actuators is usually designed in such a way that it has symmetric magnetic flux density distribution in

mx - and

my -direction, the situation for 0°

and 30° rotation is equivalent for the commutation. The rotational period p

r [º] for the general case can be calculated as:

( )pc pc pm

ppc pm pm pc pc pm

if

min mod , mod if

r r rr

r r r r r r

ìï =ïï= íï ¹ïïî, (2.32)

where pc

r and pm

r are rotational periods of the coil array and the magnet array, respectively, and mod is modulus after division. Consequently, for a lower value of the rotational period (more frequent) it is better if

pc pmr r¹ . For the COPAM,

pc pm90r r= = º, so

pr = 90º. For

the triangular coil array pc

r = 60º and pm

90r = º, so p

r = 90º mod 60º = 30º.

Because of the 30° period, the difference between the worst and the best orientation is only 15° instead of 45° as for square coil array. This does not automatically mean that the performance will be better. For example,

sc = 5/4t produces a huge change

within 15° of rotation in all three monitored quantities in Figure 2.16 - Figure 2.18. On the other hand, the worst-case current and power losses are still smaller than in case of

sc =

3/2t and the square coil array.

Contrary to the findings from the square array of coils, in the triangular array 4/3t and 3/2t coil spacings are preferable. This, again, is due to the coil spacing with respect to the orientation of the magnet array, which does not vary so much for the triangular coil array and the variation of the 4/3t and 3/2t coil spacing does not lead to a situation when the coupling between current and force is small for all the coils. Especially

sc = 4/3t (and the related coil width), which is the same one as that used in the COPAM

xm

ym

xc

yc

xmym

xc

yc

Figure 2.19: Periodicity after 30-degree rotation of the MA; alignment of the coil rows with

magnet array; Left: in x-direction for 0°-rotation, Right: in m

y -direction for 30°-rotation.


[17], have low peak current and power losses. Therefore, if this coil size and spacing is preferable due to some (other) optimization criteria, changing the coil arrangement from square to triangular array removes the obstacle of bad controllability close to 45 degrees. Moreover, when

sc = 4/3t is used the triangular array has about 20% lower peak current

and power losses also for an unrotated MA compared with present PA design based on square coil array. The performance is comparable to the best results achieved on the square coil array with

sc = 5/4t. This makes the triangular array preferable even for the

applications where rotation of the planar actuator is not required. The denser triangular coil array does not immediately mean requirement of better cooling, because the power losses per unit area and the peak currents are lower for

sc = 4/3t.

Hexagonal Halbach magnet array

As has been presented, for some coil spacings with respect to the magnet-pole pitch of the magnet array, the controllability of the system is stable for any orientation. This is partly caused by more frequent appearance of the same magnet array-coil array coordination per one revolution of the magnet array with respect to the coil array. For triangular coil array, 12 times the same coordination occurs in contrast to the 4-time occurrence for the square array. As the smallest relative rotation of coils and magnet arrays for the same orientation symmetry is relevant, such a feature can be used also for inversed design. If square coil array is important, the magnet array can be modified to obtain the same properties. An idea of a hexagonal Halbach magnet array is proposed (see Figure 2.20). The modeling of such a magnet array has not yet been done, but the general idea is that the magnetic field produced should allow for the same periodicity properties. From the figure, it might seem that the magnetic flux density produced by this hexagonal Halbach MA will be lower than by the conventional square Halbach MA, because of large

S

N

S

N

S

N

S

N

S

N

S

N

S

N

S

N

S

S

N

S

N

S

N

S

N

S

N

S

N

N

S

N

S

N

N

S

N

S

N

S

S

N

Figure 2.20: Drawing of hexagonal Halbach magnet array.


empty areas without magnets. If the dimensions of the magnets from Appendix A are used, the empty space in the square Halbach MA occupies 13 % of the surface whereas in the hexagonal Halbach MA the empty area covers 12 % of the surface, so slightly less. On the other hand, these finding are strongly dependent on the ratio between the magnet width

wm and the magnet-pole pitch t.

Other topologies for coils and magnets

In the previous paragraphs, the triangular coil array and the hexagonal Halbach magnet array were proposed as alternatives to the square coil and magnet arrays. It can be proven that only a limited number of reasonable topologies exist if only highly symmetric coil or magnet arrays are considered.

From the theory of tessellation (or tiling), only three regular tessellations made up of congruent regular polygons exist: those made up of equilateral triangles, squares, or hexagons [48]. If we use coil arrays with rounded coils, the only wise choice is to use triangular or square grid. If a different grid is used, the gap between the coils will be too wide and the wire-per-area ratio will be low leading to high currents and to high power losses. For example, the hexagonal coil array can be made from the triangular coil array by removing each third coil in the row. Obviously, reducing the number of coils makes no sense. If also the semi-regular tiling is considered, a grid made of squares and triangles can be used (e.g. snub square tiling or elongated triangular tiling, see Figure 2.21). Periodicity of the coil array can be reduced down to the point of completely removing periodicity as in Penrose tiling [93]. Controllability and overall performance investigation are of course much more difficult, because it must be performed on the whole workspace of the PA. This also complicates the optimization of such a coil array.

Several choices for the magnet array also exist. Even for the Halbach magnet array, all regular polygons are possible. By allowing other shapes of the magnets to be used, various arrays can be produced (e.g. [106, 82]). The presented hexagonal Halbach magnet array is not made up of congruent regular polygons, although the regular polygons can be obtained by setting the magnet width at half the magnet-pole pitch. Generally, the

Figure 2.21: Left: Snub square tiling, Right: Elongated triangular tiling.


ratio between the magnet width and the magnet pole-pitch belongs to the set of optimization variables [17].

Sensor system for yaw rotation

There also is another reason why full rotation of a PA has not been tested yet in practice. The reason is the sensor system. For application areas like wafer operation in the lithography industry or for pick-and-place machines, the demands on accuracy are very high, typically in sub-micrometer resolution combined with desired range of up to several hundreds of millimeters. The only commercially available sensor system nowadays that can satisfy such requirements is laser interferometry. One problem associated with laser interferometers is its limited rotational range.

A novel measurement system for detecting translational and rotational movement (primarily of a wafer stage) was described in [64]. This system uses two diffraction beams and diffraction patterns. With specific coordination of the measurement system, long-range translational and rotational movements can be detected. Moreover, information is made available on the absolute position of the measured body. However, no information about resolution, accuracy or bandwidth is available.

Another, cheaper solution can be found in 3-DOF optical position sensors [29, 70]. The sensor combines one or more CCD- or CMOS-based optical sensors reading a pattern on a measured planar surface. Such a sensor is capable of detecting two translational and one rotational movement of the surface. A 3-DOF optical precision sensor is presented in [29] that uses only one CCD chip and can detect the absolute position of the measured surface. The authors developed a prototype that has an accuracy of 0.4 μm and 0.14 mrad for translational and rotational movement, respectively. The disadvantage of the sensor is that the pattern needs to be drilled through the measured surface and the light source for the sensor is on the opposite side of the measured surface. Since there is no possibility to drill a microscopic pattern into the magnet array, such a system is not directly applicable to the PA. Another disadvantage is in the processing time of about 250 ms on a personal computer. In [70] the light source is on the same side of the surface as the CCD sensor and hence no drilling is necessary. This 3-DOF sensor is a combination of two CCD sensors, which are the same type as those that can be found in an optical mouse for a personal computer. Each sensor can detect only translational movement in two directions, but combining their measurement one can also detect rotational movement. The advantages of this sensor system are: relatively low price, no specific pattern reqiured, output of the sensors already in the position increments and relatively low delay. The resolution of the sensors used in [70] was 500 cpi (50 μm) and the delay was 4 ms. The best commercially available optical mouse already has a sensor with a resolution of 5600 cpi (4.5 μm), the

maximum tracking speed of 5 m.s-1, acceleration of 500 m.s-2, delay of 1 ms, and update rate of 1 kHz [97].

2.4 Commutation Techniques and Full Rotation on the Model of the COPAM 37

2.3.3 Conclusions

In this section, possible bottlenecks for the full rotation of moving-magnet planar actuators have been described. From many existing PA designs, it has been found that only those with rounded coils are suitable for full rotation, because the commutation algorithm does not have to be of increased complexity. This is in contrast to the square- or rectangular- coil PA, which must include the orientation of a single coil into the commutation algorithm. It was also found that the coil width and spacing optimized for translational movement can produce severe controllability problems when the magnet array is rotated about 45 degrees, causing the maximum current and power losses to increase a few times.

A new, triangular, array of the coils instead of the square array is proposed, which can have better properties for any orientation of the magnet array for given coil dimensions and spacing. Moreover, the performance can be better or at least comparable to the square coil array for some coil spacings even without rotation of the PA. This makes the triangular array preferable also for the applications where rotation of the planar actuator is not required. It also has been shown that a limited number of highly symmetric topologies exist for the array of rounded coils. However, many magnet array topologies can be found. Another interesting option for the full rotation might be a hexagonal Halbach magnet array in combination with a square array of coils, which also has higher rotational periodicity that the standard design.

The sensor system for rotating planar actuator is the main remaining difficulty. No sensor system exits for such a planar actuator today that could satisfy industrial requirements for resolution and range. Research on new types of 3-DOF optical sensors similar to optical sensors for computer pointers is proposed.

2.4 Commutation Techniques and Full Rotation on the Model of the COPAM

In Sections 2.2 and 0, both, commutation methods and full rotation of a magnetically levitated planar actuator have been discussed. The l2-norm, l∞-norm, and

clipped l2-norm based commutations (L2C, LiC, and CL2C, respectively) have been tested on the model of the COPAM and a few examples have been shown in Section 2.2.4. Extensive comparison of commutation methods in combination with full rotation on the model of the COPAM (see Figure 2.22) is the topic of this section. The commutation methods and the rotation of the magnet array were compared on several quantities: maximum current and voltages, dissipated power, controllability, and maximum acceleration. Due to the nonlinear nature of the coupling K, it is hard to find an analytical expression for the investigated quantities. Therefore, it is more convenient to obtain the values numerically from the simulations on the model of the COPAM.

For comparison, a new model with the triangular coil array (TCA) (see Figure 2.23), as proposed in Section 2.3.2, was also used. The TCA consists of 110 coils (11 10 coils) while the square coil array (SCA) from the COPAM has 99 coils (11 9 coils).


Figure 2.22: Model of the COPAM with square coil array (SCA) and magnet array in [x, y] = [0,

0] position. Dashed rectangle represents workspace of the magnet array; the dark coils are made of litz wire.

Figure 2.23: Modified model of the COPAM with triangular coil array (TCA) and magnet array

in [x, y] = [0, 0] position. Dashed rectangle represents workspace of the magnet array; the dark coils are made of litz wire.

The same coils were used and the number of the coils was chosen to cover the workspace of the magnet array.

The dark circles in the drawings in Figure 2.22 and Figure 2.23 represent the coils made of litz wire. Those coils have, besides the force generation, also a second function that is energy transfer to a secondary coil on the moving part of the setup [13]. Both modes, force generation and energy transfer, are not active at the same time, but are switched according to the actual position of the magnet array. These litz-wire coils have the same external dimension as the other coils made of solid wire. Since the litz-wire has


fewer turns of the wire in the coil, the coupling K is by a factor 2.85 smaller [13]. The resistances are also different. The solid-wire coils have resistance

swR = 4.4 W, where in the

litz-wire coils lw

R = 1.0 W (see also Appendix A). The amount of litz-wire coils in the TCA was chosen to satisfy the purpose of energy transfer as in the original design. The other simulation parameters were taken from the dimensions of the COPAM (see Appendix A) or can be found in Appendix C.1.

2.4.1 Maximum Current

The first investigated physical quantity was the maximum current in the coils. The calculations were performed for the workspace of the magnet array, which is 167 mm 67 mm, for presented norm-based commutation techniques and for two wrench vectors. The first wrench vector T[0, 0, , 0, 0, 0]w mg= represents steady-state levitation of the PA, and the second one T[ , , , 0, 0, 0]w mg mg mg= levitation with acceleration of 1g simultaneously in x- and y-direction. All angles were set to zero and the levitation height was 1 mm.

Since the coil array in the COPAM consists of two types of coils (solid-wire and litz-wire) with by factor 2.85 different couplings, two types of current amplifiers have been used to cope with different loads. For the solid-wire coils, the current amplifiers can deliver ±1.8 A of continuous current and ±6 A of peak current, and for the litz-wire coils the amplifier specifications are ±7 A of continuous and ±20 A of peak current [13]. This defines the limits on the maximum current during the steady-state levitation and the acceleration phase.

The inverse mapping from (2.11) for the L2C is calculated in such a way, that it minimizes the dissipated power. Since the coils have different resistances, the inverse mapping needs to include the weighting matrix of resistances R to obtain the minimum dissipated power. The maximum currents (over all participating coils) calculated with the L2C for the SCA (COPAM) depending on position of the magnet array are shown in Figure 2.24. The plots are for steady-state levitation ( T[0, 0, , 0, 0, 0]w mg= ) and are separated for the maximum current in the solid-wire and litz-wire coils. From the plots, one can see that the maximum current in the solid-wire coils is over the continuous-current limit of the assigned current amplifiers in some positions of the magnet array.

There are two ways of reducing the current in the solid-wire coils. The first approach is to change the weighting matrix N in (2.11), e.g. by exponentiation of each

element of R by an exponent 3/2. The second approach is to compute the commutation as if all the coils are solid-wire. Next, the currents for the litz-wire coils are multiplied by the factor 2.85, because for the wrench production this is the only difference between the solid-wire and the litz-wire coils.

Both solutions result in reduced solid-wire maximum currents, while an increased dissipated power is obtained. The aspects of the increased dissipated power are discussed in the next section. Exponentiation of the elements of R by the factor 3/2 leads to approximately the same result for both approaches when the L2C is used. Anyhow, two circumstances benefit the second one. The first one is that the current amplifiers for the litz-wire coils were chosen to handle three times higher currents than the solid-wire ones,


Figure 2.24: Square coil array (SCA); position-dependent maximum current calculated with l2-

norm based commutation with ºN R for steady state levitation T[0, 0, , 0, 0, 0]w mg= ; Left: maximum current of the solid-wire coils; Right: the litz-wire coils.

which is approximately the factor of the coupling difference. The second circumstance is the use of different commutation algorithms. The LiC minimizes the maximum current regardless of the differences in the coupling or resistance. Hence, a weighting needs to be introduced in this kind of commutation to satisfy the different needs for different types of the coils. This can be done in the same way as for the L2C by multiplication of the calculated litz-wire coils by the factor of the coupling difference. In addition, the clipping current

clipi in CL2C can then be also easily chosen.

From now on, the plots of the maximum current are valid for the solid-wire coils. The plots for litz-wire coils, which have the maximum current at most the factor 2.85 higher, are in Appendix C.2.

The calculated maximum currents depending of magnet-array position for three norm-based commutation techniques and for two wrench vectors are shown in Figure 2.25 and Figure 2.26 for SCA and TCA, respectively. The clipping current

clipi in the CL2C has

been chosen 1.1 A for the steady-state levitation and 2.2 A for the acceleration example (

clipi must be always greater than the maximum LiC current).

From the comparison of the plots for steady-state levitation T[0, 0, , 0, 0, 0]w mg= in Figure 2.25, one can see that whereas for the L2C the maximum current varies between about 0.7 A and 1.6 A, for the LiC the maximum current fluctuates between 0.5A and 1.0 A. For the levitation with acceleration ( T[ , , , 0, 0, 0]w mg mg mg= ), the peak currents are higher, but the ratio between the L2C and LiC is similar. Besides, it is evident how the CL2C limits the peak current in some positions of the magnet array, whereas in other positions the current corresponds to the L2C.

While comparing the results for the square coil array (SCA) in Figure 2.25 with the triangular coil array (TCA) in Figure 2.26, the most visible is the difference in the profile of the maximum current variation. Whereas for the SCA the maximum current profile varies equally in both directions, the TCA together with the square Halbach magnet


Figure 2.25: Position-dependent maximum current calculated with l2-norm, clipped l2-norm, and

l∞-norm based commutation for steady state levitation T[0,0, ,0,0,0]w mg= and levitation with x, y acceleration T[ , , ,0,0,0]w mg mg mg= ; SCA.


Figure 2.26: Position-dependent maximum current calculated with l2-norm, clipped l2-norm, and

l∞-norm based commutation for steady state levitation T[0,0, ,0,0,0]w mg= and levitation with x, y acceleration T[ , , ,0,0,0]w mg mg mg= ; TCA.


array causes almost no change in the maximum current along the x-axis. The second difference is the maximum current over all the coils and the whole workspace. For the TCA, the maximum current is 10 % - 25 % lower than for the SCA. This can be explained by about 15% higher wire-filling factor in the TCA as was shown in Section 2.3.1.

The maximum current depending on orientation of the magnet array is plotted in Figure 2.27. The maximum over all the coils and the whole magnet array workspace is calculated by using the L2C and LiC commutation for square and triangular coil array, and for two wrench vectors T[0, 0, , 0, 0, 0]w mg= and T[ , , , 0, 0, 0]w mg mg mg= . The profile of the plots is not surprising and comparable to those in Figure 2.12 and Figure 2.16 where the orientation-dependent maximum current is calculated with the L2C. The values in Figure 2.27 are higher, because the magnet array exceeds the coil array when it is rotated by about 45°. From the comparison of the commutation methods, LiC gives a factor three smaller the worst-case maximum current. Since the maximum continuous current of the current amplifiers for the solid-wire coils is 1.8 A [13], it is not possible to levitate the 45°-rotated magnet array even with the l∞-norm based commutation in the whole workspace. On the other hand, the region of allowed steady state levitated positions is much larger (see Figure 2.28) for the LiC. From the comparison of the coil arrays in Figure 2.27, the TCA shows no problems for any orientation of the magnet array. Although the maximum current of the TCA is higher for some angles of the accelerating magnet array ( T[ , , , 0, 0, 0]w mg mg mg= ), the worst-case value is a factor eight smaller than for the SCA. This again proves the effectiveness of the proposed triangular coil array. Similar results were obtained when the wrench had nonzero torque components (active rotation). All the plots with the L2C and LiC serve as the margins of the CL2C performance.

-80 -60 -40 -20 0 20 40 60 80

1

2

3

4

5

6

7

8

9

Orientation (deg)

Ma

x. c

urr

en

t (A

)

L2C, square arrayLiC, square arrayL2C, triang. arrayLiC, triang. array

-80 -60 -40 -20 0 20 40 60 80

5

10

15

20

25

30

35

40

Orientation (deg)


Figure 2.27: Maximum current over all the coils and positions calculated with l2-norm and l∞-

norm based commutation for the square and triangular coil array depending on the orientation of the magnet array. Left: steady state levitation T[0,0, ,0,0,0]w mg= , Right: levitation with acceleration T[ , , ,0,0,0]w mg mg mg= .


pos. x (m)

po

s. y

(m

)

0 0.05 0.1 0.150

0.02

0.04

0.06

pos. x (m)0 0.05 0.1 0.15

0

0.02

0.04

0.06

Figure 2.28: SCA; the dark areas shows where the maximum current is less than 1.8 A for the

45° rotated magnet array in steady-state levitation ( T[0,0, ,0,0,0]w mg= ); Left: current vector

calculated by l2-norm based commutation (L2C); Right: by l∞-norm based commutation (LiC).

2.4.2 Maximum Voltage

Voltage specifications are mostly important for the amplifier specification. The terminal voltages of the coils can be described by using (2.12) as

dd

( )d d

qiu i q

t tL= + +R L J , (2.33)

where R and L are the matrices of resistances and inductances, respectively, and ( )qLJ is the Jacobian of the flux of the permanent-magnet array linked by the respective coils.

Voltage equation (2.33) can be expressed by using (2.8) as:

T

T

d ( , ) d( , ) ( )

d dd dd

( , ) ( ) ( ) ( ) .d d d

q w qu q w q

t tq qw

q w q w qt t t

- -

--

-

K K

K= K + +

= K + + +

R L K

R LJ LJ K

(2.34)

For the L2C, (2.34) can be simplified by using the linear wrench-current relation (2.11) as [56]:

T22 2

d ( ) dd( ) ( ) ( )

d d d

q qwu q w w q q

t t t

-- -= + + +

KRK L LK K . (2.35)

Equations (2.33)-(2.35) consist of resistive and inductive terms, and the back-EMF (electromotive force). For the COPAM, the resistances are

swR = 4.4 W and

lwR = 1.0 W for

solid-wire and litz-wire coils, respectively, the self-inductances are sw

L = 8.1 mH and lw

L = 1.0 mH for solid-wire and litz-wire coils, respectively, and the mutual inductance M is expected to be at most one decade smaller [56]. The maximum terminal voltage of the Prodrive amplifiers in the COPAM is 52 V.

Since (2.34) is dependent on many different parameters (position, velocity, acceleration/wrench and jerk/wrench derivative), this investigation focuses on the maximum contribution of each term of (2.34) to the terminal voltage. The maxima are calculated over all xy-positions of the PA and over all the coils.


Resistive term

The value of the resistive term is determined by the nonzero wrench vector (acceleration). The same wrench vectors ( T[0, 0, , 0, 0, 0]w mg= and T[ , , , 0, 0, 0]

xyw mg mg mg= )

and the same clipping currents as in the previous section were investigated. Moreover an acceleration in x- and y-direction ( T[ , 0, , 0, 0, 0]

xw mg mg= and T[0, , , 0, 0, 0]

yw mg mg= ,

respectively) is added. The maximum voltages for different commutations and coil topologies are in Table 2.4.

Table 2.4: Maximum voltages; resistive term

max( ( ))i qR w wx or wy xyw

L2C 7.3 V 12.3 V 13.6 V

CL2C 4.8 V a 8.8 V b 9.7 V c SCA

LiC 4.6 V 7.9 V 8.1 V

L2C 5.7 V 10.3 V 11.4 V

CL2C 4.8 V a 8.8 V b 9.7 V c TCA

LiC 3.6 V 6.1 V 6.3 V

a clipi = 1.1 A, b

clipi = 2.0 A, c

clipi = 2.2 A

Inductive term due to the position change (d dq t) of the ( , )q w-K

The next term is inductive due to the position change (d dq t) of ( , )q w-K with nonzero and constant wrench (acceleration). For L2C, this term can be expressed

independently of wrench as 2d ( )

d

q

tw

-KL . The maximum values are calculated for the force-free

movement ( T[0, 0, , 0, 0, 0]w mg= ), acceleration in x- and y-direction ( T[ , 0, , 0, 0, 0]x

w mg mg= and T[0, , , 0, 0, 0]

yw mg mg= , respectively) and for the diagonal acceleration

( T[ , , , 0, 0, 0]xy

w mg mg mg= ). The values are calculated for the worst-case velocity vector in x-, y- or diagonal direction. The calculated values shown in Table 2.5 express the generated

Table 2.5: Maximum voltages per unit velocity; inductive term with constant acceleration

max ( ( ))w const

q-K=LJ w

wx or wy

wxy

L2C 0.97 V/(m.s-1) 1.9 V/(m.s-1) 2.5 V/(m.s-1)

CL2C 2.3 V/(m.s-1) a 4.1 V/(m.s-1) b 5.1 V/(m.s-1) c SCA

LiC > 17 V/(m.s-1) > 30 V/(m.s-1) > 30 V/(m.s-1)

L2C 1.4 V/(m.s-1) 2.1 V/(m.s-1) 2.4 V/(m.s-1)

CL2C 1.5 V/(m.s-1) a 2.1 V/(m.s-1) b 2.6 V/(m.s-1) c TCA

LiC > 13 V/(m.s-1) > 23 V/(m.s-1) > 23 V/(m.s-1)

a clipi = 1.1 A, b

clipi = 2.0 A, c

clipi = 2.2 A


voltages per unit velocity in x/y or diagonal direction.

Inductive due to the wrench change (dw dt) of the ( , )q w-K

The third term is inductive due to the wrench change (dw dt) of the ( , )q w-K with constant position q . This term expresses the voltage generated during the jerk phase of the trajectory. For L2C, the term can be expressed independently on wrench as d

2 d( ) w

tq-LK .

Since for CL2C and LiC commutation methods the values are dependent of the actual wrench (acceleration), the same two situations (steady state T[0, 0, , 0, 0, 0]w mg= and diagonal acceleration T[ , , , 0, 0, 0]

xyw mg mg mg= ) are studied in combination with jerk in x/y

and diagonal direction. The values are again calculated over all positions of the PA and all the coils. The calculated values shown in Table 2.6 express the generated voltages per unit jerk in x/y or diagonal direction. The relation between time-derivative of the wrench vector

and the jerk vector j (m.s-3 and rad.s-3) is: d d d dw t a t j= =M M , where a (m.s-2 and rad.s-2) is the acceleration vector and M is the diagonal mass/inertia matrix.

Table 2.6: Maximum voltages per unit jerk; inductive term with constant position

w

wxy

max( ( ) )w-KLJ M

jx and jy jxy jx and jy jxy

L2C 1.86 mV/(m.s-3) 1.32 mV/(m.s-3) 1.86 mV/(m.s-3) 1.32 mV/(m.s-3)

CL2C 8.56 mV/(m.s-3)a 7.73 mV/(m.s-3)a 8.74 mV/(m.s-3)b 7.95 mV/(m.s-3)b SCA

LiC 157 mV/(m.s-3) 157 mV/(m.s-3) 295 mV/(m.s-3) 280 mV/(m.s-3)

L2C 1.62 mV/(m.s-3) 1.16 mV/(m.s-3) 1.62 mV/(m.s-3) 1.16 mV/(m.s-3)

CL2C 1.86 mV/(m.s-3)a 1.35 mV/(m.s-3)a 2.02 mV/(m.s-3)b 1.50 mV/(m.s-3)b TCA

LiC 138 mV/(m.s-3) 138 mV/(m.s-3) 246 mV/(m.s-3) 239 mV/(m.s-3)

a clipi = 1.1 A, b

clipi = 2.2 A

Back-EMF

The fourth term represents back-EMF. This term is independent of the wrench vector, the commutation method, and, in case of the same coil dimensions, the coil

topology. The calculated value is at maximum 8.31 V/(m.s-1). Hence, back-EMF is at most 8.31 V for the speed of 1 m.s-1.

Comparing quantitatively the described four terms of (2.34), one can see that the TCA is providing the same or better performance than the SCA. When comparing the commutation methods, the CL2C does not cause much worse voltage peaks than the L2C. On the other hand, the LiC can cause problem due to current discontinuity and, consequently, the very high slew rate di/dt.

The upper bound for voltages for vmax = 2 m.s-1, amax = 10 m.s-2 and jmax = 1000

m.s-3 in any coil and any position of the PA are shown in Table 2.7. For a velocity of


1 m.s-1, the values are at least 10 V lower. It should also be stressed that these values are very conservative, based on the assumption that all the maxima occur in the same moment.

Table 2.7: Maximum voltages for vmax = 2 m.s-1, amax = 10 m.s-2 and jmax = 1000 m.s-3 in any coil

and any position of the PA

max( )u x- or y-direction

diagonal

L2C 34.5 V 37.1 V

CL2C 42.1 V a 45.2 V b SCA

LiC > 250 V > 360 V

L2C 32.7 V 34.4 V

CL2C 31.5 V a 33.5 V b TCA

LiC > 210 V > 310 V

a clipi = 2.0 A, b

clipi = 2.2 A

2.4.3 Dissipated Power

In this subsection, the dissipated power is investigated. The calculations were performed for the same conditions as in the maximum current investigation: for the workspace of the magnet array, for three norm-based commutation techniques, for two wrench vectors, and for two coil arrays. The dissipated power P (W) is calculated from the Ohmic losses as

TP i i= R , (2.36)

where i is the column vector of the currents in the coils and R is the diagonal matrix of

the coil resistances.

As was discussed in the previous section, two types of the coils used in the COPAM result in the use of a weighting matrix for the commutation calculation. Since the L2C minimizes dissipated power if the weighting matrix is equal to the diagonal matrix of resistances ºN R, this would be the obvious choice. It was shown in the previous section that this solution causes some of the solid-wire coils to require more continuous current than the amplifier can deliver for some positions of the steady-state levitated magnet array. It was decided to omit the weighting matrix and the different coupling factor of the litz-wire coils instead. Then the factor of the coupling difference is used as a gain for the litz-wire currents after the commutation is calculated. It was also mentioned that such a solution leads to increased dissipated power. Calculated minimum dissipated power regardless of the maximum current ( ºN R) depending on the position of the magnet array is shown in Figure 2.29 (corresponds to the plots of the maximum currents in Figure 2.24).


Figure 2.29: Position-dependent dissipated power calculated with l2-norm based commutation

with ºN R. Left: steady state levitation T[0,0, ,0,0,0]w mg= , Right: levitation with x, y acceleration T[ , , ,0,0,0]w mg mg mg= .

The results in Figure 2.30 and Figure 2.31 correspond to the situations in Figure 2.25 and Figure 2.26, respectively.

While comparing Figure 2.29 with the uppermost plots in Figure 2.30, one can see that the difference in the dissipated power is negligible (< 5 %). Such a result was obtained thanks to the much lower resistance of the litz-wire coils that was 1.0 W whereas the soil-wire coils have 4.4 W. This leads to the conclusion that the chosen multiplication of the litz-wire coil currents, to obtain the same coupling as in the solid-wire coils, was a good choice.

Comparison of the different commutation algorithms in Figure 2.30 shows how the current in the LiC is reduced at the expense of the increased dissipated power. It is important to point out how the increase of the dissipated power is related to the reduction of the maximum current in the CL2C (see Figure 2.25). If the maximum current is reduced by a few percent (see the saturation at the edges of the workspace in Figure 2.25), the power increase is almost undetectable. On the other hand, for heavily reduced maximum current (near the middle of the workspace) the increased power dissipation needs to be taken into account.

Position-dependent dissipated power for the triangular coil array (TCA) is shown in Figure 2.31. The plots have the same profile variation as in the current maximum case (see Figure 2.26). While comparing the dissipated power of the SCA in Figure 2.25 with the TCA in Figure 2.26, the maxima in the TCA are for the L2C about 10 % lower and more than 30 % lower for LiC. Such a result was expected because the TCA utilizes more coils for the action with less maximum current. Since the produced force increases with the current linearly and principle of superposition is applicable, whereas the dissipated power increases with square of the current, more coils with equally distributed load will have higher efficiency.

The maximum dissipated power depending on orientation of the magnet array is plotted in Figure 2.32. The maximum over the whole magnet array workspace is calculated


Figure 2.30: Position-dependent dissipated power calculated with l2-norm, clipped l2-norm, and

l∞-norm based commutation for steady state levitation T[0,0, ,0,0,0]w mg= and levitation with x, y acceleration T[ , , ,0,0,0]w mg mg mg= ; square coil array.


Figure 2.31: Position-dependent dissipated power calculated with l2-norm, clipped l2-norm, and

l∞-norm based commutation for steady state levitation T[0,0, ,0,0,0]w mg= and levitation with x, y acceleration T[ , , ,0,0,0]w mg mg mg= ; triangular coil array.


-80 -60 -40 -20 0 20 40 60 8010

2

103

Orientation (deg)

Ma

x. d

issi

pa

ted

po

we

r (W

)


-80 -60 -40 -20 0 20 40 60 80

103

104

Orientation (deg)


Figure 2.32: Maximum dissipated power calculated with l2-norm and l∞-norm based commutation

for the square and triangular coil array depending on the orientation of the magnet array. Left: steady state levitation T[0,0, ,0,0,0]w mg= , Right: levitation with acceleration

T[ , , ,0,0,0]w mg mg mg= .

by using the L2C and LiC for the SCA and TCA, and for two wrench vectors T[0, 0, , 0, 0, 0]w mg= and T[ , , , 0, 0, 0]w mg mg mg= as in Figure 2.27 for the maximum current.

The profile of the plots is again comparable to those in Figure 2.13 and Figure 2.17. The different scale is caused by calculating the real dissipated power with included resistance of the coils. The worst-case dissipated power of the TCA for an arbitrary orientation of the magnet array is, at most, two times worse than for the initial (zero) orientation, whereas for the SCA it is more than 30 times worse.

2.4.4 Error Sensitivity

Decoupling of the forces and torques in the planar actuator by using one of the commutation methods described in Section 2.2 is guaranteed only if

( ) ( , )adm

w q q w q S-» K " ÎK . (2.37)

The “»” sign is used, because the calculation of the inverse mapping -K is based on the model of the system, whereas K is a real coupling matrix. The equality sign could be used

if the model would exactly correspond to the real system. Therefore there is a distinction between the desired wrench

desw and the real wrench applied on the planar actuator (w or

realw ) in this thesis:

real des

( ) ( , )w q q w-= KK . (2.38)

Obviously, the desired situation is to keep the wrench error real des

w w wD = - close to zero 0wD . Since the wrench error is sensitive to the differences between the modeled and the real coupling matrix, this section quantifies this sensitivity.


Condition number

Condition number k is a classical measure for the investigation of the error sensitivity (see Section 2.3.1). The normalized condition number * *

2 2( , )-k K K depending on

the position of the magnet array for the square and triangular coil array (SCA and TCA,

respectively) is plotted in Figure 2.33. The commutation is calculated by minimizing the l2-norm of the current vector (L2C). Both coil arrays keep the normalized condition number very low. Even a 10% error of the coupling matrix will not produce singularity in the commutation. A similar test was performed for rotation of the magnet array. Worst-case (maximum over all positions) normalized condition number depending on the orientation of the magnet array is shown in Figure 2.34. For the SCA, the normalized condition number

Figure 2.33: Position-dependent normalized condition number. Left: Square coil array, Right:

Triangular coil array.

-80 -60 -40 -20 0 20 40 60 800

2

4

6

8

10

12

14

16

18

20

Orientation (deg)

No

rm. M

ax. C

ond

ition

Nu

m. (

-)

square arraytriang. array

Figure 2.34: Worst-case normalized condition number for the square and triangular coil array

depending on the orientation of the magnet array. The worst case is calculated over the whole workspace of the magnet array.


increases for 45°-rotated magnet array significantly. Still, if the errors are limited to a few percent as presented in [13], the wrench can be safely decoupled.

Absolute wrench

The condition number can be used to investigate if some coil array/magnet array topology is controllable and how sensitive the solution is to errors of the coupling matrix. However, various commutation methods cannot be compared in this way since for the condition number calculation the inversion of the coupling matrix -K is necessary, where

-K should be independent of des

w .

In this section, a different measure that can be used to compare the commutation methods is presented. The goal is to determine how sensitive the wrench, produced by various commutation methods, is to the errors in the coupling matrix K. To do that, the

absolute wrench abs

w is introduced and calculated as

des

11 1T

abs 1 2 abs

1

abs abs abs

( , ),

( ) ( )

, , , , ( ) ,

( ) ( )

( ) .

n

n

m mn

i q w

K q K q

i i i i q

K q K q

w q i

-= Ké ùê úê úé ù= = ê úê úë û ê úê úë û

=

K

K

(2.39)

Absolute wrench abs

w is calculated from the absolute values of the coupling matrix

absK and the current vector

absi . From the physical point of view, it can be interpreted that

all the forces and torques produced by the individual coils act in the same direction. Alternatively, the absolute wrench represents how large forces and torques are present in the translator when

desw is applied. In an ideal situation, absolute value of the desired

wrench des,abs

w = [| |,x

F | |,y

F | |,z

F | |,x

T | |,y

T T| |]z

T should be equal to abs

w . A difference between

absw and

des,absw represents forces and torques (which are in mechanics known as non-

conservative) that are acting against and canceling each other out. If those forces are large, a small error in the coupling matrix can produce a large wrench error due to violated equilibrium between these forces. The translator is supposed to be rigid, which allows superposition of the individual forces. On the other hand, manufacturing tolerances cause a difference between the modeled and real K.

Any commutation method applicable on the system produces the same final wrench w when there are no errors in K. Their main difference is in the magnitude and

distribution of the forces in the translator.

The worst-case (maximum) absolute wrench abs

w calculated over the whole workspace of the magnet array

waS can be calculated as

abs abs abs

des

abs, abs,

( )( , )

max , 1, , .wa

j jq Sw q i

i q w

w w j m

-

" Î=

=

= =K

K

(2.40)

Table 2.8 shows calculated worst-case absolute wrench abs

w for the L2C and the SCA. Each row represents

absw for unit

desw acting only in one degree of freedom. For

example, the first row represents abs

w when Tdes

[1, 0, 0, 0, 0, 0]w = . Another example: for


Table 2.8: l2-norm based commutation, square coil array

Worst-case absolute wrench per unit des

w [-] abs

w

des

w ,x abs

F ,y abs

F ,z abs

F ,x abs

T ,y abs

T ,z abs

T

xF 1.005 0.614 1.141 0.124 0.123 0.143

yF 0.614 1.005 1.141 0.123 0.124 0.143

zF 0.552 0.552 1.010 0.112 0.112 0.082

xT 5.425 6.199 10.80 1.005 1.205 0.888

yT 6.199 5.427 10.80 1.205 1.005 0.888

zT 6.566 6.567 6.633 0.740 0.739 1.004

lifting force z

F = 208 N, there are absolute forces of 208 × 0.552 = 114.8 N acting in x- and y-direction. This means that there are forces of 57.4 N in opposite directions that should cancel themselves out. For the desired force in x- or y -direction the ratio is even worse. Clearly, the magnetically levitated planar actuator is a fragile equilibrium of large non-conservative forces and torques. Any difference between the modeled and real coupling matrix K causes this equilibrium to be violated.

For comparison of various commutation methods and coil arrays, it is better to normalize the wrench vectors to the comparable quantities as in the condition number calculation in the previous section:

1des

* 1abs abs abs* *

abs des,abs

( , ),

( ) ,

ˆ ,

i q w

w q i

w w w

- -

-

= K

=

= -

NM

MN K (2.41)

where *w is the wrench difference. For the L2C and the SCA the values of the difference in the normalized wrench are in Table 2.9. The same quantity for the LiC and the SCA is in Table 2.10. From comparison of both tables is it immediately clear that the LiC creates

Table 2.9: l2-norm based commutation, square coil array

Worst-case normalized absolute wrench diff. [-] *w

desw *ˆ

xF

*ˆy

F *ˆz

F *ˆx

T *ˆ

yT *ˆ

zT

xF 0.005 0.614 1.141 0.798 0.787 0.728

yF 0.614 0.005 1.141 0.787 0.798 0.728

zF 0.552 0.552 0.010 0.719 0.719 0.416

xT 0.847 0.968 1.687 0.005 1.205 0.702

yT 0.968 0.847 1.687 1.205 0.005 0.702

zT 1.297 1.297 1.310 0.936 0.936 0.004


Table 2.10: l∞-norm based commutation, square coil array

Worst-case normalized absolute wrench diff. [-] *w

desw *ˆ

xF *ˆ

yF *ˆ

zF *ˆ

xT *ˆ

yT *ˆ

zT

xF 0.141 1.958 1.915 1.338 1.312 1.306

yF 1.981 0.141 1.923 1.312 1.337 1.317

zF 1.076 1.075 0.146 0.810 0.810 0.733

xT 1.632 1.704 1.734 0.088 1.274 1.260

yT 1.701 1.633 1.734 1.272 0.088 1.251

zT 1.532 1.543 1.885 1.328 1.302 0.070

much more non-conservative forces and torques. This is not surprising as LiC produces higher dissipative power (see Figure 2.30). The tables for the TCA with the L2C and LiC are in Appendix C.3.

For easier comparison, the whole table on the normalized wrench should be also summarized into one number. Many approaches exist, e.g. taking the maximum value, mean, or some kind of norm. In Table 2.11, the Frobenius norm is used. The Frobenius norm of a matrix A is calculated as 2 1/2( )

ijFa= åA . The norm is zero if the wrench

difference is zero for all unit des

w , and increases with the abs

w difference between abs

w and

des,absw .

From Table 2.11, the triangular coil array can produce slightly more of these forces than the square coil array. As a result, it will be also more sensitive to the errors of the coupling matrix. This finding is in agreement with the results from the condition number investigation in Figure 2.33. On the other hand, the effect between the TCA and SCA is not as high as the difference between the commutation techniques. When the LiC is used, a lot of additional non-conservative forces are produced. Hence, the coupling matrix errors can produce significant wrench error in the case of the LiC.

Table 2.11: Two commutations and coil arrays compared with the Frobenius norm

*ˆF

w square coil array triangular coil array

l2-norm comm. 5.34 5.77

l∞-norm comm. 8.07 8.58

2.4.5 Lower Bound of the Maximum Acceleration and Velocity

Acceleration

Acceleration of the magnet array, or more precisely, the lower bound (smallest) of the maximum acceleration, which can be achieved in the xy-plane with the constraint on


the maximum current is investigated in this section. The peak current for the solid-wire and litz-wire coils are 6 A and 20 A, respectively. These two values limit the acceleration. Moreover, part of the power is necessary for gravity compensation

zF mg= . The lower

bound is found from the lower bound of the amplitude of the force ,xy wc

F in any xy-direction [76]:

T

T

max

, 2( , )[ , ,0,0,0]

[ , ,0.001,0,0,0]

arg min , xy

xy

xy wc xy waq w iF

w F mgq x y

i i

F F x y S-

¥

K ==

==

= " Ì , (2.42)

where [ , ]xy x y

F F F= are forces in the xy-plane, -K is the inverse mapping calculated with one of the commutation methods,

maxi is the current limit value, and

waS is an admissible

set of coordinates of the workspace of the magnet array. The minimization (2.42) searches for the 2-norm minimal

xyF vector, which would cause that at least one of the currents

reaches the maxi limit. The lower bound of the maximum acceleration in the xy-plane can be

derived from the worst-case forces by dividing them by the mass of the translator m.

The lower bound of the maximum acceleration was investigated on the square coil array (SCA) of the original COPAM model as well as on the modified model with the triangular coil array (TCA). Moreover, two commutation methods (L2C and LiC) have been used. The levitated mass is 21.2 kg with the levitation height of 1 mm. All the angles are set to zero if not said otherwise.

The results for the SCA for the L2C and LiC are in Figure 2.35 and Figure 2.36, respectively. From the figures, the lower bound of the maximum acceleration for L2C is

about 30 m.s-2, and for LiC more than 44 m.s-2. The increased acceleration in the LiC is, of course, at the price of a huge dissipated power. From the comparison, it is visible that the topology of the lower bound of the maximum acceleration remains approximately the same for both commutation methods.

In Figure 2.37 and Figure 2.38, the plots of the lower bound of the maximum acceleration on the TCA for L2C and LiC, respectively, are presented. As expected, the results are about 15-20 % higher than for the SCA. More interesting is the fact that the difference between the minimum and maximum point of the plots is much less than for the SCA. In other words, the performance is less dependent on the position of the magnet array.

The plots for the magnet array 45°-rotated above the SCA and 15°-rotated above the TCA are in Figure 2.39 and Figure 2.40, respectively. These two angles represent the worst-case orientation of the magnet array.

For 45° and the SCA, only LiC can be used with the lower bound of the maximum

acceleration of only 4 m.s-2. For 15° and the TCA, the minimum value is about 20 m.s-2 for the L2C. For the other orientations, inversion of the plot of the maximum currents during the acceleration phase (see Figure 2.27 right) can be indicative.


pos. x (m)

pos.

y (

m)

Low er bound on the maximum acceleration (m.s-2 )

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.01

0.02

0.03

0.04

0.05

0.06

30 32 34 36 38 40 42 44 46

Figure 2.35: The lower bound of the maximum xy-plane acceleration (magnitude and direction)

depending on the magnet-array position for the SCA and L2C.

pos. x (m)

pos.

y (

m)


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.01

0.02

0.03

0.04

0.05

0.06

45 50 55 60 65


depending on the magnet-array position for the SCA and LiC.


pos. x (m)

pos.

y (

m)


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.01

0.02

0.03

0.04

0.05

0.06

35 36 37 38 39 40 41 42 43 44


depending on the magnet-array position for the TCA and L2C.

pos. x (m)

pos.

y (

m)


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.01

0.02

0.03

0.04

0.05

0.06

56 58 60 62 64 66 68 70 72 74 76


depending on the magnet-array position for the TCA and LiC.


pos. x (m)

pos.

y (

m)


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.01

0.02

0.03

0.04

0.05

0.06

5 10 15 20 25 30 35 40 45


depending on the 45°-rotated magnet-array position for the SCA and LiC.

pos. x (m)

pos.

y (

m)


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.01

0.02

0.03

0.04

0.05

0.06

20 25 30 35 40 45


depending on the 15°-rotated magnet-array position for the TCA and L2C.


Velocity

Next to the acceleration, maximum velocity is another important design factor. The maximum achievable velocity is dependent on the required voltage, which is limited by the amplifier specifications (see Section 2.4.2). However, the coil terminal voltage is also given by the actual acceleration and jerk. The maximum acceleration considered in this section uses up to 6 A of the current (solid-wire coils). This gives the resistive voltage of Ri = 4.4 W 6 A = 26.4 V. With the maximum voltage of 52 V, there is about 25 V left for

velocity and jerk. As was shown in Table 2.6, a jerk of 1000 m.s-3 requires a maximum voltage of less than 2 V (with the L2C). By combining the results presented in Table

2.5 with the calculated back-EMF, the minimum achievable velocity with the L2C is about 2 m.s-1.

With the CL2C, the value slowly decreases when the current limit is pushed to the lower values. For the presented example of the CL2C applied on the SCA and TCA (see Section 2.4.1 and 2.4.3), the minimum achievable velocity with the same acceleration and jerk is about 1.2 m.s-1 and 2 m.s-1 on the SCA and TCA, respectively.

2.5 Summary and Conclusions

In this chapter, several topics related to commutation of magnetically levitated planar actuator with moving magnets were addressed.

In Section 2.2, three norm-based commutation methods were compared: the l2-

norm, l∞-norm, and clipped l2-norm based commutations (L2C, LiC, and CL2C, respectively). The L2C was developed and successfully used in the past for control of magnetically levitated planar actuators. Its main benefit is in the minimization of dissipated power and fast analytical solution. The drawback is its inability to constrain the maximum current, which can cause saturation of the amplifiers and consequently the difference between the desired and the real forces and torques. Therefore, two other commutation techniques have been developed. The benefit of the LiC is that the maximal current is always minimized. On the other hand, since almost all active coils are set to the same maximum current level, although they can have minimal influence on the final force/torque, the power usage increases dramatically. Therefore, a novel, third kind of norm-based commutation is proposed which puts constraints on the maximal current while

the non-saturated coils are minimized in the l2 norm. By varying the clipping (saturation) value, the obtained solution is closer to either the L2C or LiC. The applied algorithm for the CL2C does not necessarily lead to the optimal solution, but the performed tests

indicate that the optimal solution in l2-norm sense can be obtained in the majority of cases whereas the suboptimal solution is still very close to the optimal one and practically indistinguishable in the real system. The benefit of this algorithm is in calculation time, which is reduced considerably in comparison to the LiC or optimal CL2C obtained via quadratic programming. Continuous current-variation with position and wrench variation can be guaranteed in the L2C and can be achieved in the CL2C if the solution is optimal. Nevertheless, as was shown in the example, the current discontinuity caused by the CL2C will not introduce high voltage peaks in the current amplifiers.


In Section 0 of this chapter, possible bottlenecks for full rotation of moving-magnet planar actuators were described. From many existing PA designs, it was found that only those with rounded coils are suitable for full rotation, because the commutation algorithm does not have to be of increased complexity. This is in contrast to the square- or rectangular- coil PA, which must include the orientation of a single coil into the commutation algorithm. It was also found that the coil width and spacing optimized for translational movement can produce severe controllability problems when the magnet array is rotated about 45 degrees, leading to a few-times increased maximum current and power losses.

A new, triangular, array of the coils instead of the square array was proposed, which can have better properties for any orientation of the magnet array for given coil dimensions and spacing. Moreover, the performance can be better or at least comparable to the square coil array for some coil spacings even without rotation of the PA. This makes the triangular array preferable also for the applications where rotation of the planar actuator is not required.

It also has been shown that a limited number of highly symmetric topologies for the array of rounded coils exist. However, many magnet array topologies can be found. Another interesting option for the full rotation might be a hexagonal Halbach magnet array in combination with a square array of coils, which also has higher rotational periodicity than the standard design.

The sensor system for rotating planar actuator is the main remaining difficulty. No sensor system for such a planar actuator exist today that could satisfy industrial requirements for resolution and range. Research of new types of 3-DOF optical sensors similar to optical sensors for computer pointers is proposed.

In Section 2.4, the L2C, LiC, and CL2C with square and triangular coil arrays were compared on the model of the COPAM. Various variables were compared, e.g. maximum current and voltage, dissipated power, error sensitivity and worst-case acceleration and velocity. The comparison of the commutation methods showed that LiC could reduce peak currents significantly, but at the price of much higher dissipated power and required voltages. As a compromise, CL2C can be used when occasional current limitation is necessary.

From the comparison of different coil arrays, less peak current and power is required with the triangular coil array (TCA). Moreover, the TCA is less sensitive to the orientation of the magnet array with respect to the coil array. On the other hand, these findings are valid only for the specified coil spacing and width as was shown in Section 2.3.1. Moreover, rotation on the square coil array (SCA) without increased power requirements is possible up to ± 30 degrees.

From the error sensitivity investigation, both coil arrays produce a well-conditioned coupling matrix. A new approach for investigation of the error sensitivity of the planar actuators was proposed. The desired wrench is compared with the absolute wrench calculated with the absolute values of the coupling matrix and the absolute values


of the applied current vector. By using the absolute wrench, it was shown that a magnetically levitated planar actuator is a fragile equilibrium of large non-conservative forces and torques. It was also confirmed that LiC can almost double these forces.

Investigation of the lower bound of the maximum acceleration showed high potential in the rounded-coil topology. The maximum possible acceleration in the whole

working range with L2C is about 30 m.s-2. Since the acceleration is mainly limited by the peak currents, LiC can increase the worst-case acceleration by almost 50% (at the price of increased dissipated power).

Study of the velocity showed that even with the maximum possible acceleration, a

minimum velocity of 2 m.s-1 can be achieved with the L2C. With the CL2C, the value slowly decreases with pushing the current limit to the lower values. For the presented example of the CL2C applied on the SCA and TCA, the minimum achievable velocity with the same acceleration and jerk is about 1.2 m.s-1 and 2 m.s-1 on the SCA and TCA, respectively.

As was shown, TCA can provide better performance than SCA in almost all aspects. TCA allows for full rotation of the PA within the current limits. Moreover, the peak current, voltage and dissipated power are about 10-20 % lower also for the zero orientation angle and, hence, higher accelerations and velocities can be achieved. This makes the triangular array preferable even for the applications where rotation of PA is not required. Since the peak current as well as the power losses are lower, demands on cooling for TCA should not be higher than for SCA. Nevertheless, investigation of the thermal properties of TCA and other parameters necessary for full design of a PA is out of the scope of this thesis. The only major drawback of the triangular array of coils found can be about 15% more coils and current amplifiers required per unit area.

63

Chapter 3

Errors in Commutation

3.1 Introduction

The sensitivity of the wrench decoupling on the errors in the commutation was discussed in Chapter 2, Section 2.4.4. The main conclusion from that section is that the magnetically levitated planar actuator is a fragile equilibrium of large forces and torques acting against each other and, in an ideal situation, cancelling each other out. Precise cancellation can be achieved only if the model of the coupling matrix, used for the commutation calculation, corresponds as much as possible to the real coupling between the currents in the coils and produced wrench. If there is a difference in the coupling, the forces are not exactly cancelled and their difference appears as a disturbing force producing acceleration of the translator.

The sensitivity of the wrench error to the coupling errors, when various commutation methods are used, was discussed in Section 2.4.4. In this chapter, sources of errors in the coupling matrix K and sensitivity of K on these errors are discussed.

Moreover, a procedure for estimation and elimination of the most severe source of the errors is described and its stability is investigated. The findings and results presented in this chapter have been published in [43, 42].

Since the coupling matrix K has n times more elements, where n represents the

number of the coils, which can all vary with the errors, than the wrench vector w, it is more practical to show influence of the various error sources on the wrench decoupling

error. Hence, a commutation needs to be chosen. As l2-norm based commutation (L2C) is a common commutation method used in various planar actuator designs [95, 78, 13], it will be used also here. Its benefits are the least dissipative power (see Section 2.4.3) and a low sensitivity to the coupling matrix errors (see Section 2.4.4) in comparison to other norm-based commutation methods. Conclusions based on L2C can also be extended on l∞-norm based commutation (LiC) when the results are combined with those from Section 2.4.4.

64 Chapter 3. Errors in Commutation

Generally there are two kinds of errors in commutation:

Hardware-related - result of trade off between price and accuracy. Software-related - even if one can measure everything, everything can not be

included in the commutation model due to limited calculation time and memory.

Calculations made in this section are based on a surface-charge model of the magnetic field and Lorentz force method described in Chapter 2. The model allows for relatively fast calculation of the forces and torques acting among the coils and the magnet array. The simulation parameters were taken from the dimensions of the COPAM (see Appendix A) or can be found in Appendix C.1.

3.2 Software-related Errors

3.2.1 Numerical Precision of the Modeled Coupling

The combined analytical-numerical Lorentz force model [56] (see Section 2.1) used for the calculation of the coupling uses simplifying assumptions that lead to a reduced

accuracy of the model while the calculation time is reduced by a factor 104 in comparison to the FEM model [16]. The first simplification is a relative permeability

rm equal to 1.0 in

the surface-charge model of the magnets. Since the high-quality NdFeB magnets are used with 1.03

rm = , the relative error is about 1.5 % [57]. The second simplification is by

neglecting reluctance forces due to the permanent magnets with 1r

m ¹ , which introduces an additional error of < 0.3 % [56].

3.2.2 Look-Up-Table-Based Description of the Coupling

For the real-time control of the PA, it is necessary to calculate the commutation for each position very fast. Even with up-to-date computers, it is not possible to calculate the commutation analytically with very high precision in time. It is possible to reduce the analytical model and design the coil topology in such a way that the analytical calculation of the coupling in real time is possible, but at the price of limits in design and performance such as neglected end effects of the magnet array [58, 112].

The way to obtain a more accurate coupling in real time is to pre-compute inversion of the coupling matrix K off-line and store it in as a look-up-table (LUT) for a dense grid of positions. Then the commutation algorithm in the real time for each coil just picks up values from the LUT. The problem is with the size of the LUT. An accurate description of the position-dependent coupling in LUT can easily occupy a few gigabytes of memory [18]. Therefore, a reduction of the LUT has been developed in [18]:

Use is made of symmetries in the magnetic flux density distribution under the magnet array. For each coil position with respect to the magnet array (

MAq S" Î ),

the coupling vector K between single coil and the magnet array is stored in the LUT. Later, combination of all the coils will approximate the coupling matrix K.

3.2 Software-related Errors 65

Consequently, computing the inversion beforehand is no longer possible, but the LUT is largely reduced.

Three small areas of the coupling vector K are used to represent the whole area of the position-dependent K (see Figure 3.1). The first one is corner area of 2t 2t magnet-pole pitch sizes, the second is picked from the side area of 2t t and the third one is t t from the inner area where the periodicity is high.

The symmetry between the different forces and torques is used. Three-dimensional (3D) coupling LUT is approximated by 2D xy-coupling with

exponential decay in z-direction. Angles about the x- and y-axis are assumed to be small enough to be neglected,

i.e. < 5 mrad.

This reduces the precision of the LUT, but the resultant size is a few hundreds of kilobytes. Moreover, the method allows for many different topologies and LUT commutation takes into account end effects of the magnet array. In [18] the precision of the LUT was tested only for a few parts of the K. Here, a more detailed view on coupling errors is presented. Figure 3.2 and Figure 3.3 show the difference between the full (see Figure 2.1 and Figure 2.2) and the reduced LUT-based coupling for

FzK and

TxK

components, respectively. It is visible that the LUT does cover most areas of K perfectly, but there are small areas around the corners that are neither periodic nor symmetric and consequently cannot be described by this LUT perfectly. A possible solution would be to increase the size of the corner LUT to enclose that area if the memory requirement allows that. The question remains what force and torque ripples acting on the translator this

τ

τ

2τ

2τ

Figure 3.1: Principle of reduction of the LUT coupling size by using three small areas of the

position-dependent coupling vectorK.


Figure 3.2: Absolute error between exact- and LUT-calculated coupling for z

F .

Figure 3.3: Absolute error between exact- and LUT-calculated coupling for x

T .

might produce. To simulate this, precise coupling was used in the model together with L2C calculated from the LUT coupling. An example of the position-dependent ripples

zFD and

xTD when a force in

xF is applied is shown in Figure 3.4. The maximum error for

zF is

about 0.25 % of the x

F force. All maximum errors are summarized in Table 3.1. To make the values with the different quantities comparable, the entries in the table are normalized with the effective arm (2.31) as was described in Section 2.3.1. The same is done with all the tables in this section. For example, the value in the first row and second column in Table 3.1 shows that applying the desired

xF (e.g. 100 N) can cause maximum ripple

yFD =

0.035 % x

F = 0.00035 x

F (0.035 N).


Figure 3.4: Example of relative position dependent errors for commutation calculated by using

LUT; Left: z

FD when force in x

F is applied; Right: *x

TD when force in x

F is applied.

Table 3.1: Errors due to the LUT commutation

Normalized maximum relative errors *wD /w (%) *wD

w *x

FD *

yFD *

zFD *

xTD

*y

TD *z

TD

xF 0.044 0.035 0.251 0.025 0.307 0.020

yF 0.035 0.044 0.251 0.307 0.025 0.020

zF 0.064 0.064 0.031 0.032 0.032 0.010

xT 0.018 0.020 0.075 0.136 0.021 0.077

yT 0.020 0.018 0.075 0.021 0.136 0.077

zT 0.060 0.060 0.042 0.309 0.309 0.073

From the table it is clear that the errors are less than 0.4 % and mostly less than 0.1 %. It should be stated that the errors are linearly dependent on the wrench component.

3.3 Hardware-related Errors

3.3.1 Coil Position

Due to mechanical tolerances, each coil can only be placed with a limited precision. To imagine the magnitude of the wrench error that such an alignment can produce, the following calculation has been made. Wrench error

jw produced by a position

offset j

qD of the coil j at a particular position j

q can be approximated by using the first order term of the Taylor expansion:

( )

( ) ( , ) ,jj K q j j j j j

w q q i q q iD = D = DJ J (3.1)


where ( )

( )jK q

qJ is the Jacobian matrix of K in the position j

q with respect to the coordinates

q :

( )

( ) ( )

( )

( ) ( )

x x

j

z z

F F

K q

T T

jq q

K q K q

xq

K q K q

x =

é ù¶ ¶ê úê ú

¶ ¶fê úê ú= ê úê ú¶ ¶ê úê úê ú¶ ¶fë û

J

. (3.2)

Then the total wrench error is the summation over all coils

1 1 1 1 2

( , ), , ( , ) ( , ), , ( , ) ( )n n n n

w q q q q i q q q q q w-é ù é ùD = D D = D Dê ú ê úë û ë ûJ J J J K . (3.3)

This relation represents the position-dependent wrench error as the contribution of all misaligned coils. At this point, it should be clear that there are infinitely many combinations of position offsets of the coil set. Therefore, it is better to restrict the problem to the maximally possible contribution to the error of one coil in any possible position q . In Figure 3.5, two examples of the position dependent error are shown. Each example is for one coil that produces the biggest error of the investigated error-wrench vector element for the particular applied wrench. An overview of maximum error that any coil in any position can produce if it has an offset in the x-direction is shown in Table 3.2. For reasons of symmetry, the same holds for a small offset in the y-direction. The results for an offset in the z-direction (vertical) are in Table 3.3. The values in the tables are calculated as errors per millimeter offset. For small offsets in comparison to the magnet-pole pitch, the errors are assumed linear with the offsets. Usual mechanical tolerances are 0.1 mm, so the maximum errors are 10 times smaller and the worst-cases are presented. On the other hand, these errors are just for one misaligned coil; hence, summation over all coils will give worse numbers although averaging effect of randomly distributed errors will cause reduction of the final error.

Figure 3.5: Example of position-dependent error in one coil, which produces the biggest error;

Left: z

FD when force in x

F is applied and coil no. 34 has offset in x-direction; Right: x

TD when torque in

xT is applied and coil nr. 65 has offset in z-direction.


Table 3.2: Coil position offset in x-direction

Normalized maximum *wD /w/Δx errors (%/mm) Δx

*

xFD

*

yFD *

zFD *

xTD

*

yTD *

zTD

xF 0.387 0.436 0.332 0.313 0.243 0.323

yF 0.387 0.438 1.011 0.838 0.819 0.288

zF 0.174 0.585 0.466 0.377 0.358 0.374

xT 0.357 1.022 0.960 1.032 0.808 0.684

yT 0.362 1.004 0.767 0.651 0.639 0.675

zT 0.665 0.504 1.093 1.137 1.138 0.518

Table 3.3: Coil position offset in z-direction

Normalized maximum *wD /w/Δz errors (%/mm) Δz

*

xFD

*

yFD *

zFD *

xTD

*

yTD *

zTD

xF 0.792 0.248 0.666 0.531 0.486 0.531

yF 0.243 0.792 0.666 0.486 0.531 0.531

zF 0.343 0.343 0.906 0.697 0.697 0.242

xT 0.576 0.742 1.548 1.250 1.262 0.519

yT 0.742 0.576 1.548 1.262 1.250 0.519

zT 0.863 0.863 0.762 0.846 0.846 0.752

Excursion: By looking at (3.3), one could think of reconstruction of the exact coil positions

from measurements. To solve (3.3) with , 1, ,j

q j nD = as variables, the wrench error wD , applied current i and the real coupling ( )K q are required. The wrench error

real desw w wD = - can be obtained in steady state levitation when the real wrench only

compensates gravity T

real[0, 0, , 0, 0, 0]w mg= and

desw is the known applied wrench. The

applied current vector is also available and the exact model of the coupling could be obtained by measuring the wrench produced by one active coil moving over the whole area of the magnet array.

Since dim( )wD is much smaller than the number of searched variables that is equal to dim( )q nD ´ , at least several measurements of the wrench error are required. Each measurement must be performed for a different current vector i or a different position of the magnet array. Since a consistent system of linear equation is necessary, varying the position of the magnet array over the whole workspace is preferable in order to achieve activation of all the coils. It is also better to obtain more measurements and solve over-determined system (3.3) in the least squares sense to reduce the influence of the measurement noise.


The process of finding the exact coil position is iterative: 1k k kj j j

q q q+ = D + , 1, ,j n= . In each step, a better approximation of

jq should be obtained.

Simulation of the proposed algorithm was performed first on the commutation model of the COPAM without any additional noise or errors in the model. The coils were misaligned in x-, y- and z-direction by a random offset of the value between 0 and 0.1 mm. From the simulation, the following was found:

The algorithm converges to the real coil positions only if ( )

( )jK q

qJ is very accurate.

A difference of 0.1 % can cause an error in the estimated position offset of several hundreds of percents.

A more precise solution can be found if the number of the measurements is much higher than the number of variables and the highly over-determined linear system of equation is solved in the least squares sense.

The number of necessary measurements needed for an approximate solution increases with the number of coils exponentially. E.g. for four misaligned coils about five measurements were enough, but even 200 measurements did not decrease the offset of 36 misaligned coils.

The conclusion is that it is NOT possible to estimate the exact coil positions in this way, because the estimated position-offsets are very sensitive to the wrench errors. Therefore, the accuracy of the modeled coupling must be very high. Moreover, in the real system the calculation is influenced by other sources of errors discussed further. On the other hand, the previous conclusion also means that the coupling error is less sensitive to the precise placement of the coils.

3.3.2 Coil Dimensions

In general, coil dimensions are another source of possible errors in commutation. The most significant is usually the variation in coil width, due to the winding process, that in our case was 51 ± 1.2 mm. A difference between minimum (or maximum) and nominal coil width can cause a difference in coupling of about 0.15 N/A for

FxK and 0.25 N/A for

the entries in Fz

K . Table 3.4 again presents maximum errors that any coil with minimal or maximum dimensions in any position can produce. Once more, it should be pointed out that the table presents the maximum contribution to the errors of only one coil.

3.3.3 Magnet Array Bending

If the translator is not ideally stiff, the magnet array on the bottom side can bend. There can be two reasons for the bending. The first cause is static: gravity, tension due to the gluing process, and forces between the magnets in the array. The second cause is dynamic: vibration amplification at the resonance frequency of the translator. Bending of the center of the magnet array (due to the forces in the Halbach magnet array [57]), about the diagonal and about one of the horizontal axes was simulated (see Figure 3.6). The metric dimension of bending means the difference between the highest and the lowest point


Table 3.4: Coil dimension deviation ± 2.5%

Normalized maximum *wD /w errors (%)

*

xFD

*

yFD *

zFD *

xTD

*

yTD *

zTD

xF 0.146 0.146 0.240 0.249 0.249 0.146

yF 0.146 0.146 0.240 0.249 0.249 0.146

zF 0.114 0.114 0.187 0.198 0.198 0.116

xT 0.206 0.206 0.338 0.356 0.356 0.205

yT 0.206 0.206 0.338 0.356 0.356 0.205

zT 0.185 0.185 0.305 0.321 0.321 0.186

Figure 3.6: Simulated bending of the magnet array; central, diagonal and axial bending.

of the bended magnet array. The maximum wrench errors for the central bending c

z , bending about the diagonal

dz and about the x-axis

xz are presented in Table 3.5, Table 3.6

and Table 3.7, respectively. The bending can be mainly seen as varying vertical positions of the individual coils. Since the coupling vector K changes only its magnitude in the vertical direction, the bending mainly causes the difference in the wrench gain, but not too much cross-coupling. This can be seen as higher diagonal terms in the tables. Estimated bending

Table 3.5: Magnet array central bending

Normalized maximum *wD /w/Δc

z errors (%/mm) Δ

cz

*

xFD

*

yFD *

zFD *

xTD

*

yTD *

zTD

xF 3.970 0.106 0.570 0.016 0.387 0.184

yF 0.107 3.970 0.580 0.388 0.016 0.164

zF 0.290 0.290 4.100 0.172 0.170 0.005

xT 0.015 0.275 0.309 3.695 0.040 0.246

yT 0.275 0.015 0.268 0.040 3.695 0.248

zT 0.375 0.375 0.013 0.449 0.450 3.116


Table 3.6: Magnet array bending about diagonal

Normalized maximum *wD /w/Δd

z errors (%/mm) Δ

dz

*

xFD

*

yFD *

zFD *

xTD

*

yTD *

zTD

xF 4.796 0.061 0.301 0.179 0.288 0.131

yF 0.061 4.824 0.302 0.294 0.179 0.227

zF 0.146 0.149 4.890 0.089 0.089 0.086

xT 0.139 0.143 0.233 4.697 1.838 0.564

yT 0.143 0.174 0.248 1.839 4.613 0.545

zT 0.262 0.408 0.268 0.547 0.540 4.385

Table 3.7: Magnet array bending about x-axis

Normalized maximum *wD /w/Δx

z errors (%/mm) Δ

xz

*

xFD

*

yFD *

zFD *

xTD

*

yTD *

zTD

xF 3.037 0.212 1.490 0.038 1.325 0.242

yF 0.059 3.767 0.025 0.236 0.006 0.263

zF 0.743 0.049 3.589 0.012 0.441 0.020

xT 0.020 0.045 0.037 3.591 0.020 0.494

yT 0.714 0.045 0.695 0.079 2.126 0.124

zT 0.044 0.474 0.025 1.118 0.037 2.316

in our setup is about 0.1 mm, while the translator is controlled just about 1 mm above the surface. In that case the maximum values are 10 times smaller and they are total (no summation or averaging as in the case of the coil errors).

3.3.4 Magnets

The magnets inside the magnet array are also not ideal. The magnets are manufactured and placed with some tolerance, usually 0.1 mm or better. Magnetization in magnets is not uniform and the magnetization vector is not completely perpendicular to the magnet surface. Moreover, the magnetization decreases with the increased temperature by approximately 1 % per 10 K. As some of the magnets in the magnet array are more heated than others (local heating), the magnetic field produced by the magnet array has local variances. All these inaccuracies are very time consuming to simulate. Since the magnets are the second component that together with the coils produces the wrench, it can be expected that the tolerances in the placement and dimensions of the magnets cause similar errors to the tolerances of the coils.


3.3.5 Measurement System

An adjusted and reliable measurement system is an important part of commutation. If there is an offset between the real and the estimated position of the

magnet array with respect to the coil array, the commutation will produce another unwanted wrench error. Table 3.8 shows an example of individual maximum force and torque errors per millimeter offset in the x-direction. From the table it is clear that even 1 mm offset can produce more than 10 % error in some direction. An explanation of such a large error is in the sum of all the coil-position errors without the averaging effect. As these errors were studied carefully, it was found that offset in some direction produces big and almost constant errors only in some particular directions. It has been found, that, if only

zF

is applied the produced wrench error is mainly in the same direction as the direction of the offset. This relation is summarized in Table 3.9. Based on this idea an algorithm was developed that can iteratively estimate and eliminate these offsets [13]. In Section 3.5, the principle of the algorithm is described and proof of convergence is given.

Table 3.8: Measurement system, x-direction offset

Normalized maximum *wD /w/off

x errors (%/mm) off

x *

xFD

*

yFD *

zFD *

xTD

*

yTD *

zTD

xF 0.921 1.491 11.42 0.422 0.691 0.096

yF 0.222 1.738 0.127 0.256 0.083 0.653

zF 6.028 0.209 0.554 0.083 0.601 0.070

xT 0.208 0.119 0.257 1.320 0.121 4.929

yT 0.359 0.089 0.478 0.814 4.271 0.156

zT 0.121 1.425 0.377 10.73 0.221 3.353

Table 3.9: Measurement system, only z

F applied

Norm. max. *wD /z

F /off

q errors (%/(mm or mrad)) z

F *

xFD

*

yFD *

zFD *

xTD

*

yTD *

zTD

offx 6.028 0.209 0.554 0.013 0.094 0.014

offy 0.200 6.017 0.517 0.105 0.011 0.010

offz 0.002 0.003 10.35 0.001 0.001 0.000

offq 0.240 0.164 0.578 0.232 0.066 0.020

offy 0.057 0.025 0.035 0.004 0.187 0.002

offf 0.243 0.094 0.583 0.050 0.067 0.207


Another aspect is noise of the measurement, which is usually not a problem for the

commutation, because if the sensor-noise amplitude is < 1 μm, the errors from the tables are at least a factor 1000 smaller. For example in steady-state levitation when .

zF m g= »

200 N and sensor noise off

z = ±1 μm, the worst-case error amplitude due to the sensor

noise is z

FD = 200 N 103.5 10-6 = ±0.02 N = 0.01 % of z

F . Usually, the sensor noise is at least a factor 10 smaller, which reduces the wrench noise even more.

3.3.6 Current amplifiers

Current amplifiers are the last significant source of errors. Gain differences (linearity), offsets and drift of those quantities can contribute to the errors. The drift causes that even when the amplifiers is tuned and inversed to high precision at the beginning, it can cause a several-orders-of-magnitude worse errors after some time. The error owing to a current offset depends on the coupling K. Limited bandwidth and delay cause errors during transient operation. A delay

dt (s) causes to the PA running at actual

horizontal velocity xy

v (m.s-1) a position offset of xy d xy

q t v= (m). The produced wrench errors can be calculated by using Table 3.8. This finding is valid for all the delays in the control loop. On the other hand, when the delay and the trajectory are known in advance, the position offset can easily be predicted and corrected in commutation. A high-quality output filter is another, not less important, part of the current amplifier, because pulse-width modulation (PWM) produces many high-frequency signal components that become signal noise.

The amplifiers in the COPAM have bandwidth of about 3 kHz with about 0.3 ms delay due slow communication with the controller. When assuming the maximum velocity

of 1 m.s-1, the position offset due to the delay can be up to 0.3 mm. With the coupling factor of 10 N/A, the current noise in amperes produces up to ten times higher noise in newtons. Anyhow, the amplifiers in the COPAM have RMS noise level less than 1 mA, so RMS force noise level is less than 10 mN.

3.4 Comparison and Summary

The estimated importance of the software-related error causes are as follows:

1) Analytical-numerical Lorentz-force model – Estimated relative error of about 2-3 % [56].

2) Look-up-table description of coupling [13] – The maximum error of 0.5% is small in comparison to the other sources of the errors.

The same for the hardware-related error causes:

1) Measurement system offset – One millimeter of the position offset can produce 10 % of the wrench difference. Nevertheless, these can be estimated and eliminated by using an algorithm described in Section 3.5.

2) Coils position/dimensions – The magnitude of the maximum error that one coil can produce is not severe, but more coils can produce higher error. Although an


averaging effect reduces the magnitude of the total error, it can be locally increased for some coordinates of the translator.

2) Magnet placement/dimensions – Magnets are the second component that together with coils produces the wrench, therefore their tolerances are as important as the tolerances of the coils.

3) Magnet plate bending – Since it produces mostly gain differences, and almost no cross-coupling, it is less severe. Moreover, the error is total (so summation of individual components as in the case of the coils/magnets).

4) Measurement system noise – Should be the smallest problem for the commutation. If the noise amplitude of the sensor is less than 1 μm, the produced wrench-noise magnitude is less than -80 dB of the wrench signal.

–) Current amplifiers – They might be severe or not depending on the linearity, noise, delay, bandwidth and other parameters. The current amplifiers in the COPAM exhibit excellent linearity and noise level in comparison to the rest of the system. On the other hand, the delay and limited bandwidth reduce the performance during transient operation.

Other sources of errors exist which are either difficult to quantify (e.g. magnet magnetization uniformity, perpendicularity of the magnetization vector) or have minor influence on the commutation precision (e.g. temperature variation of the parameters). Combining all the sources of errors one should expect a total error of a few percents when the mechanical tolerances of the components are about 0.1 mm. The results from the wrench error measurements on the COPAM prototype showed the peak wrench error of 3 % and RMS errors of about 1.5 % when tolerances on the coils and magnets were 0.1 mm and 0.05 mm, respectively [15, 13] (also see Chapter 4, Figure 4.5).

As the findings have been supported by simulation results based on a model of commutation of the COPAM, the relative effect of the errors can be indicative for similar planar actuators.

3.5 Auto-Alignment of the Measurement System

This section presents and investigates the stability of the procedure originally introduced in [13], which can estimate and eliminate offsets of the measurement system of the magnetically levitated planar actuator with moving magnets. Moreover, an extension to different coil topologies is made. Simulation and experimental results, based on the Contactless Planar Actuator with Manipulator (COPAM) project will be presented as examples. As was presented in Chapter 2, ironless magnetically levitated planar actuators

can be described by a linear relation between a current j

i applied to the jth-coil in the stator and a wrench vector

jw . The final wrench acting on the platform is a superposition

of the contributions of all active coils. This relation can be described with a coupling matrix K between current vector i applied to the coils and the final wrench vector acting

on the platform: ( )w q i= K , where T[ , , , , , ]q x y z= q y f is the position and orientation vector of the levitated platform with magnet array with respect to the coordinate system of the


stator (see Section 2.1). The wrench vector w contains three force and three torque components: T[ , , , , , ]

x y z x y zw F F F T T T= . Because the number of active coils is much larger

than the dimension of the wrench vector, the commutation needs to distribute the desired wrench over many coils. Therefore, there is design freedom for optimization. Usually, the criterion for the setup is minimal power consumption which leads to the minimization of the 2-norm of the current vector. Such a condition is satisfied when the commutation is

calculated using the pseudo-inversion of the coupling matrix ( )1

T T2

-- =K K KK (see Section

2.2.1). The current vector 2 est des( )i q w-= K , where

desw is the desired wrench and

estq is the

estimated position of the PA. So the real wrench real

w is calculated as:

real 2 est des

( ) ( ) ( )w q i q q w-= =K K K . (3.4)

As was shown in Section 3.3, the errors in the measurement system are more severe than the accuracy of the calculated coupling matrix. Therefore, for now it will be assumed that

2-K is calculated based upon true K. If also

estq qº , one get:

( )1

T Treal des des

( ) ( ) ( ) ( )w q q q q w w-

= =K K K K (3.5)

3.5.1 Consequence of Offsets in the Measurement System

In reality, if the measurement system of a magnetically levitated planar actuator (PA) is not well calibrated, it produces an offset

offq on the position and orientation q and

consequently errors on the real wrench (see Figure 3.7). These errors can produce quite severe force and torque ripples. Table 3.8 showed an example of such errors produced by offset in the x-direction of the measurement system. As illustrated in Figure 3.7, the

current vector i was calculated by the l2-norm based commutation algorithm with an offset on the position estimate. Next, i was fed into the model of the coupling without that offset to predict what forces and torques would act on the PA, and what the difference between the desired and real values is:

real 2 off des

real des

( ) ( ) ,

.

w q q q w

w w w

-= +D = -

K K (3.6)

As these errors were studied carefully, it was found that an offset in a particular direction produces big and almost constant (independent on position) errors only in some directions. It has been found that the measurement offsets mainly affect the corresponding wrench

Figure 3.7: Commutation algorithm.

P(s)

( )qK

2 off( )q q K

C wdes wreal i

ref q

offq

Planar actuator


-40

-20

0

20

40F

orc

e e

rro

r (N

)

Fx

Fy

Fz

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-3

-1

-0.5

0

0.5

1

offset in x-direction (m)

To

rqu

e e

rro

r (N

m)

Tx

Ty

Tz

-1

-0.5

0

0.5

1

Fo

rce

err

or

(N)

Fx

Fy

Fz

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-3

-1.5-1

-0.50

0.51

1.5

offset in -orientation (rad)

To

rqu

e e

rro

r (N

m)

Tx

Ty

Tz

-40

-20

0

20

40

Fo

rce

err

or

(N)

Fx

Fy

Fz

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-3

-1

-0.5

0

0.5

1

offset in y-direction (m)

To

rqu

e e

rro

r (N

m)

Tx

Ty

Tz

-1

-0.5

0

0.5

1

Fo

rce

err

or

(N)

Fx

Fy

Fz

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-3

-1.5-1

-0.50

0.51

1.5


To

rqu

e e

rro

r (N

m)

Tx

Ty

Tz

-10

-5

0

5

10

Fo

rce

err

or

(N)

Fx

Fy

Fz

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-4

-0.2

-0.1

0

0.1

0.2

offset in z-direction (m)

To

rqu

e e

rro

r (N

m)

Tx

Ty

Tz

-1

-0.5

0

0.5

1

Fo

rce

err

or

(N)

Fx

Fy

Fz

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-3

-1.5-1

-0.50

0.51

1.5


To

rqu

e e

rro

r (N

m)

Tx

Ty

Tz

Figure 3.8: Simulation of wrench errors wD dependening on measurement offsets in separate

directions (the other five offsets are equal to zero) in one randomly-chosen and constant position q , while T

real[0,0, ,0,0,0]w mg= ; the plots have similar shape for other positions q .

errors wD (see Figure 3.8) in the situation that only the vertical force z

F is dominantly unequal to zero (see Table 3.9). This is the default situation in steady state of the PA as

zF

should compensate gravity. For example, an offset of 1 mm in the x-direction can produce an error on

xF of more than 6 % of the applied

zF force. Based on this, an algorithm was


developed which can iteratively estimate and eliminate these offsets [13]. The idea is explained in the next section.

3.5.2 Auto-Alignment Procedure

In the previous section it was presented that offset in the measurement system can cause a wrench error mainly in the same direction in steady state. The wrench error is given as the difference between the real and desired wrench

real desw w wD = - . Since in the

real system is not possible to directly measure real

w applied on the PA, the only way to estimate

realw is in steady state when the only active force should be

zF for gravity force

compensation so Treal,ss

[0, 0, , 0, 0, 0]w mg= . Then the wrench error wD expresses the difference between the real wrench in steady-state

real,ssw and the desired wrench

des,ssw

which is necessary to obtain the steady-state levitation. This des,ss

w is the output of the controller C (see Figure 3.7) and deviates from

real,ssw just because of the offset

offq . This

assumes that z

F acts exactly in the center of the gravity of the PA, otherwise nonzero torques

xT or

yT are necessary to stabilize the PA. When the wrench error wD in the steady

state is estimated, the measurement offset off

q can be estimated with the help of the nonlinear mapping

off( , )w q qD = G based upon the simulated behavior:

( ) 1

real,ss des,ss off 2 off real,ss off( , ) ( ) ( ) ( , ).w w w q q q q q w q q

--é ù

D = - = - + = Gê úê úë ûI K K (3.7)

The mapping off

( , )q qG is considered for some particular and constant position q . For the sake of readability, the particular and constant position q will not be mentioned in the text until further notice. To obtain

off( )qG a simulation is necessary, because it would be

difficult to find an exact mathematical description of the wrench-error dependency on the offset. This is caused by the nature of the coupling matrix K and its pseudo-inverted

2-K ,

which are calculated using numerical tools. The mapping off

( )qG is generally nonlinear but with diagonal dominance and it has been found from simulations that the wD dependence on

offq is mostly linear for reasonably small

offq (see Figure 3.8). Hence a first-order

approximation G of the mapping G can be derived so that off

w qD » G .

Then an auto-alignment (AA) procedure is used in the system to eliminate the offset by subtracting its estimate 1

offˆ (diag( ))q w-= DG from the measured positions and

angles (see Figure 3.9). As G is nonlinear, the algorithm is executed iteratively so that it accomplishes the gradient descent or “steepest descent” method (see e.g. [4]).

Figure 3.9: Principle of Auto-alignment procedure (AA).

P(s)

( )qK

2( )q d- +K

C

AA

wdes

wreal i

ref q

offqoff

q

Planar actuator

q d+


Since the algorithm requires steady state, the dynamics of the offset correction is slowed down by a gain 1k , to force the alignment to be gradual and not interfere with the controller-planar actuator dynamics (see Figure 3.10). Once

offq was converged to its

steady-state value, the AA stops further adaptation to prevent variation of the estimates in normal operation of the PA.

The dynamics of the AA can be described as

AA

( ) ( ( ))d t d t= GC , (3.8)

where the offset difference d is defined as the difference between the real offset off

q and the estimated offset

offq :

off offˆd q q= - and represents the residual offset acting on the system.

AAC is a constant diagonal gain matrix of the AA procedure

AA=C 1(diag( ))k -- G . The goal

of the algorithm is to direct the d from an unknown initial state to the origin where

off offˆq q= .

In reality, the coupling ( )qK and the dynamics of the PA are effected by the real system while the commutation inversion

2( )q d- +K together with controller C are

calculated in the computer. If real

w has the expected value, then the only necessary information for the algorithm is

desw in steady state.

0 2 4 6 8 10 12 14-6

-4

-2

0

2

4

6x 10

-4

Est

ima

ted

offs

et (

m o

r ra

d)

Time (s)

offx

offy

offz

off

off

off

Figure 3.10: Elimination of position and orientation errors with auto-alignment procedure; data from the COPAM.

3.5.3 Investigation of Stability of the Auto-Alignment Procedure

To investigate the convergence of the system (3.8) to the origin ( 0d º ) it is necessary to obtain a mathematical description of ( )dG suitable for the investigation of convergence. Since it has been found from the simulations that the

jwD dependence on

jd is

mostly linear for reasonably large d , a first order approximation of the mapping has been used. The reasonably large offsets are defined as ±5 mm in x- and y-direction, ±0.5 mm in z-direction, and ±5 mrad about all axis. The problem is that the gains

ijg of the gain

matrix G (the first order approximations of w d¶D ¶ ) are influenced by the other offsets and therefore varies within the state space of d . Dealing with this problem is possible either

0 2 4 6 8 10-1

-0.5

0

0.5

1

1.5

2

Wre

nch

offs

et (

N o

r N

m)

Time (s)

Fx

Fy

Fz

Tx

Ty

Tz

Fx

Fy

Fz


by trying to describe the parameter dependence ( )dG so ( )w d dD = G , which would lead to the loss of linear description, or to find boundaries of the gain variations

minG and

maxG (see

Figure 3.11) and transform the problem into a robust stability investigation.

When the second approach is used, the gain 66 matrix G can be expressed as

( ) ( )G

t t= + ⋅G G G

Δ , (3.9)

-40

-20

0

20

40

Fo

rce

err

or

(N)

Fx

Fy

Fz

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-3

-1

-0.5

0

0.5

1

offset in x-direction (m)

To

rqu

e e

rro

r (N

m)

Tx

Ty

Tz

-1

-0.5

0

0.5

1

Fo

rce

err

or

(N)

Fx

Fy

Fz

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-3

-1.5-1

-0.50

0.51

1.5


To

rqu

e e

rro

r (N

m)

Tx

Ty

Tz

-40

-20

0

20

40

Fo

rce

err

or

(N)

Fx

Fy

Fz

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-3

-1

-0.5

0

0.5

1

offset in y-direction (m)

To

rqu

e e

rro

r (N

m)

Tx

Ty

Tz

-1

-0.5

0

0.5

1

Fo

rce

err

or

(N)

Fx

Fy

Fz

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-3

-1.5-1

-0.50

0.51

1.5


To

rqu

e e

rro

r (N

m)

Tx

Ty

Tz

-10

-5

0

5

10

Fo

rce

err

or

(N)

Fx

Fy

Fz

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-4

-0.2

-0.1

0

0.1

0.2

offset in z-direction (m)

To

rqu

e e

rro

r (N

m)

Tx

Ty

Tz

-1

-0.5

0

0.5

1

Fo

rce

err

or

(N)

Fx

Fy

Fz

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-3

-1.5-1

-0.50

0.51

1.5


To

rqu

e e

rro

r (N

m)

Tx

Ty

Tz

Figure 3.11: Bounds on gain matrix G; How much G varies in the space of other five offsets for the same position q as in Figure 3.8.


where G is a 6 6´ matrix of average values min max

( ) / 2= +G G G , G

is a 6 6´ matrix of maximal parameter variations

max min( ) / 2= -G G G

, ( )t

GΔ is a time-varying uncertainty

6 6´ matrix with all elements | | 1ijd < and “⋅” is the element-by-element multiplicative

operator. The controller, the constant diagonal gain matrix AA

C , is obtained as the inversion of the diagonal terms of G : 1

AA(diag( ))k -= -C G . The system (3.8) can now be

transformed into:

1

1

( ) (diag( )) ( ) ( ),

( ) (diag( )) ( ( )) ( ).

d t k t d t

d t k t d t

-

-

= -

=- + ⋅G

G G

G G G

Δ

(3.10)

Because (3.10) represents a linear time-varying (LTV) system or a system with uncertain parameters, the Linear Fractional Transformation (LFT) has been used to convert the system into the LFT form:

( ) 1

11 12 22 21( ) ( ) ( ) ( )d t t t d t

-é ù= + -ê úê úë û

M M I M M Δ Δ , (3.11)

where ( )tΔ is a time-varying diagonal matrix with 36 independent elements on the diagonal and with ( ) 1t £Δ .

11 22, ,M M are submatrices of a constant matrix M. A block scheme,

which represents (3.11), is in Figure 3.12 left. Comparing (3.10) with (3.11) shows the following relations:

111 22

1 6 6

6 61 212 21

6 66

(diag( )) , ,

0 0

0 0(diag( )) , ,

0 0

j x

xj

xj

k

k

-

-

= ºé ù é ùê ú ê úê ú ê úê ú ê ú= =ê ú ê úê ú ê úê ú ê úê ú ê úë ûë û

M G G M 0

G I

IGM G M

IG

(3.12)

where 1j

G

represents the first row of G

, and 6 6x

I is the identity matrix 6 6´ .

Using LFT the uncertainty matrix was removed from the model. Now, define a linear time-invariant (LTI) system H(s):

( ) 1

22 21 11 12( )s s

-é ù= + -ê úê úë û

H M M I M M . (3.13)

Together with the time-varying uncertainty ( )tΔ the system ( )sH represents a basic perturbation model (see Figure 3.12 right).

Figure 3.12: Left: Block diagram of LFT representation of (3.11). Right: Block diagram of basic perturbation model.

M

( )tΔ

d d

H

( )tΔ


By Shamma [102] the LTI system ( )sH is exponentially stable for arbitrarily fast time-varying ( )tΔ as defined above if and only if

1

admisibleinf 1- <

DDHD , (3.14)

where D is any constant invertible diagonal gain matrix which commutes with any

admissible time-varying ( )tΔ . The norm 1

admisibleinf -

DDHD is an upper bound for the

structured singular value ( )m H and it is usually computed via convex programming tools as mussv in Matlab. Hence if the inequality (3.14) is satisfied for all frequencies:

1

admisiblesup inf 1-

wÎ<

DDHD

, (3.15)

the exponential stability of the algorithm is proven. Figure 3.13 shows how the norm of the system varies over the frequency range of 10-3 rad to 103 rad. From the figure, it is clear that the system is stable.

Until now offset variations and stability around one particular position of the PA has been investigated. To prove that the algorithm will always converge to the real values the stability condition needs to be calculated for offset variations around all positions in the workspace of the PA. Because it was not possible to prove stability on the simulated model with one set of bounds on G that would cover the whole workspace of the PA, the bounds have been calculated separately for a dense grid of the PA positions. It was found that the algorithm is stable for all investigated positions (see Figure 3.14). Because the gain matrix varies smoothly with the position with continuous first derivative, the procedure is expected to be stable everywhere.

10-3

10-2

10-1

100

101

102

103

0

0.2

0.4

0.6

0.8

1

Freqency [rad]

Mag

nitu

de [

-]

limit

(H(j))

Figure 3.13: Frequency-dependent norm of the system for one position of the PA.

3.5.4 Measurements on the Experimental Setup

The simulation in the previous section was executed in the situation where all coils and magnets had exactly the designed dimensions and they were infinitely accurately positioned in the grid. This is not the case in a real system where the errors on the wrench vector from all other sources cause the procedure to converge not to the exact position but


10-3

10-2

10-1

100

101

102

103

0

0.2

0.4

0.6

0.8

1

Freqency [rad]

Mag

nitu

de [

-]

limit

(x,y)

Figure 3.14: Frequency-dependent of system for different positions of the PA.

to some estimated position given by all these production inaccuracies and measuring errors. To vindicate the usefulness of the presented procedure, tests on the COPAM experimental setup have been performed. One result of the offset estimation was already presented in Figure 3.10.

Repeatability of the procedure for one position of the planar actuator was examined initially. The procedure was turned on for the time during which the offset estimates changed. Then it was turned off, the found values were reset and the planar actuator was moved along the surface and back to the same initial position. Then the test was repeated. Ten repetitions were performed. It was found that the offset estimates had repeatability of 0.2 μm and 1 μrad.

Since the decoupling of the wrench is not perfect and it varies with the position of the planar actuator (see Chapter 4, Figure 4.5), the second experiment examined influence of the imprecise decoupling on the offset estimation. The test was performed for a grid of positions with a step of 10 mm. The initial wrench errors and the final estimated offsets are shown in Figure 3.15. The figure shows that the other sources of the errors which also produce the residual cross-coupling of the wrench influence the precision of the offset estimation. Still, the offsets in x- and y-DOF are clearly visible. Moreover, the estimates can be averaged over the measured positions to improve the accuracy to about 50 μm and 0.1 mrad.

3.5.5 Discussion and Extension

In the model of COPAM, coils are distributed in an orthogonal grid and the l2-norm based commutation was used. Further simulations were made which revealed that the procedure is not restricted to a specific commutation. The same results were also obtained with the l∞-norm based commutation (see Section 2.2.2).

Another simulation was performed with the coils distributed in a triangular array (as defined in Section 2.3.2). The procedure could still remove the measurement system offsets in most positions of the PA (see Figure 3.16). The area for [0.02, 0.035]y Î m did not cause instability of the algorithm, but convergence to wrong estimates. As the same region


0 10 20 30-2

0

2

4

6

8x 10

-4

Est

imat

ed o

ffse

t (m

)

Position (-)

offx

offy

offz

0 10 20 30-2

0

2

4

6

8

Wre

nch

offs

et (

N)

Position (-)

Fx

Fy

Fz

0 10 20 30-1

-0.5

0

0.5

1

1.5x 10

-3

Est

imat

ed o

ffse

t (r

ad)

Position (-)

off

off

off

0 10 20 30-0.4

-0.2

0

0.2

0.4

Wre

nch

offs

et (

Nm

)

Position (-)

Tx

Ty

Tz

Figure 3.15: Initial wrench errors (bottom) and the final estimated offsets (top) for different positions of the planar actuator.

causes a slightly increased condition number value (see Figure 2.33) and lowest maximum possible acceleration (see Figure 2.37), it seems that this area is disadvantageous for the triangular coil array with the particular dimensions. However, since theoretically only one applicable position is necessary, the algorithm can also be used for this kind of coil array. Although the procedure successfully estimated any combination of the offsets in those positions, the stability of the algorithm in those positions could not been proven. The most probable reason is that the representation of ( )dG by the bounds

minG and

maxG is too

conservative.

The last test was performed on a model of a special magnetically levitated planar actuator with moving magnets [56, 76]. This planar actuator utilizes Halbach magnet array with rectangular coils distributed in a herringbone pattern. The coils have one side which is more than three times longer than the other. Moreover, they are rotated by 45 degrees with respect to the magnet array. Therefore, this PA represents a completely different coil topology. Although again it was not possible to prove the convergence, the procedure could still successfully estimate any combination of the measurement system offsets in all positions of the PA.

From all the simulations on the models of different planar actuators it seems that the relation between the offsets and the produced wrench error is caused mainly by the magnetic flux density distribution of Halbach magnet array, which was used in all the cases. Since the distribution has dominant first spatial harmonics (see e.g. Figure 2.1 and

3.6 Conclusions 85

2.2), the position derivative has also dominant first spatial harmonics. This gives the dependency of the wrench error of a single coil on the position offset and the current applied to the coil. The total wrench error is then the summation of the spatial harmonics over all active coils, which again has dominant first spatial harmonics. This is most probably the reason for the wrench error behavior (see Figure 3.8).

pos. x (m)

po

s. y

(m

)

0 0.02 0.04 0.06 0.08 0.10

0.01

0.02

0.03

0.04

0.05

0.06

Figure 3.16: Test of the auto-alignment procedure on the triangular coil array. The dark areas represent positions where the procedure correctly estimated any combination of the measurement system offsets.

3.6 Conclusions

In this chapter, different sources of commutation errors of the magnetically levitated planar actuator with moving magnets have been presented. When the errors from different sources are compared, the most severe seem to be errors from the offsets in measurement system. Nevertheless, these errors can be estimated and eliminated by using the procedure developed in [13]. An investigation of the stability of this procedure was performed in this chapter for the coil topology used in the COPAM. Although it could not be proven that the procedure is stable for the whole workspace of the PA, the convergence was proven at least for all the investigated points. From the investigation, convergence for any position in the workspace of the PA is expected. It was found that it is possible to use the procedure also with different topologies and with different commutations. Although it was not possible to prove the convergence at least in the tested points, successful tests were performed with rounded coils distributed in a triangular array and with rectangular

coils in a herringbone pattern. Furthermore, the same results were obtained with l2-norm and l∞-norm based commutations. From all these results it seems that the relation between offset and wrench errors is caused mainly by the Halbach magnet array. If true, the procedure offers a powerful tool to remove measurement offsets in many different kinds of planar actuators with moving Halbach magnet array.


87

Chapter 4

Control

4.1 Introduction

In this chapter, the control of the magnetically levitated planar actuator with the manipulator on top of it is discussed.

Planar actuators are used for motion in the horizontal xy-plane. Other degrees of freedom are stabilized or fixed by some kind of (ball, air, magnetic, etc.) bearing. Although yaw-rotation about the vertical axis of the planar actuator (PA) has been discussed in Section 2.3, common designs only allow translational movement in x- and y-direction with limited translational vertical and rotational movements. Then, control of the 6-DOF magnetically levitated planar actuator consists of tracking control of two DOF’s (x- and y-direction) and stabilization of the other four DOF’s for accomplishing a stiff bearing.

The manipulator on top of the platform has two ironless linear actuators attached to a beam. In the center of the beam, a rotary motor is assembled with an arm attached to it. The tip of this arm can be positioned anywhere in the xy-plane between the two horizontal linear legs by combining the translation of the beam and the rotary movement of the arm. Modeling of the manipulator is extensively described in [13].

This chapter is organized as follows. Derivation of the model of the planar actuator that includes commutation errors is described in Section 4.2. This model has to mimic the real plant as closely as possible in order to test the designed controllers in simulation reliably. Section 4.3 contains a description of the reference trajectory generation and a feedforward control for the PA without control of the manipulator. Different control strategies for controlling the PA and simulation results are presented in Section 4.4. The last section (4.5) discusses combined control of the PA with the manipulator. The results from the control of the real experimental setup will be presented in Chapter 6.

88 Chapter 4. Control

4.2 Model of the Planar Actuator

In magnetically levitated ironless planar actuators with moving magnets, the coupling matrix K, which relates the currents in the coils i to the final wrench w =

T[ , , , , , ]x y z x y z

F F F T T T acting on the translator, is nonlinear with the position q = T[ , , , , , ]x y z q y f of the PA (see Section 2.1). Therefore, Chapter 2 was devoted to linearizing

and decoupling the wrench applied on the PA. Also the output of the PA, represented by sensor readings s , needs to be decoupled (see Figure 4.1) since the goal is to control the six-degree-of-freedom (6-DOF) MIMO PA as 6-times a SISO system (see Figure 4.2). The approach for the position reconstruction from the sensor readings is dependent on the particular design of the PA and, therefore, it will be discussed in Chapter 6, which is devoted to the COPAM prototype.

In an ideal situation, a magnetically levitated planar actuator can be modeled in each DOF as a double-integrator system:

Figure 4.1: Linearization and decoupling of PA.

Figure 4.2: Ideal 6-DOF PA represented as six linear SISO systems.

2

1

ms

Linearized and decoupled system

Fx

Fy

Fz

Tx

Ty

Tz

x

y

z

θ

2

1

ms

2

1

ms

2

1

xJ s

2

1

yJ s

2

1

zJ s

Planar

actuator P

Commutation

uT

Position reconstruction

yT


Fx

Fy

Fz

Tx

Ty

Tz

x

y

z

θ

s1

s2

.

.

.

sm

i1 i2 i3 .

.

.

in


x y . . .


2

1( )P s

as= , (4.1)

where a is representing mass m in translational DOF’s or inertias x

J , y

J , and z

J in rotational DOF’s. For simulation purposes, the values are taken from the CAD model of

the COPAM and measurements presented in [13]: m = 20.5 kg, x

J = y

J = 0.4 kgm2, and

zJ = 0.6 kgm2.

Since the commutation -K is calculated based on a model M

K of the coupling matrix K of the real system (see Figure 4.3 and Section 2.4.4), linearization and decoupling is never exact. Consequently, a residual cross coupling exists between different degrees of freedom.

A model of the planar actuator of the COPAM was developed for simulation purposes (see Figure 4.4) and consists of a model of the coupling matrix

MK in series with

six separate (diagonal) SISO systems ( , , , , ,x y z

P P P P P Pq y f) represented by (4.1). The disturbing residual wrench cross coupling is modeled with a position-dependent error coupling matrix ( )qX between the currents applied to the coils i and the produced wrench error

errw . The real wrench acting on the system is then simulated as

( )real M est des( ) ( ) ( , )w q q q w-= + KK X , (4.2)

where the difference des M est des

( ( ) ( , ))w q q w-- KK represents the wrench errors due to the sensor noise and the delay in the system, whereas

des est des( ( ) ( , ))w q q w-- KX represents the

wrench errors due to the imprecise model of the coupling matrix. The principle of disturbance modeling is explained in the following paragraph.

Figure 4.3: Block scheme of a planar actuator in control loop.

Figure 4.4: Block scheme of the model of the planar actuator in the control loop.

( )

( )

xP s

P sf

est( )q-K

C(s) wdes

i

ref q Commutation

( )qX white noise

M( )qK

white noise

werr

Model of planar actuator

Delay

estq

P(s)

( )qK

est( )q

C(s) wdes wreal

i

ref q Planar actuator Commutation

Actuator noise

Sensor noise

Delay est

q


From measurements between the desired (applied) wrench

desw and the real wrench

realw acting on the PA of COPAM in dependence on xy-position [15] (see Figure 4.5) it was found that the gain of these disturbing (cross-)terms varies with the position of the PA. Moreover, from the figures one can see that the variation has one dominant spatial frequency component, which corresponds to the spacing of the coils in the stationary coil array. The problem was that only one set of position-dependent measurements of

realw for

one vector des

w = [50 N, 50 N, 50 N, 0 Nm, 0 Nm, 0 Nm] was available. Therefore, to develop at least a rough model, which would mimic the real residual cross coupling, an assumption has been made that all the errors are caused by the mechanical tolerances of the coil array. Effects of the dimensional and placement tolerances of one coil in the magnetic field of the magnet array were simulated for the calculations presented in Section 3.3. It was found that those tolerances resulted in the error coupling ( )

jX q between the

current in one coil and produced wrench error err

w having similar position-dependent behavior as the primary coupling ( )

jK q between the current in that coil and the produced

wrench (see Figure 4.6). Both kinds of tolerances have dominant first spatial harmonics in the xy-plane (corresponding to the coil spacing) and both are similarly influenced by the end-effects of the magnet array. The main difference is in the phase and amplitude of the first harmonic. The principle of superposition can be applied and the total error coupling of one coil produced by combination of its different tolerances can be again expressed as a harmonic model that has particular amplitude and phase and with included end-effects.

The error coupling X for jth coil is then modeled as:

, , , ,

, , , ,

, , , ,

,

sin sin ,

sin sin ,

sin sin ,

j j

Fx j Fx j Fx j Fx j

j j

Fy j Fy j Fy j Fy j

j j

Fz j Fy j Fz j Fz j

Tx j T

x yX g

x yX g

x yX g

X g

æ ö æ ö÷ ÷ç ç÷ ÷ç ç= + b + g÷ ÷ç ç÷ ÷ç çt t÷ ÷ç çè ø è øæ ö æ ö÷ ÷ç ç÷ ÷ç ç= + b + g÷ ÷ç ç÷ ÷ç çt t÷ ÷ç çè ø è øæ ö æ ö÷ ÷ç ç÷ ÷ç ç= + b + g÷ ÷ç ç÷ ÷ç çt t÷ ÷ç çè ø è ø

= ( )

( )

( )( )

, MA , ,

, , MA , ,

, , MA MA , ,

sin sin ,

sin sin ,

sin sin

j j

x j j Tx j Tx j

j j

Ty j Ty j j Ty j Ty j

j j

Tz j Tz j j j Tz j Tz j

x yy y

x yX g x x

x yX g x x y y

æ ö æ ö÷ ÷ç ç÷ ÷ç ç- + b + g÷ ÷ç ç÷ ÷ç çt t÷ ÷ç çè ø è øæ ö æ ö÷ ÷ç ç÷ ÷ç ç= - + b + g÷ ÷ç ç÷ ÷ç çt t÷ ÷ç çè ø è ø

æ ö æ÷ç ç÷ç ç= - - + b + g÷ç ç÷ç çt t÷ç çè ø è,ö÷÷÷÷÷ø

(4.3)

where j

x and j

y represent the position of the jth coil with respect to the magnet array, t is magnet-pole pitch,

MAx and

MAy represent the center of gravity of the magnet array, and

,Fx jg , …,

,Tz jg ,

,Fx jb , …,

,Tz jb , and

,Fx jg , …,

,Tz jg are the gain and phase parameters of the jth

coil, respectively. Moreover, the expressions (4.3) are exponentially decreased at the perimeter of the magnet array to mimic the decay of the magnetic field there. As a result, there are three parameters

, , ,( , , )

j j jg⋅ ⋅ ⋅b g per one coil and one DOF. Since the plots in

Figure 4.5 consist of about 1200 measured values in each DOF while 99 coils have been active with known current in those coils, there are enough measurements to estimate 99 ×


Figure 4.5: Planar actuator of COPAM; measured real wrench real

w when des

w = [50 N, 50 N, 50

N, 0 Nm, 0 Nm, 0 Nm].


Figure 4.6: Position dependent coupling (top) and error coupling (middle and bottom) of one coil

in the magnetic field of the magnet array; for the coil with position error of dx = 1 mm (middle) and dimension error of 2.5 % (bottom); Left: F

z component and Right: T

x component.


3 = 297 parameters. Matlab LSQNONLIN function was used to estimate the parameters. The function solves non-linear least squares problems. The results from estimation are depicted in Figure 4.7. Although the parameters found ensure a good approximation of the measured data (with an RMS errors of 0.12 N for the forces and 0.014 Nm for the torques compared to Figure 4.5), the parameter values does not have to reflect the real behavior of the error coupling. To ensure realistic values of the parameters, regularization is used. Then the minimized cost function for one DOF is written as

2 2

, ,min ( , , )g

y f g gj y

æ ö÷ç - b g +a ÷ç ÷çè ø , (4.4)

where y represents the measured data, f is vector of the data from the model, and a is the regularization parameter. In this way, all the gains

,X jg had similar small value as was

expected. In Figure 4.8, there are plots from the simulation of the error coupling between the applied desired wrench

desw and the produced error wrench calculated as

err 2 des( ) ( )w q q w-= X K . From the figure, one can see that the produced wrench error is at

most about 6% of the applied wrench and that the error has a dominant harmonic shape as expected. This makes the model of PA from Figure 4.4 suitable for mimicking the behavior of the real PA from COPAM.

It is clear that the model does match the real wrench-error behavior, but this was not the intention since insufficient data from the setup were available. Still, the model can be used for realistic simulation of the designed controllers. The easiest way to obtain a more precise model would be to apply a wrench on the real PA in each degree of freedom separately and measure the wrench error over the whole working area of the PA as shown by the simulation data in Figure 4.8.

In addition to the error coupling, white noise is inserted at the plant input and output to model the noise of the actuators and sensors. The real actuator noise is less than 1 mA RMS and the sensor noise is about 0.1 μm RMS and 1 μrad RMS for position and orientation, respectively [13].

The sampling frequency of the COPAM prototype is 3 kHz and it is limited by the transfer of the calculated current setpoints from the dSpace to the current amplifiers [13]. There is also a delay in the system of approximately four sample times: control calculation in the dSpace (0.33 ms), transfer of the current setpoints to the current amplifiers (0.33 ms), the current amplifiers (0.2 ms), and the sensors (0.4 ms) [13].


Figure 4.7: Simulated errors of the commutation calculated for des

w = [50 N, 50 N, 50 N, 0, 0, 0].


Figure 4.8: Simulation of the error coupling between the applied desired wrench des

w and

produced wrench error err

w in working range x 0,016é ùÎ ê úë û m, y 0,0.06é ùÎ ê úë û m.


4.3 Trajectory Generation and Feedforward Design

As described in the introduction section of this chapter, planar actuators are usually propelled in two DOF’s. For these two degrees of freedom, a trajectory must be planned and trajectory profile generated. Planning of the trajectory must take into account electromechanical constraints of the system. Trajectory profile generation, on the other hand, means representation of the trajectory in a suitable form. Once the trajectory information is available, feedforward control can use this information for calculation of the control signals. Therefore, trajectory generation and feedforward control are closely connected.

Planar actuators are mainly designed for use in pick-and-place machines or wafer scanners. Hence, point-to-point motion with eventual constant velocity part is desirable. Common trajectory planning and feedforward control for a rigid body is based on second-order feedforward (also called rigid-body feedforward). Consider a simple motion system:

mx dx kx F+ + = , (4.5)

where x, x , and x are the position, velocity, and acceleration of the system, m, d and k are the mass, damping and stiffness, respectively, and F is the applied force. For this kind of system, a second order trajectory can be generated as depicted in Figure 4.9. The trajectory satisfies constraints on velocity and acceleration and construction of the trajectory profile is straightforward. Second-order feedforward then uses information on position, velocity, and acceleration to generate the feedforward control signal (force).

There are several reasons why the performance of a real motion system with second-order feedforward can be unsatisfactory:

0 0.05 0.1 0.15 0.2 0.25 0.3-10

0

10

acc

(m.s

-2)

0 0.05 0.1 0.15 0.2 0.25 0.3

0

0.2

0.4

vel (

m.s

-1)

0 0.05 0.1 0.15 0.2 0.25 0.3

0

0.05

0.1

pos

(m)

t (s)

Figure 4.9: Second-order trajectory profiles.


A: The first reason is that in the real system, not only acceleration, but also jerk must be limited. As the acceleration is represented by the current i in the actuator, jerk is represented by the time-derivative of the current di/dt which limit is given by amplifier specifications. Therefore, at least third-order trajectory with a constraint on the jerk is necessary.

B: The second reason is that a mass-only MIMO system, as in Figure 4.1, is statically decoupled with rigid body decoupling represented by the transformations

uT

(commutation) and y

T (sensor decoupling). Such a system is typically constructed to be light and stiff, so the resonance modes are far above the bandwidth of the controller. The plant can then be expressed as [47]:

rb flex2

1( ) ( ) ( )s s s

s= +P P P , (4.6)

where 21

rb( )

ssP represents the rigid body part and

flex( )sP the flexible body part. A simplified

model of the plant for the frequencies below the first resonance can be written as:

rb flex2

1ˆ ˆ( ) ( )s ss

= +P P P , (4.7)

where flex

P is the constant matrix representing all modal contributions in low frequencies added to the model. After the rigid body decoupling transformations

uT and

yT , the plant

has the form:

rb flex rb flex2 2

1 1ˆ ˆ( ) ( ) ( ) ( )y u y u y u

s s s ss s

= = + = +P T P T T P T T P T P P . (4.8)

It was shown in [20] that if the second-order (acceleration) feedforward 1 2rb

( )s s-=F P is used on the system (4.8), the tracking error can be expressed as

1 2flex rb

( )o

e s s r-= -S P P , (4.9)

where r is the reference trajectory and ( )osS is the output sensitivity of the closed-loop

system. It was also shown there that without feedback control (o=S I) the tracking error

equals the acceleration profile scaled with the factor 1flex rb

-P P . With feedback control applied, the output sensitivity function has at least slope +2 at low frequencies, so the tracking error peaks during non-zero jerk phase of the trajectory. Hence, the acceleration feedforward cannot improve low frequency tracking and reduce cross-talks. Therefore, a multivariable feedforward with jerk derivative is proposed there. The feedforward controller is a fourth-order Taylor approximate of the inverse plant from (4.8):

1rb flex2

1 2 1 1 2rb flex rb

1 2 1 1 4rb rb flex rb

1( ) ( ( ) )

( )

(6).

s ss

s s

s s

-

- - -

- - -

= +

= +

= - +

F P P

I P P P

P P P P

(4.10)

Then the fourth-order (jerk derivative) feedforward controller is

2 4 1 2 1 1 4acc djerk rb rb flex rb

( )s s s s s- - -= + = -F F F P P P P . (4.11)

Now, the tracking error without feedback controller equals to

1 1 4flex rb flex rb

e s r- -= P P P P . (4.12)


It should be mentioned that as rigid body decoupling is applied, only the jerk derivative feedforward part will be non-diagonal. The residuals in (4.12) are quadratically smaller in comparison to (4.9). From the feedforward controller defined in (4.11), fourth-order trajectory with limit on the jerk derivative is necessary.

C: The third reason is that higher-order trajectories have lower energy content at higher frequencies (see Figure 4.10) [100, 67], which results in lower high-frequency content of the tracking error signal.

100

101

102

103

10-40

10-30

10-20

10-10

100

Frequency (Hz)

Po

we

r S

pe

ctra

l De

nsity

(m2

/Hz)

2nd-order3rd-order4th-order

Figure 4.10: Power spectral density of second-, third- and fourth-order trajectory.

Higher-order trajectories have a disadvantage in increased trajectory execution time, but this disadvantage is usually compensated by the reduced settling time [67]. If the same execution time is required (minimal number of movements per minute must be satisfied) without considering the settling time, a trajectory with higher acceleration and/or velocity limits will be required. Then, the dissipated power will be higher [27].

An effective algorithm for calculation of fourth-order trajectory is described in [67]. The trajectory-planning algorithm is based on construction of a jerk derivative profile that is integrated four times to obtain fourth-order position trajectory (see Figure 4.11). The jerk-derivative profile is calculated based on bounds on velocity, acceleration, jerk and derivative of jerk.

Figure 4.11: Trajectory profile generation from jerk-derivative profile.

1

s 1

s 1

s 1

s

d. jerk jerk acc. vel.

pos.


The algorithm also takes into account implementation issues, like discrete time calculations or synchronization of the profiles. The calculated time intervals must be rounded-off towards a multiple of the sampling time

ST . This must be done such that the

given bounds are not violated, but at the same time approximated as closely as possible.

On the other hand, each discrete-time integrator causes one sample time delay. To synchronize all the profiles, the jerk derivative, jerk, acceleration and velocity profile are delayed by 2 ×

ST , 1.5 ×

ST , 1 ×

ST , and 0.5 ×

ST , respectively. An example of fourth-

order trajectory with bounds on velocity, acceleration, jerk and derivative of jerk is in Figure 4.12.

For the simulation purposes and based on the knowledge of the prototype specifications, the trajectory parameters for each axis (x, y) have been chosen as follows:

maxv = 0.5 m.s-1,

maxa = 10 m.s-2,

maxj = 500 m.s-3 and

maxd = 1×105 m.s-4, for the velocity,

acceleration, jerk and jerk derivative, respectively. The trajectory profiles for x- and y-direction used for the simulations in the next section are presented in Figure 4.13. Since the limiting values are related to the axis, the actual maximum values can be up to a factor 2 higher, which is the case of part 5 and 7 of the trajectory in Figure 4.13 right.

0 0.05 0.1 0.15 0.2 0.25 0.3-1

0

1x 10

5

d.je

rk (

m.s

-4)

0 0.05 0.1 0.15 0.2 0.25 0.3-500

0

500

jerk

(m

.s-3

)

0 0.05 0.1 0.15 0.2 0.25 0.3-10

0

10

acc

(m.s

-2)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

vel (

m.s

-1)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

pos

(m)

t (s)

Figure 4.12: Fourth-order trajectory profiles.


0 2 4 6 8

0

0.01

0.02

0.03

0.04

0.05

t (s)

po

s (m

)

xy

Figure 4.13: Left: x, y trajectory position profile, Right: trajectory profile in xy-plane.

4.4 Control of the Planar Actuator

4.4.1 PID Control

Background

The first kind of control tested on the model of the PA was classical PID control. The structure of the controller is the same in each DOF as it is used in [76]. Each controller consists of an integrator for low frequencies, a lead-lag filter for the bandwidth frequencies and a roll-off filter for high frequencies:

( ) ( ) 1

( )( ) ( )

i llPID bw

ll lp

s z s zC s k

s s p s p

+ +=

+ + , (4.13)

where bw

k is the gain, i

z is the zero of the integrator, ll

z and ll

p are the zero and pole of the lead-lag filter, respectively, and

lpp is the pole of the low-pass roll-off filter. Assuming a

plant transfer function consisting only of a double integrator (as defined in (4.1)), the parameter values are derived as follows:

2,3 2

, , ,22,,

, ,

, ,

111 ,

11

, ,

, ,

p ll

bw bw z i z ll p lp

z llz i

ll bw z ll ll bw p ll

i bw z i lp bw p lp

ck ac c c c

cc

z c c p c c

z c c p c c

+= +

++

= == =

(4.14)

where a is the mass m or moment of inertia J of the particular DOF, bw

c is the bandwidth defined as the frequency, where the amplitude of the open-loop transfer function crosses 0dB point for the first time. The parameters

,z ic ,

,z llc ,

,p llc , and

,p lpc are used for the loop

shaping of the system independently of the final bandwidth bw

c .

-0.01 0 0.01 0.02 0.03 0.04 0.05

0

0.01

0.02

0.03

0.04

0.05

x (m)

y (m

)

2

3,6

1

45

7

8


Simulations

The parameter values have been chosen as ,z i

c = 6.0, ,z ll

c = 6.0, ,p ll

c = 4.0, and ,p lp

c = 8.0. Bode plot of the open-loop system with

bwc = 300 rad/s is presented in Figure 4.14.

The delay margin can be calculated as 180p

mf /

bwc = 1.57 ms » 1/(2

bwc ), where

mf is the phase

margin. Increasing the bandwidth or the stability margins was difficult due to the phase lag from the delay (4 × 0.33 ms » 1.3 ms).

100

101

102

103

104

-50

0

50

100

Ma

gn

itud

e (

dB

)

Gain margin 6.3 dB @ 554 rad/s

100

101

102

103

104

-270

-225

-180

-135

Ph

ase

(d

eg

)

Frequency (rad/s)

Phase margin 27 deg @ 300 rad/s

Figure 4.14: Bode plot of the open-loop system PID

L PC= with bw

c = 300 rad/s.

The controller was tested together with the model of PA presented in section 4.2 in several situations. Since the model does not include flexible modes, only the acceleration (rigid-body) feedforward was used. In the case of no disturbing error coupling, the acceleration feedforward should allow for the optimal control of the PA. The reference position for x- and y-direction was delayed four times since the delay between the applied feedforward and the position reading is four sample times too. The maximum achieved bandwidth without oscillations in the system was 47 Hz due to the delay of about 1.3 ms, which causes phase lag of 23 degrees at that frequency. The errors in all the DOF’s during the tracking of the trajectory from Figure 4.13 and corresponding force and torque outputs of the feedback controllers (without feedforward part) are plotted in Figure 4.15 with gray color.

The delay in the system causes the commutation to have significant position offset

at high velocities. At the maximum velocity of 0.5 m.s-1 with the delay of 4 × 0.33 ms in the closed loop, the commutation has a position offset of about 0.65 mm. This can be seen as the measurement system offset, which was discussed in Section 3.3.5. From Table 3.8 in that section, one can calculate that the x-direction offset of 0.65 mm can produce

xF force

disturbance of up to 4% of z

F (» 8 N). Moreover, it can also produce z

F force disturbance

of up to 7.5% of x

F (» 15 N for acceleration of 10 m.s-2). To reduce this effect, x- and y- position of the PA for the commutation are taken from the reference trajectory positions (without delay), which can be used as a predictor. Since the tracking error is only several


micrometers, the position offsets in the commutation during high-velocity movement were reduced at least by factor 100. The resulting tracking errors are plotted in Figure 4.15 using black. From now on, the prediction in x- and y-direction is used in all other simulations.

The second simulation was carried out with offsets in estimated values of the center of gravity (COG) of the PA model, its mass and inertias. While the model values for

the mass, inertias and COG were m = 20.5 kg, x

J = y

J = 0.4 kgm2, and z

J = 0.6 kgm2, and [

cm cm cm, ,x y z ] = [0.01 m, 0.0 m, 0.025 m], the estimated values used for the controllers,

feedforward and decoupling were m = 21 kg, x

J = 0.35, y

J = 0.45 kgm2, z

J = 0.65 kgm2, and [

cm cm cm, ,x y z ] = [0.015 m, 0.05 m, 0.02 m]. In Figure 4.16, the results of the simulation

are represented by gray and can be compared to the ideal situation (without offsets) represented by black (the same as in Figure 4.15).


Figure 4.15: PID control; the tracking errors and feedback controller outputs during tracking of

the trajectory from Figure 4.13; without (gray) and with (black) the position prediction for the commutation.


Figure 4.16: PID control; the tracking errors and feedback controller outputs during tracking of

the trajectory from Figure 4.13; with (gray) and without (black) offsets in the mass, moments of inertias and center of gravity estimation.


4.4.2 Sliding-Mode Control

Background

Sliding(-mode) control is generally applied to nonlinear systems, but its main benefit is tolerance for model imprecision [115, 103]. The imprecision can be caused by actual uncertainty about the plant or by a simplified representation of the system’s dynamics. Therefore, it has also been used to control high-precision linear or planar motors stages (e.g. [61, 105, 75, 81, 19]). Two other major approaches for dealing with model uncertainty are robust control and adaptive control. Sliding-mode (SM) control is a simple

approach to robust control. It is based on rewriting nth-order systems to equivalent 1st-order systems, which are easier to control. “Ideal” performance can, in principle, be achieved, but at the price of excessive control activity.

For a single-input dynamic system:

( ) ( ) ( )nx f x b x u= + , (4.15)

where the scalar x is the output of the system of interest (position), u is the control input, ( 1) T[ , , , ]nx x x x -= is the state vector and the ( )f x and ( )b x are generally nonlinear functions

and not exactly known. For the 1-DOF model of the PA defined in (4.1), n = 2 and

1

( ) 0, ( )f x b xa

= = . (4.16)

The control problem is to get the state x to track a specific time-varying state ( 1) T[ , , , ]n

d d d dx x x x -= in the presence of model imprecision on ( )f x and ( )b x . The time-varying state is obtained from the generated trajectory profiles defined in Section 4.3:

T[ , , , , . ]d

x pos vel acc jerk d jerk= . Next, a time-varying surface S(t) in the state-space n by the scalar equation ( , ) 0x ts = is defined, where

1d

( , )d

n

x t xt

-æ ö÷ç ÷s = + lç ÷ç ÷çè ø, (4.17)

l is a strictly positive constant which will be explained later, and x is the tracking error vector

dx x x= - . For the system (4.1) with n = 2,

x xs = + l , (4.18)

which is a weighted sum of the position and velocity error. Hence, the problem of tracking

the n-dimensional vector d

x (nth-order tracking problem) can be reduced to that of keeping

the scalar quantity s zero (1st-order stabilization problem).

The dynamics, while in SM, can be written as 0s = . For the second-order system (4.15) one gets:

0d d

x x x f bu x xs = - + l = + - + l = . (4.19)

The control law is then defined as:

( )1ˆ ˆ sgn( )d

u b f x x k-= - + -l - s , (4.20)


where b and f are the approximations of b and f, respectively, sgn() is the sign function, and

k = F + h , (4.21)

where, F is some known function ( , )F F x x= which bounds the estimation errors f f-

F£ and h defines the speed of convergence to the sliding surface:

21 d

2 dts £ h s . (4.22)

From (4.20)-(4.22) one can see that choosing h high leads to fast convergence to the sliding surface at the price of high control action. The control law (4.20) does not include integral action (can be seen as a PD controller with acceleration feedforward). To include also

integral control (PID), 0

( ( )d )t

x r rò should be the variable of interest. The system is now

one order higher and

2

2

0 0

d( )d 2 ( )d

d

t t

x r r x x x r rt

æ ö æ ö÷ç ÷ç÷s = + l = + l + lç ÷ç÷ç ÷ç÷ç è øè ø ò ò (4.23)

with the modified control law:

( )1 2ˆ ˆ sgn( )d

u b f x x x k-= - + -l -l - s . (4.24)

The main drawback of SM control is the so-called chattering effect caused by the sgn() function in (4.20) or (4.24) once s reaches the sliding surface (s = 0). To avoid this problem, the sgn() function is often replaced by saturation sat() or arctangent atan() function. As a result, chattering is eliminated, but perfect tracking is exchanged for tracking to within a guaranteed precision. Nevertheless, the trade-off between the tracking precision and robustness to unmodeled dynamics can be well controlled by a strictly positive parameter d in those functions: sat(s/d) or atan(s/d). The smaller the parameter d is, the closer the behavior of those functions is to the behavior of saturation.

The parameter l is called control bandwidth and it is typically limited by three factors in mechanical systems [103]:

Structural resonance modes: l must be smaller than the frequency R

v of the lowest unmodeled structural resonance mode; classically 2

3 Rvpl £ .

Neglected time delays: 1(3 )A

T -l £ , where A

T is the largest unmodeled time delay. Sampling rate: 1

S(5 )T -l £ where

AT is the sampling time.

With the sampling rate of 3 kHz and four delays, the control bandwidth should be l £ 250. If expected resonances above 100 Hz are included, the control bandwidth should be l £ 200.

The SM control discussed so far is also known as standard sliding-mode control. Due to the problems with chattering higher-order sliding-mode control (HOSM) has been developed over the past few years [7, 8, 71, 73]. HOSM generalizes the standard SM notion

to the case, when the discontinuity appears for the first time in the rth total time-derivative ( )rs . A motion keeping s = 0 is called that case rth-order sliding mode with


( ) 0rs = s = s = = s = (4.25)

r-sliding point set [71]. The controller has the form:

( 1)1,

( , , , )rr r

u --= -a Y s s s , (4.26)

where a is positive controller parameter and 1,r r-Y specifies the type of the controller. Two

known types of controllers exist. Nested r-sliding control is defined as [74]:

( ), , 1,

0,/( 1)/ /( 1) ( 1) ( )/

,

sgn( ) 1, , 1,

sgn( ),

( ) ,

ii r i i r i r

rp r ip r p r i r i p

i r

N i r

N

-

- +- - -

Y = s + b Y = -Y = s

= s + s + + s

(4.27)

where p being at least a common multiple of 1, 2,…, r, and 1 1, ,

r-b b are properly chosen positive parameters. Then the controllers for r £ 4 and

ib taken from [74] are

1. sgn( )u =-a s ,

2. 1/2

sgn( sgn( ))u =-a s+ s s ,

3. 2/32 1/6sgn[ 2( ) sgn( sgn( ))]u =-a s+ s + s s+ s s ,

4. 3 3 3/46 3 1/12 4 1/6sgn 3( ) sgn[ ( ) sgn( 0.5 sgn( ))]u =-a s + s + s + s s+ s + s s+ s s .

Later on, quasi-continuous controllers have been developed [72], which have discontinuity only in ( ) 0rs = s = s = = s = . The controllers can be obtained by substituting

, , ,( ) ( 1)/( 1)

, 1, 1,

( ) ( 1)/( 1), 1,

0, 0,

/ 1, , 1,

,

,

,

i r i r i ri r r i

i r i i r i r

i r r ii r i i r

r r

N i r

N

N N

N

- - +- -

- - +-

Y = j = -

j = s + b Y

= s + b

j = s = s

(4.28)

in (4.26). The quasi-continuous controllers for r £ 4 and i

b , taken from [72], have form:

1. sgn( )u =-a s ,

2.

1/2

1/2

sgn( )

( )u

s+ s s=-a

s + s

,

3.

2/3 2/31/2

2/3 1/2

2( ) ( sgn( ))

( 2( ) )u

-s + s + s s+ s s= -a

s + s + s

,

4.

3/4 3/4 3/41/3 2/3 1/2

3/4 2/3 1/2

3[ ( 0.5 ) ( 0.5 sgn( ))][ ( 0.5 ) ]

3[ ( 0.5 ) ]u

- -s + s+ s + s s+ s s s + s + s=-a

s + s + s + s

.


In HOSM, s is not calculated according to (4.17) or (4.23), but it is defined only

as a difference between the output of the dynamic system and its reference:

d

x x xs = - = . (4.29)

Moreover, the controller (4.26) requires real-time exact calculation or direct measurement

of s,s ,…, ( 1)r-s . A homogenous real-time robust (r-1)th-order differentiator of s is presented in [71]:

( 1)/1/0 0 0 0 0 0 1

( 2)/( 1)1/( 1)1 1 1 1 1 0 1 0 2

1/21/22 2 2 2 2 3 2 3 1

1 1 1 2

, sgn( ) ,

, sgn( ) ,

, sgn( ) ,

sgn( ),

r rr

r rr

r r r r r r r r r

r r r r

z v v L z z z

z v v L z v z v z

z v v L z v z v z

z L z v

-

- --

- - - - - - - - -

- - - -

= =-g -s -s +

= =-g - - +

= =-g - - +=-g -

(4.30)

where the parameters of the differentiator are chosen according to the condition ( )r Ls £ , L

is to satisfy M

L C K³ +a , where ( )m M

0 r

uK K¶

¶< £ s £ and ( )

0

r

uC

=s £ . The values of

ig

can be chosen in advance. In [72], the following values are proposed: 0

g = 1.1, 1

g = 1.5, 2

g = 2,

3g = 3,

4g = 5, and

5g = 8. The differentiator values can be easily changed. The

larger the parameters are, the faster the convergence and the higher the sensitivity to input noises and the sampling step is.

According to [71], it is advantageous to use differentiators of higher-order, because they track more accurately also lower derivatives of the real signal (s ,s, …) than the lower-order differentiators do. The drawback is its increasing demands on the calculation precision with increasing order of the differentiator. Already a fifth-order differentiator needs long double calculation precision (128 bits) [71] whereas the calculation precision in Matlab is only double (64 bits).

The controller (4.26) is now constructed from the outputs of the differentiator:

1, 0 1 1

( , , , )r r r

u z z z- -= -aY . (4.31)

From the structure of the controller (nested or quasi-continuous) one can see that a detailed mathematical model of the controlled plant is not necessary. This makes the controller suitable for controlling nonlinear systems and robust against model inaccuracies. On the other hand, limited accuracy can be expected.

In HOSM as presented in [71-74], the controller does not include a feedforward part as in standard SM since it assumes that the reference signal is not known in advance. Anyhow, one is free to add a feedforward controller to cooperate with HOSM controller.

It should be also pointed out that for LTI systems like (4.1), standard sliding-mode control and back-stepping control lead to the same controller structure [118]!

Simulations

Several simulations were performed with sliding-mode controllers. First, standard, first-order sliding-mode controllers with integral action were designed with saturation


instead of sgn function to avoid chattering. These SM controllers need information on the velocity the error:

dx x x= - . Since the simple differentiation of the error signal caused

high noise on the velocity error estimation, two types of estimators were used to estimate the velocity: Kalman filter and third-order differentiator from (4.30). The results from simulations for the standard SM controllers with Kalman filter and third-order differentiator are in Figure 4.17 and Figure 4.18, respectively. Both figures contain results from the simulations with the exact plant values (black) and with offsets in the estimated values of the COG, mass and inertias (gray) (see Section 4.4.1). The parameter values of the SM controller with the Kalman filter were l = 150, d = 1/60, and k = 10, and the SM controller with the differentiator: l = 250, d = 1/5, k = 10, and L = 1500. From comparison of Figure 4.17 and Figure 4.18 one can see that with the differentiator slightly better tracking performance in translational DOF’s was achieved, but at the price of higher steady-state noise and control activity. Compared to the PID controller performance from Figure 4.16, the SM controllers provide (slightly) better tracking performance. More importantly, from those figures it is evident that the SM controllers are more robust against the parameter uncertainty (gray plots).

Third-order quasi-continuous HOSM controllers with third-order differentiators were also tested. Since the trajectory is known in advance, the acceleration feedforward was added. The simulation results are in Figure 4.19 for the exact plant values (black) and the values with offsets (gray). The parameter values were a = 600 and L = 1500. A fourth-order differentiator was also tested without any improvement of the error tracking. Overall achieved performance was much worse than with the first-order SM controllers. One of the reasons is that the HOSM controllers react more to the signs of s and its derivatives rather than to their actual values. Therefore, in situation close to s = s = s = ... = 0, still a lot of chattering (which can be for a limited sampling rate seen as discontinuity) is visible in the control signal. Consequently, the chattering together with the delay in the loop caused worse tracking errors. On the other hand, first-order SM controller (4.24) combines linear controller with nonlinear switching which was for 0s linearized (saturation instead of sign). In the case of a highly nonlinear plant, the HOSM controller would probably have performed better with less sensitivity to the parameter uncertainty, but the PA is generally a linear system with minor nonlinearities and parameter uncertainties. Therefore, the first-order SM controller is able to achieve better performance.


Figure 4.17: First-order SM controller with Kalman filter for the velocity estimation; the tracking

errors and feedback controller outputs during tracking of the trajectory from Figure 4.13; with

(gray) and without (black) offsets in the mass, moments of inertias and center of gravity estimation.


Figure 4.18: First-order SM controller with the differentiator (4.30); the tracking errors and

feedback controller outputs during tracking of the trajectory from Figure 4.13; with (gray) and

without (black) offsets in the mass, moments of inertias and center of gravity estimation.


Figure 4.19: Third-order HOSM controller with third-order differentiator (4.30); the tracking

errors and feedback controller outputs during tracking of the trajectory from Figure 4.13; with (gray) and without (black) offsets in the mass, moments of inertias and center of gravity estimation.


4.4.3 Iterative Learning Control

Background

Iterative learning control belongs to the category of learning control. A learning controller is a control system that comprises a function approximator of which the input-output mapping is adapted during control in such a way that a desired behavior of the controlled system is obtained. Three main categories of learning controllers exist: Iterative learning control (ILC), Repetitive control (RC) and Learning feedforward control (LFFC). ILC and RC are designed for systems that perform the same repetitive task each time. The main difference is that ILC assumes the same initial conditions each time, whereas RC expects continuous repetitive motion where the initial states of the next run are equal to the states at the end of the previous run. Therefore, RC is mostly applied on the system subject to periodic disturbances (e.g. from rotating machine) [50, 110]. If the disturbance production is related to the performed task (e.g. position-dependent), ILC is preferable. In case of repetitive tasks, LFFC is similar to ILC and RC [80, 116]. The main difference is in the implementation of a B-spline neural network, which works as a function approximator. LFFC for repetitive tasks uses a time-indexed memory of the feedforward signal. If the memory is input-indexed, LFFC can store a feedforward signal also for non-periodic motions. Input-indexed memory means that the feedforward signal is stored for each combination of the reference signal and its derivatives (e.g. position, velocity and acceleration) used as inputs. This is also drawback of LFFC known as curse of dimensionality (the amount of necessary memory as well as the learning time increases exponentially with the number of inputs) [116]. Therefore, input-indexed LFFC is not suitable for MIMO systems. Since the COPAM was developed for point-to-point operations, the initial conditions at the beginning of each cycle can be assumed to be the same. Therefore, ILC has been chosen to be tested on the system.

There are two categories of ILC: standard [113, 5, 9] and lifted [30, 111, 36]. The block diagram of a feedback controlled system with standard ILC added is shown in Figure 4.20. The standard ILC uses the tracking error off-line to calculate a feedforward

Figure 4.20: Block representation of standard ILC.

P C r k

y

L

Q

Memory 1k

f +

kf k

e

ILC


signal that reduces the error between subsequent iterations. For this, the error

ke from kth

iteration is filtered by a learning filter L and added to the feedforward kf . The sum of the

filtered error and the feedforward is filtered by a robustness filter Q to obtain the feedforward signal of the next iteration

1kf + . The performance and robustness of ILC is

determined by the design of the filters L and Q.

The feedback error with respect to the reference r and the feedforward signal of the kth iteration can be written as:

pk k

e Sr S f= - , (4.32)

where S is the sensitivity and p

S the process sensitivity. The learning update from one to the next iteration is

1

( )k k kf Q f Le+ = + . (4.33)

By using (4.32) and (4.33) the error of (k+1)th iteration can be written as:

1 p

(1 )k k

e Q LS e+ = - . (4.34)

Then the error converges if

p

(1 ) 1,Q LS- < "w. (4.35)

From (4.35) it follows that the optimal filter is given by 1p

L S -= . However, the ideal L filter is difficult to derive since 1

pS - is often a non-causal transfer function,

especially for motion systems. Another difficulty arises if p

S is the non-minimum phase, the inverse of

pS will then have instable poles and numerical problems can arise. Therefore in

practical cases, an approximation of 1p

S - is usually calculated using the zero phase error tracking (ZPETC) algorithm [109]. This algorithm calculates an alternative for the exact inverse of the

pS by mapping the non-minimum phase zeros to the left half plane. The

algorithm assumes that a discrete model of the p

S and a closed-loop stabilizing controller C are available. The inverse is calculated with zero phase-error. The non-causality of the L filter is eliminated by shifting the filtered signal back in time, which can be done since the feedforward signal is calculated off-line.

The robustness filter Q is used to satisfy (4.35) subject to the modeling errors and non-repetitive signals at high frequencies. The Q filter is usually designed as a low-pass filter whose cut-off frequency is chosen in such a way that (4.35) holds.

The remaining tracking error, when the learning has converged, can be expressed as:

1

(1 )(1 )

Qe r

Q PC PQL¥

-=

- + +. (4.36)

The Q filter is 1» at low frequencies and 0Q » at high frequencies. From (4.36), the tracking error at low frequencies is 0e¥ » , while on high-frequencies

1

1e r Sr

PC¥ » =+

. (4.37)


From (4.37) one can see that the reference trajectory r should have limited high-frequency components (smoothness, see Figure 4.10) to achieve small tracking errors. It should also be pointed out that the final tracking error combines (4.36) with additional non-repetitive disturbances (e.g. sensor noise). These disturbances deteriorate the performance of ILC. It was shown in [83] that the additional error due to the non-repetitive disturbances is approximately equal twice the disturbance amplitude. Several techniques have been developed to reduce the influence of the tracking error on these disturbances. An ILC algorithm for systems that are subject to measurement noise is introduced in [85]. The algorithm is based on a multi-loop control interpretation of the signal flow in ILC to derive the necessary and sufficient condition for convergence. The input is updated according to an estimate of the Markov parameters [85]. A phase-lead compensated ILC scheme is presented in [119]. It has a significantly broader learnable frequency band, which results in a higher precision. A disturbance observer in the iteration domain can be used as a pre-filter to improve performance of the ILC scheme [23]. Another approach to prefiltering is wavelet filtering of the error signal [84]. A time-varying Q filter can be designed using a time-frequency analysis of the reference and feedforward signal [37, 124, 123].

In the lifted ILC, the system dynamics is considered to be a static map, which describes the system behavior along a finite time interval [30, 111, 36]. In the lifted representation, the reference trajectory r is considered as a disturbance that does not change from iteration to iteration and needs to be suppressed. The filter L becomes the feedback gain matrix, which stabilizes the system given by the process sensitivity function

pS (see Figure 4.21).

Since the PA is meta-stable system, its stabilization is the priority. Therefore, from the practical point of view, it is safer to apply standard ILC as the feedforward control to the existing stabilizing controller than to combine design of stabilizing and learning controller into one. Moreover, the performance of the standard and lifted ILC are comparable [101].

Figure 4.21: Block representation of lifted ILC.

Simulations

For the simulations, the same PID controller as in Section 4.4.1 has been used. Since the focus of this simulation is on the feedforward (represented by ILC), the open-loop bandwidth of the controller was decreased to 30 Hz to emphasize performance of the ILC. Moreover, the simulation has been performed with offsets in the estimated values of the COG, mass and inertias.

Sp L r

Q

Memory

1kf +

kf

ke

lifted ILC

S


First, the learning filer L was designed as the approximate inverse of the process

sensitivity 1p

L S -» by using the ZPECT algorithm (see Figure 4.22). Then the simulation of the system with the 30Hz PID controller and the reference trajectory was executed 10 times. Each time, the tracking error e(t) was captured. A mean value of the error over ten

simulations was calculated: 101

10 1( ) ( )

iie t e t

== å . The mean value represents the repetitive

part of the error and ( ) ( )e t e t- represents the non-repetitive part. From the plots of the power spectral density (see Figure 4.23) of those errors, one can see that trying to remove errors above the frequency of about 100 Hz makes no sense. For that reason, the robustness

100

101

102

103

-200

-100

0

100

200

Ma

gn

itud

e (

dB

)

L

Sp

100

101

102

103

-400

-200

0

200

400

Ph

ase

(d

eg

)

Frequency (rad/s)

Figure 4.22: Bode plots of the process sensitivity Sp (gray) and the learning filter L (black); for

the translational DOF’s.

100

101

102

103

10-24

10-22

10-20

10-18

10-16

10-14

10-12

10-10

Frequency (Hz)

Po

we

r S

pe

ctra

l De

nsity

(m2

/Hz)

repetitivenon-repetitive

Figure 4.23: Power spectral density of the error in x-direction; repetitive (black) and non-

repetitive part (gray).


filter Q has been designed as 4th-order low-pass Butterworth filter with cut-off frequency of 100Hz. However, low-pass filters introduce a phase shift to the filtered signal, which results in a reduced tracking performance. Therefore, a filtering procedure is used that first filters a signal normally and successively restores its phase by applying the same filter backwards in time (MATLAB’s filtfilt function). To smooth the learning process and to reduce sensitivity to the non-repetitive disturbances, a learning gain (0,1c Î is commonly used in (4.33):

1( )

k k kf Q f cLe+ = + . If c = 1, theoretically only one iteration step would be necessary.

Since the ILC is applied to a MIMO system which has some residual cross-coupling, c = 1 could destabilize the learning process. Therefore, after several initial simulations c = 0.7 was chosen. The only disadvantage of reducing the learning gain is increased number of necessary iterations.

Twenty learning iterations were executed while the tracking error already converged in seven iterations (see Figure 4.24). The final tracking error and the controller action (including the ILC feedforward without acceleration feedforward) are shown in Figure 4.25. From the plots, it is clear that the trajectory tracking errors are not explicitly visible anymore, but the final noise level of the error signal is about twice the sensor noise level (see Section 4.2) as was expected. Nevertheless, the noise level is comparable or even smaller than what was achieved with the PID or SM control in steady-state levitation.

Anyhow, if even higher tracking-error suppression was necessary, one of the methods for non-repetitive error reduction described in the theoretical part of this section could be applied.

2 4 6 8 10 12 14 16 18 20

10-7

10-6

10-5

10-4

10-3

Ma

x. e

rro

r (m

) o

r (r

ad

)

Iterations

xyz

Figure 4.24: Maximum tracking errors in different DOF’s.


Figure 4.25: Final tracking errors and controller outputs (feedback + ILC without the

acceleration feedforward) during tracking of the trajectory from Figure 4.13; with offsets in the mass, moments of inertias and center of gravity estimation.


4.5 Control of Planar Actuator with Manipulator

As was mentioned in the introduction section of this chapter, there is a manipulator on top of the planar actuator. Generally, the manipulator can be used to increase the number of degrees of freedom of the system or to decrease the dissipative power by actuation of a small, light manipulator on the platform for some short-stroke operations. It was also described in the introduction Chapter 1, that the purpose of the manipulator in this particular setup is to test the combination of three contactless technologies: electromagnetic bearing, contactless energy transfer and wireless control.

The manipulator on top of the platform has two ironless linear actuators attached to a stiff beam. The bearing of the beam is assumed ideal, therefore, only 1-DOF movement of the beam is considered possible. In the center of the beam, a rotary motor is assembled with an arm attached to it. The tip of this arm can be positioned anywhere in the xy-plane between the linear motors by combining the translation of the beam and the rotary movement of the arm. The linear and rotary actuators are 3-phase, synchronous ironless and slotless motors, respectively. Therefore, they do not suffer from cogging or reluctance effects.

Commutation for the linear and rotary motor is provided by the FPGA (Filed-Programmable Gate Array) which is also situated on the moving platform. The controller for the manipulator is implemented in the dSpace system next to the setup. Infrared wireless data transfer is used for communication between the dSpace and the FPGA (see Chapter 5). The developed wireless link causes no measureable delay in comparison to the sampling rate of the system. Therefore, it does not have to be considered during the control design.

The resolutions of the linear and rotary motor encoders are 1 μm and 20 μrad, respectively, which is about a factor 10 worse that the resolution of the sensors of the PA. Therefore, the goal of controlling the total system is to reduce disturbances acting between the manipulator and the planar actuator rather than trying to achieve the best tracking performance of the end-tip of the manipulator with respect to the fixed world. The general block diagram of the system is shown in Figure 4.26. Without considering the wireless link, the control diagram can be rewritten using one multi-body system as shown in Figure 4.27. For the sake of simplicity, the error-coupling ( )X q block and delay (see Figure 4.4) are not shown in the figure although they are included in the model. Previously, in the case of the single-body PA, the commutation block combined transformation of the desired wrench vector into the current vector with decoupling of the center of gravity of the PA. Now, the total decoupling of the multi-body system is in separate block. In this way, separate SISO controllers can be again applied on each DOF.

Modeling and identification of the manipulator together with generation of the model of multi-body dynamics of the total system is extensively described in [13]. Here, a short summary is given.


Figure 4.26: Control diagram of the COPAM.

Figure 4.27: Control diagram of a multi-body model of the COPAM; des

w is the desired wrench of

the decoupled 8-DOF system, des

t is the desired wrench for the actuators (PA and manipulator (MA)), i is the current vector applied to the actuators, ( )qK represents the electromechanical coupling between the currents and produced wrench, and t is the real wrench applied on the multi-body system.

4.5.1 Multi-body Model

The manipulator on top of the planar actuator adds two DOF’s to the six DOF’s of the planar actuator. Position and orientation of the total, 8-DOF system can be described by eight generalized coordinates:

T

LM RM, , , , , , ,q x y z xé ù= q y f fê úë û , (4.38)

where , , , ,x y z q y, and f represent the positions and orientations of the geometric center of the magnet array,

LMx is the displacement of the beam with respect to its center position,

and RM

f is the orientation of the rotary motor.

A multi-body model of the COPAM by using the equation of Lagrange (e.g. [66]) was first developed in [13]. The model combined mechanical and electromagnetic domain, which allowed merging the dynamic equations of the coil currents and the

PPA(s)

1PA-K

CPA(s) wPA refPA

PAq

Planar actuator

Sensor noise

PMA(s)

CMA(s) wMA refMA MA

q Manipulator

Commutation

1MA-K

Commutation

Wireless

wMA

wPA

MAi

PAi

PAK

MAK

Real disturbing

forces

Compensation and decoupling

Estimated disturbing and feedforward forces

Wireless

Multi-body Dynamics

1est

( )q-K

C(s)

dest

ref q Planar actuator + Manipulator

Sensor noise

Commutation

i

( )qK

est est( , )D q q

desw t

Decoupling 8x SISO

Feedforward


electromechanical interaction between coil current and the moving magnets into a single set of dynamic equations together with the equations of the multi-body dynamics. However, this extensive model with 98 DOF’s was too big to be used for simulations and the full multi-body model was never used to decouple the COPAM. Moreover, the model presented in [12] wrongly assumed no relevant friction in the system. Therefore, a new multi-body model is derived here.

The multi-body model is also derived by using the equation of Lagrange:

( )T

ncd

d

T T VQ

t q q q

æ öæ ö¶ ¶ ¶ ÷ç ÷ç ÷÷ç ç - + =÷÷ç ç ÷÷ç ÷ç ¶ ¶ ¶ ÷ç è øè ø, (4.39)

where T and V represent the kinetic and potential energy of the system, respectively, and ncQ is the vector of generalized non-conservative forces.

The kinetic energy of the system is the sum of the translational and rotational kinetic energies of each body. In the case of three moving bodies:

( )3

1

1

2 i CMi CMi i CMi ii

T m r r=

= ⋅ + w ⋅ wå J , (4.40)

where i

m is the mass of ith body, CM

r is the velocity vector of the center of mass, CM

J is the

inertia matrix of the body with respect to its center of mass, w is the angular velocity vector with respect to the fixed world, and the operator “⋅” represents the inner product.

Since the system is modeled as a rigid-body with no springs, the potential energy is caused only by the gravitational forces acting on the bodies:

3

1i CMi

i

V m g r=

= - ⋅å , (4.41)

where g = [0, 0, -9.81] m.s-2 is the gravitational acceleration vector.

The non-conservative forces represent the external forces and torques acting on the system, and the losses in the system due to the friction or damping. Since the PA is a contact-less system, no friction is present and the damping due to the air is negligible. However, there is a friction in the bearings in the manipulator. Although the resulting force due to the friction acting on the total system is zero (Newton’s third law), its modeling is important, because it will cause the difference between the applied force and the force causing acceleration of the body. The friction in the rotary motor is assumed to be negligible. The vector of non-conservative forces can be written as:

T

ncLM f LM RM

, , , , , , sgn( ),x y z x y z

Q F F F T T T F F x Té ù= -ê úë û , (4.42)

where x

F ,…,z

T are the forces and toques produced by the planar actuator, LM

F is the force produced by the linear motors,

RMT is the torque from the rotary motor, and

fF is the

friction force due to the beam bearing.

The Lagrange equation (4.39) can be rewritten in the following form:

( ) ( , )q q H q q+ = tM , (4.43)


where t is the vector of the applied forces and torques, ( )qM is the mass matrix dependent on the generalized coordinates, ( , )H q q is the vector combining centripetal, Coriolis, friction and gravitational terms. The generalized accelerations are then derived as:

( )1( ) ( , )q q H q q-= t -M . (4.44)

To realize the decoupling of (4.43), inversed relation D is necessary:

1des des

( , , ) ( ) ( , )D q q w q w H q q-t = = +M M , (4.45)

where des

w is the wrench vector of desired forces and torques of the decoupled system, M is the diagonal mass matrix and t represents the real applied forces and torques. By combining (4.45) with (4.44), the decoupled system is obtained:

1des des

, orq w q w-= =M M . (4.46)

The artificially added diagonal mass matrix M can be also omitted and, instead of forces and torques, desired acceleration can be output of the controllers.

To realize the decoupling from (4.45), the states of the system ( , )q q are necessary. With a priori knowledge of the system, the number of necessary states can be reduced. The planar actuator is designed for long-stroke movements in the horizontal plane with stabilization of the rotations. From the simulations in Section 4.4, one can see that the maximum angle deviation is only several tens of μrad. Therefore, the angles are assumed to be zero in the decoupling: 0q = y = f = . Moreover, because the arm connected to the rotary motor has very low mass in comparison with the rest of the system, the information about the orientation and velocity of the arm is also excluded from the decoupling. Then the position and velocity of the beam

LM LM( , )x x and angular velocities of the PA ( , , )q y f are

the only necessary information.

4.5.2 Simulations

For the following simulations, the reference trajectories for the linear and rotary part of the manipulator have been created (see Figure 4.28). The reference trajectories for the xy movement of the planar actuator remained the same. The trajectory parameters for

the linear motors are the same as for the trajectory of the PA: max

v = 0.5 m.s-1, max

a = 10

m.s-2, max

j = 500 m.s-3, max

d = 1×105 m.s-4. The trajectory parameters for the rotary motor

are as follows: max

v = 100 rad.s-1, max

a = 500 rad.s-2, max

j = 3×105 rad.s-3, max

d = 5×106

rad.s-4.

Three controllers have been tested on the multi-body model of the COPAM: PID, first-order sliding mode with Kalman filter, and iterative learning controller. The controller parameters remained the same. The same controllers were also then used for controlling the manipulator.

During simulations, it was found that for the translational movement of the platform, the centripetal and Coriolis forces are negligible and, therefore, they do not have to be included in the decoupling. This reduces the number of the relevant parameters in the decoupling to just two: the position and velocity of the beam

LM LM( , )x x . It was also


0 1 2 3 4 5 6 7 80

0.02

0.04

pos

(m

)

x

y

0 1 2 3 4 5 6 7 8-0.1

0

0.1

x LM (

m)

0 1 2 3 4 5 6 7 8-5

0

5

t (s)

RM

(rad

)

Figure 4.28: Reference trajectory profiles for the planar actuator with the manipulator.

found that the offsets in the estimated values of the masses, inertias and the center of gravity of the multi-body model produce approximately the same errors as in the situation of the single-body planar actuator (see Figure 4.16, Figure 4.17, and Figure 4.25). Therefore, the following simulation results focus on the two new parameters in the decoupling

LM LM( , )x x .

In the first test, variation of the beam position LM

x is not included in the decoupling. The results with the PID controllers are in Figure 4.29. Since the accuracy of the linear and rotary motor tracking is not influenced in this test, those plots for linear and rotary motors in Figure 4.29 show the capability of their controllers in the ideal situation. From the figure, it is clear that the beam position mainly affects the orientation DOF’s of the PA with the peak error of about 30 μrad.

In the second test, an offset of 2 N in the estimated dynamic friction f

F is made. The results with PID controllers are in Figure 4.30. The plots show that the friction offset can produce significant errors in the related directions. Since the friction was modeled as a dynamic friction, which produces constant resistive force during nonzero velocity, only the acceleration phase of the trajectory was influenced. In the case of more nonlinear friction behavior, also the constant-velocity phase of the trajectory would be influenced.

The results from the same two tests with the sliding-mode controllers are in Figure 4.31 and Figure 4.32. The influence of the beam position and the friction on the tracking precision is similar, only the magnitude of the error is about half thanks to the sliding-mode controllers.

The iterative learning controllers exhibit in both cases no visible tracking errors. The number of necessary iterations to achieve removal of the disturbances remained approximately the same as in the situation with the single-body PA (see Figure 4.24 and Figure 4.25), so about seven.


Figure 4.29: PID control; the tracking errors and feedback controller outputs during tracking of the trajectory from Figure 4.28; with (black) and without (gray) included variation of the beam position

LMx in the decoupling. Only the relevant DOF’s shown.

4.6 Summary and Conclusions

In this chapter, two control situations were considered. At first, only the control of the planar actuator was investigated. The planar actuator was modeled as 6-DOF rigid-body mass-only system. To include the electromechanical tolerances in the model of the commutation, a model of the error coupling was developed, mapping the applied currents in the coils to the wrench (forces and torques) disturbances acting on the PA. This error


Figure 4.30: PID control; the tracking errors and feedback controller outputs during tracking of the trajectory from Figure 4.28; with (gray) and without (black) an offset in the friction estimation in the decoupling. Only the relevant DOF’s shown.

coupling caused residual cross-coupling between the different DOF’s. The model does not match the real error-wrench behavior, but this was not intention since insufficient data from the setup were available. Still, the model mimics the disturbance behavior of the real PA from the COPAM with its magnitudes and position-dependency, which makes it suitable for simulation of the designed controllers.

The planar actuator is propelled in two DOF’s. For these two degrees of freedom a trajectory must be planned and trajectory profile generated. Planar actuators are mainly designed for use in pick-and-place machines or wafer scanners. Hence, point-to-point motion with an eventual constant velocity part is desirable. To satisfy electromechanical constraints on the system and to reduce excitation of the flexible modes, fourth-order trajectory generation was used.

Three different kinds of controllers were tested on the model of the PA: classical PID, sliding mode and iterative learning control.

With the PID control, the resulting peak tracking errors in the x- and y-direction were about 3 μm and 4 μm, respectively, while in the z-direction the error was at most 7 μm. The peak orientation error was for each axis about 20 μrad. These errors are the result of the delay in the system and the error coupling in the commutation. When offsets in the estimated values of the mass, inertias and the center of gravity of the PA were included,



errors and feedback controller outputs during tracking of the trajectory from Figure 4.28; with (black) and without (gray) included variation of the beam position

LMx in the decoupling. Only

the relevant DOF’s shown.

the tracking errors increased to about 5 μm, but mainly the errors in the rotational DOF’s increased to about 70 μrad peak.

Several sliding-mode controllers were tested. The standard (first-order) sliding-mode (SM) controller for the second-order system needs information on the velocity error. Two approaches were tested for estimation of the velocity: Kalman filter and a differentiator developed for higher-order SM controllers [71]. Saturation instead of sign



errors and feedback controller outputs during tracking of the trajectory from Figure 4.28; with

(gray) and without (black) an offset in the friction estimation in the decoupling. Only the relevant DOF’s shown.

function was used in the SM controller to eliminate the chattering effect. A higher-order SM (HOSM) controller was also investigated as an alternative for reduction of the chattering effect in the standard SM. From the simulation results, the best performance was achieved with standard (first-order) SM controllers in combination with Kalman filters for the velocity estimation. These controllers were able to reduce the tracking error by about 30 % and improve the orientation stabilization by 50 % in comparison to the PID. The improvement was similar also in the case of the offsets in the estimated values. The tracking errors of the SM controller with differentiator were even more reduced, but at the price of high control activity. The worst performance was achieved with HOSM represented by a third-order SM controller. The errors were about a factor 10 higher than in the previous cases. The reason for this result is the general control structure of HOSM controller, which does not take into account the plant model. The only information necessary to construct the controller is the plant order. This makes the HOSM suitable for stabilization of highly nonlinear plants, but on the other hand, high tracking precision cannot be achieved when the plant dynamics is not taken into account.

Iterative learning controller (ILC) was applied as the feedforward to the system controlled by PID. By using this control technique, the errors were reduced to the noise level within several iterations. The final peak errors were about 0.5 μm for the translational DOF’s and 5 μrad for the rotational DOF’s.


In the second situation to be investigated, the manipulator was included. The 8-

DOF system, the 6-DOF planar actuator with the 2-DOF manipulator, was described with a multi-body model by using the Lagrange’s equation. The system was decoupled to allow for separate SISO control of each DOF again. Since in the simulation almost exact decoupling can be achieved (except for the delay and the noise in the states), the aim of the investigation was to compare the relevance of the parameters for the decoupling of the system. It was found that since the rotational degrees of the PA are well stabilized, their influence as well as the centripetal and Coriolis forces are negligible for the translational trajectory. Moreover, the small arm connected to the rotary motor of the manipulator is very light in comparison to the rest of the system. Therefore, the information about its position and velocity hardly affect precision of the decoupling. The only relevant parameters necessary for good decoupling are the position and velocity of the manipulator’s beam. The position of the beam gives information about the varying center of the gravity of the whole system and the velocity information is necessary for counteraction of the friction forces in the beam’s bearing.

Three controllers have been tested on the multi-body model of the COPAM: PID, first-order sliding mode with Kalman filter, and iterative learning controller. The controller parameters remained the same as for the control of the planar actuator alone. The same controllers were also used for controlling the manipulator. The results from simulations are similar to the results made on the single-body planar actuator. The SM controller again outperformed the PID by several tens of percents, whereas with the ILC, the errors were reduced to the noise levels within several iterations.

Since the best performance was achieved with the feedback SM and with the feedforward ILC (with PID feedback), one can think about combining feedforward ILC for repetitive system behavior with feedback SM for random disturbance attenuation. This combination has already been investigated for some time [105, 121, 99, 38]. However, in those articles, due to the nonlinearity of the SM controller the learning filter L in the ILC is not based on the inversion of the process sensitivity 1

pS - , but as a simple gain. Therefore,

in case of LTI system with dominant deterministic behavior (as the COPAM), this approach cannot mach the performance produced by the frequency-domain based ILC (where 1

pL S -= ).

129

Chapter 5

Wireless Communication

5.1 Introduction

The manipulator on top of the contactless planar actuator (COPAM) is powered wirelessly using energy transfer from the coils in the baseplate to the secondary coil placed on the moving platform (see Chapter 1). The goal of this chapter is to find the most suitable wireless communication technique for real-time closed-loop control of the manipulator. The chapter is organized as follows. The demands on the wireless control are presented in the next section. Section 5.3 contains a comparison of several wireless solutions that seem most suitable for the task and a new infrared (IR) wireless data link for real-time control is presented. This wireless solution is described in detail in Section 5.4. Section 5.5 contains results of the IR wireless link performance. This work has been published in [45].

5.2 Desired Specifications of the Wireless Link

For control systems, the requirements on wireless connection are not just high speeds over long distances. Requirement for short delay (latency) is very often on the first

place together with reliability of the connection. Short delay is a relative expression,

depending on the controlled system. When the temperature in a building with time constants of seconds or minutes is controlled, a delay of hundreds of millisecond is sufficient. On the other hand, when a fast motion system is controlled, the delay of the link should be kept in microseconds. In such a system, delays on the sensors, controller and actuators can already significantly reduce the performance of the system and therefore all additional delays should be reduced as much as possible. Reliability of the link is also of great importance for control systems. When the delay of the link is very small in

130 Chapter 5. Wireless Communication

comparison to the sampling rate and time constants of the system, the corrupted data packet can be retransmitted, which is the usual behavior of most of the wireless communication techniques. This is not possible in a system with a high sampling rate and small time-constants like a motion system. Here, retransmission of the data packet is undesirable because the controller or actuator needs newest data samples and older data are irrelevant. This gives two additional conditions on such a wireless link: no

retransmission of the packets and very small error rates. Another property that should be

taken into account is what kind of connection between sensors, controller and actuators is necessary. If the sensors and/or actuators are distributed in some area, multipoint connection is desirable. This is usually true for distributed multiple-input multiple-output (MIMO) systems. On the other hand, multipoint connection means that one carrying medium is shared between several points and hence the communication needs to be controlled with some scheduler. This situation is partly valid also for point-to-point connection when such a link is in the area of another communication link working in the same range of the used medium. Otherwise, in the case of single-input single-output (SISO) systems or MIMO systems with concentrated sensors and actuators, as in case of the COPAM, point-to-point connection is more beneficial because there is no need for

communication scheduling.

Since there is no cable connection between the base plate and the platform, the platform with manipulator can theoretically move freely in horizontal plane. In reality, the long-stroke movement is limited by the area of the baseplate filled with coils. The area of free movement for the planar actuator is a few hundred millimeters in each direction. This puts another condition on the wireless link: sufficient range of about one meter. The

manipulator involves three three-phase motors. Encoders are used as the sensors. Since a commutation algorithm for all three motors is executed in an FPGA (field-programmable gate array) on the platform, only the current amplitude is needed as a control signal for each motor. The desired resolution of the control signal for each motor is 20 bits. The position of each motor is represented by 21bit counters, which are connected to the encoders. Apart from that, there are also some control and state signals. Finally, the data

packet has been chosen to be 64 bits for both ways. Since the sampling rate of the

controller in the dSpace is about 3 kHz, the sampling rate of the wireless link should be at least the same or better in order to obtain new data each sample time. This gives absolute minimum bandwidth of 192 kb/s. Because such a data link causes an additional delay of two samples, it is desirable to reduce the delay to negligible values so the wireless connection will provide approximately the same performance as the wired one. For that reason, the limit on the delay or latency of the link was chosen to be less than 50 μs in each direction. As a result, the additional delay due to the wireless link will be only 100 μs of the sample time or 36 degrees of phase delay at 1 kHz in closed loop. This gives the minimum bandwidth of 1.28 Mb/s, when no additional bits are used in the packet.

Summarized requirements:

Latency: < 50 μs Low packet-loss ratio with no data retransmission Range: > 1 m Effective bandwidth (throughput) for 64-bit data packet: > 1.28 Mb/s

5.3 Overview of Standardized Wireless Solutions 131

Point-to-point connection Preferable no communication scheduling

5.3 Overview of Standardized Wireless Solutions

In this section, an overview of existing wireless communication techniques that seem suitable for real-time control is presented. There are two main types of wireless link. The first type transmits data on radio waves. Usually the license-free frequency around 2.4 GHz is used. This is the case of Zigbee [125], IEEE 802.11b WLAN (UDP/IP) [54] and Bluetooth [12]. The second type transmits data using light in either a visible or invisible spectrum. Two main technologies belong to this category: on infrared light based VFIR [117], and on laser light based Laser link [68]. Zigbee, Bluetooth and Laser link are designed as reliable network connections, which use retransmission of the data. This can increase the delay. Even though the principle of retransmission does not seem efficient in the case of delay specification, several measurements were done in the past concerning control with techniques using retransmission protocols. From these studies, it should be possible to derive a certain general delay for each specific wireless solution. A short comparison of the mentioned wireless technologies is given in Table 5.1.

Table 5.1: Wireless links overview

Technology Raw data

rate Maximum

throughputa

Minimum delay

Indoor range

Zigbee 250 kb/s 160 kb/s 6 ms 75 m

802.11b UDP/IP 11 Mb/s 7.7 Mb/s 1.78 ms 38 m

Bluetooth 1-3 Mb/s 2.1 Mb/s 7 ms 10 m

VFIR 16 Mb/s > 15 Mb/s < 100 μs 1 m

Laser link >100 Mb/s > 80 Mb/s ~ ms > 100 m

a With optimal data packet size.

Zigbee was developed for low-power digital radios in wireless personal area networks (WPAN’s) for applications that require a low data rate, long battery life and secure networking. The maximum real throughput is about 160 kb/s and a delay of 6 ms for 122 bytes of data [69]. Therefore, neither bandwidth nor latency is sufficient for the setup.

Bluetooth uses a radio technology called frequency hopping spread spectrum. It has also been developed for WPAN for easy connection of multiple devices. It has sufficient real bandwidth of 2.1 Mb/s, but the minimum delay is too large.

IEEE 802.11b UDP/IP (User Datagram Protocol) is a simple, message-based connection protocol. This protocol does not resend packets and therefore the delay is small [94], but still more than 35 times larger than is desired. In practice, the maximum throughput that can be achieved is about 7.7 Mb/s for large data packets but only about


250kb/s for 64bit data packets (e.g. [52])! This is caused by a lot of overhead in the packet and the way of processing the signal.

VFIR (Very Fast Infra Red) is the last standard of infrared data association (IrDA) [55]. It is a point-to-point, narrow-angle (30° cone), ad-hoc data transmission standard. VFIR has promising bandwidth as well as delay.

Laser link is one of the solutions using a laser beam for high-bandwidth point-to-point data transmission for distances from a few meters to several kilometers. Its disadvantage is high price and the need for precise aiming of the transceivers. Moreover, the delay is not suitable for a motion system control.

There is one thing common to all presented wireless solutions, which is an extensive packet with a lot of information necessary for networking, but useless in point-to-point real-time control. There is also the possibility of buying a custom transceiver that has custom protocol and sufficient bandwidth. One of such solutions is presented in [14], where 2.4GHz transceivers are used with very simple protocol and maximum bandwidth of 1.5 Mb/s. Packet size of 123 bits contained 64 bits of data, so the average latency of 175 μs was achieved. Unfortunately, the packet-loss ratio increased significantly with the

bandwidth up to more that 10-2 at 1.5 Mb/s in ideal situation*. If there was retransmission in the packet, the link would be reliable, but the delay would increase. This is the case with technology like Zigbee or Bluetooth. The problem of the trade-off between the packet-loss ratio and the delay is common for all radio-frequency (RF) based links working at 2.4GHz frequency. This is no surprise since this frequency is occupied by many different wireless technologies. For that reason, it is better to look for a wireless communication medium where the link is not shared with others. In the case of point-to-point connection in a defined environment without obstacles in the path, light seems to be the most suitable candidate. The main difference between laser and IR connection is that laser radiates in narrow beam to the long distance contrary to IR, which uses infrared-light emitting diodes producing a conic light beam. The intensity of the conic light beam is inversely proportional to the square of the distance, and limits the range of the IR data link. Despite this, VFIR has the best properties from the overview of existing solutions. The necessity of a direct path without obstacles and a short range is no problem for the presented motion system.

VFIR physical layer (IrPHY) specifications define the maximum latency of 100 μs. This latency is defined for the receiver as the maximum time for processing of the incoming signal. There is also another delay caused by packet definition. In the case of sending only small amount of a data in the packet, which is a common situation for control signals, many waste bits are sent as well. For example, 64 bits of data creates a packet of 360 bits. Then the real throughput is reduced to 2.8 Mb/s and the efficiency is less than 0.18. Therefore, a new IR wireless link has been developed based on modification of the IrDA VFIR standard. The modification includes a custom processing of sent and received signals and a new packet definition more suitable for real-time control applications.

* Active contactless energy transfer at the setup causes high electromagnetic interference with the custom RF wireless link. Consequently, the packet-loss ratio increased to 0.7 (70 %)! This is discussed in Chapter 6.

5.4 Infrared Wireless Link for Real-Time Motion Control 133

5.4 Infrared Wireless Link for Real-Time Motion Control

Since the novel IR wireless link uses the same principle as VFIR, the same transceivers are used. The transceiver modules itself are very simple, they either work as transmitters, and convert the incoming electrical pulses to the light pulses, or they are in receiving mode where the incoming light pulses are converted to the electrical pulses. Such transceiver module is for example TFDU8108 from Vishay [107]. This device is very small and needs a minimum amount of additional discrete parts. For control of these modules and processing of incoming and outgoing signal, one FPGA Xilinx Spartan IIE [104] on each side is used. The FPGA has been chosen since it allows parallel execution of several threads at different clock rates. Moreover, it has very fast integer arithmetic due to the use of interconnected logic gates.

Because the IR wireless link has been designed to behave as a replacement of the wired connection, the whole link should look from the outside like a digital cable with 64 conductors. Moreover, there are two of these links working simultaneously (full duplex). Simultaneous operation is achieved by two pairs of the transceivers on each side (see Figure 5.1). It would be possible to use just one pair of transceivers for two-way communication, but not in full-duplex mode and switching between sending and receiving mode in the transmitters would cause additional delays.

Figure 5.1: Full-duplex IR wireless link for RT control.

5.4.1 Data Processing

The data processing inside the FPGAs contains several blocks that are built according to the standard VFIR physical layer definition [117]. VFIR has a defined data rate of 16 Mb/s, but the data are modulated with a code rate of 2/3, which means that the data are sent at 24 MHz coded as chips or pulses. Each two bits are encoded as three chips (or pulses). HHH(1, 13) is used as the modulation code to achieve the specified data rate. The HHH(1, 13) ((d, k) = (1, 13), run-length limited) code guarantees for at least one empty chip and at most 13 empty chips between chips containing pulses in the transmitted IR. The encoder/decoder system is enhanced with simple scrambler/descrambler functions [117]. Scrambler/descrambler is a simple mathematical operation with the passing data. The primitive polynomial

8 4 3 21x x x xÅ Å Å Å is used, where Å indicates a modulo-2

RX

IR tran

smitter

IR receiver

IR r

ecei

ver

IR t

ransm

itte

r

FPGA

TX

Init

FPGA 2 x 64 bits

2 x 64 bits


addition or, equivalently, a logic exclusive OR (XOR) operation. By enhancing the system with scrambling/descrambling functions during data transmission/reception, one achieves generally better duty cycle statistics in the HHH-coded channel chip stream.

At the transmitter side of the FPGA, the input parallel data signals are captured in an input buffer (see Figure 5.2). Then the data are processed serially. The cyclic redundancy check (CRC) is calculated from the passing data and its checksum is added to the packet at the end of the data sequence. The data are scrambled and processed in the HHH encoder. The output of the encoder is then fed into IR transmitter.

At the receiver side, the input serial data from the IR receiver are processed in the HHH decoder and then descrambled. After that, the CRC checksum is again calculated from the incoming data. If the calculated CRC checksum matches the one in the incoming packet, the data are copied to the output buffer. Since there is no data correction sequence in the packet, the output signal stays unchanged in the case of error in the data. Clock signal in the receiver part of the FPGA is recovered in Digital Phase Loop Lock (DPLL) block from the receiving data. This clock signal is used to synchronize receiver blocks (decoder, descrambler).

The last block, which is not connected to the incoming or outgoing data processing, is used for initialization of the IR transceivers after powering them up (see Figure 5.2).

Figure 5.2: Data processing inside the FPGA.

5.4.2 Custom Communication Protocol

The main difference between standard VFIR and the novel IR wireless link is the communication protocol. A custom communication protocol is designed in such a way that there is a continuous stream of a synchronization sequence followed by a packet. The synchronization sequence (SS) contains 24 chips (equivalent to 16 bits) to synchronize the internal clock of the receiver and to recognize the beginning of the packet. This part is not

Scrambler

HHH encoder

CRC calculation

Descrambler

HHH decoder

CRC calculation

Input buffer

DPLL

RX TX

FPGA Output buffer

CRC check

Init IR


encoded by the HHH. The packet contains three fields: Data, CRC and Flush byte (FB) (see Figure 5.3). The data field consists of 64 bits of data, but arbitrary length is possible. The CRC has 16 bits and it is followed by 8-bit Flush byte to enable complete decoding in the HHH. The packet has been created without any data correction sequence. The reason for that choice will be shown later. SS is sent again immediately after the packet. In total, the packet together with SS contains 104 bits or 156 chips with a net transmission time of 6.5 μs, a sampling frequency of the parallel input/output of about 150 kHz and a

throughput of 9.8 Mb/s. This is about 3.5 times faster than the standard VFIR communication protocol! This improvement was achieved by skipping a lot of information unnecessary for point-to-point connection when the transceivers and the data frame are known in advance.

The total transmission time from the parallel input port of the first FPGA to the parallel output port of the second FPGA is a little bit longer (~ 6.8 μs) due to encoding/decoding of the signal.

Figure 5.3: The frame structure of the custom packet.

5.5 IR Wireless Link Performance

In the previous chapter, an IR wireless link for real-time control was presented. The link, as it was presented, can be used as a replacement of the cable on short distance. The delay of less than 7 μs makes it comparable to the wired solution for most applications. For implementation into the presented motion-system control loop, some adjustments have been made. The FPGA on one side is connected to a dSpace system, which is used as a controller (see Figure 5.4 and Figure 5.5). On the manipulator’s side, the second FPGA is used to run the commutation algorithm for the motors as well as ensure IR wireless communication. The two connections from dSpace digital I/O port to the FPGA and back are created by two 32-conductor cables instead of 64-conductor ones, because it would be too expensive and complicated to use 128 digital I/O signals. Since the parallel-link width was reduced twice, the data from or to dSpace cannot be sent at once, but two 32-bit words are sent/received one after another. Sending/receiving of the data between the dSpace and FPGA causes additional delays. The main delay is on the side of digital input to dSpace where total time of 1.8 μs is needed to read two consecutive data words. Communication with the digital output is much faster and both 32-bit words are sent in 0.6 μs. On the side of the manipulator, no external input or output is used, because the sent/received data are used internally inside the FPGA for control of the manipulator.

DATA SS CRC FB SS FB

16 bits 16 bits 8 b. 64 bits

DATA

6.5 μs


Figure 5.4: Photo of prototype full-duplex IR wireless link.

A few tests were performed on the presented IR wireless link without manipulator in the control loop (see Figure 5.4). Therefore, instead of the commutation algorithm, the data received by second FPGA were just resent back to test bi-directional communication.

In an “open-loop” test, the delay in one-way communication between the manipulator’s FPGA and dSpace was tested. Since it is difficult to measure the exact time between, for example, the start of the output cycle in dSpace and the moment the FPGA received the complete data, a small “trick“ was used. Since the transmitting and receiving links inside each FPGA are mutually independent, the test was done by using just one FPGA and a mirror to reflect the beam back to the same FPGA (see Figure 5.5). Output data where then sent back to dSpace. For the reason that dSpace is not synchronized with the FPGA (the wireless link is continuously sending the data from the buffer regardless of when the buffer was updated), the delay varied from minimum delay of the link to the double time as you can see in Figure 5.6. The figure shows a variation in the delay from 9.2 μs to 16 μs. If the delay of the digital I/O interface of dSpace is subtracted, 1.8 plus 0.6 μs, the values correspond to the variation from minimum wireless-link delay of 6.8 μs to the double delay of 13.6 μs.

The second test was performed in “closed loop”. The signal was sent from dSpace wirelessly to the remote FPGA and back (see Figure 5.7). Again, due to no synchronization of the wireless links, the delay varies from 15.9 μs (twice the link plus dSpace) to 29.6 μs (twice the link plus dSpace plus twice the buffer delay) (see Figure 5.8). It is clear that unsynchronized communication causes performance degradation. The test presented, fortunately, does not correspond to the real usage of the IR wireless link. If the manipulator control will be introduced into the loop, there is no delay in that FPGA, because the actual position from the counters will always be transmitted as soon as possible regardless of the control input. Therefore, the maximum closed-loop delay of the

IR Transceivers

FPGAs

to/from dSpace


Figure 5.5: Simulated and real delay measurement in “open loop”; since the transmitting and

receiving links inside each FPGA are mutually independent, the real test has been done by using just one FPGA and a mirror to reflect the beam back to the same FPGA.

0 0.005 0.01 0.015 0.02 0.0250.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7x 10

-5

Time (s)

Del

ay (

s)

Figure 5.6: “Open-loop” delays measured in dSpace.

wireless link will be less than 23 μs (29.6 μs - 6.8 μs). The second part of the buffering delay can be solved by synchronization of the transmission start with dSpace if necessary.

IR tran

sm. IR

rec

eive

r

FPGA

TX

FPGA

dSpace

simulated “open-loop” delay meas.

RX

RX

IR tran

sm.

IR receiver

FPGA

TX dSpace

Mirror

real delay meas.

32 b.

32 b.

32 b.

Buffer


Figure 5.7: Delay measurement in “closed loop”.

0 0.5 1 1.5 2 2.5 31.5

2

2.5

3x 10

-5

Time (s)

Del

ay (

s)

Figure 5.8: “Closed-loop” delays measured in dSpace.

The range and error rate of the IR wireless link were also tested (see Figure 5.9). According to the specifications of the VFIR standard, wireless communication over distance of 1 m should be possible. From the measurement, the maximum reliable distance is only about 0.7 m. Wireless communication using light beams is also sensitive to orientation of both transceivers. To indicate in which area the IR wireless link can be used, for different link can be seen from Figure 5.10. In the inner area, no CRC error was

detected during two minutes of measurement. That indicates an error rate better that 10-7. That is the real reason why no data-correction field was included in the packet. When the modules are within the safe distance, no packet is lost. In addition, even when there is loss of one packet, another will come in just 7 μs. Outside of the reliable area packet-loss ratio increases rapidly, therefore a correction mechanism would not help to improve working distance significantly. For the manipulator of the COPAM, the reliable area is sufficient.

dSpace

RX

IR tran

sm.

IR receiver

IR r

ecei

ver

IR t

ransm

.

FPGA

TX FPGA

32 b. Buffer

Buffer 32 b.

delay meas.


Figure 5.9: Illustration of the radius measurement.

Figure 5.10: Position dependent packet-loss ratio; the non-convexity in the several regions is

caused by the interpolation of the values measured in a grid with a step of 100 mm.

Moreover, in the case we need to cover a wider area by the wireless link, it is possible to mount one of the transceivers on a simple manipulator to approximately follow the second module. Because the IR transceiver weighs only a few grams, a simple servomechanism can do the job, since directional data are available, in principle, from the control and sensor signals of both planar actuator and manipulator.

The IR wireless link performance was tested also on the manipulator of the COPAM. The rotary motor transfer function was measured while connected with wired and wireless connection (see Figure 5.11). Since the wireless link causes a phase lag of only 8 degrees at 1 kHz, both frequency responses are practically indistinguishable. The phase difference at the frequencies 100-500 Hz is caused more by the identification accuracy than by the wireless-link delay.

> 10-2

10-3

10-4

10-5

10-6

Distance (m)

Tangen

tial d

ista

nce

(m)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.3

-0.2

-0.1

0

0.1

0.2

0.3

IR tran

sm.

IR r

ecei

ver

Distance

Tangential Distance


In Figure 5.12, four different wireless solutions are elucidated. The frequency-dependent phase lags in closed loop caused by UDP/IP [54], custom RF [14], standard VFIR [117] and modified IR wireless link are compared. The figure gives a clear idea of each wireless link capability. If, for example, additional phase lag of at most 30 degrees is possible in the control of a motion system, UDP/IP will achieve this phase lag at 25 Hz, custom RF at 280 Hz, standard VFIR at 1.2 kHz and the modified IR link at 3.5 kHz. Of course, delays of sensors, control and actuators will also contribute to the total delay and the marginal phase lag will be obtained at lower frequency, but the contribution of the connection is significantly reduced.

101

102

103

-150

-100

-50

0

Mag

nitu

de (

dB)

Wired

IR Wireless

101

102

103

-400

-350

-300

-250

-200

Frequency (Hz)

Pha

se (

deg)

Figure 5.11: Measured manipulator’s rotary motor transfer functions of torque to position with

wired and wireless connection.

100

101

102

103

104

-180

-150

-120

-90

-60

-30

0

Frequency (Hz)

Ph

ase

lag

(d

eg

)

WiredUDP/IP 3.6 msCustom RF 300 sVFIR 70 sCustom IR 23 s

Figure 5.12: Comparison of different wireless solutions; phase lags in closed-loop control.

5.6 Conclusions 141

5.6 Conclusions

In this chapter, several wireless solutions were compared for their suitability for real-time control of a motion system. From the comparison, VFIR is an attractive solution for the wirelessly controlled manipulator presented. However, efficiency of the VFIR decreases to less than 0.18 (18 %) when the small data packets are transmitted. Therefore, a new IR wireless link was developed based on modification of the VFIR standard.

The presented IR wireless link has a transmission time of less than 7 μs for 64 data bits with sampling rate of about 150 kHz. This time was achieved by a custom packet and data processing. The closed-loop delay produced by the wireless link is then less than 23 μs. Such a delay makes the wireless link indistinguishable from a wired link up to closed-loop bandwidth of a few kilohertz. The IR wireless link provides more that 100 times shorter delay than IEEE 802.11b UDP/IP, more than 13 times shorter than the custom RF link developed in [14], and about 3.5 times shorter than standard VFIR protocol.

The reliability of the link has a CRC error rate of less than 10-7 in the working area, without retransmission of the data. This makes the presented wireless solution superior to any RF wireless link.

The maximum reliable distance is about 0.7 m, which might be a drawback of the link, but this depends on the application. However, IrDA is already developing Ultra Fast Infrared (UFIR) and Giga-IR standards for data transfer speeds of up to 100 Mb/s and 1 Gb/s, respectively, over larger distances [55].

Directionality of the transported signal is an advantage as well as a drawback of the link. The advantage is in the possibility to use more of these links at one place without interference problems (unlike wireless radio links). On the other hand, it is a drawback for the applications if one of the modules is moving in plane or even rotating. A possible solution to this problem can be placing one of the IR transmitters on a simple servomechanism that would approximately follow the trajectory of the second module.


143

Chapter 6

Experiments

6.1 Introduction

The Contactless Planar Actuator with Manipulator (COPAM) project consists of two parts. In the first part, described in [13], the experimental setup, combining three contactless technologies (electromagnetic suspension and propulsion, contactless energy transfer and wireless communication), was designed and built. The second part of the project (this one) focuses on the control of the prototype.

The COPAM is very complex system and since the two parts of the project (hardware building and control design) partially overlapped in time, the final prototype was not ready at the beginning of this part of the project. Therefore, a pre-prototype with limited complexity had to be built to test the control initially. This pre-prototype is presented in Appendix D.

When the first part of the project was finished, all three contactless technologies were integrated into single system. However, the preliminary control prohibited the full performance of the system from being shown. The tracking errors were in tens of

micrometers while the setup was accelerating only at 1 m.s-2. Moreover, the problems with the sensor system reduced the trajectories of the planar actuator (PA) to just one long-stroke degree-of-freedom (DOF).

In this chapter, the theory and findings presented in the previous chapters have been used to improve the performance of this prototype.

This chapter is organized as follows. First, improvements to the hardware of the setup are discussed in Section 6.2. Section 6.3 is devoted to identification of the setup. Finally, the results from control of the experimental setup are presented in Section 6.4. Part of this work has been published in [41, 44, 46].

144 Chapter 6. Experiments

6.2 Improvements of the Experimental Setup

Although the main focus of this thesis is on the control part of the project, the hardware of the final experimental setup of the COPAM has also been improved. Both, the measurement system, the contactless energy transfer and the wireless control have been redesigned and improved.

6.2.1 Measurement System

The originally designed measurement system for reading the position and orientation of the planar actuator (PA) consisted of three eddy-current sensors (from Lion Precision) and three laser triangulation sensors (from Micro-Epsilon). The eddy-current sensors (U8B) have limited range of 2 mm, resolution of 0.16 μm and 5 μm linearity. They were used for measuring the vertical z-position and two titling angles ( ,q y) of the PA. The laser triangulation sensor (ILD 2200-200) for measuring the x-position had a range of 200 mm, 10 μm resolution and 60 μm linearity. Two sensors (ILD 2200-100) for measuring the y-position and (yaw) f-orientation had a range of 100 mm, 5 μm resolution and 30 μm linearity (see Figure 6.1). The laser triangulation sensors were chosen for their ability to measure position on a surface tilted by several degrees. With the tilt angle of 5º, the error is less than 0.12% of the measuring range. Since the design of the PA together with the commutation allows for control of rotated PA (see Chapter 2 and [13]), such sensors were used to prove the rotational capability of the setup (at least for a limited rotation).

Figure 6.1: Previous measurement system for the planar actuator consisting of three eddy-

current sensors and three laser triangulation sensors.

Unfortunately, when the PA was lifted and controlled in the closed-loop for the first time, it started to vibrate. The cause of the vibration was due to the laser triangulation sensors. The problem was in the sensitivity of the sensors to the perpendicular movement of the measured targets. This is best explained in Figure 6.2, where three additional eddy-current sensors were used in the control loop of the PA, while the signals of laser triangulation sensors were also captured. The PA was moved in x-direction by small steps of 100 μm. From the figure, it is visible that the laser sensors in the perpendicular y-direction recorded a position change of almost 30 μm, which was not

x,θ y,ψ

z,f

Eddy current sensors

Laser triangulation sensors


0 0.5 1 1.5 2 2.5 3 3.5

-100

0

100

200

x (

m)

lasereddy

0 0.5 1 1.5 2 2.5 3 3.5

-10

0

10

20

30

y (

m)

0 0.5 1 1.5 2 2.5 3 3.5

-200

-100

0

100

200

(

rad)

t (s)

Figure 6.2: Problems with the laser triangulation sensors; in the control loop, the position of the

PA is measured by the eddy-current sensors (gray) which give real position values; the position reconstructed from the laser triangulation sensors is plotted in black; the PA is moved in 100 μm steps in x-direction.

there. The same happened to the yaw orientation measurement. The problem was not in the actual position error, which was within the linearity specifications of those sensors, but in the fast variation of the error with the perpendicular movement of the target. Even 10 μm step in one direction could cause measurement error of about 20 μm in the perpendicular direction. Obviously, such a coupling between different DOF’s can easily cause observed vibrations.

To get rid of the vibrations, the two laser triangulation sensors ILD 2200-100 were temporarily replaced by two eddy-current sensors. As a result, long-stroke translational movement of the PA in x-direction was only possible. Without the vibrations present, the principle of the COPAM system with three contactless technologies could be proved, albeit only with one translational DOF for the PA [13].

Evidently, the first task in order to improve the hardware was to design a new measurement system that would show the full potential of the setup. The requirements for the sensors were contactless measurement, operation in vacuum, range > 200 mm, resolution and linearity < 10 μm, bandwidth of several kHz, insensitivity to magnetic field and possibly to the rotation of the PA. None of the sensors available on the market satisfied all the requirements. The closest were the laser triangulation sensors and laser interferometry. Since the laser triangulation sensors caused vibration of the PA, the only alternative was the interferometer system. The problem of the latter is the necessity of precise aiming of the reflected laser beam. Consequently, yaw-rotational DOF of the PA is lost and, moreover, the orientation of the PA must be kept within 1 mrad. Furthermore,


laser interferometry measures the relative position (like encoders) and, therefore, an additional arrangement for acquiring the absolute position is necessary. The other specifications of the interferometer system are more than sufficient for the system: range > 1 m, resolution and linearity < 10 nm (depending on the air stability) and bandwidth > 1 MHz.

The concept of the sensor system allowing full yaw rotation of the PA was discussed in Section 2.3.2 of Chapter 2. However, the solution presented there is still not commercially available in the desired resolution and reliability.

During the design of the new measurement system with the interferometers, several circumstances had to be considered. First, the vertical position and tilting angles were sufficiently covered by the eddy-current sensors. In addition, acquiring the vertical position in the systems with planar movement requires a complex mirror system. Furthermore, the cost of full 6-DOF interferometer system was beyond the available budget. Therefore, the interferometer system was used only for three DOF’s (x, y, and f) (see Figure 6.3 and Figure 6.4). The maximum velocity captured by the sensors is given by the laser source, receivers and processing electronics. For the reason of significant cost reduction, the system with a maximum velocity of 0.5 m.s-1 and a bandwidth of 20 MHz was chosen.

To obtain the absolute position of the PA, three additional eddy-current sensors were used. These sensors are placed to cover the area close to [x, y] = [0, 0] position of the PA (see Figure 6.3 and Figure 6.4). The system is started by lifting the PA in the range of these sensors. Once the steady state levitation with orientation close to zero is achieved, the sensor system is switched to the laser interferometers. One can imagine that, although the interferometers are very accurate, the repeatability of the xy-movement is limited by the accuracy of the eddy-current sensors. Therefore, the reading from the eddy sensors is averaged over period of time before it is used as the reference point. Anyhow, since the position of the mirrors is not in the center of gravity of the PA, the noise from the three eddy-current sensors for the vertical position and tilting angles is also propagated to the reconstructed xy-position. Consequently, the resolution of the x- and y-position is reduced to about 100 nm and 50 nm, respectively, on the contrary, the yaw f-orientation is reconstructed only from laser interferometers with the resolution of about 10 nrad.

Figure 6.3: New measurement system for the planar actuator consisting of six eddy-current

sensors and three laser interferometers.

x,θ y,ψ

z,f

Mirrors

Laser

Interferometers Eddy sensors for reference


Figure 6.4: Photos of the experimental setup with the interferometer measurement system.

6.2.2 Contactless Energy Transfer

The prototype is a complex system combining three contactless technologies, which were designed to operate in three separate electromagnetic frequency bands [13]: electromagnetic suspension and propulsion (0 – 104 Hz), contactless energy transfer (~105

Hz), and wireless communication (~109 Hz). Proper design of each of them is necessary for independent operation without interfering each other. In particular the contactless energy transfer (CET) works at high frequencies and energy levels. Although the prototype was successfully tested with all contactless technologies integrated, the CET caused disturbances to several parts of the setup. Two different types of disturbances were found.

Laser source

CET unit and secondary coil

FPGA

Manipulator

Reference eddy sensors

Amplifiers

Interferometers

Mirrors

Eddy sensors


The first one was temporal, appearing for several hundreds of microseconds during switching between different primary coils. It was caused by the electromechanical relays used for the switching. This situation is depicted in Figure 6.5. The bottom plots show signals in the electronics for one primary coil. When the particular primary coil is activated (CET-enable signal is high), an electronic oscillator starts to generate high-frequency signal, which is then amplified. There is also an electromechanical relay in the circuit, which is used for connecting the particular coil either to the CET electronics or to the current amplifier (for the force production). An electromechanical relay introduces switching noise by its mechanical contacts, which connects-to and disconnects-from the coil several times in very short time-period. This is shown in the figure with the gray color (measured with a DC power source and an oscilloscope connected in series to the relay contacts). Consequently, a lot of high-energy noise is produced in the electromagnetic field near the coils (see Figure 6.5 top). It was found that not only the wireless communication, but also the eddy-current sensors are disturbed by this phenomenon. This was observed as a pulse in the position reading with a width of one sample and with an absolute value of more than 50 % of the sensor range! The problem was solved by delaying the oscillator start-up time after the settling time of the relay (see Figure 6.6). The consequent slightly increased time of switching between different primary coils did not cause any problem to the continuity of the energy transfer.

-2 0 2 4 6 8 10 12 14-0.04

-0.02

0

0.02

0.04

EM

C p

robe

(V

)

t (ms)

-2 0 2 4 6 8 10 12 14

0

0.2

0.4

0.6

0.8

1

t (ms)

CET enableCET signalRelay contacts

Figure 6.5: Original contactless energy transfer (CET); Bottom: Signals in electronics for one

primary coil; CET enable signal (dashed), CET oscillator signal (black solid), noise at the relay contacts (gray solid); Top: Produced disturbance in the electromagnetic field.


-2 0 2 4 6 8 10 12 14 16 18

0

0.2

0.4

0.6

0.8

1

t (ms)

CET enableCET signalRelay contacts

Figure 6.6: Improved contactless energy transfer (CET); signals in electronics for one primary

coil; CET enable signal (dashed), delayed CET oscillator signal (black solid), noise at the relay contacts (gray solid).

The second type of the disturbance of the CET was found in the electromagnetic interference mainly to the eddy-current sensors. The magnitude of the disturbance was much lower than in the previous case, however, it is always present when the CET is active (steady-state effect). The reason for the interference is in the oscillation frequency of 1 MHz of the eddy-current sensors. This frequency is less than one decade away from the CET working frequency of about 150 kHz. The noise levels of one eddy-current sensor when the CET is enabled and disabled are shown in Figure 6.7. The top plot shows the signal in the time domain whereas the bottom one shows the spectra of the signals

0 5 10 15 20 25 30 35 40-0.04

-0.02

0

0.02

0.04

Edd

y-se

nsor

sig

nal (

V)

Time (ms)

0 20 40 60 80 100 120-140

-120

-100

-80

-60

-40

Spe

ctru

m o

f th

e si

gnal

s (d

B)

Frequency (kHz)

CET offCET on

Figure 6.7: The noise levels of one eddy-current sensor when the CET is enabled and disabled;

Top: Time-domain data; Bottom: Spectra of the signals obtained by an oscilloscope with FFT module.


measured by an oscilloscope with FFT module. From the time-domain plot, one can see that the noise level is increased about ten times. On the other hand, the frequency-domain plot shows that it is not possible to filter-out the disturbance completely since it appears in the frequency range from 0 Hz to 40 kHz. The magnitude of the noise level caused by the CET is dependent on the actual distance of the particular sensor from the active primary coil. The presented example shows typical values. It was not possible to solve this problem without complete redesigning of the CET or replacing the eddy-current sensors, which was not possible due to the time and money limits. The effect on the sensors was partly reduced by additional shielding of the cables and by decreasing the bandwidth of the anti-aliasing filter, so by reducing closed-loop bandwidth.

6.2.3 Wireless Control

The original custom-made wireless link developed for the setup used radio-frequency modules at 2.4 GHz with an open-loop delay of 150 μs [13]. The preliminary results with this wireless link showed a packet error rate of about 10-2. However, when the link was integrated to the system, the error rate due to the CET increased rapidly. One example is shown in Figure 6.8, where the top plot shows position readings of the manipulator’s beam received by the dSpace. The transition from one position to another should be smooth, but each lost packet caused holding of the previous position reading. Consequently, instead of smooth transition, “teeth” are visible in the plot. The bottom plot

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0.04

0.045

0.05

0.055

0.06

0.065

0.07

x LM

(m

)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

No

Yes

t (s)

Pac

ket

OK

Figure 6.8: Previous wireless control with custom RF communication link [13]; Top: Transition

of the manipulator’s beam as seen from the dSpace side; each lost packet caused holding of the previous value; Bottom: Each dot represents if the received packet was OK or not.


shows how many packets have been lost. Unexpectedly, due to the CET, the packet-loss

ratio increased to more than 0.7 or 70 %! Such a system does not allow for precise and, more importantly, reliable control.

Therefore, a novel infrared communication link for wireless real-time control of a fast motion system was developed (see Figure 6.9). A detailed description of the link is in

Chapter 4. Briefly, the new wireless link has packet-loss ratio < 10-7, an open-loop delay of 7 μs, the data are sampled at 150 kHz, and it is immune to the CET electromagnetic interference. This makes the infrared wireless link indistinguishable from a wired connection in this setup.

Figure 6.9: Photo of the infrared wireless communication modules.

6.3 Identification

In this section, the identification results of the planar actuator are presented. The identification results of the manipulator and its interaction with the planar actuator were presented in [13].

Since the identification problem was the same as in the project of Herringbone Pattern Planar Actuator (HPPA) [76], a similar procedure was used. The identification was done in closed-loop by using the signal and system analyzer SigLab (from Spectral Dynamics, Inc.). As an input signal source, Gaussian white noise was used and the frequency response of the output sensitivity was identified. Six SISO PID controllers were used to stabilize the PA. The structure of the controllers was presented in Section 4.4.1. The parameters are:

,z ic = 5.0,

,z llc = 3.0,

,p llc = 4.0,

,p lpc = 60 and

bwc = 60 rad/s. The

controller transfer function can be written as:

7 ( 12) ( 20) 1( ) 3.31 10

( 240) ( 3600)PID

s sC s

s s s

+ += ⋅

+ +. (6.1)

Fixed IR module

Translator IR module

Planar Actuator

Stator


As in [76], the maximum frequency, which can be properly identified, is limited by the double-integrator behavior of the identified plant. The behavior of the plant at higher frequencies needs an increasing amount of input power to be visible. Nevertheless, the input power had to be limited to satisfy the thermal limits of the active coils and to preserve the structural integrity of the translator, which consists of an array of magnets glued together. The maximum identification stroke of the translator was also limited. The limitation in the vertical direction is given by the end stops and for the orientations by the interferometer measurement system, which necessitates keeping the orientations within about 1 mrad (depending on the actual distance between the interferometer and the mirror). The frequency response of the sensitivity 1( ) ( ( ) ( ))j j j -w = + w wS I P C has been obtained from Siglab, which correlated the inserted position (or orientation) noise at the inputs with the measured position and orientation error. ( )jwS is the 6 6´ matrix of the frequency responses from the each input to the each output. With the known frequency response of the controller ( )jwC , the plant frequency response was reconstructed as

( ) 1 1( ) ( ) ( )j j j- -w = w - wP S I C . (6.2)

To be able to compare the cross-terms and the diagonal terms of ( )jwP of different physical units, normalization has also been used:

1* ( ) ( )j j -w = wP NP N , (6.3)

where N is the normalization matrix defined in Section 2.3.1 and [76].

The identification was performed at the position [x, y, z] = [5, 5, 1] mm with zero orientation. The Bode plots of the identified, normalized transfer functions * ( )jwP are shown in Figure 6.10 - Figure 6.15. To show exclusively the relevant data, only the values with a coherency of at least 0.7 are drawn. The lowest significant resonance modes at about 65 Hz and 220 Hz are caused by the FPGA module and by the CET module with the secondary coil connected to the side of the translator (see e.g. Figure 6.4). For each diagonal term, a linear approximation by the second-order transfer function ˆ* ( )P s with 3

ST

(= 1 ms) delay are shown in the figures:

2

2

2

2

ˆ ˆ* ( ) * ( ) 1 / ,ˆ* ( ) 1 / ( 25 ),ˆ ˆ* ( ) * ( ) 1 / ( 7 ),ˆ* ( ) 1 / .

x y

z

P s P s s

P s s s

P s P s s s

P s sq y

f

= =

= +

= = +

=

(6.4)

As one can immediately see, the approximated transfer functions (6.4) for vertical z-direction and two tilting DOF’s contain damping terms. This is mostly visible in the phase plots as increased phase at low frequencies. The reason for such behavior is in the low clearance between the stator and the translator, and not testing in vacuum. The clearance is even more reduced by a rubber plate with thickness of 0.4 mm, which protects the coil array in the case of malfunction. Then, the levitation height of 1.0 mm corresponds to the clearance of about 0.6 mm. Consequently, the air between the stator and translator cannot get in or out easily to compensate the pressure changes and causes the air-damper behavior. A test was performed where ( )

zP jw was identified for different levitation heights


Figure 6.10: Normalized Bode plot of x x , its approximation by the second-order transfer

function ˆ*x

P with the delay and the amplitudes of the cross-terms.

Figure 6.11: Normalized Bode plot of y y , its approximation by the second-order transfer

function ˆ*y

P with the delay and the amplitudes of the cross-terms.


Figure 6.12: Normalized Bode plot of z z , its approximation by the second-order transfer function ˆ*

zP with the delay and the amplitudes of the cross-terms.

Figure 6.13: Normalized Bode plot of q q , its approximation by the second-order transfer function ˆ*Pq with the delay and the amplitudes of the cross-terms.


Figure 6.14: Normalized Bode plot of y y , its approximation by the second-order transfer

function ˆ*Py with the delay and the amplitudes of the cross-terms.

Figure 6.15: Normalized Bode plot of f f , its approximation by the second-order transfer

function ˆ*Pf with the delay and the amplitudes of the cross-terms.


Figure 6.16: Normalized Bode plots of z z for different levitation heights and its approximation by the second-order transfer functions ˆ*

zP .

(see Figure 6.16). The Bode plots in the figure confirm that the damping effect is less visible with increased levitation height. The identified transfer functions are:

2

2

2

1 / ( 80 ), 0.8 mm,ˆ* ( ) 1 / ( 25 ), 1.0 mm,

1 / ( 15 ), 1.2 mm.z

s s z

P s s s z

s s z

ìï + =ïïïï= + =íïïï + =ïïî

(6.5)

This effect was not observed during the identification of HPPA [76], where the clearance of 1.2 mm shifted the damping effect to very low frequencies. The effect is also not relevant for planar actuators operating in vacuum.

Another experiment was performed to test the precision of the commutation decoupling at different xy-positions. At four different positions [x, y] = [[5, 5], [5, 40], [40, 45], [45, 5]] mm, the identification was performed. Since 6 6 4´ ´ frequency responses were captured, only the magnitude parts of the Bode plots are shown in Figure 6.17 and Figure 6.18. The data are again plotted for the coherency > 0.7. The plots show that although the commutation does not decouple the system exactly, its performance is stable in all tested positions. The plots also clarify which parts of the plant dynamics are caused by the commutation (position-dependent) and which are caused by the flexible modes of the system (position-independent).

The last identification experiment tested the influence of the position of the manipulator’s beam on the decoupling of the system. The identification was performed for

6.4 Control 157

three positions of the beam: the most left (-80 mm), the middle (0 mm) and the most right (80 mm), while the position information was not included in the decoupling. Since the beam moves in the x-direction (

LMx ), the cross-terms y « f and z « y should be the most

influenced. The change of the mass distribution also changes the inertias for y- and f-DOF:

p b

p b b

p b b

LMb b

2b b b

2b b b b

0 0 0 0 0

0 0 0 0

0 0 0 0( )

0 0 0

0 0 ( )

0 0 ( )

x xy xz

xy y b yz

xz yz z

m m

m m m x

m m m xx

J J m x y J m x z

m x J m x y J m x z J

m x J m x z J J m x y

é ù+ê úê ú+ê úê ú+ -ê ú

= ê ú- -ê úê úê ú- - + +ê úê ú- + +ê úë û

M

LM

LM

LM b LM b2

LM LM b LM2

LM LM b LM

,

(6.6)

where p

m and b

m are the masses of the platform and the beam, ,...,x yz

J J are the inertias of the platform, [

LMx ,

by ,

bz ] is the position of the beam with respect to the center of rotation

of the PA. The terms due to shifted center of gravity of the static part of the PA are not shown in (6.6). The identification results are shown in Figure 6.19. Only the relevant diagonal and cross-terms are shown. The plots confirm the influence of the beam position on the decoupling of y « f and z « y, where for the outer positions of the beam the coupling is magnified by a factor 10. On the other hand, the influence of the beam position on the inertias is minimal. Since

b by z , the cross-terms q « y are much less influenced by

the beam position than q « f (see (6.6) and Figure 6.19).

6.4 Control

The control of the prototype can be split into three categories: Control of the planar actuator, control of the manipulator and control of the combined 8-DOF system.

6.4.1 Control of the Planar Actuator

The control of the planar actuator (PA) alone was tested with the manipulator’s beam being fixed at its center position. Since the PA does not need contactless energy transfer (CET) for its operation, this unit was switched off during the tests. It was shown in Section 6.2.2 that the CET increases the noise level of the eddy-current sensors. Hence, the performance of the prototype with the present measurement system can be best tested without the CET active.

Three different controllers were applied: PID, first-order sliding-mode (SM), and iterative learning controllers (ILC) with PID. The description of the controllers and their parameters are presented in Chapter 4 with a few differences. The first difference is in an additional notch filter

2

2

1 0.001 / 220 ( / 220)( )

1 2 / 220 ( / 220)z

s sN s

s s

+ +=

+ + (6.7)


Figure 6.17: Normalized magnitudes of frequency response functions q q for four different xy-positions: [x, y] = [[5, 5], [5, 40], [40, 45], [45, 5]] mm (from light gray to black); part 1.

6.4 Control 159

Figure 6.18: Normalized magnitudes of frequency response functions q q for four different xy-positions: [x, y] = [[5, 5], [5, 40], [40, 45], [45, 5]] mm (from light gray to black); part 2.


Figure 6.19: Normalized magnitudes of frequency response functions q q for three different beam positions (left, middle, right).

6.4 Control 161

in the controller of the vertical z-position, which prevented the controller from exciting the lowest resonance mode at 220 Hz (see Figure 6.12). This resonance caused instability of the system for the z-position closed-loop bandwidth above 25 Hz. The other resonances at the frequencies above 400 Hz did not cause any stability problems since these frequencies are sufficiently damped with the high-frequency roll-off filters.

The other difference is in the value of l parameter for the SM controllers. The new measurement system has lower noise levels in several DOF’s in comparison with the expected values used for the simulations in Chapter 4. Therefore, the value of l was tuned separately for each axis to achieve the best performance without instability or too high steady-state control action (due to the sensor noise). For , , , , ,x y z q y f DOF, the parameter l was set to 280, 300, 170, 180, 180, 200. The value of

zl is much lower than

xl or

yl , because the noise level is higher than in those DOF’s, however the value is still

better than expected during the simulations in Chapter 4.

The same reference trajectory as in Section 4.3 was used (see Figure 4.13).

Although the current amplifiers allow for the accelerations of the PA of more than 30 m.s-2

(see Section 2.4.5), the limit was set to 10 m.s-2 for each axis (14 m.s-2 during the diagonal movement) for safety reasons (the prototype was not designed and tested for such high accelerations). The maximum velocity was limited by the interferometer sensor system to

0.4 m.s-1 for each axis (0.6 m.s-1 during the diagonal movement).

The control results for the PID and SM control are shown in Figure 6.20. The results are similar or even better than the results obtained by the simulations (see Chapter 4, Figures 4.16 and 4.17). The SM control was in the most cases able to reduce the error by factor 2 compared to the PID control (that is better than in the simulation thanks to higher l). The only exception is the vertical z-position where the performance of the PID and SM controllers were approximately the same. The cause is in the measurement of the vertical position with the three eddy-current sensors. These sensors measure the position against the surface of the planar actuator (see Figure 6.1) the flatness of which is at best 100 μm. Therefore, the PA is actually tracking the surface irregularity in the vertical DOF. Increasing the tracking precision of that DOF would require even more control action than it is shown in the

zF plot of the Figure 6.20.

With the ILC, the final tracking errors are close to the sensor noise-levels (see Figure 6.21). The final peak-errors in the x- and y-translational DOF’s are caused by the measurement-system coupled noise from the eddy-current sensors (as was explained in Section 6.2.1). Figure 6.22 shows how the tracking errors converged to their minima during the iteration process. The y-rotational DOF suffers the most from sensor noise. Therefore, the error reduction in this DOF took the highest number of iterations.

For planar actuator used in lithography industry, tracking errors during constant-velocity phase of the trajectory are important. Since the trajectory defined in Chapter 4, Figure 4.13 had very limited part of the trajectory with constant velocity, a test with

reduced maximum velocity (max

v = 0.14 m.s-1 during diagonal movement of the PA) was

performed (see Figure 6.23). The figure shows only the 5th part of the trajectory defined in Figure 4.13 right. The velocity is constant between t = 4.4 and 4.6 s. The tracking results clearly indicate better performance of the SM control (than the PID).


Figure 6.20: Control of the PA with the PID (gray) and SM (black); the manipulator is off; the tracking errors and controller outputs (without the acceleration feedforward) during tracking of the trajectory from Figure 4.13.

6.4 Control 163

Figure 6.21: Control of the PA with ILC; the manipulator is off; the final tracking errors and controller outputs (PID + ILC without the acceleration feedforward) during tracking of the trajectory from Figure 4.13.


0 5 10 15 2010

-7

10-6

10-5

10-4

Ma

x. e

rro

r (m

) o

r (r

ad

)

Iterations

xyz

Figure 6.22: Control of the PA with ILC; maximum tracking errors in different DOF’s depending on the number of iterations.

Figure 6.23: Control of the PA with the PID (gray) and SM (black); the tracking errors during

tracking of the 5th part of the trajectory from Figure 4.13 with max

v = 0.14 m.s-1.

6.4 Control 165

6.4.2 Control of the Manipulator

The manipulator consists of two linear motors joined with the beam (linear movement) and the rotary motor in the middle of the beam with the small arm connected to the rotary motor (rotary movement, see Figure 1.1). It was explained in the introduction, Chapter 1, that the manipulator on top of the platform was not designed to achieve the highest accuracy, but to show the principle of the wirelessly powered and controlled robot on top of the magnetically levitated platform. As the resolutions of the linear and rotary encoders are about a factor 10 worse than the resolution of the PA measurement system, spending time on the precise control of the manipulator would not make any sense. Therefore, only the PID control was applied on the manipulator.

One sample-time delay in the control loop of the PA is caused by the communication delay between the dSpace system and 99 current amplifiers for the stator coils. Since this is not the case in the control loop of the manipulator, the total closed-loop delay is shorter. The delay due to the wireless communication is negligible (less than 23 μs in the closed loop). This allowed increasing the bandwidth of the PID controllers up to 80 Hz.

In the trajectory generation for the beam, the maximal acceleration had to be

decreased to max

a = 5 m.s-2, because the higher acceleration profile together with the compensation of the forces from the moving PA caused limitation of the currents in the

amplifiers of the linear motors. Then, for the beam (linear motors): max

v = 0.5 m.s-1, max

a =

5 m.s-2, max

j = 500 m.s-3, and max

d = 1×105 m.s-4. For the rotating arm (rotary motor) the

values stay the same as defined in Section 4.5.2: max

v = 100 rad.s-1, max

a = 500 rad.s-2, max

j =

3×105 rad.s-3, and max

d = 5×106 rad.s-4.

The results of the manipulator trajectory tracking are elucidated in Figure 6.24. The residual tracking error of the beam (

LMx ) is mostly caused by the friction in the

bearing. The period of two trajectory cycles is plotted to show that the tracking error has stochastic behavior as a consequence of the time-varying value of the static friction. Moreover, the friction causes non-zero steady state applied force (

LMF ). The tracking error

of the rotary movement (RM

f ) is caused by the high velocity and acceleration in combination with limited bandwidth and sensor resolution. The tracking precision of the rotary movement can be compared to the maximum velocity:

maxerr /

maxv = (400 μrad)/(100

rad/s) = 4 μs. This can be seen as a time-offset (-error) of the motion. The value proves

the effectiveness of the wireless link for real-time control of a fast motion system.

6.4.3 Control of the Planar Actuator with Manipulator

The results of the combined control of the planar actuator and the manipulator are presented in this section. The experiments were again performed with the PID, SM, and PID+ILC controllers. This time, the active contactless energy transfer (CET) unit disturbs the measurement of the eddy-current sensors (see Figure 6.7). To reduce the noise at least partially, the bandwidth of the anti-aliasing filter was reduced from 2 kHz to 1.5 kHz. Due to the additional delay of the filter, the PID controller bandwidth had to be


Figure 6.24: Control of the manipulator with the PID; the reference trajectory, tracking errors and controller outputs.

decreased from 47 Hz to 42 Hz. Moreover, since the noise levels on the different DOF’s were still about five-times higher than without active CET, the SM controller parameter l also had to be reduced to 220, 250, 100, 100, 100, 140.

Statically levitated planar actuator with moving manipulator

The first set of the experiments tested attenuation of the disturbance acting on the statically levitated planar actuator caused by the moving manipulator. The manipulator was controlled by the PID controller as in the previous section, following the trajectory shown in Figure 6.24. The results with the PID and SM control applied on the PA are in Figure 6.25. The residual stabilization errors are partially caused by the friction in the beam bearings (as in the previous section). The problem of the friction estimation caused wrong estimation of the real force causing acceleration of the beam and, at the same time, acceleration of the platform in the opposite direction. As a side effect, one can immediately see benefits of the frictionless magnetically levitated planar actuator. Figure 6.25 also shows how much the sensor noise is increased in different DOF’s due to the CET (compare to Figure 6.20).

The results with the ILC are shown in Figure 6.26. Figure 6.27 again shows how the tracking errors converged to their minima during the iteration process. From both figures, it can be observed that the stochastic behavior of the friction caused the ILC

6.4 Control 167

Figure 6.25: Control of the statically levitated PA with the PID (gray) and SM (black), and with the manipulator following the trajectory from Figure 6.24; the position and orientation errors and controller outputs.


Figure 6.26: Control of the statically levitated PA with the ILC, and with the manipulator following the trajectory from Figure 6.24; the final position and orientation errors and controller outputs.

6.4 Control 169

Figure 6.27: Control of the statically levitated PA with the ILC, and with the manipulator following the trajectory from Figure 6.24; maximum tracking errors in different DOF’s depending on the number of iterations.

iteration process to fail to achieve better performance than the SM control. On the other hand, this proves that the decoupling of the forces and torques between the platform and manipulator works well.

Moving planar actuator with the manipulator

The last set of control experiments tested the control precision of the moving planar actuator with the moving manipulator. The manipulator was following the trajectory shown in Figure 6.24, while the PA was tracking the same trajectory as in Section 6.4.1 (also see Figure 4.13). The results with the PID and SM control are in Figure 6.28. Since the total tracking errors are caused by the combination of the errors from the tracking of the PA trajectory and the errors from the disturbing manipulator, while the controller bandwidth had to be reduced, the peak errors are about 30-50 % worse than in case of the moving PA alone (see Figure 6.20). When compared to the results with the ILC shown in Figure 6.29, it can be observed that a significant part of the residual errors is caused by the sensor noise and by the friction in the manipulator’s beam.

Figure 6.30 again shows how the tracking errors converged to their minima during the iteration process. In this case, the ILC was able to reduce the errors, but the final errors are far worse than in the case of the PA alone (see Figure 6.22). The errors are closer to the statically levitated PA with the moving manipulator (see Figure 6.27).

0 1 2 3 4 5 6 7 8 910

-7

10-6

10-5

10-4

Ma

x. e

rro

r (m

) o

r (r

ad

)

Iterations

xyz


Figure 6.28: Control of the PA with the PID (gray) and SM (black); the manipulator is active and following the trajectory from Figure 6.24 while the PA is tracking the trajectory from Figure 4.13; the tracking errors and controller outputs (without the acceleration feedforward).

6.4 Control 171

Figure 6.29: Control of the PA with the ILC; the manipulator is active and following the trajectory from Figure 6.24 while the PA is tracking the trajectory from Figure 4.13; the final tracking errors and controller outputs (PID + ILC without the acceleration feedforward shown).


0 5 10 1510

-7

10-6

10-5

10-4

Ma

x. e

rro

r (m

) o

r (r

ad

)

Iterations

xyz

Figure 6.30: Control of the PA with the ILC; the manipulator is active and following the trajectory from Figure 6.24 while the PA is tracking of the trajectory from Figure 4.13; maximum tracking errors in different DOF’s depending on the number of iterations.

6.4.4 Clipped l2-norm based commutation

In order to verify correctness of the clipped l2-norm based commutation (CL2C),

the following experiments were performed. The CL2C was compared with the l2-norm based commutation (L2C), which was used in the previous experiments. Two situations

were studied: slow (max

v = 0.14 m.s-1, max

a = 1.4 m.s-2) and fast (max

v = 0.6 m.s-1, max

a = 14

m.s-2) movement of the PA. In both situations, the PA followed the trajectory from Figure 5.13 controlled with the PID controllers. For the slow and fast movement, the clipping (saturation) current

clipi in the CL2C was set to 1.2 A and 1.8 A, respectively.

Since the calculation time of the control with the CL2C algorithm was up to 480 ms (for five iterations), the sampling rate of the system had to be decreased from 330 ms to 500 ms. Due to the increased delay, the bandwidth of the PID had to be reduced to 35 Hz. The results of the slow and fast movement are in Figure 6.31 and Figure 6.32, respectively. The plots compare the CL2C with the original L2C and with L2C with artificially saturated currents (the same saturation values as for the CL2C are used). The L2C with the saturated currents demonstrates the system behavior in the event of saturated current amplifiers.

In the case of the L2C with saturated amplifiers, the plots with x- and y-position errors show tracking errors several times increased. The large tracking errors are caused by the total applied wrench, which does not correspond to the desired wrench when the currents are saturated. On the other hand, the tracking errors for the CL2C and L2C are almost identical. This proves that the total wrench vectors produced by the both commutation algorithms are equivalent. The plots with the maximum current show by how much the peak current was reduced (up to 30 % and 45 %, respectively). The current

6.5 Conclusions 173

profiles in all the coils are also shown. The plots with the maximum absolute current slew

rate (1

max | / | max | ( ) / |k k

i t i i t-D D = - D ) show no visible discontinuity in the solution.

The increased current change is only caused by the necessity of faster variation of the non-saturated currents. The clipping current limits (1.2 A and 1.8 A) were chosen to obtain the solution in the maximum of five iterations, while up to 12 and 20 currents, respectively, were saturated. The plots also show that for the CL2C the dissipative power increased only by about 6 % and 8 %, respectively, in comparison to the L2C. Although from the plots it seems that the L2C with saturation still consumed less power than CL2C, the energy E (J) consumed by the setup during movement was 597 J, 598 J, and 603 J for the fast trajectory when the L2C, CL2C and L2C with saturation, respectively, were used. The higher energy consumption was caused by the longer period of the high-control activity when the L2C with saturated currents was used. The energy E (J, Ws) over the trajectory period was calculated as

( )end

start

t

tE P t dt= ò . (6.8)

6.5 Conclusions

With the improved measurement system, full and accurate planar motion of the translator has been made possible. Although the rotational capability of the setup has been lost, the resolution and linearity of the sensors in all three degrees of freedom (DOF’s) were greatly increased. With the new infrared wireless link, the control of the manipulator is reliable without noticeable delay. Although the problem of switching between two primary coils of the contactless energy transfer (CET) was fixed, the CET is still causing increased noise levels in the eddy-current sensors. Solving this problem would require redesigning of the CET or replacing the eddy-current sensors by e.g. laser interferometers.

The identification of the PA showed that due to the small levitation height, the air between the stator and translator caused damping in the vertical (z) and two tilting ( ,q y) DOF’s. However, since these DOF’s were only stabilized, the low-frequency damping did not cause any problems to the control. If necessary, the damping effect can be reduced by increasing the levitation height (at the price of a loss of efficiency) or when operating the system in vacuum. The identification at four different positions showed that although the commutation does not decouple the system exactly, its performance is stable in all tested positions.

During the control of the PA alone (with the manipulator’s position fixed and the CET unit switched off), the tracking errors of the xy-movement with the PID controller were less than 3 μm and the orientation was stabilized within 30 μrad. The SM reduced the errors by an additional 50 %. The best performance was achieved when adding ILC feedforward, which reduced the errors to the maximum of 400 nm and 3 μrad.

The control of the separate manipulator showed limitation of the performance in the friction of the beam bearings, which was partly non-repetitive. The tracking accuracy for the linear movement was mostly better than 20 μm, but a peak in the error appears randomly as a consequence of the varying value of the static friction. The tracking error of


Figure 6.31: Comparison of commutation methods; L2C (green), L2C with saturated amplifiers

(red) and CL2C (blue); tracking of the trajectory from Figure 4.13 at max

v = 0.14 m.s-1 and max

a

= 1.4 m.s-2.

6.5 Conclusions 175

Figure 6.32: Comparison of commutation methods; L2C (green), L2C with saturated amplifiers

(red) and CL2C (blue); tracking of the trajectory from Figure 4.13 at max

v = 0.6 m.s-1 and max

a =

14 m.s-2.


the rotary movement was less than 400 μrad, caused by the high angular velocity and acceleration in combination with limited bandwidth and sensor resolution. However, the time-offset of the rotary-motor trajectory tracking was calculated to be less than 4 μs, which proved the effectiveness of the infrared wireless link for real-time control of a fast motion system.

For the combined control of the planar actuator with the manipulator, the PID and SM controller bandwidths had to be reduced due to the increased noise in the measurement system due to the CET unit. The test with statically levitated PA and moving manipulator showed good decoupling of the forces and torques between the manipulator and the platform. The residual stabilization errors were mostly caused by the friction in the beam bearings (as in the previous test), which was difficult to estimate. This was proved by the applied ILC, which failed to improve the performance beyond the results obtained with the SM.

The last test combined control of the moving planar actuator with the moving manipulator. Since the total tracking errors were caused by the combination of the errors from the tracking of the PA trajectory, the errors from the disturbing manipulator and the reduced controller bandwidths, the peak errors were about 30-50 % worse than in case of the moving PA alone. The results with the ILC showed that a significant part of the residual errors was caused by the sensor noise and by the friction in the manipulator’s beam.

Clipped l2-norm based commutation (CL2C) was successfully tested on the prototype. The CL2C was able to reduce the peak current by up to 45%, while the maximum increase of the dissipated power was only 8 % in comparison to the l2-norm based commutation (L2C). This commutation method allows achieving higher accelerations with the same hardware configuration or using less-powerful (cheaper) current amplifiers while maintaining the same performance as with the L2C.

From all these observations, several conclusions can be made: The contactless energy transfer needs to be re-designed. Otherwise, it can easily interfere with other parts of the device. If a low-order band-pass filter for the CET is used, the frequency gap between the CET and the eddy-current sensors have to be at least 3 decades. The friction in the manipulator reduces the precision of the forces and torques decoupling. The performance of the PA with the feedback control (PID, SM) can be improved by decreasing the delay (caused by the calculation, communication with the amplifiers, amplifier bandwidth, sensor bandwidth and anti-aliasing filter) in the control loop (so the bandwidth can be increased). With feed-forward control (ILC) (or a high-bandwidth feedback control), the most important component to increase performance is a low-noise measurement system (e.g. 6-DOF laser interferometry). Since the residual position-dependent cross-coupling due to the mechanical and electrical tolerances has low-frequency characteristic (~ 20 Hz) during the movement of the translator (at 1 m.s-1), it can be successfully suppressed by sufficient bandwidth of the controller (for random motion) or by application of some kind of learning control (for repetitive motion).

177

Chapter 7

Conclusions and Recommendations

This thesis has explored the opportunities for improving the performance of ironless magnetically levitated planar actuators with moving magnets and with a wirelessly powered and controlled manipulator on top of the translator.

7.1 Commutation

Two novel norm-based commutation methods have been presented in this thesis:

the l∞-norm, and the clipped l2-norm based commutations (LiC, and CL2C, respectively).

Contrary to the l2-norm based commutation (L2C), which was proposed in [78, 95], the two novel methods put constraints on the maximum current. The benefit of the LiC is that the maximal current is always minimized. On the other hand, since almost all active coils are set to the same maximum current level, the power usage increases dramatically. Therefore, CL2C was developed which puts constraints on the maximal current, whereas the non-

saturated coils are minimized in the l2 norm. By varying the clipping (saturation) value, the solution obtained is closer to either L2C or LiC. Solving the CL2C problem with quadratic programming in real time is not fast enough. A faster, approximate solution of the CL2C problem is suited for real-time implementation, but does not necessarily always

lead to the optimal solution. Tests indicate that the optimal solution in l2-norm sense can be obtained in the majority of cases, whereas the suboptimal solution is still very close to the optimal one and practically indistinguishable in the real system. The benefit of this algorithm is in calculation time, which is reduced considerably in comparison to the LiC or optimal CL2C. Continuous current-variation with position and wrench variation can be guaranteed in the L2C and can be achieved in the CL2C if the solution is optimal. As was shown in the example, the partial current discontinuity caused by the approximate CL2C will not cause high voltage peaks in the current amplifiers.

178 Chapter 7. Conclusions and Recommendations

Bottlenecks that limit controllability of moving-magnet planar actuators with full rotation have been described. From many existing PA designs, it was found that only those with rounded coils are suitable for full rotation, because the commutation algorithm does not have to be of increased complexity. This is in contrast to the square- or rectangular- coil PA, which must include the orientation of a single coil into the commutation algorithm. It was also found that the coil width and spacing optimized for translational movement can produce severe controllability problems when the magnet array is rotated about 45 degrees, which causes the maximum current and power losses to be increased a few times.

A new, triangular, array of the coils (TCA) instead of the square array (SCA) has been proposed, which can have better properties for any orientation of the magnet array for given coil dimensions and spacing. It also has been shown that a limited number of highly symmetric topologies for the array of rounded coils exist. However, many magnet array topologies can be found. For example, for full rotation, a hexagonal Halbach magnet array in combination with a square array of coils can be interesting.

The three commutation methods have been compared on the model of the contactless planar actuator with the manipulator (COPAM) with square and triangular coil arrays. Various variables have been compared, e.g. maximum current and voltage, dissipated power, error sensitivity and worst-case acceleration and velocity. The comparison of the commutation methods showed that LiC could reduce peak currents significantly, but at the price of much higher dissipated power and voltage. As a compromise, CL2C can be used when occasional current limitation is necessary.

From the error sensitivity investigation, both coil arrays produce a well-conditioned coupling matrix. A new approach for investigation of the error sensitivity of the planar actuators is proposed. The desired wrench is compared with the absolute wrench calculated with the absolute values of the coupling matrix and the absolute values of the applied current vector. By using the absolute wrench, it has been shown that a magnetically levitated planar actuator is a fragile equilibrium of large non-conservative forces and torques. It has been also confirmed that LiC can almost double these forces.

Investigation of the lower bound of the maximum acceleration, possible for bounded currents, showed high potential in the rounded-coil topology. The worst-case

(smallest) maximum acceleration in the whole working range with L2C is about 30 m.s-2. Since the acceleration is mainly limited by the peak currents, LiC can increase the worst-case acceleration by almost 50% (at the price of increased dissipated power).

Study of the velocity showed that even with the maximum possible acceleration, a

minimum velocity of 2 m.s-1 can still be achieved with L2C. With CL2C, the value slowly decreases with pushing the current limit to the lower values. It was shown that TCA could provide better performance than SCA in almost all aspects, when TCA utilizes the same coil dimensions designed for the COPAM. TCA allows for full rotation of the PA within the current limits. Moreover, the peak current, voltage and dissipated power are about 10-20 % lower also for a zero orientation angle and, hence, higher accelerations can be achieved. This makes the triangular array preferable even for the applications where rotation of the PA is not required. Since the peak current as well as the power losses are

7.3 Control 179

lower, demands on cooling for TCA should not be higher than for SCA. Nevertheless, investigation of the thermal properties of TCA and other parameters necessary for full design of a PA was out of the scope of this thesis. A drawback of the triangular array of coils is that 15% more coils and current amplifiers per unit area can be required.

7.2 Errors in Commutation

Different sources of commutation errors of the magnetically levitated planar actuator with moving magnets have been presented. When the errors from different sources were compared, the most severe were found errors from the offsets in the measurement system. These errors can cause wrench cross-coupling of more that 10 % per one millimeter of the position offset. On the other hand, the cross-coupling caused by the look-up-table based calculation of the coupling matrix was found to be less than 0.5 %. This is much less than the errors caused by the electromechanical tolerances of the system.

A procedure to eliminate offsets of the measurement system was developed and successfully tested on COPAM in [13]. In Chapter 3, an investigation of the stability of this procedure was performed for the coil topology used in COPAM. Although it could not be proved that the procedure was stable for the whole workspace of the PA, the convergence was proved at least for all the investigated points. From the investigation, convergence is expected for any position in the workspace of the PA. It was found that it is possible also to use the procedure with different topologies and with different commutations. Although it was not possible to prove the convergence at least in the tested points, successful tests were performed with rounded coils distributed in a triangular array and with rectangular coils in a herringbone pattern. Furthermore, the same results were

obtained with l2-norm and l∞-norm based commutations. It is hypothesized that the relation between offset and wrench errors is caused mainly by the Halbach magnet array.

7.3 Control

During the control design of for the COPAM, two situations were considered. At first, only the control of the planar actuator (PA) was investigated. The planar actuator was modeled as a 6-degree-of-freedom (6-DOF) rigid-body mass-only system. To include the electromechanical tolerances in the model of the commutation, a model of the error coupling was developed, mapping the applied currents in the coils to the wrench (forces and torques) disturbances acting on the PA. To satisfy the electromechanical constraints of the system and to reduce excitation of the flexible modes, fourth-order point-to-point trajectory generation was used.

From a comparison of different kinds of controllers tested on the model of the PA, it was found that a (standard) first-order sliding-mode (SM) controller can achieve about 30-50% better performance than the classical PID. To eliminate the chattering effect, saturation instead of sign function was used in the SM controller. A higher-order SM (HOSM) controller was also investigated as an alternative for reduction of the chattering effect in the standard SM, but the performance was about a factor 10 worse than with the first-order SM. The reason for this result is the general control structure of the HOSM


controller, which does not take into account the plant model. Next, an iterative-learning controller (ILC) was applied as feedforward control. By using this control technique for repetitive trajectories, the errors were reduced to the noise level within several iterations.

In the second situation we investigated, the manipulator was included. The 8-DOF system, the 6-DOF planar actuator with the 2-DOF manipulator, was described with a multi-body model by using the Lagrange’s equation. The system was decoupled to allow for separate SISO control of each DOF. It was found that the only relevant parameters necessary for good decoupling are the position and velocity of the manipulator’s beam. The position of the beam gives information about the varying center of the gravity of the whole system and the velocity information is necessary for counteraction of the friction forces in the beam’s bearing.

The results from the simulations with the same controllers are similar to the results obtained on the single-body planar actuator. The SM controller again outperformed the PID by several tens of percents, whereas with the ILC, the errors for repetitive trajectories were reduced to the noise levels within several iterations.

7.4 Wireless Communication

After comparing several existing wireless solutions to determine their suitability for real-time control of a motion system, very fast infrared (VFIR) proved to be an attractive solution for the wirelessly controlled manipulator. However, the efficiency of the VFIR decreases to less than 18 % when small data packets are transmitted. Therefore, a new infrared (IR) wireless link was developed based on a modification of the VFIR standard.

The novel IR wireless link has a transmission time of less than 7 μs for 64 data bits with sampling rate of about 150 kHz. This time was achieved by a custom packet and data processing. The closed-loop delay produced by the wireless link is less than 23 μs. Such a delay makes the wireless link indistinguishable from a wired link up to closed-loop bandwidth of a few kilohertz. The IR wireless link provides more than 100 times shorter delay than IEEE 802.11b UDP/IP, more than 13 times shorter than the custom radio-frequency (RF) link developed in [13], and about 3.5 times shorter than standard VFIR protocol.

The reliability of the link has a CRC error rate of less than 10-7 in the working area, without retransmission of the data. This makes the wireless solution presented superior to any RF wireless link.

The maximum reliable distance is about 0.7 m, which might be a drawback of the link, but this depends on the application. However, new standards for higher speeds over larger distances are already in development [55].

The directionality of the transported signal is an advantage as well as a drawback of the link. The advantage is the possibility to use more of these links at one place without interference problems (unlike wireless radio links). On the other hand, it is a drawback for the applications if one of the modules is moving in plane or even rotating. A possible

7.5 Experimental Verification 181

solution to this problem can be to place one of the IR transmitters on a simple, cheap servomechanism that would approximately follow the trajectory of the second module.

7.5 Experimental Verification

The prototype was a complex system, which, for the first time, integrated three contactless technologies into one multi-level system. A few problems appeared during building and testing of the setup, so, improvements to the experimental setup had to be done. The original measurement system, consisting of a combination of short-range eddy-current sensors and long-range laser triangulation sensors, introduced vibration of the levitated planar actuator. The reason for the vibration was the sensitivity of the laser triangulation sensors to the perpendicular movement of the measured target. As the first improvement, the laser triangulation sensors were replaced by laser interferometers. Although the rotational capability of the setup was lost, the vibrations disappeared and the resolution and linearity of the sensors in the associated degrees of freedom (DOF’s) were greatly increased.

The second improvement was done on the contactless energy transfer (CET). The CET introduced electromagnetic disturbances during switching between two primary coils. This disturbance caused spikes in the position reading from the eddy-current sensors and in the wireless communication. This problem was solved by improving the start-up sequence in the electronics of the CET. However, active CET was still causing increased noise levels in the eddy-current sensors. The base frequencies of the CET and eddy-current sensors were too close to avoid interference. Solving this problem would require redesigning the CET or replacing the eddy-current sensors by e.g. laser interferometers.

The third improvement of the hardware was the implementation of the novel infrared wireless communication link. Although the original custom-made radio-frequency

wireless link was tested with the packet-loss ratio of 10-2 » 1 % [13], the practical results

showed a packet-loss ratio of up to 0.7 » 70 %. The cause was the CET. The new infrared

wireless link was successfully tested in the system with the packet-loss ratio always better

than 10-7, as expected.

The identification of the PA showed that due to the small levitation height, the air between the stator and translator causes damping in the vertical (z) and two tilting ( , ) DOF’s. Since these DOF’s are only stabilized, the low-frequency damping does not cause any control problems. The damping effect is reduced by increasing the levitation height or when operating the system in vacuum. The identification was also repeated for four different positions of the levitated PA. The results presented show that although the commutation does not decouple the system exactly, its performance is stable in all tested positions.

The control of the system was tested with three controllers: PID, Sliding-Mode (SM) and Iterative-Learning Control (ILC) with PID, in three different situations: Control of the planar actuator alone, control of the manipulator alone, and control of the combined 8-DOF system.


During the control of the PA alone, the manipulator position was fixed and the

CET unit was switched off. Although the acceleration of the PA of more than 30 m.s-2

could be used, the limit was set to 14 m.s-2 (diagonal movement) for safety reasons. The

maximum velocity was limited by the interferometer sensor system to 0.6 m.s-1 (diagonal movement). When the PA was controlled with the PID, the tracking errors of the xy-movement were less than 3 μm and the orientation was stabilized within 30 μrad. The SM halved these errors. The best performance was achieved by adding the ILC to the PID, which reduced the errors to a maximum of 400 nm and 3 μrad.

The control of the separate manipulator showed a limitation of the performance in the friction of the beam bearings, which was partly non-repetitive. The tracking accuracy for the linear movement was mostly better than 20 μm, but a peak in the error appears randomly as a consequence of the varying value of the static friction. The tracking error of the rotary movement was less than 400 μrad, caused by the high angular velocity and acceleration in combination with limited bandwidth and sensor resolution. The time-offset of the rotary-motor trajectory tracking was calculated to be less than 4 μs, which proves the effectiveness of the infrared wireless link for real-time control of a fast motion system.

For the combined control of the planar actuator with the manipulator, the PID and SM controller bandwidths had to be reduced due to the increased sensor noise caused by the CET unit. The test with the statically levitated PA and the moving manipulator showed good decoupling of the forces and torques between the manipulator and the platform. The residual stabilization errors were mostly caused by the friction in the beam bearings, which was difficult to estimate. This was proved by the applied ILC, which failed to improve the performance beyond the results obtained with the SM.

In the test with the combined control of the moving planar actuator with the moving manipulator, the total tracking errors were caused by the combination of the errors from the tracking of the PA trajectory, the errors from the disturbing manipulator and the reduced controller bandwidths. Therefore the peak errors were about 30-50 % worse than in case of the moving PA alone. In comparison with the previous test, the ILC was able to reduce the errors this time, but the final errors are still far worse than in the case of the PA alone. The results with the ILC show that a significant part of the residual errors is caused by the sensor noise and by the friction in the manipulator’s beam. Summarization of the results achieved on the experimental setup can be found in Table 7.1.

From the experiments on the experimental setup, several conclusions can be drawn: The contactless energy transfer needs to be well designed with respect to electromagnetic interference, otherwise, it can easily interfere with other parts of the device. The less reproducible friction in the manipulator reduces the precision of the forces and torques decoupling. The performance of the PA with the feedback control (PID, SM) can be improved by decreasing the delay in the control loop (to increase the bandwidth). With the feed-forward control (ILC) (or a high-bandwidth feedback control), the most important is low-noise measurement system (e.g. 6-DOF laser interferometry). Since the residual position-dependent cross-coupling due to the mechanical and electrical tolerances has a low-frequency characteristic during the movement of the translator, it can be successfully suppressed by sufficient bandwidth of the controller or by the application of some kind of learning control (as was shown with the ILC).

7.6 Thesis Contributions 183

Table 7.1: Achieved performance of the experimental setup

Trajectory x,y z ,q y f

Random 1.5 μm 6 μm 10 μrad 2 μrad PA

Repetitive 0.3 μm 0.4 μm 3 μrad 0.3 μrad

Random 2 μm 10 μm 50 μrad 7 μrad

Tracking

accuracya

PA with

manipulatorc Repetitive 2 μm 6 μm 30 μradb 2 μrad

Link delay 7 μs

Closed-loop delay < 23 μs

Update rate 150 kHz

Infrared

wireless link

Packet error rate < 10-7

Sampling frequency 3 kHz

Closed-loop delay 1.2 ms

Mechanical tolerances 100 μm

Amplifier noise < 1 mA (≈ < 10 mN)

Laser interf. 0.15 nm Sensor resolution

Eddy current 160 nm

CET off on

Laser interf. < 10 nm < 10 nm

Important

system

parameters

Sensor noise

Eddy current 200 nm ~ 1.5 μmd

a With the max. acceleration of 14 m.s-2 and max. velocity of 0.6 m.s-1 b A peak of 40 μrad appears randomly due to the friction in the linear motor bearings c The worse results are mostly caused by the contactless energy transfer (CET) d Average, the actual noise level depends on the distance from the CET coil to the sensor

7.6 Thesis Contributions

The major contributions of this thesis are summarized as follows:

The derivation of two novel commutation methods for improving the performance of overactuated planar actuators. The methods can put bounds on the maximum currents in the coils to prevent saturation of the current amplifiers. Amplifier saturation prevents sufficient accurate linearization and decoupling of the forces and torques. Moreover, with these new commutation methods, a higher acceleration of the planar actuator can be achieved and/or less powerful (cheaper) current amplifiers can be utilized.


The investigation and optimization of moving-magnets planar actuator for long-range yaw rotation. The possibility of a completely propelled and controlled rotation about the vertical axis instead of just stabilizing it for bearing has been analyzed from a control point of view. Based on this investigation, a novel coil array with a triangular grid of rounded coils has been proposed for better controllability in any orientation of the PA. In addition, other coil and magnet topologies have been studied from a control point of view for their suitability for full rotation.

A characterization and quantification of the causes that reduces the decoupling of the forces and torques in magnetically levitated planar actuators. It was found that the measurement system has the most significant impact on the wrench decoupling in the prototype. Proof of correctness of the procedure to estimate and eliminate offsets of the measurement system was given. Moreover, an extension of the procedure to be applied on different coil topologies was successfully tested.

A wireless communication link for real-time control of a fast motion system. The link, based on infrared transceivers, provides reliable (very low error rate) and high-performance (low-latency, high-sampling rate) connection for the wireless control of systems with a closed-loop bandwidth of several kilohertz.

7.7 Recommendations for Future Development

Based on the results and findings in this thesis, the following recommendations for future development are emphasized.

7.7.1 New Commutations

The clipped l2-norm based commutation is a promising form of commutation, which can satisfy constraints on the maximum current while still minimizing the dissipative power. However, a fast, accurate, real-time implementation is still lacking and a boundary on the current discontinuity of an approximate, but real-time algorithm has not yet been found. If a boundary condition is found or the real-time algorithm is improved to guarantee the current continuity while keeping the low execution time, the method has a real added value in over-actuated control.

7.7.2 New Topologies

As was found during this research, a triangular array of rounded coils is promising topology for future planar actuators. Therefore, deeper investigation of its applicability including temperature profiles should be done. A new magnet-array topology for planar actuator can yield better rotational capability.

7.7 Recommendations for Future Development 185

7.7.3 New Measurement Systems

Long-range rotating planar actuator

Research into a new measurement system for industrial applications that allows long-stroke translational and rotational movement of the planar actuator is desirable. The focus should be concentrated on new types of 3-DOF optical sensors, similar to optical sensors for computer pointers as a cheap substitution for the method presented in [64]. If a new type of the optical sensor were to be developed together with specific measured surface (target), the resolution, reliability and delay could be much improved in comparison to what is available on the market. Moreover, absolute position and orientation reading could be possible with a well-designed pattern on the measured surface.

Wireless sensing

The infrared wireless link presented in this thesis allows for wireless transmission of a sensor data with very low latencies. This gives the opportunity to place several sensors on the translator without the necessity of the cable connection. Such a sensor system can be used for example to measure local deformations or heating of the translator, or to simplify measurement of the vertical position. There are many possibilities for improving the PA performance by utilizing these wireless sensors.

7.7.4 New Planar Actuators

Industrial systems

Multiple-DOF reluctance actuators are difficult to decouple since the produced force is proportional to the square of the current and inversely with the square of the distance. Moreover, permanent attractive force, flux-driven power amplifiers and parasitic phenomena (nonlinearities, slow change of flux, eddy currents, hysteresis) prevent using the reluctance actuators for high-precision 6-DOF controlled planar actuators. However, for industrial systems, reluctance planar actuators with their higher density of power are interesting options for further development.

High-precision systems

For high-precision systems, Lorentz-force actuators are preferable. Over-actuated design can be utilized for control of deformation of the translator [2]. An opposite idea is to use over-actuation to deliberately deform a levitated stage-membrane. This membrane can be e.g. a mirror, which will receive variable optical properties.

7.7.5 Elevation systems

Elevation systems or ceiling actuators are being researched [90, 88]. These systems use reluctance actuators for their permanent attractive force and, therefore, provide safety in the event of energy dropout. However, reluctance actuators are more difficult to


decouple and control. Therefore, research into new high-precision elevation systems based on Lorentz-force actuators is recommended.

187

Appendix A

Dimensions of the Experimental Setup

xy

xz

z ch

mh

y

z

OPA

OPA

Figure A.1: Drawing of the coil and magnet array of the planar actuator (PA); top and side

view; distribution of the solid-wire (light-gray) and litz-wire (dark-gray and black) coils; the “black” litz-wire coils are for the contactless energy transfer (CET), whereas the “dark-gray” litz-wire coils are to reduce the coupling between the stator coils during active CET; the dashed

rectangle represents the workspace of the magnet array; the PA is in [x,y] = [0,0] position; OPA

represents the center of rotation of the PA.

Solid-wire coils

Litz-wire coils Litz-wire coils used for contactless energy transfer

Secondary coil

188 Appendix A. Dimensions of the Experimental Setup

τ

τ

cs=4/3τ

cw

cbw

mw

Figure A.2: Definition of dimensions of the planar actuator.

OPA OMA MA≈COG

OPA

xLM

zb

COGMA

xz

y

xy

z z xy

Figure A.3: Drawing of the planar actuator with the manipulator.

Manipulator’s arm

Manipulator’s beam

Static part of the manipulator

Aluminum plate

Aluminum support plate

Magnet array

189

Table A.1: Dimensions of the experimental setup

Parameter Symbol Dimension

Magnet array

10 × 10 magnet poles

Magnets VACODYM 655 HR

Magnet-pole pitch t 40 mm

Halbach magnet width wm 26 mm

Magnet height hm 10.3 mm

Magnet permeability r

m 1.03

Coil array (for levitation) 9 × 11 coils

Solid- (litz-) wire coils 84 (15) coils

Coil pitch sc 53.3 mm

Coil width wc 51 mm

Coil bundle width bwc 21.1 mm

Coil height hc 11.4 mm

Coil turns – solid (litz) wire 684 (240) turns

Coil resistance Rsw (Rlw) 4.4 (1.0) Ω

Coil inductance 8.1 (1.0) mH

Max. continuous current 1.8 (7.0) A

Max. peak current 6.0 (20) A

Amplifier bandwidth 3 kHz

Levitated massa m

20.5 kg

Beam massb bm 1.68 kg

Arm mass am

0.28 kg

Translator sizec 425 × 425 mm

Max. horizontal displacement x,y 167 × 67 mm

Max. vertical displacement z 2 mm

Max. tilting angles θ, ψ ± 1.5 mrad

Max. vertical (yaw) rotation ± 0.2 rad

Laser triangulation sensors Micro-Epsion

ILD 2200-100 (ILD 2200-200) for x-position (y-position and -rotation)

Range

100 (200) mm

Resolution

5 (10) μm

Linearity

30 (60) μm

Bandwidth

2.5 kHz

190 Appendix A. Dimensions of the Experimental Setup


Eddy-current sensor Lion Precision

ECL-100 U8B for z-position and θ, ψ-orientation (tilting)

Range 2 mm

Resolution 0.16 μm

Linearity 5 μm

Bandwidth

10 kHz

Contactless energy transfer

Max. power > 100 W

Efficiency ~ 70 %

Frequency 150 kHz

Wireless communication Micro-Linear + Xillinx

ML2724SK-02 + FPGA Spartan IIE XC2S300E

Frequency 2.4 GHz

Data rate 1.28 Mbps

Max. latency 170 μs

Packet-loss ratio 10-2

Manipulator

Beam displacement LMx

± 80 mm

Linear motors cont. force (each) LMF 10 N

Linear motors peak force (each)

35 N

Linear encoder resolution

1 μm

Rotary motor cont. torque RMT 0.28 Nm

Rotary motor peak torque

1.68 Nm

Rotary encoder resolution

20 μrad

a Including the static part of the manipulator b Including the arm of the rotary motor c The square part with the magnet-array underneath

191

Appendix B

List of Symbols and Abbreviations

Table B.1: List of symbols

Symbol Unit Quantity

a a vector

A a matrix

q m, rad Coordinate vector

x, y, z m Cartesian coordinates

, , rad Rotation about x- , y-, and z-axis, respectively

i A Current vector

F N Force

T Nm Torque

w N, Nm Wrench vector consisting of force and torque

components

K N/A, Nm/A Coupling vector

K N/A, Nm/A Coupling matrix

Κ- A/N, A/Nm Inverse coupling vector mapping

Κ- A/N, A/Nm Inverse coupling matrix

X N/A, Nm/A Error coupling matrix

admS Admissible set of coordinates

192 Appendix B. List of Symbols and Abbreviations

Symbol Unit Quantity

g 9.81 m.s-2 Gravity constant

t s Time

TS s Sampling time

m Magnet-pole pitch

κ Condition number

R Ω Resistance

L H Inductance

P W Power

p m Position

v m.s-1 Velocity

a m.s-2 Acceleration

j m.s-3 Jerk

d m.s-4 Jerk derivative

w rad.s-1 Radial frequency

f Hz Frequency

Table B.2: List of abbreviations

Abbreviation Full name

AA Auto-alignment

CCD Charge-coupled device

CET Contactless energy transfer

CL2C Clipped l2-norm based commutation

CMOS Complementary metal–oxide–semiconductor

COG Center of gravity

COPAM Contactless planar actuator with manipulator

CRC Cyclic redundancy check

DOF Degree of freedom

DPLL Digital phase lock loop

FEM Finite element method

FPGA Field-programmable gate array

HOSM Higher-order sliding mode (control)

HPPA Herringbone pattern planar actuator

ILC Iterative-learning control

193

Abbreviation Full name

IR Infrared

IrDA Infrared data association

L2C l2-norm based commutation

LFFC Learning feedforward control

LFT Linear fractional transformation

LiC l∞-norm based commutation

LTI Linear time-invariant

LTV Linear time-varying

LUT Look-up table

MA Magnet array

MIMO Multiple-input multiple-output

PA Planar actuator

PID Proportional-integral-derivative (control)

PWM Pulse-width modulation

RC Repetitive control

RF Radio frequency

RMS Root mean square

SCA Square coil array

SISO Single-input single-output

SM Sliding-mode (control)

TCA Triangular coil array

UDP/IP User datagram protocol/Internet protocol

VFIR Very fast infrared

WLAN Wireless local area network

WPAN Wireless personal area network

ZPECT Zero phase error tracking

194 Appendix B. List of Symbols and Abbreviations

195

Appendix C

Additional Simulation Data

C.1 Additional Simulation Parameters

The magnetic flux density underneath the magnet array was calculated with a resolution of 0.5 mm in a volume of x × y × z = 0.48 m × 0.48 m × 0.014 m. The coil model had a resolution of 0.5 mm. The full and reduced LUT coupling matrixes had a resolution of 0.5 mm.

The maximum current, dissipative power, error sensitivity and acceleration were investigated in the working area of the translator of 0.167 m × 0.067 m with a resolution of 1 mm. The rotational step was 1 deg.

The maximum commutation errors in Section 2.5 were calculated over the working area of the translator with a resolution of 2 mm.

In all the calculations, it was verified that a factor 2 higher resolution does not change the simulation results by more that 5 %.

C.2 Maximum Currents in the Litz-Wire Coils

196 Appendix C. Additional Simulation Data

Figure C.1: Position-dependent maximum current in the litz-wire coils calculated with l2-norm,

clipped l2-norm, and l∞-norm based commutation for steady state levitation T[0,0, ,0,0,0]w mg=

and levitation with x, y acceleration T[ , , ,0,0,0]w mg mg mg= ; SCA.

C.2 Maximum Currents in the Litz-Wire Coils 197

Figure C.2: Position-dependent maximum current in the litz-wire coils calculated with l2-norm,

clipped l2-norm, and l∞-norm based commutation for steady state levitation T[0,0, ,0,0,0]w mg=

and levitation with x, y acceleration T[ , , ,0,0,0]w mg mg mg= ; TCA.

198 Appendix C. Additional Simulation Data

C.3 Normalized Absolute Wrench Difference

Table C.1: l2-norm based commutation, triangular coil array.

Worst-case normalized absolute wrench [-] abs

w

des

w ,x abs

F ,y abs

F ,z abs

F ,x abs

T ,y abs

T ,z abs

T

xF 1.003 0.599 0.987 0.843 0.646 0.762

yF 0.580 1.017 1.161 0.869 0.773 0.711

zF 0.494 0.571 1.006 0.879 0.656 0.431

xT 0.976 2.462 2.111 1.010 1.363 1.330

yT 0.601 0.650 1.222 1.037 1.006 0.582

zT 1.015 1.356 1.135 0.965 0.911 1.005

Table C.2: l∞-norm based commutation, triangular coil array.

Worst-case normalized absolute wrench [-] abs

w

des

w ,x abs

F ,y abs

F ,z abs

F ,x abs

T ,y abs

T ,z abs

T

xF 1.081 1.653 1.659 1.330 1.102 0.969

yF 1.664 1.157 1.999 1.661 1.400 1.431

zF 0.954 1.072 1.197 0.906 0.794 0.765

xT 2.081 2.686 2.661 1.259 1.858 1.734

yT 1.272 1.571 1.616 1.320 1.055 1.070

zT 2.045 2.218 1.933 1.612 1.518 1.908

199

Appendix D

(3+2)-DOF Experimental Setup

D.1 Introduction

This appendix describes the modeling, identification and control results of a (3+2)-DOF experimental setup (3-DOF platform with 2-DOF manipulator) which was built as a pre-prototype of a fully 6-DOF planar actuator with a 2-DOF manipulator on top.

In the final 6-DOF planar actuator, levitation and propulsion is provided with coils in standstill base-plate together with permanent magnets in the moving platform [13, 56, 76]. The manipulator on the platform is controlled and powered wirelessly [13]. To test and validate the control design for this final contactless planar actuator with manipulator, an experimental setup was designed, which yields only three DOF’s for the platform. Precise control of the platform is necessary, because in the final prototype the platform is lifted and operated just about 1 mm or less above the surface. Such a working condition is a trade-off between the operational height and power needed to lift the platform with the manipulator in the final prototype. In this pre-study, the translations for propulsion and the rotation about the vertical axis were fixed.

A description of the experimental setup is provided in Section D.2. Section D.3 describes a model of the experimental setup. In section D.4, the identification procedure and the results are presented. The results from control of the experimental setup are presented in Section D.5. These control results include the infrared wireless communication link for the control of the manipulator (see Chapter 5).

200 Appendix D. (3+2)-DOF Experimental Setup

D.2 Experimental setup

The experimental setup consists of two main parts: a 3-DOF platform actuated by nine voice coils and a 2-DOF manipulator on top of it (see Figure D.1 and Figure D.2).

The platform, that mimics the final magnetic array, is suspended such that it can only move in three DOF’s: vertically and over two tilting angles. The other three DOF’s are fixed by a suspension system consisting of three rods connecting the platform

Figure D.1: Drawing of the experimental setup; the base-frame is made partly transparent to

show the voice coils here.

Figure D.2: Photo of the experimental setup.

Beam Rotary motor with arm underneath

2-DOF manipulator

Linear motors

3-DOF platform

9 voice-coils + 9 sensors underneath

Suspension by 3 rods

Base-frame

z

y

x

Rod

D.2 Experimental setup 201

with the base-frame. Because the final planar actuator is magnetically levitated and controlled in six DOF’s, it has zero spring constants and zero damping. Therefore, the suspension system was designed in a way to achieve minimal spring constants and minimal damping in the three free DOF’s as well, but with very high stiffness in the other three fixed DOF’s. The option of built-in leaf springs at the ends of the rods has been chosen. The rods can easily bend in two directions at its ends but they cannot shrink or extend, so they behave as a spring with low spring constant in two directions and with high spring constant elsewhere. This feature of the suspension rods was unfortunately also their drawback, as will be shown later.

The platform is actuated by nine voice coils underneath, which are distributed into a regular array to simulate the topology of the final planar actuator and are used for gravity compensation and for manipulator platform control in three DOF’s. The voice coils are custom-made type NCC05-18-045-2A from H2W Technologies, Inc. They are driven by current amplifiers LCAM from Quanser. Using this over-actuated platform allows the distribution of the necessary force for lifting the platform as well as compensating the disturbances from the manipulator. The total force produced by the nine voice coils is about 180 N. Part of this force is used for gravity compensation. With a total weight of the platform with manipulator of about 7 kg, the force available for control is approximately equal to 110 N. Unfortunately, due to the actuators shape and the reduced air gap between the magnet and the coil of 1 mm, the voice coils also behave as air dampers.

Furthermore, inductive position sensors (type U8B + driver ECL100 from Lion Precision) are mounted under each voice coil in such a way that we have measurements at the same position as the actuator acts (see Figure D.3).

The manipulator on top of the platform is essentially an H-bridge with small arm connected to the rotary motor on the beam. A detailed description of the manipulator can be found in Chapter 1 and [13]. The dimensions of the system are listed in Table D.1 and indicated in Figure D.4.

Figure D.3: Drawing of the experimental setup; cross-section.

Coil

Al. target for position sensor

Inductive position sensor

Manipulator

Voice coil Magnet

Platform


Table D.1: Dimensions of the (3+2)-DOF experimental setup


Platform massa pm

5.65 kg

Beam massb b

m 1.68 kg

Arm mass am

0.28 kg

Actuator and sensor spacing vc

r 80 mm

Spring fixation distance l 150 mm

Beam vertical positionc

bz

102 mm

Max. vertical displacementd z ± 1 mm

Max. tilting angles θ, ψ ± 6 mrad

Beam displacementc LMx

± 80 mm

Max. continuous force per coil i

F 20 N

Peak force per coil 60 N

Motor constant 20.43 N/A

Max. amplifier cont. current

7 A

Amplifier bandwidth 10 kHz

Inductive sensor resolution @ 10kHz 0.16 μm

Inductive sensor linearity 5 μm

a Including the static part of the manipulator, b Including the arm of the rotary motor, c See Figure D.5, d Zero vertical position is defined 1 mm above the bottom end-stops

Figure D.4: Distribution of actuators and springs on the platform.

F9 F8

F7

F6 F5

F4

F3 F2

F1

x z

y

rvc

θ,Tx

ψ,Ty

k

k

k

Base-plate

Platform

l

l

rvc

D.3 Model 203

D.3 Model

The model is based on the multi-body model presented in Section 4.5.1 with the following:

Horizontal x- and y-axis of the platform as well as yaw-rotation f about the vertical z-axis are suspended and thus constant (zero).

The effect of the rotary motor with connected bar is neglected, because it produces small disturbing torque round z-axis which is suspended and the effect on the other DOF’s is very small.

Only small deviations around the zero working point of the 3-DOF platform are assumed.

The three rods connected to the platform add stiffness and mass to the system. The voice coils have low air gap between the coil and the magnet, which causes

air-damper behavior (see Figure D.3).

Using these assumptions, equations of motions only for 4-DOF system

(T

LM, , ,q z xé ù= q yê úë û ) can be linearized with respect to the angles and the position of the beam

LMx of the manipulator appears as the only relevant parameter:

LM

( ) ( , )x q H q q+ = tM . (D.1)

Applied wrench (force and torques) column T

LM, , ,

z x yF T T Fé ùt = ê úë û consists of the force applied

in vertical z-direction, the two torques applied about x- and y-axis and the force applied to the linear motors of the manipulator. The wrench vector t is the input of the system whereas position vector q is the output.

The mass matrix with linearized angles is then written as:

p b b LM2

b b2 2LM

b LM b b LM b b

b b b

0 0

0 0( ) ,

( )

0 0

x xy

xy y

m m m x

J m z Jx

m x J J m z x m z

m z m

é ù+ê úê ú+ê ú= ê ú+ + -ê úê ú

-ê úë û

M (D.2)

where LM

x and b

z are relative positions of the manipulator’s beam center of mass with respect to the origin and centre of rotation O in x- and z-direction (see Figure D.5).

pm and

bm are the masses of the platform (together with the static part of the manipulator) and the beam (with the rotary arm), respectively, and

xJ and

yJ are moments of inertia of the

whole system. The center of the mass of the platform is not exactly in the origin of the coordinate system because of weight of the voice coils, attached suspending rods and static part of the manipulator. These terms are also included in the mass matrix, but for simplicity not shown.


Figure D.5: Drawing of the simplified model of the platform with moving beam and with

indicated center of the rotation O.

The linearized joined vector is given by:

p b2

b b vc2

b LM b vc

b

( ) ( ) 9

( ) 6( , )

( ) ( ) 6

zm m g k q dz

k q z m g r dH q q

k q m g x z r d

m g

q

y

é ù+ + +ê úê ú- q + qê ú= ê ú+ - y + yê úê ú

yê úë û

, (D.3)

where d is the damping coefficient of the voice coils, vc

r is the arm of the voice coils/sensors (see Figure D.4). Gravity constant g = 9.81 m·s-2, and zk , k q, and k y are row vectors containing spring constants

Lk ,

TLk , and

THk derived from the fact that the suspending rods

behave as a spring with different spring constants in each direction (see Figure D.6):

2

3 1 3L TL TL L TL2 44

21 1TL L TH TL TL2 2

23 1 9L TL TL L TH TL4 2 4

( ) (3 ) ( ) ,

( ) (2 2 ) ,

( ) ( ) ( ) ,

z l ll

l

l

k q k k z k lk k

k q k z l k k k k

k q lk k z k l k k kq

y

= + + q + + y

= + + + q - y

= + - q + + + y

(D.4)

where l is the length from the center to the side of the platform (see Figure D.4). The rods have been designed to have a low spring constant for movement in vertical direction (

Lk

and TL

k ). Unfortunately, the rods also twist when the platform is tilted. This causes a high torque spring constant (

THk ) for tilting about x- and y-axis.

Figure D.6: Suspending rod; linear and torque spring behavior.

The joined vector can be split into a stiffness matrix:

z

x y

Beam

Platform

xLM

O

mp

mb

zb

kTH

kL

suspending rod

leaf springs

kTL

D.3 Model 205

2

3 1 3L TL TL L TL2 44

21 1TL L TH TL b b TL2 2

23 1 9L TL TL L TH TL b b b4 2 4

b

3 0

2 2 0

0 0 0

l ll

l

l

H

q

k k k lk k

k l k k k m gz k

lk k k l k k k m gz m g

m g

¶= =

¶é ù+ +ê úê ú+ + - -ê ú= ê ú+ - + + -ê úê úê úë û

K

, (D.5)

and a damping matrix:

2vc

2vc

6 0 0 0

0 6 0 0

0 0 9 0

0 0 0 0

r d

r dH

dq

é ùê úê ú¶ ê ú= = ê ú¶ ê úê úê úë û

D

. (D.6)

Then ( , )H q q q q C= + +K D , where C contains all the static terms, and the equation of motion (D.1) can be rewritten as linear parameter-varying (LPV) system in the “parameter”

LMx (while

bz is fixed as a constant):

LM

( ) .x q q q C+ + + =M D K τ (D.7)

The system can also be described with a state space model:

LM LM( ) ( ) ,

,

x x x x

q x

= + t=

A B

C

(D.8)

where:

1

LMLM

1

LMLM

( ) ,( )

( ) ,( )

[ ],

xx

xx

-

-

é ù é ùê ú ê ú= ê ú ê ú- -ê ú ê úë û ë ûé ù é ùê ú ê ú= ê ú ê úê ú ê úë û ë û

=

I 0 0 IA

0 M K D

I 0 0B

0 M I

C I 0

(D.9)

the identity matrix I has dimension 4 × 4 and the state vector

TT

LM LM, , , , , , , ,x q q z x z xé ùé ù= = q y q yê ú ê úë û ë û

. (D.10)

The platform itself is a system with 9 actuators and 9 sensors, but with just three

DOF’s ( , ,z q y). Therefore, a transformation from T

PA , ,z x y

F T Té ùt = ê úë û to the vector u of the 9

actuator forces 1 9

F - according to u PA

u = tT :

vc vc vc vc vc vc

vc vc vc vc vc vc

T1 1 1 1 1 1

6 6 6 6 6 6

1 1 1 1 1 1u 6 6 6 6 6 6

1 1 1 1 1 1 1 1 19 9 9 9 9 9 9 9 9

0 0 0

0 0 0r r r r r r

r r r r r r

- - -

- - -

é ùê úê ú= ê úê úê úë û

T . (D.11)


A second transformation is from the vertical position sensors outputs y to the column of coordinates

PA yq y= T , where T

y u=T T and T

PA[ , , ]q z= q y . These linearized transformation

matrices are not unique. The first one has been calculated to minimize the power losses of the actuators (minimized 2-norm of the current vector) and the second one to reduce the noise from the sensors.

D.4 Identification

Identification of the system was necessary because the values of the spring and damping constants were not known. The identification results of the manipulator were presented in [13]. Therefore, the identification here deals with the suspended platform. Since the platform is an almost meta-stable system (ideally double integration in each DOF), identification should be performed on a closed-loop system. The identification was done using two different indirect approaches. In the first approach, time-domain data of the system were processed by using Ho-Kalman’s algorithm [53]. The second approach was based on frequency-domain system identification, where data were obtained from the system by using signal and system analyzer SigLab.

A controller, consisting of three separate SISO PI controllers, was used to stabilize the system. So the three DOF’s were controlled separately. The integrative components were used to remove steady-state errors. For each method, the position of the beam (

LMx ) –

the parameter of the system - was set to three different locations: the most left (-80 mm), middle (0 mm) and the most right (80 mm). The sampling rate of the system was set to 8 kHz.

In the first approach of the closed-loop identification, the transfer function of the process sensitivity G was identified. Gaussian white noise was applied as a disturbance

each time in turn at one of the inputs of the plant P. All three outputs of the plant were

measured to obtain direct- and cross-transfer functions of G, where 1( )-= +G P I CP .

From the measured time-domain data, cross-correlation functions from the each input to each output were derived. Because the input signal was a white noise process, estimating the cross-correlation function directly provides us with information on the pulse response of the system [89]. The identified pulse response sequence

0, , ( )

tg t = ¥ was used in

Ho-Kalman’s algorithm as the sequence of Markov parameters of the system

0( ) ( ) t

tG z g t z

¥ -

==å .

The algorithm is based on the Hankel matrix

(1) (2) (3) ( )

(2) (3) (4)

(3) (4) (5)( )

( ) ( 1)

c

r c r

g g g g n

g g g

g g gG

g n g n n

é ùê úê úê úê ú= ê úê úê úê ú

+ -ê úë û

H

, (D.12)

which has the same elements on the skew diagonals. The matrices (ˆ ˆ ˆ ˆ, , ,A B C D) of the estimated state space model G of the system G, in the form


ˆ ˆ( 1) ( ) ( ),ˆ : ˆ ˆ( ) ( ) ( ),

x t x t u t

y t x t u t

+ = +

= +

A BG

C D (D.13)

can be constructed applying singular value decomposition on the Hankel matrix [122]:

=H U VΣ (D.14)

Then we can derive the matrices of the state space model as

1/2 T 1/2 1/2

1/2 T

ˆ ˆ, ,ˆ ˆ, (0),g

- -= ⋅ ⋅ =

= =

A U H V B U

C V D

Σ Σ Σ

Σ (D.15)

where H

is the shifted Hankel matrix by one block column to the left. A valuable feature of the Ho-Kalman’s algorithm is that it can also handle MIMO systems. Thus each g(i) element of the H represents a m p´ matrix, which corresponds to the pulse response of the

MIMO system with p inputs and m outputs at a time instant it . For the 3-DOF system,

sub-matrices g(i) have dimensions of 3 3´ elements. The reconstructed state-space model consists of states, which are common for whole MIMO model. The size of the Hankel matrix H was chosen as

r cn n= = 1500, which corresponds to 0.5 s of the signal sampled at

1 kHz. Because the measured data are not exact, it is necessary to reduce the order of the model by choosing number of relevant singular values in Σ matrix. From the magnitude of the singular values (see Figure D.7) it was verified that the identified model G has nine main states, which corresponds to the six states of the plant with the three states of the

controller. Now the plant transfer functions estimate HC

P can be reconstructed from estimate G as

( ) 1

HCˆ ˆ ˆ -

= -P G I CG , (D.16)

which can be calculated because the controller C is known. Bode magnitude plots of the

frequency response of the identified plant HC

P are shown in Figure D.8.

In the second approach, frequency-domain data were obtained from the system by using signal and system analyzer SigLab (from Spectral Dynamics, Inc.). As a source, Gaussian white noise was used again, but the frequency response of the input sensitivity

1S

ˆ ˆ( ) ( ( ) ( ))j j j -w = + w wS I C P was identified. The frequency response of the plant can be reconstructed as follows:

( ) 11

Sˆ ˆ( ) ( ) ( )j j j

--w = w w -P C S I (D.17)

which poses no problems as all matrices are 3 3´ . The sensitivity matrix ˆ( )jwS is constructed from the 3 3´ responses of the system and ( )jwC is again a diagonal matrix containing 3 SISO PI controllers. Bode magnitude plots of the frequency response of the identified plant

Sˆ ( )jwP for left, center and right position of the beam are shown in

Figure D.9.

From Figure D.8 and Figure D.9, the results obtained by both identification

techniques are similar and the platform can be modeled with 6th-order MIMO model up to frequency of about 60 Hz. The resonances visible in Figure D.9 are probably caused by


backslash in the manipulator bearings and by flexible modes in the system at higher frequencies.

After the identification part, the LPV model described in Section D.3 was fine-tuned to match the results obtained by the identification. In particular the spring constants (

Lk ,

TLk ,

THk ) and the damping d, which were estimated initially, had to be fine-

tuned to match the identification. In Figure D.10, one can see the resulting bode magnitude plot of the model, which matches the identified transfer functions. In order to check whether this LPV model (D.7) fits the closed-loop behavior of the real system, a test was performed. From the simulated and measured values in Figure D.11, it can be seen that the simulation and the measurement of the real setup match well. The difference between the simulated and measured values is caused by linearization of the model and uncompensated position-dependent friction in the beam bearings (see the next section).

0 2 4 6 8 10 1210

-6

10-4

10-2

100

Singular values

n

Figure D.7: Largest 12 singular values of the system.


Figure D.8: Bode magnitude plot of HC

P for 3 positions of the beam (left, middle, right).

Figure D.9: Bode magnitude plot of S

P for 3 positions of the beam (left, middle, right).

MiddleLeft Right

Middle Left Right


Figure D.10: Bode magnitude plot of the 6th-order MIMO model of the platform based on

Figure D.8 and Figure D.9; three positions of the beam.

-20-10

010

z (

m)

Measured Predicted

-10-505

(

rad)

-1

0

1

(

mra

d)

0.2 0.4 0.6 0.8 1 1.2 1.4

-0.05

0

0.05

Time (s)

x LM (

m)

Figure D.11: Time plot of the system perturbed by moving beam LM

( )x t ; simulated and measured

values.

Middle Left Right

D.6 Summary and Conclusions 211

D.5 Control

The identified parameters of the system were used to decouple the multi-body system (see Section 4.5.1). Then, the controllers designed in Chapter 4 were applied. Since the commutation algorithm for the platform contains only the static matrix and only nine amplifiers are utilized, this experimental setup is capable of higher sampling rates and lower delays than the final prototype. A sampling frequency of 10 kHz with two delays was achieved. To mimic the final prototype behavior better, the sampling frequency of the system was set to 3 kHz and three additional delays were added to the control loop.

From the final prototype specifications, where the levitation height is about 1 mm above the surface, the combined control precision must satisfy:

err err err2 2

1 mmr rz + q + y < , (D.18)

where err

z , errq , and

erry are the position and orientation errors, respectively, and r is the

length of the platform. For the expected length r = 0.4 m and allowing each term in (D.18) 1/3 of the error, the absolute limits are:

errz < 330 μm,

errq < 1.6 mrad,

erry < 1.6 mrad.

The results with PID and sliding-mode (SM) controllers are shown in Figure D.12. During the tests, the manipulator beam followed the trajectory shown in the top plot of the figure. Already with the PID controllers, the position and orientation errors are less than 0.5 μm and 10 μrad. With the SM controllers, also the error of the y-orientation is close to the sensor noise level, while higher control effort in all DOF’s is applied. The beam position was controlled with the PID controller. The residual position and orientation errors were mainly caused by the position-dependent friction variation in the beam bearings (see Figure D.12 bottom).

D.6 Summary and Conclusions

In this appendix, the pre-prototype of the final planar actuator with manipulator was presented. The pre-prototype consists of the 3-DOF platform with the same 2-DOF manipulator as in the final prototype positioned on top of it. It was built to test the control algorithms and the real-time wireless link. The results with the wireless link have already been described in Chapter 5.

The platform that mimics the final magnetic array is suspended such that it can only move in three DOF’s: vertically and over two tilting angles. The other three DOF’s are fixed by the suspension system consisting of three rods connecting the platform with the base-frame. The platform is actuated by nine voice coils underneath.

Based on the modeling and simulation results presented in Chapter 4, an LPV model of system was created. Since the suspending rods and the voice coils act as springs and dampers, respectively, their parameters had to be included in the model. Two identification methods were used to obtain the plant transfer functions with similar results. The results were used to estimate the values of the spring constants and the damping, and to decouple the system consequently.


Figure D.12: Position and orientation errors of the platform disturbed by the moving beam (top plot) and related control action; PID (gray) and Sliding-mode (black) control.

The results with the PID controllers confirmed that the designed control applied on the decoupled system is able to keep the disturbances produced by the operating manipulator within 300 nm and 10 μrad. With the SM controllers, the final orientation errors were reduced to 3 μrad, close to the sensor noise levels. This is more than a factor 500 smaller than the maximum tolerable errors.

The main conclusion from this investigation is that the required performance for good stabilization of the platform with the operating manipulator is achievable.

213

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223

Summary

Controlling a Contactless Planar Actuator with Manipulator

An existing magnetically levitated planar actuator with manipulator has been studied and improved from a control point of view. This prototype consists of a magnetically levitated six-degree-of-freedom (6-DOF) planar actuator with moving magnets, with a 2-DOF manipulator on top of it. This system contains three different contactless technologies: contactless bearing and propulsion of the planar actuator, wireless powering of the manipulator, and wireless communication and control of the manipulator.

The planar actuator (PA) consists of a Halbach magnet array, which is levitated and controlled in all six DOF’s above a stationary coil array. The PA is propelled in two horizontal translational DOF’s while the other four DOF’s are stabilized to accomplish a stiff bearing. Each active coil contributes to the production of forces and torques acting on the magnet array. Since the number of active coils is much larger than the number of DOF’s, the desired force production can be distributed over many coils. Therefore, a commutation algorithm has to be used to invert the mapping of the forces and torques exerted by the set of active coils as a function of the coil currents and the position and orientation of the translator. One method for linearization and decoupling of the forces and torques was developed in the past. The method is called direct wrench decoupling and guaranties minimal dissipation of energy. However, no constraints on the maximum current can be given. This study proposes two novel, norm-based commutation methods: l∞-norm and clipped l2-norm based commutation. Both methods can put bounds on the maximum currents in the coils to prevent saturation of the current amplifiers. The first method focuses on minimization of the maximum current whereas the second method limits the peak current while it minimizes the power losses. Consequently, a higher acceleration of the translator can be achieved and/or less powerful (cheaper) current amplifiers can be utilized and/or fewer commutation errors arise.

224 Summary

Only a long-stroke translational movement of the moving magnet planar actuators has been considered in the past. The possibility of a completely propelled and controlled rotation about the vertical axis instead of just stabilizing it for bearing has been analyzed in this thesis from a control point of view. Enhancing the planar actuator with a long-range rotation will increase its utility value and opens new application areas. Based on this investigation, a novel coil array with a triangular grid of rounded coils has been proposed for better controllability in any orientation of the PA. In addition, other coil and magnet topologies have been studied from a control point of view for their suitability for full rotation.

The influence of different kinds of error-causes on the commutation precision has been studied. From this investigation, it has been found that the offsets of the measurement system have the highest influence on the precision of the commutation. Investigation of the convergence of the procedure for estimation and elimination of these offsets has been performed. Although it was not proven that the procedure could be applied on the whole workspace of the PA, the convergence has been shown at least for all the investigated points. From this investigation, convergence for any position in the workspace of the PA is expected. It was found that it is possible to use the procedure also with different topologies and with different commutations.

A novel wireless link has been developed for the real-time control of a fast motion system. The wireless link communicates via infrared-light transceivers and the link has a delay and a packet-loss ratio almost indistinguishable from the wired connection for the bandwidth of the system up to several kilohertz.

The clipped l2-norm based commutation method has been successfully tested on the experimental setup after improving the measurement system, the contactless energy transfer and the wireless communication. With a new, interferometer sensor system, a well-controlled PA with two long-stroke DOF’s has become available. Improved contactless energy transfer does not cause increased electromagnetic interference during switching between the primary coils any more and the wireless connection using the infrared link provides a reliable and sufficiently fast communication channel between the manipulator and the fixed world.

Several control approaches have been tested on the experimental setup. Both, the classical PID control, Sliding-mode control and Iterative learning control have been implemented. Each controller brought better performance than the previous one. Also, a fourth-order trajectory and enhanced feedforward control helped to improve performance.

The tracking errors, in comparison to the initial situation, were reduced by a factor 10 (and even more than by a factor 50 with deactivated contactless energy transfer) while the velocity and acceleration of the system were a factor 4 and 14, respectively, higher.

225

Samenvatting

Het regelen van een vrij bewegende planaire actuator met manipulator

Er is een studie gemaakt van een bestaande magnetisch geleviteerde planaire (in een vlak bewegende) actuator met manipulator die heeft geresulteerd in een verbeterd geregeld systeem. Het bestudeerde prototype bevat een magnetisch geleviteerde planaire actuator, met bewegende magneten en zes mechanische vrijheidsgraden, waarop een twee vrijheidsgraden manipulator is gemonteerd. In het systeem worden drie verschillende contactloze technologieën verenigd: contactloze lagering en voortstuwing van de translator van de planaire actuator, draadloze energievoorziening naar de manipulator en draadloze communicatie en regeling van de manipulator.

De planaire actuator (PA) bestaat uit een translator met permanente magneten met een quasi-Halbach magnetisatie dat wordt geleviteerd en geregeld in alle zes de vrijheidsgraden boven een spoelenpatroon dat zich op de vaste wereld bevindt. De translator van de PA kan met een lange-slag worden voortbewogen in het horizontale vlak, dus met twee vrijheidsgraden, terwijl de overige vier mechanische vrijheidsgraden worden gestabiliseerd om zo een stijve actieve magnetische lagering te realiseren. Elke actieve spoel draagt bij aan de productie van kracht en koppel op de magneet structuur. Aangezien het aantal actieve spoelen veel groter is dan het aantal mechanische vrijheidsgraden, kan de gewenste krachtproductie worden gedistribueerd over een groot aantal spoelen. Het gevolg is dat een commutatie algoritme moet worden gebruikt om een inverse afbeelding te creëren van de krachten en koppels die geproduceerd worden door de actieve spoelen als functie van de spoelstromen en de positie en oriëntatie van de translator. Eerdere onderzoeksresultaten hebben geleid tot een methode voor het lineariseren en ontkoppelen van de krachten en koppels. Deze methode garandeert minimale energie dissipatie. Er kunnen echter geen beperkingen worden opgelegd aan de maximale stroom. Het onderzoek beschreven in dit proefschrift heeft geleid tot de volgende twee nieuwe normgebaseerde commutatie methoden: een l∞-norm en een afgekapte l2-norm commutatie. Beide methodes

226 Samenvatting

zijn in staat om grenzen te stellen aan de maximale spoelstromen om zo verzadiging van de stroomversterkers tegen te gaan. De eerste methode richt zich op het minimaliseren van de maximale stroom, terwijl de tweede methode naast het minimaliseren van de vermogensverliezen ook de piekstromen begrenst. Het gevolg is dat een hogere acceleratie van de translator kan worden gerealiseerd en/of een minder krachtige (goedkopere) stroomversterker kan worden toegepast en/of een kleinere commutatie fout word gemaakt.

In het verleden werd er binnen het onderzoek naar planaire actuatoren alleen naar lange-slag translaties gekeken. In dit proefschrift is, vanuit een regeltechnisch standpunt, een analyse gemaakt van de mogelijkheid een volledig geactueerde en geregelde rotatie rond de verticale as te realiseren. Het uitbreiden van de PA met een lange-slag rotatie vergroot de gebruikswaarde en opent de weg naar nieuwe toepassingsgebieden. Dit onderzoek heeft geleid tot een voorstel voor een nieuwe spoeltopologie met een driehoekige plaatsingsstructuur van ronde spoelen waarbij de regelbaarheid is toegenomen over de volledige oriëntatie van de PA. Daarnaast zijn er, vanuit een regeltechnisch standpunt, ook alternatieve spoelen magneettopologieën onderzocht op hun geschiktheid voor volledige rotatie.

De invloed van verschillende foutoorzaken op de commutatienauwkeurigheid is bestudeerd. Uit deze studie is naar voren gekomen dat de offset van het meetsysteem de grootste invloed heeft. Verder is er onderzoek naar de convergentie van de procedure voor het schatten en elimineren van deze offsets uitgevoerd. Hoewel het niet is bewezen dat de procedure kan worden toegepast in het volledige werkgebied van de PA, is het convergentiebewijs geleverd voor alle bestudeerde werkpunten. De verwachting is dat alle posities in het werkgebied van de PA zullen convergeren. Het is mogelijk om deze procedure ook toe te passen op andere topologieën en commutatiemethoden.

Er is een nieuwe draadloze verbinding ontwikkeld voor het regelen van een snel bewegingssysteem. De draadloze verbinding communiceert via infrarood licht zender/ontvangers en heeft een vertraging en pakketverlies die bijna niet zijn te onderscheiden van de bedrade verbinding voor een systeembandbreedte tot enkele kilohertzen.

De afgekapte l2-norm commutatie methode is succesvol getest op de experimentele opstelling na het verbeteren van het meetsysteem, de contactloze energie overdracht en de draadloze communicatie. Door het toevoegen van een laser interferometer systeem is een goed geregelde PA met twee lange-slag bewegingsvrijheden beschikbaar gekomen. De verbeterde contactloze energie uitwisseling veroorzaakt geen elektro-magnetische interferentie meer tijdens het schakelen tussen primaire spoelen en de infrarood verbinding zorgt voor een betrouwbare en voldoende snelle communicatie tussen de manipulator en de vaste wereld.

Verschillende regelstrategieën zijn getest op de experimentele opstelling. Zowel de klassieke PID regelstrategie, de “sliding-mode” strategie als de iteratief lerende regelstrategie zijn geïmplementeerd, elk met toenemende prestaties. Verder heeft het invoeren van een vierde orde traject en verbeterde voorwaarts koppeling geleid tot een verbeterde prestatie.

De volgfouten in vergelijking tot de initiële situatie zijn gereduceerd met een factor 10, en bij het deactiveren van de contactloze energie uitwisseling met meer dan een factor 50, terwijl de snelheid en de acceleratie respectievelijk met een factor 4 en 14 zijn verhoogd.

227

Acknowledgements

The research which led to this thesis was performed at the Control Systems group at the Eindhoven University of Technology. It was part of an IOP-EMVT program funded by SenterNovem, an agency of the Dutch Ministry of Economic Affairs. I would like first to express my gratitude for the funding received through SenterNovem.

The work described in this thesis was accomplished with the kind support of a number of people. Here I would like to thank to everyone who, with good advice and a kind word, made my life easier during these four years.

My thanks go to my supervisors Ad Damen, Paul van den Bosch and Elena Lomonova for transferring their knowledge and wisdom to me. Ad, thank you for the endless discussions we had and for your, often completely different, point of view, which helped me to see the things from another perspective. Paul, thank you especially for giving me this opportunity to work on this research topic. I am also thankful to you for believing in me the whole time. Your moral support was crucial. Elena, without your very constructive remarks and criticism, this thesis would be of much less value. Thank you for bringing order to the chaos.

I would like to thank the members of my reading committee Jan van Eijk, John Compter and Okko Bosgra, whose constructive remarks and suggestions helped me a lot in the final steps of writing this thesis.

Building and rebuilding the experimental setups would not have been possible without technical support of Will Hendrix and Marijn Uyt de Willigen. You were always friendly and helpful anytime I needed you. I would also like to thank people from GTD (Gemeenschappelijke Technische Dienst) for building the experimental systems. Moreover, I thank Andre Veltman for helping us with the electronics.

Next, I would like to thank Nelis and Helm for their scientific support and many friendly advises.

228 Acknowledgements

I also thank all my old and new colleagues for providing such a nice atmosphere in the group: Ana, Andelko, Andrej, Can, Carmen, Chenyang, Femke, Jaron, Jasper, Jochem, John, Joris, Leyla, Mark, Maarten, Mircea, Mohammed, Patrick, Ralph, Rob, Satya, Siep, Thomas, Veaceslav. I thank Mircea, Satya, Chenyang and Andelko for being my good international friends. Satya, I will never forget your great Indian curry. Andrej, Patrick, Jaron, Femke, Leyla, Jochem, you never said no when I needed your help.

Finally, I would like to thank my family for their understanding and support, my mother, father, my sisters and their families for being there for me, and especially Klara, my girlfriend and best friend on my way through life, who was always there. I could not have done it without you!

Michal

229

Curriculum Vitae

Michal Gajdušek was born in Frýdek-Místek, Czech Republic, on the 5th of September in 1979. After finishing his secondary education at Gymnasium Petra Bezruče in Frýdek-Místek, he started his study at the faculty of Electrical Engineering at Brno University of Technology, Czech Republic in 1998. He did his master project “Position location of micro-robot MIROSOT by a CCD camera” in the group of Control and Instrumentation and obtained his M.Sc. (cum laude) degree in electrical engineering in June 2004. He also received a prize awarded by the Dean of the faculty of Electrical Engineering at Brno University of Technology for excellent study and scientific results. From July 2004 to May 2006, he was employed as a research assistant in the same group. His tasks were path planning and trajectory tracking of a differentially-driven mobile robot with a robot soccer application. From September 2005 to May 2006, he visited the Control Systems group of the Electrical Engineering department at the Eindhoven University of Technology. From June 2006 to May 2010, he pursued his Ph.D. degree in the same group at the Eindhoven University of Technology. The topic of his Ph.D. research was “Controlling a Contactless Planar Actuator with Manipulator” which resulted in this thesis. During his Ph.D. research he also received the course certificate of the Dutch Institute of Systems and Control (DISC) for completing the required number of postgraduate courses.

Documents

Controlling a contactless planar actuator with manipulatoroutperform the old one. Magnetically levitated planar actuators with moving magnets [62, 51, 114, 108, 26, 95, 56, 76, 13]