Control Moment Gyro actuator for small satellite ... · Control Moment Gyro Actuator for Small Satellite Applications R. Berner Department of Electrical & Electronic Engineering University

Control Moment Gyro Actuator for Small SatelliteApplications

by

Reimer Berner

Thesis presented at the University of Stellenboschin partial fulfilment of the requirements for the

degree of

Master of Science in Electrical & Electronic Engineering

Department of Electrical & Electronic EngineeringUniversity of Stellenbosch

Private Bag X1, 7602 Matieland, South Africa

Study leader: Prof W.H. Steyn

April 2005

Copyright © 2005 University of StellenboschAll rights reserved.

Declaration

I, the undersigned, hereby declare that the work contained in this thesis is my own originalwork and that I have not previously in its entirety or in part submitted it at any universityfor a degree.

Signature: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .R. Berner

Date: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

Abstract

Control Moment Gyro Actuator for Small Satellite Applications

R. Berner

Department of Electrical & Electronic EngineeringUniversity of Stellenbosch

Private Bag X1, 7602 Matieland, South Africa

Thesis: M Sc Eng (E & E)

April 2005

The aim of the thesis is to design a Control Moment Gyro (CMG) actuator which can beused in small satellite applications. The hardware and software of the CMG has to bedesigned according to specifications given. A satellite fitted with these CMGs has to beable to do a 30 degree rotation within 10 seconds.

A mathematical model of a satellite fitted with six CMGs was designed for simulationpurposes. This model was then extended to include a 3-axis control algorithm which con-trol the angular momentum vectors of the CMGs. An imaging sequence, which describesthe attitude and angular rate of the satellite at any point in time, was also implementedinto the design to produce a smooth attitude function during pointing maneuvers. Thisimaging sequence is used as input to the 3-axis control algorithm to ensure high precisionpointing of the satellite.

The CMG was tested on an air bearing table and the results of the tests were comparedto the mathematical simulations. The results of these tests were as expected and thefunctionality of the CMG was verified.

iii

Uittreksel

Control Moment Gyro Actuator for Small Satellite Applications

R. Berner

Departement Elektriese & Elektroniese IngenieursweseUniversiteit van Stellenbosch

Privaatsak X1, 7602 Matieland, Suid Afrika

Tesis: M Sc Ing (E & E)

April 2005

Die doel van die tesis is om ’n Beheer Moment Giro (BMG) aktueerder te ontwerp wat opklein satelliete gebruik kan word. Die hardeware en sagteware van die BMG is ontwerpvolgens gegewe spesifikasies. ’n Satelliet wat met hierdie BMGs toegerus is, moet ’n 30grade rotasie binne 10 sekondes afhandel.

’n Wiskundige model van ’n satelliet met ses BMGs was ontwerp vir simulasie doeleindes.Hierdie model is uitgebrei om ’n 3-as beheer algoritme in te sluit wat die hoekmomen-tum vektore van die BMGs beheer. ’n Beeldafneem sekwensie wat die posisie en diehoeksnelheid van die satelliet op enige gegewe oomblik beskryf, is ook geïmplementeerin die ontwerp om sodoende ’n gladde rotasie funksie te verkry wanneer die satelliet inverskillende rigtings moet mik. Hierdie beeldafneem sekwensie dien as intree tot die 3-asbeheerder om sodoende die satelliet akkuraat te kan rig.

Die BMG is getoets op ’n luglaertafel en die resultate van die toetse is vergelyk met diewiskundige simulasies. Die resultate van hierdie toetse is soos verwag en die funksionaliteitvan die BMG is bevestig.

iv

Acknowledgements

I would like to thank the following for their help and assistance in the successful completionof this thesis:

• Thank You Almighty God for Your guiding hand in my life.

• Prof. W.H. Steyn for his guidance and support throughout this thesis.

• Xandri Farr for all his help, and the design and layout of the microcontroller pcb.

• Mnr. J. Treurnicht and Corne van Daalen for their help on the air bearing table.

• Mnr. J. Blom and the SMD group for the mechanical design of the CMG.

• Eckhardt Kuhn, who helped with all the practical tests.

• My parents for their support and my education.

• All my friends for encouragement during the writing of my thesis.

v

Contents

List of Figures x

List of Tables xiii

List of Acronyms and Abbreviations xiv

1 Introduction 1

1.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Overview of the Design Approach . . . . . . . . . . . . . . . . . . . . . . . 6

2 CMG and Satellite theory 7

2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Attitude parameterization . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Control Moment Gyros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Single Gimbal CMG Model . . . . . . . . . . . . . . . . . . . . . . 11

2.2.4 Mathematical Model of Satellite with SGCMGs . . . . . . . . . . . 13

2.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Control Moment Gyro Design 19

3.1 System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

vi

CONTENTS vii

3.2 Mechanical System Overview and Design . . . . . . . . . . . . . . . . . . . 20

3.3 Motor Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Brushless DC Motor . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.2 Stepper Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 BLDC Motor Power Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Stepper Motor Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.6 BLDC Motor Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6.1 Controller Component Values . . . . . . . . . . . . . . . . . . . . . 25

3.6.2 Current Loop Design . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.6.3 Digital Speed Loop Design . . . . . . . . . . . . . . . . . . . . . . . 27

3.6.4 Speed Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.7 Microcontroller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.8 Computer Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Satellite Attitude Control 35

4.1 Single-Axis Attitude Control . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Three-Axis Quaternion Feedback Control . . . . . . . . . . . . . . . . . . . 38

4.2.1 Quaternion Feedback Control Logic . . . . . . . . . . . . . . . . . . 38

4.2.2 Eigenaxis Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.3 Rotation Under Slew Rate Constraint and Control Input Saturation 41


4.3.1 Single-Axis Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.2 Three-Axis Quaternion Feedback Rotation . . . . . . . . . . . . . . 43

5 Control Demand for Imaging Sequence 46

5.1 Imaging Sequence Objective . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Detailed Slew Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3 Attitude Demand Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 49

CONTENTS viii

5.3.1 Slew 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3.2 Slew 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3.3 Slew 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3.4 Slew 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


6 Measurements and Results 60

6.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.1.1 Gimbal Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.1.2 Moment of Inertia Calculation . . . . . . . . . . . . . . . . . . . . . 61

6.1.3 Glass Surface Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.1.4 Constant Angular Rate Test . . . . . . . . . . . . . . . . . . . . . . 64

6.2 CMG Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2.1 Rest-to-Rest Slew . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2.2 Moving Demand Test . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7 Conclusions 68

7.1 Results Obtained . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.2 Additional Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Bibliography 70

A CMG Machine Drawing 73

B Schematics of Brushless DC Motor Electronics 75

C CMG Setup for Practical Tests 82

D CMG Interface Program 85

D.1 Stepper Motor Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

D.2 BLDCM Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

CONTENTS ix

D.3 Data Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

D.4 Data Display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

E Matlab Simulation Design and S-function Code 89

F RF Link 94

G Datasheets 98

G.1 Brushless DC Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

G.2 Stepper Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

G.3 Stepper Motor Gearhead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

G.4 Stepper Motor Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

List of Figures

1.1 Agile small satellites will increase the amount and quality of data collected 1

1.2 Pyramid configuration containing four SGCMGs . . . . . . . . . . . . . . . 3

1.3 Picture of a Control Moment Gyro . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 A 2-1-3 Euler Angle Rotation . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 The Euler Angles: Roll φ, Pitch θ and Yaw ψ . . . . . . . . . . . . . . . . 9

2.4 Torque, acceleration and angle diagrams for a small satellite executing a

30o maneuver in 10 seconds . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 A Single Gimbal CMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 SGCMG Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.7 Three Set SGCMG Configuration . . . . . . . . . . . . . . . . . . . . . . . 13

2.8 Simulation with constant CMG Torque . . . . . . . . . . . . . . . . . . . . 17

2.9 Simulation with constant Gimbal Rate . . . . . . . . . . . . . . . . . . . . 18

3.1 Block Diagram of CMG System . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Picture of the CMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Gyro Torque acting back to the system . . . . . . . . . . . . . . . . . . . . 23

3.4 BLDC Motor Power Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Analogue PI Controller Circuit . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Measured acceleration per second for 1000 units input . . . . . . . . . . . . 28

3.7 Discrete Closed-loop speed control . . . . . . . . . . . . . . . . . . . . . . . 29

x

LIST OF FIGURES xi

3.8 Program flow diagram of Microcontroller . . . . . . . . . . . . . . . . . . . 32

4.1 Simulation results of the PID Saturation Control Logic in one axis . . . . . 42

4.2 Simulation results of the Quaternion Feedback Control Logic in one axis . 43

4.3 Simulation results of the Quaternion Feedback Control Logic in one axis . 44

4.4 Simulation results of the Quaternion Feedback Control Logic in three axis . 45

4.5 Simulation results of the Quaternion Feedback Control Logic in three axis . 45

5.1 Image Sequence Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2 Quaternion Demand for Imaging Sequence . . . . . . . . . . . . . . . . . . 56

5.3 Body Rate Demand for Imaging Sequence . . . . . . . . . . . . . . . . . . 57

5.4 Simulation Result for Imaging Sequence - Torque Command . . . . . . . . 57

5.5 Simulation Result for Imaging Sequence - Euler Angles . . . . . . . . . . . 58

5.6 Simulation Result for Imaging Sequence - Gimbal Rate . . . . . . . . . . . 58

5.7 Simulation Result for Imaging Sequence - Euler Angle Error . . . . . . . . 59

6.1 Measuring the gimbal’s step size . . . . . . . . . . . . . . . . . . . . . . . . 60

6.2 Angular acceleration of Momentum Wheel and Cart (Gyro) . . . . . . . . 62

6.3 Diagram of the CMG setup . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.4 Measured results of ’Mirror’ Test . . . . . . . . . . . . . . . . . . . . . . . 63

6.5 Measurements for constant angular rate input . . . . . . . . . . . . . . . . 64

6.6 Measured results of the Rest-to-Rest Slew . . . . . . . . . . . . . . . . . . 66

6.7 Measured results of the Moving Demand . . . . . . . . . . . . . . . . . . . 67

A.1 Drawing of CMG stand with Gimbal . . . . . . . . . . . . . . . . . . . . . 74

C.1 Picture of aluminium frame with gas canister and nozzles . . . . . . . . . . 82

C.2 Picture of the table with glass surface . . . . . . . . . . . . . . . . . . . . . 83

C.3 Diagram of the CMG setup . . . . . . . . . . . . . . . . . . . . . . . . . . 84

C.4 Picture of the CMG setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

LIST OF FIGURES xii

D.1 Picture of the CMGprobe control panel . . . . . . . . . . . . . . . . . . . . 88

E.1 Picture of Matlab Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 90

F.1 TX2 Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

F.2 RX2 Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

F.3 Voltage Regulator Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

F.4 Picture of the RF Link Unit . . . . . . . . . . . . . . . . . . . . . . . . . . 97

List of Tables

1.1 Satellite Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Minimum and maximum values for the two simulations . . . . . . . . . . . 18

3.1 CMG Command Packet IDs . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Packet frame of data transmitted from microcontroller . . . . . . . . . . . 33

3.3 Packet frame of data transmitted from PC . . . . . . . . . . . . . . . . . . 34

5.1 Mission Sequence of Attitude Demand . . . . . . . . . . . . . . . . . . . . 48

6.1 Results from Moment of Inertia Tests . . . . . . . . . . . . . . . . . . . . . 61

xiii

List of Acronyms and Abbreviations

A Ampere

Acc Accelerate

ACS Attitude Control System

ADCS Attitude Determination And Control Systems

BLDCM Brushless Direct Current Motor

CAN Controller Area Network

cm Centimetre

CMG Control Moment Gyro

CMOS Complementary Metal-Oxide-Semiconductor

CCW Counter Clockwise

CW Clockwise

DC Direct Current

Deg Degrees

DGCMG Double Gimble Control Moment Gyro

EPROM Erasable Programmable Read Only Memory

HB High Byte

I/O Input/Output

IC Integrated Circuit

ID Identification

kbit Kilobit

kHz Kilohertz

LB Low Byte

m Milli

Max Maximum

Mbit Megabit

MHz Megahertz

Min Minimum

mm Millimetre

MOSFET Metal Oxide Semiconductor Field Effect Transistor

Nm Newton Metre

xiv

LIST OF ACRONYMS AND ABBREVIATIONS xv

PC Personal Computer

PCB PC Board

PID Proportional, Integral and Derivative

PWM Pulse Width Modulation

RF Radio Frequency

ROM Read Only Memory

RPM Revolutions Per Minute

RW Reaction Wheel

s Seconds

Sat Saturate

SGCMG Single Gimble Control Moment Gyro

Sgn Signum

SUNSAT Stellenbosch UNiversity SATellite

V Volt

VSCMG Variable Speed Control Moment Gyro

XOR eXclusive OR

Chapter 1

Introduction

Satellites are required to have more rapid rotational maneuverability and agility than

before. These satellites, known as agile satellites, need attitude control systems (ACS)

that can provide rapid multi-target pointing and tracking capabilities [2]. An agile satellite

is much more efficient and functional, and it’s return of data is substantially increased by

it’s agility.

Future satellite applications, such as missile-tracking, imaging and the tracking of ground

moving targets will, as a necessity, require the ability to do rapid rotational maneuvers.

For instance, the next-generation commercial Earth imaging satellites would rather move

the whole spacecraft body rapidly than to sweep only the imaging system from side to

side. This ensures improved stability and high-resolution images with better definition.

Earth

Satellite ground track

Imaging

target

Figure 1.1: Agile small satellites will increase the amount and quality of data collected

Rapid retargeting maneuvers however, are subjected to the physical limits of sensors,

satellite’s structural rigidity, actuators and mission constraints. For a satellite to be agile,

it requires fast slew maneuvers in the range of 1-10o/s. Designing a high performance

ACS is also constrained by the physical size of a satellite, especially small satellites.

1

Chapter 1 — Introduction 2

Unfortunately, current ACS actuators such as reaction wheels and momentum wheels, are

not able to provide this degree of agility efficiently, because of their small control torque

capability. Control Moment Gyros (CMG) on the other hand are ideal, since it has much

bigger control torque output for a small torque input. CMGs have been used on various

large spacecrafts before, but yet, to date, not on small satellites.

Spacecraft missions [10] in which CMGs have been used are Skylab, MIR, ISS, KH-11

and some USSR spacecrafts. The CMGs used on these spacecrafts are large in size,

mechanically complex and very expensive. CMGs were employed as primary actuators

because of their much higher control torque capability [21] of 100 to 3000 Nm maximum

torque, whereas large reaction wheels have a maximum torque of only 1.5 Nm.

Due to the large size and weight of these CMGs it was not possible to use them on smaller

satellites. This situation however, has changed recently since the development of lighter

and more compact CMGs. Medium sized satellites use the large torque capability of

these smaller CMGs to do faster slew maneuvers. CMGs are becoming more attractive

for smaller satellites as well, such as micro satellites and nano satellites due to this torque

amplification capability.

Name Mass Class

Large satellites > 1000 kg Medium to Large

Medium satellites 500 - 1000 kg Satellites

Mini satellites 100 - 500 kg

Micro satellites 10 - 100 kg Small Satellites

Nano satellites 1 - 10 kg

Pico satellites 0.1 - 1 kg

Table 1.1: Satellite Classification

The biggest inherent problem encountered with these CMGs are the appearance of sin-

gularities [8]. These singularities are the condition when no torque can be produced for

certain gimbal angle outputs. When a gimbal encounters a singularity, it would stop (lock)

and not be able to move in any direction since that would produce the wrong torque.

This led to the development of a steering logic which will avoid singularities by steering

the CMGs away from a point with zero torque potential. Many control laws have been

developed which avoid singularities and still remain within the hardware contraints of the

system. The most popular one is for a pyramid configuration containing four SGCMGs.

Such a system has an approximate spherical momentum envelope with almost the same

momentum capability in all three axis.


The pyramid mounting arrangement is shown in Figure 1.2. Four SGCMGs are con-

strained in such a way that the gimbal axes are orthogonal to the faces of the pyramid.

The faces of the pyramid are inclined with an angle of β from the horizontal to result in

gimbal axes angles of (90o−β) degrees from the horizontal. A nearly spherical momentum

envelope is achieved when all four CMGs have the same angular momentum about their

spin-axis and the skew angle is chosen as β = 54.73 degrees [23].

H1

H2

H3

H4

βδ1

δ2δ3

δ4

x

y

z

CMG #2

Gimbal Axis

CMG #1

Gimbal Axis

CMG #3

Gimbal Axis

CMG #4

Gimbal Axis

Figure 1.2: Pyramid configuration containing four SGCMGs

A CMG consists of a momentum wheel with large, constant angular momentum of which

the angular momentum vector can be rotated with respect to the satellite’s body [22]. The

momentum wheel is mounted on a gimbal or gimbals and can be pivoted by torquing the

gimbal. This results in a precessional, gyroscopic reaction torque orthogonal to both the

rotor spin and gimbal rotation axes. A more thorough explanation is given in Section 2.2.2.

CMGs consist out of two basic types: a single gimbal control moment gyro (SGCMG) and

a double gimbal control moment gyro (DGCMG). For a SGCMG, the momentum wheel

is constrained to rotate along a circle normal to the gimbal axis. The momentum wheel

of a DGCMG is suspended inside two gimbals and therefore the momentum vector can be

aimed along any direction on a sphere. The gimbal steering logic of a DGCMG can more

easily avoid singularities since it has an extra degree of freedom. However, a SGCMG is

a lot simpler from a hardware point of view, and it has a significant cost, power, weight

and reliability advantage over a DGCMG.

Another type of CMG is the variable speed control moment gyro (VSCMG) [24]. Where

a conventional CMG is constrained to a constant wheel spin rate, a VSCMG can vary it’s

wheel spin rate. This ensures an extra degree of freedom to the VSCMG, and enables it

to achieve additional objectives such as energy storage with attitude control.

CMGs increase the agility of a satellite which again increases the amount of earth and

space science data that can be collected while using the same resources. This directly


Figure 1.3: Picture of a Control Moment Gyro

increase the scientific and commercial value of these satellites. Since the time available

for data transmission is severely limited on a small satellite, the use of CMGs will allow off-

nadir pointing for substantial larger times and thus increase the time available to transmit

data. This will also allow more frequent transmitting of data to available ground stations.

Since the wheel speed of a CMG is constant instead of varying as with reaction wheels,

the wheel induced vibrations in a satellite will decrease and the pointing accuracy of the

satellite will increase.


1.1 Problem Definition

The aim of this thesis is to design a SGCMG with controller which can be used in small

satellite applications. The function of the CMG is to increase the agility of a small satellite

by enabling it to do rapid pointing maneuvers. It is very important for these maneuvers

to be very accurate, since the quality of images, for example, directly depends on it. The

following design specifications of the CMG system and satellite is given to ensure such

rapid pointing maneuvers:

• Satellite’s moment of inertia matrix = diagonal [20 20 15] kgm2

• Satellite should be able to do a 30o rotation in 10 seconds

• The maximum CMG torque = 0.25 Nm

• Maximum gimbal rate = 10o/second

• Gimbal minimum step size < 0.02 degrees

• Angular momentum of wheel = 1.6 Nms

A mathematical model of the CMG system and satellite has to be designed for simulation

purposes. This model must then be extended to include a 3-axis control algorithm which

controls the angular momentum vectors of the CMGs to produce the correct torque out-

put. A 30o step input to the controller has to be able to generate a 30o rotation of the

satellite within 10 seconds.

An imaging sequence, which describes the attitude and rate of the satellite at any point

in time, should be implemented into the design to produce a smooth attitude function

during pointing maneuvers. This imaging sequence should be used as input to the 3-axis

control algorithm to ensure high precision pointing of the satellite.

The results from these simulations should be compared to tests done on the CMG to

verify the functionality of the hardware design.


1.2 Overview of the Design Approach

The presentation of the thesis document is given below:

Chapter 2: A brief explanation on coordinate systems and attitude parameterization

are given, followed by CMG fundamentals. Thereafter a mathematical model is derived

for the CMG system and satellite. The functionality of the mathematical model is then

verified by simulations.

Chapter 3: The design and development of the hardware and software of the CMG

system is explained. A brief explanation is given on the brushless DC motor and stepper

motor chosen for the CMG, as well as the controllers for the motors. The functions of the

microcontroller and computer interface are also described.

Chapter 4: A three-axis quaternion feedback control logic under slew rate constraint is

derived for the satellite’s attitude control. It is followed by simulations of the mathemat-

ical model with the attitude control law.

Chapter 5: The objective of this chapter is to design a control mode for the imaging phase

making use of the Moving Demand approach. This approach implements consecutive slew

maneuvers via a time varying attitude and rate demand. The attitude demand is used as

input to the control logic of Chapter 4.

Chapter 6: The CMG module designed in Chapter 3 is tested on an air bearing table.

The results obtained from these tests are compared to the theory and simulations from

Chapters 2, 4 and 5 to verify the functionality of the CMG module.

Chapter 7: The final conclusions on the CMG actuator and recommendations for future

work are discussed.

Chapter 2

CMG and Satellite theory

2.1 Background

2.1.1 Coordinate Systems

In spaceflight analysis it is important to know the position and motion of the satellite.

In order to do this, we must first select the correct coordinate system for the problem.

Any coordinate system will actually do, but choosing the correct one will increase insight

into the problem. To define a coordinate system for space applications, two characteristics

have to be specified firstly: whith respect to what the coordinate system is fixed to and the

location of the coordinate system’s center. This thesis uses the following three coordinate

systems to describe the attitude of a satellite: Body Frame, Orbit Reference Frame and

Inertial Reference Frame.

Body Frame

The origin of the body frame is located at the satellite’s center of mass and it is fixed with

respect to the satellite’s body. This frame is used to determine the satellite’s orientation

with respect to another reference frame. The coordinates XB, YB and ZB is shown in

Figure 2.1.

Orbit Reference Frame

The origin of the orbit reference frame, XO, YO, ZO, is located at the satellite’s center of

mass and it is fixed with respect to the orbit. The ZO axis is nadir pointing (i.e. pointing

towards the Earth’s center), the YO axis is in the orbit anti-normal direction and the XO

7

Chapter 2 — CMG and Satellite theory 8

axis completes the orthogonal set. In a circular orbit, the XO axis would point in the

direction of the velocity vector. The orbit reference frame is mainly used for attitude

maneuvers. See Figure 2.1.

Inertial Reference Frame

The inertial reference frame XI , YI , ZI has it’s origin at the center of the Earth and

it is fixed with respect to inertial space. The ZI axis is in the same direction as the

Earth’s geometric north pole, the XI axis is in the Vernal equinox direction and the YI

axis completes the orthogonal set. The inertial reference frame is normally used for orbit

analysis.

ZB

YB

XB

OX

OY

OZ

Vernal Equinox

Direction

ZI

YI

X I

Earth

Figure 2.1: Coordinate Systems

2.1.2 Attitude parameterization

A satellite’s attitude can be represented by a direction cosine matrix. There exist a few

alternative representations of three-axis attitude, but the two mostly used for present-

ing a satellite’s orientation are the Euler angle representation and the Euler symmetric

parameterization (quaternions). The Euler angle representation is often used in a user’s

interface during attitude computation because of it’s clear physical interpretation. It is

also useful for analysis, especially for finding closed-form solutions to the equations of

motion. It’s parameters consist of three rotation angles known as the roll angle φ, pitch

angle θ and yaw angle ψ.

There are 12 distinct representations for the Euler angle rotations (φ, θ, ψ). For instance, a

3-2-1 Euler angle rotation means that the first rotation is about the z-axis, second rotation

about the y-axis and the third rotation about the x-axis. The 2-1-3 sequence is used in


Yo

Zo'

Xo'Xo

Yo

ZoZo'

Xo'

θ

θ

Yo'

Zo''

φ

φ

Xo'

Yo'

Zo''

Yo''

Xo''

ψ

ψ

2: Y Axis Rotationo 1: X Axis Rotationo' 3: Z Axis Rotationo

''

Figure 2.2: A 2-1-3 Euler Angle Rotation

this thesis. Thus, the first rotation θ is about the initial YO axis, the second rotation φ is

around the XO’ axis and the third rotation ψ is around the ZO” axis. Figure 2.2 shows

the rotation of the axis. The 2-1-3 direction cosine matrix A is given as:

A =

c(ψ)c(θ) + s(ψ)s(φ)s(θ) s(ψ)c(φ) −c(ψ)s(θ) + s(ψ)s(φ)c(θ)

−s(ψ)c(θ) + c(ψ)s(φ)s(θ) c(ψ)c(φ) s(ψ)s(θ) + c(ψ)s(φ)c(θ)

c(φ)s(θ) −s(φ) c(φ)c(θ)

(2.1)

where c = cosine function and s = sine function. The attitude matrix A transforms an

Earth pointing satellite which is orbit referenced to satellite body coordinates. The Euler

angles φ, θ and ψ can be seen in Figure 2.3.

Roll

Pitch

Yaw

Roll

Pitch

Yaw

Flight Direction

EarthOrbit

Figure 2.3: The Euler Angles: Roll φ, Pitch θ and Yaw ψ

Euler angle representation is best when used with small angles since it encounters singu-


larities at certain large rotations. It is also computationally intensive to use in numerical

computation because of the trigonometric functions. For numerical computation Euler

symmetric parameterization (quaternions) is used, since it uses no trigonometric functions

and has no singularities. For these reasons Euler symmetric parameters are commonly

used for onboard attitude calculations. The attitude matrix A, expressed in quaternion

form, is given as:

A =

q21 − q2

2 − q23 + q2

4 2(q1q2 + q3q4) 2(q1q3 − q2q4)

2(q1q2 − q3q4) −q21 + q2

2 − q23 + q2

4 2(q2q3 + q1q4)

2(q1q3 + q2q4) 2(q2q3 − q1q4) −q21 − q2

2 + q23 + q2

4

(2.2)

2.2 Control Moment Gyros

2.2.1 Motivation

A satellite is required to do a 30 degree rotation in 10 seconds as specified in Section 1.1.

In order to do the 30o rotation within 10 seconds, an acceleration phase of 5 seconds is

needed, followed by a deceleration phase of 5 seconds as shown in Figure 2.4.

T

t(s)

105

.

t(s)5 10

6 /so

t(s)5 10

15

30

o

o

Figure 2.4: Torque, acceleration and angle diagrams for a small satellite executing a

30o maneuver in 10 seconds

The relationship between the torque T , the moment of inertia of the satellite Is and the

angular acceleration θ is given by Newton’s first law of rotation (around one axis):

T = Is θ (2.3)

Integrating this equation twice reveals the following two equations:

θ =T

Ist (2.4)


θ = 0.5T

Ist2 (2.5)

With θ = 0.2618 radians (15o), Is = 20 kgm2 and t = 5 seconds, the torque required

for this maneuver is calculated using (2.5) as T = 0.4189 Nm. From (2.4) the maximum

angular velocity of the satellite is calculated as θ = 0.1047 rad/s (6o/s).

2.2.2 Fundamentals

To be able to derive a mathematical model for a CMG, it is neccesary to understand the

fundamentals of how a CMG works. As can be seen in Figure 2.5, the CMG consist of a

spinning wheel with an angular velocity of ω rotating about the x-axis. If the spinning

wheel has a moment of inertia I, then the angular momentum produced would be h = Iω

along the x axis. Rotating the spinning wheel about the gimbal axis z at a rate of δ, would

then result in a large torque output about an axis normal to the angular momentum axis

and the gimbal axis.

z

y

x

.

h

Tin

-Tout

Figure 2.5: A Single Gimbal CMG

The torque output about the y-axis in Figure 2.5, would thus be the cosine of the gimbal

angle δ times the total output torque of the CMG. The output torque of the CMG in

vector format is then given as [6]:

Tout = h × δ (2.6)

2.2.3 Single Gimbal CMG Model

When a SGCMG is gimballed, the output torque vector changes continuously because the

direction of the angular momentum vector does not stay along the same body axis. For this


reason it is not possible to achieve a torque about only one body axis with one SGCMG.

To overcome this problem, two SGCMGs can be used together in the configuration shown

in Figure 2.6. If the wheel speeds ω1 and ω2 are in the directions shown, then the angular

momentums, h1 and h2, will be along the positive and negative y-axis respectively.

x

z

y

12

12

Figure 2.6: SGCMG Configuration

By gimballing both momentum wheels such that δ1 = −δ2 while ω1 = −ω2, the resulting

output torque will be about the x-axis only. The output torque of the two SGCMGs about

the other two axis will cancel each other since they have equal magnitudes in opposite

directions. With this configuration it is possible to increase or decrease the wheel speeds

without causing a resulting torque. As long as the angular momentum of the two wheels

are in opposite directions, the change of wheel speed, while ω1 = −ω2, will have no netto

effect on the satellite. This enables the possibility of increasing the wheel speeds for bigger

torque capabilities before large maneuvers are done.

A mathematical model of the SGCMG system is needed for analysis and to simulate

the influence it has on the satellite. The total angular momentum of the SGCMG setup

is needed for this mathematical model. Through inspection of Figure 2.6, the angular

momentum h is found as:

h =

−h1 sin δ1 + h2 sin δ2

h1 cos δ1 − h2 cos δ2

0

(2.7)

The derivative of the angular momentum is then given as

h =

−h1δ1 cos δ1 − h1 sin δ1 + h2δ2 cos δ2 + h2 sin δ2

h1 cos δ1 − h1δ1 sin δ1 − h2 cos δ2 + h2δ2 sin δ2

0

(2.8)


with h1 = Iw ω1 and h2 = Iw ω2 with Iw the moment of inertia of the momentum wheel.

A satellite has to be controlled about all three axes. The SGCMG system shown in

Figure 2.6 is then extended to a model with three of these systems, one for every axis.

The three sets of the SGCMG configuration are shown in Figure 2.7.

x

z

y

1x2x

2x 1x

x

z

y

1y

1y

2y

2yx

z

y

1z

2z

1z

2z

Figure 2.7: Three Set SGCMG Configuration

The total angular momentum of all three sets together is:

hT =

−hx1 sin δx1 + hx2 sin δx2 + hz1 cos δz1 − hz2 cos δz2

hx1 cos δx1 − hx2 cos δx2 − hy1 sin δy1 + hy2 sin δy2

hy1 cos δy1 − hy2 cos δy2 − hz1 sin δz1 + hz2 sin δz2

(2.9)

2.2.4 Mathematical Model of Satellite with SGCMGs

It is necessary to develop a simple mathematical model for the steering and attitude

control of satellites with CMGs. This section develops the dynamic model of the satellite

with three SGCMG sets for agile steering. The rotational equation of motion of a satellite

equipped with CMGs is given by [9, 23]:

HIs + ωI

s × HIs = Text (2.10)

with HIs the angular momentum vector of the system expressed in the inertial reference

frame and ωIs = [ωxi ωyi ωzi]

T the body angular velocity vector of the satellite also

inertially referenced. The external torque vector Text expresses the solar pressure, gravity-

gradient and aerodynamic torques in the same body frame. The angular momentum vector

of the system HIs consists of the total angular momentum of the SGCMGs configuration

(2.9) and of the satellite body angular momentum. The angular momentum of the system

can thus be expressed as:

HIs = Is ωI

s + hT (2.11)


where Is is the total inertia matrix of the whole satellite and hT is the total momentum

vector of the SGCMGs configuration (2.9). From (2.10) and (2.11) we obtain:

IsωIs + hT + ωI

s × (Is ωIs + hT ) = Text (2.12)

This equation can be rewritten in terms of the SGCMG control torque vector, u =

(ux, uy, uz), which gives:

IsωIs + ωI

s × Is ωIs = u + Text (2.13)

with

u = −hT − ωIs × hT (2.14)

The required control torque vector u is the input to the steering logic of the satellite and

can be assumed to be known. In Chapter 4 a non-linear feedback control law is derived

and implemented to compute the torque needed to maintain the correct attitude. Since

the control torque vector is known, Equation 2.14 can be written in terms of the desired

SGCMG angular momentum rate as:

hT = −u − ωIs × hT (2.15)

The angular momentum vector h of the SGCMG configuration is a function of the gimbal

angles δ = (δx1, δx2, δy1, δy2, δz1, δz2) and can be written as:

hT = hT (δ) (2.16)

Instead of finding an inverse for this equation to solve for h, the differential relationship

between the gimbal angles and the angular momentum vector of the SGCMG configuration

is used. The reason for choosing this method is because it offers less singularities for this

specific SGCMG configuration. The time derivative of h is obtained as:

hT = B(δ)δ (2.17)

where

B =∂hT

∂δ(2.18)

=

−hx1 cos δx1 hx2 cos δx2 0 0 −hz1 sin δz1 hz2 sin δz2

−hx1 sin δx1 hx2 sin δx2 −hy1 cos δy1 hy2 cos δy2 0 0

0 0 −hy1 sin δy1 −hy2 sin δy2 −hz1 cos δz1 hz2 cos δz2


If we assume the speeds of the reaction wheels to stay constant during gimbal maneuvers

and that the gimbal rate of the two gimbals per set are of the same magnitude but opposite

sign (ωa1 = ωa2 = 0, ωa1 = ωa2, δa1 = −δa2 = δa, δa1 = −δa2 = δa, with a = x, y, z),

then the momentum rate is obtained from (2.17) and (2.18) as follows:

hT =

hx

hy

hz

=

−2hxδx cos δx

−2hyδy cos δy

−2hz δz cos δz

(2.19)

From this we achieve an equation for the gimbal rate vector, since the value of the mo-

mentum rate vector is already known from (2.15). The equation for the gimbal rate

follows:

δ =

δx

δy

δz

=

hx/[−2hx cos δx]

hy/[−2hy cos δy]

hz/[−2hz cos δz]

(2.20)

The gimbal angles are achieved by integrating the gimbal rate δ over time. These new

values are then used to determine new values for the angular momentum vector.

The angular momentum and momentum rates are fed into Euler’s equation. This basic

equation of attitude dynamics relates the time derivative of the angular momentum vector,

dH/dt, to the total external applied torque, Text, and is given as:

IsωIs = Text − ωI

s × [Is ωIs + hT ] − hT (2.21)

This equation is the same as Equation 2.12, but written in a different way. For the rest of

the thesis we will assume that no external torque disturbance is applied to the satellite,

which results in the external applied torque vector, Text, being equal to zero.

A kinematic equation is needed which relates the angular velocity of the satellite ωIs to the

attitude rate of change. For this a set of first-order differential equations are used which

specifies the time evolution of the attitude parameters. This is known as the kinematic

equations of motion and is given by:

q =1

2Ω q =

1

2Λ(q) ωo

s (2.22)

where Ω is the skew-symmetric matrix

Ω =

0 ωzo −ωyo ωxo

−ωzo 0 ωxo ωyo

ωyo −ωxo 0 ωzo

−ωxo −ωyo −ωzo 0

(2.23)


and Λ the rotation matrix

Λ =

q4 −q3 q2

q3 q4 −q1−q2 q1 q4

−q1 −q2 −q3

(2.24)

The satellite angular velocity vector ωos is orbit referenced and determined as follows:

ωos = ωI

s − Aω0 (2.25)

with A the attitude matrix (Equation 2.2) in terms of Euler’s symmetric parameters and

which has no singularities. Assuming the satellite has a near circular orbit with a constant

orbital angular rate ω0, then we have a constant orbital rate vector:

ω0 =

0

−ω0

0

(2.26)

The angular velocity vector ωos can now be determined from Equation 2.25 and used

in Equation 2.22 to define the derivative of the quaternion vector q. This vector can

then be integrated to propagate the quaternion vector q. Comparing Equation 2.1 and

Equation (2.2 results in the equations:

θ = arctan4

(2(q1q3 + q2q4)

−q21 − q2

2 + q23 + q2

4

)(2.27)

φ = arcsin2 (−2(q2q3 − q1q4)) (2.28)

ψ = arctan4

(2(q1q2 + q3q4)

−q21 + q2

2 − q23 + q2

4

)(2.29)

which is used to convert the quaternions into the Euler angles θ, φ and ψ.

2.3 Simulation Results

The simulation results of a small satellite performing a 30 degree rotation maneuver in 10

seconds is depicted in Figure 2.8. The mathematical model derived in Section 2.2.4 was

used with the design specifications of Section 1.1 for the simulation. A constant x-axis


torque, derived in Section 2.2, was used as an input to the model to generate a 30 degree

roll maneuver. A momentum wheel speed of 1000 rad/s was used .

The gimbal angle excursions, shown in Figure 2.8, does not exceed 45 degrees, while the

gimbal rates reach a maximum rate of 11.2 o/s (0.1955 rad/s). The total CMG torque

is 0.4188 Nm for the first 5 seconds (acceleration phase) and -0.4188 Nm for the next

5 seconds (deceleration phase). From Section 2.2 the maximum angular velocity of the

satellite of 6 o/s can be seen from Figure 2.8.

0 5 10 150

10

20

30

40Roll Angle

degr

ees

0 5 10 15−50

0

50Gimbal Angles

degr

ees

0 5 10 15−20

−10

0

10

20Gimbal Rates

degr

ees/

s

0 5 10 15−0.5

0

0.5Input Torque

Nm

0 5 10 15−3

−2

−1

0

1Angular Momentum

time (s)

Nm

s

0 5 10 15−5

0

5

10Angular Rates

time (s)

degr

ees/

s

Figure 2.8: Simulation with constant CMG Torque

Since the gimbal rates exceed the maximum gimbal rate according to the design speci-

fications, another simulation is done, where the gimbal rates are kept constant and the

torque varies. In Figure 2.9 the gimbal rate has a value of 0.146 rad/s (8.365 o/s). The

maximum gimbal angle excursions, angular momentum and angular rate is slightly less

in the second simulation. The maximum and minimum torque as well as gimbal rate of

the two simulations are compared in Table 2.1.

From this it can be seen that a larger initial torque for a maneuver results in smaller

excursion angles and smaller maximum satellite angular velocities. It is thus better to do

slew maneuvers as fast as possible without exceeding the specified maximum slew rate.


Max Torque Min Torque Max Gimbal rate Min Gimbal rate

Simulation (Nm) (Nm) (deg/s) (deg/s)

Const. torque 0.4188 0.4188 11.121 8.0

Const. gimbal rate 0.438 0.326 8.365 8.365

Table 2.1: Minimum and maximum values for the two simulations

In Section 4.2.3 this knowledge is used to design a saturation control law for the satellite.

These two simulations give a better understanding of the mathematical model of the

satellite with a CMG configuration. They also assist in defining the design parameters

for the SGCMG and performance for the 3-set SGCMG configuration. This model is

expanded in Chapter 4 to a model with attitude feedback control logic.

0 5 10 150

10

20

30

40Roll Angle

degr

ees

0 5 10 15−50

0

50Gimbal Angles

degr

ees

0 5 10 15−10

−5

0

5

10Gimbal Rates

degr

ees/

s

0 5 10 15−0.5

0

0.5Input Torque

Nm

0 5 10 15

−2

−1

0

Angular Momentum

time (s)

Nm

s

0 5 10 15−2

0

2

4

6Angular Rates

time (s)

degr

ees/

s

Figure 2.9: Simulation with constant Gimbal Rate

Chapter 3

Control Moment Gyro Design

3.1 System Overview

The Control Moment Gyro (CMG) module uses a Brushless DC (BLDC) Motor with Hall

effect sensors and a Stepper Motor to spin and gimbal the momentum wheel respectively.

It also consists of a BLDC motor controller, a 3-phase drive stage, a stepper motor

controller, an optical encoder module and a microcontroller. A block diagram of the

CMG system is shown in Figure 3.1.

SpeedController

CurrentController

BLDC MotorController(UC2625)

3-PhaseElectronic

Drive Stage

BLDC Motorand

ReactionWheel

OpticalCodeWheel

Microcontroller

SpeedDemand

CurrentDemand

CurrentFeedback

Hall effect position sensors tocommutation circuit in UC2625

Optical encoder speed feedback

RS232Communication

StepperMotor

Controller(AD VL M1)

StepperMotor

Figure 3.1: Block Diagram of CMG System

The microcontroller (Cygnal - C8051F041) receives control data via RS232 from a PC,

19

Chapter 3 — Control Moment Gyro Design 20

implements it and then sends control commands to the UC2625 Brushless DC motor

controller and the AD-VL-M1 Stepper Motor controller. A speed control algorithm is

implemented in the microcontroller’s software which controls the BLDC motor via the

UC2625 controller. A Hewlett Packard Optical Encoder is used for high accuracy speed

feedback to the microcontroller.

The low voltage 2 phase stepper motor driver (AD-VL-M1) receives command signals

from the microcontroller such as clock pulse, direction and inhibit. It has it’s own full

bridge drivers to drive the stepper motor. The UC2625 controller has all the necessary

functions for driving a BLDC motor, and directly controls the 3-phase drive stage which

connects to the motor. A current control loop runs internal to the speed control loop to

ensure a smoother motor current.

3.2 Mechanical System Overview and Design

The CMG has to be as compact and light as possible to satisfy the constraints of a

small satellite. These aspects were taken into account while designing the CMG. The

most significant problem encountered, which also has a great influence on the size of the

CMG, is the backlash inherent to gears. Since zero backlash is needed to ensure the

gimbal has a rotational resolution less than 0.02 degrees, specially designed gears had to

be manufactured.

The CMG platform consists of a base with a free moving platform rotating on ball-

bearings. Attached to the base is an anti-backlash gear. This anti-backlash gear consists

of two seperate flushed gears of same size which are spring-loaded to generate a thrust in

opposite directions. This would cause the teeth of the anti-backlash gear to lock into the

driving gear and thus avoid the backlash. The driving gear is attached to the gearhead

of the stepper motor, which in turn is mounted on the free moving platform. When the

stepper motor is pulsed, the driving gear rotates around the outside of the fixed anti-

backlash gear and thus cause the platform to rotate. A mechanical drawing of the CMG

platform is shown in Appendix A.

A BLDC motor is mounted on the free moving platform. An optical encoder is attached

to the one end of BLDC motor for accurate speed measurement. On the other end, a

momentum wheel with a moment of inertia of 0.0015 kgm2 is attached to the shaft using

a shaft-lock. The prototype mechanical system is shown in Figure 3.2.

The base of the CMG is attached to a stand in such a way that the gimbal axis is parallel

to the ground. This is also used in the setup for the practical tests (Chapter 6). Since

this is only a prototype for testing, the weight of the CMG was not important. The size


and weight of this prototype CMG can easily be reduced by using lighter materials and

by removing or resizing certain parts such as the platform, stand and anti-backlash gear.

Stepper Motor

& Gearhead Anti-backlash Gear

Momentum

Wheel

Optical Encoder

Driving Gear

Stand

Rotating Platform

BLDC Motor

Figure 3.2: Picture of the CMG

3.3 Motor Descriptions

The Control Moment Gyro depends on two motors for operation. The one is a brushless

DC motor which is used to spin the momentum wheel and the other one is a stepper

motor used to gimbal the momentum wheel around an axis which is perpendicular to the

spin axis.


3.3.1 Brushless DC Motor

The momentum wheel is driven by a 3-phase (star connected), 2-pole Brushless DC motor

designed by Faulhaber (Part number: 3056 036B). The motor has electronic commutation

via Hall effect sensors placed 120 mechanical degrees apart for rotor position feedback.

Since the maximum angular wheel momentum per CMG was chosen as 1.6 Nms and the

moment of inertia of the wheel is only 0.0015 kgm2, a maximum wheel speed of 10187

rpm will be needed. The nominal voltage of the BLDC motor is 36 Volt, but only 12 Volt

is applied to the power stage. Thus the maximum speed of the BLDC motor is only a

third of the no-load speed of 8840 rpm. It will not be possible to achieve the specified

wheel momentum with the current wheel inertia and speed specifications, however it is

still good enough to illustrate the theory.

3.3.2 Stepper Motor

A maximum gimbal rate of 10 degrees/second is needed according to specifications, with a

single step smaller than 0.02 degrees. A Faulhaber 2-phase stepper motor (AM 1524 V6)

with anti-backlash gearhead (15/8 Series) was chosen for this application. The stepper

motor has a nominal voltage of 6 Volts and a step angle of 15 degrees.

Since the gimbal step size should be smaller than 0.02 degrees, the total gear reduction

ratio should be bigger than 150.02

= 750 : 1. A gearhead with a 262:1 ratio was chosen. An

additional anti-backlash gear system with a 3:1 ratio is connected to the output of the

gearhead to give a total gear reduction ratio of 786:1. The maximum step rate for the

stepper motor to achieve a gimbal rate of 10 degrees/second is given:

Total gear ratio

Motor step size× gimbal rate =

786

15× 10 = 524 steps/second (3.1)

Another important consideration when sizing the CMG is the effect of the ”gyro torque”

acting back to the gimbal system. This torque can have a great effect, especially at high

angular rates. This ”gyro torque” is generated when the satellite’s inertial body rate is

in the same direction as the torque output of the CMG. The rotation of the satellite has

the same effect as a gimbal has on the momentum wheel, but the output torque due to

the satellite’s rotation acts directly onto the gears of the gimbal. The ”gyro torque” is

calculated as follows:

TGyro = ωI−i × hi (3.2)

where ωI−i is the inertial body rate of the satellite in the same direction as the torque


output of CMGi. hi is the angular momentum vector of CMGi. The ”gyro torque” for a

simulation similar to the one from Section 4.3.1 is shown in Figure 3.3. From the figure,

0 5 10 15−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12Gyro Torque

Nm

time (s)

Figure 3.3: Gyro Torque acting back to the system

the maximum torque acting back to the system is 0.112 Nm.

Since the maximum allowable continuous torque output of the gearhead is 100 mNm,

and the additional 3:1 ratio gears system has an efficiency of at least 80%, the maximum

torque the gimbal can deliver is approximately 100 × 3/0.8 = 375 mNm. The ”gyro

torque” calculated is well within the limits of the gimbal’s maximum torque and would

thus not damage the gears of the gearhead.

3.4 BLDC Motor Power Stage

The BLDC Motor power stage is in full-bridge configuration which enables 4 quadrant

motor driving. Thus the motor can accelerate and brake in both directions. N-channel

and P-channel HEXFET Power MOSFETs are used as switching devices for the power

stage. Dual high speed 1.5 A MOSFET Drivers are used for the high-side and low-side

drivers. The current sense resistors, RCS, are used to feedback the measured current in the

motor phases for the current control loop. These resistors have a value of 0.27Ω, which

are calculated in Section 3.6.1 to limit the maximum motor current. The power stage is

shown in Figure 3.4.

High frequency, fast recovery Schottky rectifiers (30BQ100) are used for the free-wheeling

diodes in the power stage. The function of these diodes are to protect the MOSFETs

when the motor is running freely acting like a generator. The phases A, B and C are

connected directly to the BLDC motor.


Phase APhase BPhase C

VCC

R CS R CS

PUCPUBPUA

PDA PDB PDC

Figure 3.4: BLDC Motor Power Stage

3.5 Stepper Motor Control

The AD-VL-M1 stepper motor driver [3] was used to drive a small 2-phase stepper motor

in full step or half step mode. These drivers are in voltage mode and can operate with

low power supply such as 3 V to 12 V. Full step mode is chosen for this application,

which results in steps of 15 degrees. Three inputs are used, namely: CW (CCW), clock

pulse and inhibit. The CW (CCW) input controls the direction of rotation of the stepper

motor. Each time a pulse is generated on the clock pulse input, the stepper motor will

take one step. The inhibit input switches off all currents to the motor, saving energy

while the motor is not stepping.

The stepper motor controller consists mainly of two sections, the power stage (HIP4020)

and the translator (COP8SAB720). The translator is an 8-Bit CMOS ROM based, one-

time programmable (OTP) microcontroller with 2k EPROM memory. Inputs from the

Cygnal microcontroller to the stepper motor controller is processed by the translator

which again controls the power stage. The power stage consists of two 0.5A full bridge

power drivers which directly drives the stepper motor.


3.6 BLDC Motor Control

The UC2625 motor controller [18] IC has most of the functions required for high perfor-

mance four quadrant, 3-phase brushless DC motor control integrated into one package.

The following are a few of the functions incorporated into the CMG design:

• Drives Power MOSFETs directly on low side

• Latched Soft Start

• High-speed Current-sense Amplifier with Ideal Diode

• Average Current Sensing

• Direction Latch for Safe Direction Reversal

• Trimmed Reference Source

• Programmable Cross-conduction Protection

• Four-Quadrant Operation

The operating temperature range of the UC2625 is from -40oC to +105oC.

3.6.1 Controller Component Values

The UC2625 controller has the functions: current-limit mode, tachometer pulse width and

oscillator frequency selection for which specific component values have to be calculated.

This enables the IC’s parameters to be adjusted for different applications.

Current-limit Mode

The current sense amplifier, internal to the UC2625, has a fixed gain of two. A peak-

current comparator allows the PWM to enter a current-limit mode when the voltage

across the current sense resistors exceeds 0.2 V. An over-current comparator provides a

fail-safe shutdown in case the voltage across the resistors exceeds 0.3 V. The current sense

resistors were chosen as 0.27 Ω which will limit the motor current to 740 mA.


Tachometer Pulse Width

A fixed-width 5V pulse is triggered on the Tach-Out pin with the average voltage on the

pin directly proportional to the speed. Each time the Tach-Out pulses, a capacitor tied

to the RC-Brake discharges from 3.33 V down to 1.67 V through a resistor. The pulse

width of the tachometer is approximately T = 0.67RTCT, with RT and CT a resistor

and capacitor from RC-Brake to ground. The maximum length of the pulse depends on

the maximum speed of the motor. With each change in the Hall effect sensor outputs,

a pulse will be generated. The maximum speed needed for the motor (Section (3.3.1) is

approximately 10200 rpm. Since the motor has 2 magnet poles and 3 Hall effect sensors,

there will be 6 pulses per revolution. Thus a pulse width of less than 0.49 ms is needed

to prevent the tachometer output from saturating. RT and CT are chosen as 150 kΩ and

4.7 nF respectively to give a pulse width of 0.47 ms.

Oscillator Frequency

The motor current can be regulated using fixed-frequency pulse width modulation (PWM)

which is set by an RC oscillator circuit. The oscillator frequency is given by:

FOSC =2

ROSC COSC

A switching frequency of approximately 30 kHz is desired, which results in ROSC and

COSC values of 68 kΩ and 1 nF respectively.

3.6.2 Current Loop Design

From [16], a first order open-loop model is determined for the 4-quadrant PWM amplifier.

It is done by breaking the feedback loop between the output pin 7 of U7B (LM358A) and

E/A OUT pin 27 of U8 (UC2625) and applying a square wave signal varying between 1.5

and 2.5 Volt at 20 Hz to E/A OUT pin 27 of U8 (See Appendix B, BLDCcurrentsense).

The open-loop output of the amplifier is measured at output pin 1 of U7A (LM358A).

The DC gain of the first order model is given by the peak-to-peak voltage of the output

and the time constant is calculated from the rise and fall times. The open-loop current

model is as follows:

GOL(s) =k

1 + Ts=

3

1 + 0.0128s=

240

s+ 80(3.3)


An analogue PI controller is used to obtain a closed loop system with unit gain and the

same time constant as the open-loop model:

GCL(s) =GOL(s)GPI(s)

1 +GOL(s)GPI(s)=

80

s+ 80(3.4)

⇒ GPI(s) =kp(s+ a)

s=

0.333(s+ 80)

s(3.5)

R

R C

VV

I 1

2

O

Figure 3.5: Analogue PI Controller Circuit

The analogue PI controller from Figure 3.5 has the following transfer function:

GPI(s) =VO

VI=

−R2

R1

(s+ 1/CR2

s

)(3.6)

Comparing Equation 3.5 and Equation 3.6, the best values for the resistors and capacitors

are selected as:

R1 = 39kΩ, R2 = 12kΩ, C = 1µF

Since the microcontroller has a maximum output voltage of 2.4V, which is applied to ISET

(See Appendix B, BLDCcurrentsense), the maximum voltage measured at IMON will also

be 2.4V. The maximum voltage measured over the 0.27Ω current resistors will then be

2.4V devided by the gain of the differential amplifier (gain=10), which gives 0.24V. Thus

the maximum current in the phases of the BLDC motor is limited to 0.24/0.27 = 0.888 A.

The current-limit mode of the BLDC motor controller is set to 740 mA, and thus would

the motor current never reach 0.888 A.

3.6.3 Digital Speed Loop Design

An integrator open-loop model is chosen for the speed control loop. The input to the

speed control loop is the D/A current (torque) demand output from the microcontroller


and the output of the speed control loop is the measured encoder speed (in 0.6*rpm units,

see Section 3.6.4). From a simple open-loop test it can be seen in Figure 3.6 that the

speed will accelerate at 48 units per second on average if the D/A output is 1000 units.

Thus, the integrator model used is:

GOL(s) =Speed(0.6 ∗ rpm)

D/A(units)=

0.048

swhere 0.048 =

48

1000(3.7)

0 5 10 15 20 25 30 35 4030

32

34

36

38

40

42

44

46

48

50

Time (sec)

Uni

ts /

sec

Wheel acceleration per second

Figure 3.6: Measured acceleration per second for 1000 units input

The discrete sampling time is chosen as Ts = 0.1 seconds. From Equation 3.7, the Z-

transform open-loop model of the speed control loop and ZOH is calculated as:

GOL(z) =0.0048

z − 1(3.8)

The following specifications are chosen for the closed-loop system:

• Bandwidth, ΩBW , is 20 times lower than the sampling frequency

• Second order system is optimally damped

The desired closed-loop poles in the discrete domain can then be calculated as:

ΩBW =2π

20Ts

= 3.1416 = Ωn; ζ = 0.707; 5%tsettling =3

ζΩn

= 1.35 seconds


⇒ sCL = −2.221 ± j2.2219

⇒ zCL = 0.7803 ± j0.1765 (3.9)

A discrete PI controller is used to ensure a zero tracking error for constant speed demands

and a transfer function expression for the closed-loop system is obtained as:

GCL(z) =GPIGOL

1 +GPIGOL=

0.0048K(z − a)

z2 − (2 − 0.0048K)z + (1 − 0.0048Ka)(3.10)

where

GPI(z) =K(z − a)

z − 1

From Equation 3.9 and Equation 3.10 the constants K and a can be calculated as:

K = 91.542

a = 0.819 (3.11)

The difference equation for the discrete PI controller can then be implemented as:

G(z−1) =U(z)

E(z)=

91.542(1 − 0.819z−1)

1 − z−1

⇒ u(k) = u(k − 1) + 91.542e(k) − 74.973e(k − 1) (3.12)

The values of K1 and K2 is respectively taken as 91 and 74 for the program code.

PI ControlD/A

ReactionWheel

ZOHe(k) u(k)+

-

ST

K(z - a)z - 1

0.048s

SpeedCommand

Wheelspeed

Figure 3.7: Discrete Closed-loop speed control


3.6.4 Speed Measurement

The two outputs of the shaft encoder are XOR’ed together to give 1000 pulses per wheel

revolution from a 500ppr code wheel. This increments a counter in the microcontroller

which is read and cleared every 100ms. The speed measurement accuracy (rpm) is then

calculated as:

∆Ωwheel =60

1000Ts

= 0.6 rpm per clock pulse

⇒ Ωwheel = ∆Ωwheel × Count rpm (3.13)

The wheel speed is expressed as an integer and it is given in 0.6 rpm units. Therefore the

commanded wheel speed has also to be converted into 0.6 rpm units before it is sent to

the microcontroller. The direction of the wheel speed is determined from the phases of

the outputs of the shaft encoder. If the clock output A is lagging clock output B by 90o,

then the wheel speed is negative, else the wheel speed is positive.

3.7 Microcontroller

The Cygnal C8051F041 microcontroller is used to control the CMG and to serve as an

interface to the rest of the spacecraft via the CAN bus. For this thesis an RS232 serial link

is used to send data packages between the microcontroller and a PC. Control commands

for the CMG is received from the PC and then implemented by the microcontroller. The

functions of the microcontroller are described as follows:

• Digital Speed Controller - A discrete PI controller is used with a sampling time of

TS = 0.1 seconds to control the wheel speed accurately to within 0.6 rpm units. The

controller has a 5% settling time of 1.35 seconds to small disturbance inputs and it

controls the speed smoothly from -10000 to +10000 rpm.

• Speed Measurement - The speed of the BLDC motor is measured by a two channel

Optical Incremental Encoder Module with a 500 cpr codewheel. The two channels

are XOR’ed together to give a resolution of 1000 pulses per revolution. This is then

fed back to the microcontroller which counts the amount of pulses per 100 ms and

uses it in the digital speed controller. The two channels are also fed into a flip-flop

to obtain the direction of rotation of the BLDC motor.

• Direction Control - The microcontroller outputs a digital signal which controls the

direction bit of UC2625. This controls the direction of rotation of the BLDC motor.


• Coast Command - This is used to shut off the power to the BLDC motor. If the

microcontroller enables the coast command, the UC2625 opens the high side and

low side MOSFETs, which shuts off the power.

• Stepper Motor Pulse - From the control data received from the PC, the microcon-

troller calculates the frequency needed to achieve the necesary gimbal step rate.

This signal is fed to the Stepper Motor Controller which steps the stepper motor.

• Stepper Motor Direction - This outputs the direction of rotation of the stepper

motor to the Stepper Motor Controller.

• Inhibit - After each gimbal rotation, the Inhibit bit is set. This informs the Stepper

Motor Controller to shut off the power to the stepper motor to save power. At the

start of a gimbal instruction, the Inhibit bit is cleared.

• Rate Gyro Data - Every 100 ms a pulse is sent to the Rate Gyro, which commands

it to send the angle increment of the past 100 ms. All the angle increments are

added together to give the total angle of rotation of the whole system. The data

transfer between the microcontroller and the Rate Gyro is via RS422.

• Serial Communications - Communication between the microcontroller and the PC

is via a RS232 link with a baudrate of 38.4 kbps. Every 100 ms data packages

containing the wheel speed, gimbal angle, package number and system rotation

angle is send to the PC. RF link modules (Radiometrix) were used for wireless

communications during tests on the air bearing table.

A flowchart describing the flow of processes in the microcontroller is shown in Figure 3.8.

The following interrupts are running parallel to the processes:

• Timer1 - Set to overflow every 1 ms. A counter increments with each overflow and

sets a flag after 100 ms. When the counter reach 97, the rate gyro is polled to sent

the latest angle increment.

• Timer4 - The overflow rate of Timer4 depends on the gimbal rate needed. With

each overflow the stepper motor controller is toggled.

• UART0 - This UART is used for communication between the microprocessor and

PC. A checksum is used to determine the validity of data received. All data is sent

twice.

• UART1 - This UART is used to poll the rate gyro and to receive the new angle

increment.


Start

New

Command received

from PC

Initialize all variables

and configure registers

Gyro data

available

Byte has

been sent

Data

packet has been sent

once

100ms flag

is set

Receive

new

parameters

Get new

Gyro angle

increment

Send next

byte

Send data a

2nd time

Speed control loop

- new current demand

- output on dac1

Send new data to PC

Yes

No

No

No

No

No

Yes

Yes

Yes

Yes

Figure 3.8: Program flow diagram of Microcontroller


3.8 Computer Interface

The PC program (RWprobe2) was written in Delphi. The CMG is fully controllable from

this program. The gimbal can be rotated by a certain amount of degrees, or it can be

stepped by a certain amount of steps. The gimbal rate is adjustable from 10 to 500 steps

per second and the direction of rotation can be selected as well.

A gimbal angle profile with the rotation angle for each 100 ms interval can also be used

to control the gimbal. The wheel speed is adjustable from -10000 to +10000 rpm. CMG

data is received from the microcontroller every 100 ms and can be written to a file. When

a new command is send to the microcontroller, an ID corresponding to the particular

command is also sent. The IDs are shown in Table 3.1.

Packet ID Description

0x02 Output current demand (Current controller)

0x03 Switch RW on/off

0x05 Wheel speed command

0x06 New controller gains

0x07 CMG gimbal rate and rotation angle

0x08 Zero gimbal angle

0x09 Step Gimbal to zero position

0x0A Step gimbal (x steps)

0x0B Stop gimbal immediately

0x0C CMG gimbal data from file

0x0D Zero Gyro angle

Table 3.1: CMG Command Packet IDs

The data packet received from the microcontroller is made up of 11 bytes. The first one

is the PC’s ID, 0xCC. The following nine bytes are telemetry containing the wheel speed,

gimbal angle, packet number and the gyro angle. The last byte is a checksum. Table 3.2

shows the packet received from the microcontroller.

Byte No 1 2 3 4 5 6 7 8 9 10 11

Data CC (HB) (LB) (HB) (LB) (HB) (LB)

Content PC ID Wheel Gimbal Packet Gyro angle Checksum

speed angle No

Table 3.2: Packet frame of data transmitted from microcontroller


The command data packet transmitted by the PC is made up of 6 bytes. The first byte

is the microcontroller’s ID. The second byte corresponds to the command packets shown

in Table 3.1 and the last four bytes are the command data. Table 3.3 displays the packet

transmitted by the PC.

Byte No 1 2 3 4 5 6

Data 7F (02-0D)

Content Microcontroller ID Packet ID Command Data

Table 3.3: Packet frame of data transmitted from PC

An RF link was used for communications when testing on the air bearing table. It consists

of two receiver modules and two transmitter modules. When a transmitter is on and thus

transmitting, both receivers would recieve the data. For this reason it was necessary to

include another ID in the data packets to differentiate between data which was meant for

the microcontroller and data for the PC.

Chapter 4

Satellite Attitude Control

In this chapter an attitude control system is developed for an agile spacecraft requiring

rapid retargeting and fast transient settling. Section 4.1 describes a single-axis attitude

control problem to illustrate a nonlinear control algorithm achieving rapid, large-angle

retargeting and robust transient settling subject to constraints such as actuator saturation

and slew rate limit. In Section 4.2 a three-axis quaternion feedback control problem is

described, which was extended from the model in Section 4.1. Simulation results of rapid,

large-angle retargeting is presented in Section 4.3.

4.1 Single-Axis Attitude Control

To illustrate a nonlinear control algorithm with rapid, large-angle retargeting and robust

transient settling, consider the single-axis attitude control problem of a rigid spacecraft

described by [21]

Iθ = u; |u(t)| ≤ U (4.1)

where I is the spacecraft moment of inertia, θ the attitude angle, and u the control torque

input with the saturation limit of ±U .

The feedback control logic which assures the time-optimal control for the commanded

constant attitude angle of θc is given by

u = −Usgn(e+

1

2aθ∣∣∣θ∣∣∣) (4.2)

where e = θ − θc and a = U/I is the maximum control acceleration. The signum

function is defined as: sgn(x) = 1 if x ≥ 0 and sgn(x) = −1 if x < 0.

35

Chapter 4 — Satellite Attitude Control 36

A direct implementation of this ideal, time-optimal switching control logic results in a

chattering problem in practice. There exist various ways of avoiding this problem which

is inherent to the ideal, time-optimal switching control logic. However, a feedback control

logic of the following form is used:

u = − satU

K sat

L(e) + Cθ

(4.3)

where e = θ− θc and K and C are respectively the attitude and attitude rate gains to be

properly determined. The saturation function is defined as

satL

(e) =

L if e ≥ L

e if |e| < L

−L if e ≤ −L

and it can also be represented as

satL

(e) = sgn(e)min |e| , L

The limiter in the attitude-error feedback loop is constraining the attitude rate to

−∣∣∣θ∣∣∣

max≤ θ ≤

∣∣∣θ∣∣∣max

where∣∣∣θ∣∣∣

max= LK/C. The proper use of the feedback control logic of Eq. (4.3) will in

most practical cases result in a ”bang-off-bang” control.

For all the possible attitude-error signals that do not saturate the actuator, the controller

gains, K = kI and C = cI can be determined as

k = ω2n and c = 2ζωn (4.4)

where ωn and ζ are respectively the specified linear control bandwidth and damping ratio.

If the maximum slew rate is specified as∣∣∣θ∣∣∣

max, the limiter in the attitude-error feedback

loop can be selected as

L =C

K


=c

k


(4.5)

Due to the actuator saturation as the attitude-error signal and the slew rate limit becomes

larger for rapid maneuvers, the overall response becomes sluggish with increased transient

overshoot. One way of still achieving rapid transient settling for large commanded attitude

angles, is to adjust the slew rate limit as developed in [22] as follows:


= min√

2a |e|, |ω|max

(4.6)


where e = θ− θc is the attitude error, |ω|max is the maximum specified slew rate and a =

U/I is the maximum control acceleration. The value a should be scaled to accommodate

uncertainties in the spacecraft inertia and actuator dynamics. The attitude-error feedback

loop limiter can then be written as

L =C

K


=c

k


=c

kmin

√2a |e|, |ω|max

and the nonlinear control logic obtained has the following form:

u = − satU

K sat

L(e) + Cθ

= − satU

kI sgn(e)min

(|e| , c

k

√2a |e|, c

k|ω|max

)+ cIθ

(4.7)

If there is a constant external disturbance present which results in a steady-state pointing

error, the use of integral control has to be introduced. The feedback control logic of

Eq. (4.7) can then be modified into the following PID (proportional-integral-derivative)

saturation control logic:

u = − satU

K sat

L(e+

1

T

∫e) + Cθ

(4.8)

where T is the time constant of the integral control and

L =c

kmin

√2a |e|, |ω|max

The gains for the PID controller in terms of the standard notation: KP , KI and KD is

KP = K, KI = K/T, KD = C

The PID controller gains can be determined as

KP = I(ω2n + 2ζωn/T ) (4.9)

KI = I(ω2n/T ) (4.10)

KD = I(2ζωn + 1/T ) (4.11)

and the integral control time constant T is normaly selected as T ≈ 10/(ζωn).


If the attitude reference input to be tracked is not a multi-step input, but a smooth

function, then a PID saturation control logic of the following form can be implemented:

u = − satU

kI sat

L(e +

1

T

∫e) + cIe

(4.12)

where e = θ − θc.

When working with PID-type controllers with actuator saturation, a phenomenon known

as ”integrator windup” causes substantial transient overshoot and control effort. This

phenomenon appears when the actuator or a limiter in the control loop saturates and the

integrator control keeps on integrating. This can easily be prevented if the controller is

implemented on a digital computer by switching off the integral action when the actuator

or any other limiter in the control loop saturates.

4.2 Three-Axis Quaternion Feedback Control

A 3-axis quaternion feedback control problem is described in this section, using a nonlinear

control algorithm from [21].

4.2.1 Quaternion Feedback Control Logic

The rotational equation of motion of a rigid spacecraft is described by

IωIs + ωI

s × IωIs = u (4.13)

where I is the inertia matrix, ωIs = [ωxi ωyi ωzi]

T the inertially referenced angular

velocity vector, and u = [uxi uyi uzi]T the control torque input vector. Angular velocity

vector components ωi along the body-fixed control axes are measured by angular rate

gyros to ensure high accuracy.

A parameterization of the attitude rotation matrix referenced to, either an inertial frame

or the orbit frame has proved to be quite useful in spacecraft attitude modeling. If the

Euler rotation unit vector is denoted by λ = [λx λy λz]T , then the four elements of

quaternions are defined as

q1 = λx sin(θ/2)

q2 = λy sin(θ/2)

q3 = λz sin(θ/2)

q4 = cos(θ/2)


where θ denotes the rotation angle about the Euler axis. The four Euler symmetric

parameters are dependant by the following constraint equation:

q21 + q2

2 + q23 + q2

4 = 1

From Equation 2.22 to Equation 2.26, the quaternion kinematic differential equations are

given by

q1

q2

q3

q4

=1

2

0 ωzo −ωyo ωxo

−ωzo 0 ωxo ωyo

ωyo −ωxo 0 ωzo

−ωxo −ωyo −ωzo 0

q1

q2

q3

q4

(4.14)

A quaternion vector q = [q1 q2 q3]T can be defined such that

q = λ sin

θ

2

and Eq. (4.14) can then be written in terms of the quaternion vector

q = −1

2ωO

s × q +

1

2q4ω

Os (4.15)

q4 = −1

2ωO T

s

q (4.16)

with

ωOs ×

q ≡

0 −ωzo ωyo

ωzo 0 −ωxo

−ωyo ωxo 0

q1

q2

q3

Since the use of quaternions are so useful for the onboard real-time computation of space-

craft orientation, it can be used in a linear state feedback controller for real-time imple-

mentation of the following form:

u = −Kq −CωO

s (4.17)

where K and C are controller gain matrices which have to be determined.


From the commanded attitude quaternions (q1c, q2c, q3c, q4c) and the current attitude

quaternions (q1, q2, q3, q4), the attitude-error quaternions (e1, e2, e3, e4) can be derived as

follows:

e1

e2

e3

e4

=

q4c q3c −q2c −q1c

−q3c q4c q1c −q2c

q2c −q1c q4c −q3c

q1c q2c q3c q4c

q1

q2

q3

q4

The control logic of Eq. (4.17) can be modified using the quaternion error vector e =

[e1 e2 e3]T into the following form:

u = −Ke −CωO

s (4.18)

4.2.2 Eigenaxis Rotation

The gyroscopic term of Euler’s rotational equation of motion is not significant for most

practical rotational maneuvers. It may in some cases however, be desirable to directly

counteract this term by applying a control torque, as follows:

u = −Ke −CωO

s + ωIs × IωI

s (4.19)

In [21] it was shown that the closed-loop system with the controller from Eq. (4.19) is

globally asymptotically stable if the matrix K−1C is positive definite. To guarantee such

conditions, K and C should be selected as follows: K = 2kI and C = cI where k and c

are positive scalar constants to be properly selected, similar to the single-axis case:

k ≈ ω2n and c ≈ 2ζωn

where ωn and ζ are respectively the desired or specified linear control bandwidth and

damping ratio of the three-axis attitude control system. Furthermore, a rigid spacecraft

with the controller

u = −I(2ke +cωO

s ) + ωIs × IωI

s (4.20)

performs a rest-to-rest reorientation maneuver about an eigenaxis along the commanded

quaternion vector qc. If needed, an integral control can also be added to the quaternion-

error feedback control logic of Eq. (4.20) as follows:

u = −I(2ke +

2k

T

∫e +cωO

s ) + ωIs × IωI

s (4.21)


where T is the time constant of quaternion-error integral control. The gyroscopic decou-

pling control term, ωIs × IωI

s , from Eq. (4.21) is not significant during most practical

rotational maneuvers [21], and the term can be neglected without much impact on per-

formance and stability.

4.2.3 Rotation Under Slew Rate Constraint and Control Input

Saturation

To demonstrate the slew rate constraint and control saturation, first consider a rigid

spacecraft which is required to maneuver about an orbit fixed axis. The maneuver should

be as fast as possible, but it should not exceed the specified maximum slew rate about

that eigenaxis. The following saturation control logic provides such a rest-to-rest eigenaxis

rotation under a slew rate constraint [21]:

u = −K satLi

(e +

1

T

∫e) − CωO

s

= −I2k satLi

(e +

1

T

∫e) + cωO

s (4.22)

where e = [e1 e2 e3]T is the quaternion-error vector, K = 2kI, C = cI and the saturation

limits Li are determined as:

Li =c

2k|ωi|max (4.23)

where |ωi|max is the specified maximum angular rate about each axis. Let the control

torque input for each axis be constrained such that

−U ≤ ui(t) ≤ +U ; i = 1, 2, 3

where U is the saturation limit of each control input. This can then be used to set up

a control logic with a control torque input which saturates at a maximum value U , but

that still provides an eigenaxis rotation under a slew rate constraint. The control logic

can be expressed as

τ = −I2k sat

Li

(e +

1

T

∫e) + cωO

s

(4.24)

u = satU

(τ ) =

τ if ‖τ‖∞ < U

U τ‖τ‖∞ if ‖τ‖∞ ≥ U

(4.25)


where ‖τ‖∞ = max |τ1| , |τ2| , |τ3|. The slew rate limit needs to be adjusted to achieve

robust, rapid transient settlings for large attitude-error signals. This is similar to the

single-axis attitude control case discussed in Section (4.1). The saturation limits Li are

Li =c

2kmin

√4ai |ei|, |ωi|max

where ai = U/Iii are the maximum control acceleration about the ith control axis.


4.3.1 Single-Axis Rotation

The single-axis attitude control law of Section 4.1 is implemented in the mathematical

model of Section 2.2.4. The maximum slew rate is assumed as |ω|max = 6 deg/s and the

maximum control torque command U is limited to 0.42 Nm to prevent the gimbal rate

from exceeding 10 degrees/s. A maximum control acceleration a is chosen as 70% of U/J

to accommodate the actuator dynamics and the nonlinearity of quaternion kinematics.

The angular velocity of the momentum wheels are assumed as 1000 rad/s.

0 5 10 150

5

10

15

20

25

30

35Attitude Angle

degr

ees

0 5 10 15−1

0

1

2

3

4

5

6Attitude Rate

degr

ees/

s

0 5 10 15−0.5

0

0.5Control Torque

Nm

time (s)0 5 10 15

−15

−10

−5

0

5

10

15Gimbal Rate

degr

ees/

s

time (s)

Figure 4.1: Simulation results of the PID Saturation Control Logic in one axis


Choosing the slew maneuver 5% settling time ts = 2 seconds and the damping ratio

ζ = 0.9, gives the undamped natural frequency ωn = 3/ζts = 1.667 rad/s and integral

control time constant T = 10 seconds. The controller gains are derived as:

k = ω2n + 2ζωn/T = 3.078

c = 2ζωn + 1/T = 3.1

A commanded attitude rotation angle of 30 degrees is assumed for the slew maneuver.

The simulation results are shown in Figure 4.1. The time taken for the satellite to do a

30 degree rotation to within 5% is less than 10 seconds with a maximum attitude rate of

5.5 degrees/s. The maximum gimbal rate is kept below 10 degrees/s while the maximum

control torque is 0.42 Nm.

4.3.2 Three-Axis Quaternion Feedback Rotation

The three-axis quaternion feedback control law of Section 4.2.3 is now implemented in

the mathematical model of Section 2.2.4. The parameters are assumed to be the same as

in the previous simulation.

0 10 20 30−1

−0.5

0

0.5

1q1

0 10 20 300

0.1

0.2

0.3

0.4

0.5q2

0 10 20 30−1

−0.5

0

0.5

1q3

time (s)0 10 20 30

0

10

20

30

40

50

60Pitch Angle

degr

ees

time (s)

Figure 4.2: Simulation results of the Quaternion Feedback Control Logic in one axis


Figure 4.2 and Figure 4.3 show the results for two successive rest-to-rest maneuvers for

the commanded attitude angles: θc = 30 degrees for 0 ≤ t < 15 seconds and θc = 50

degrees for 15 ≤ t < 30 seconds.

From Figure 4.2 one can see the pitch angle rotation and the corresponding quaternions.

The pitch rotation is smooth with almost no overshoot and very fast settling time. The

gimbal rates, shown in Figure 4.3, are within the permitted 10 degrees/s parameter for

the two successive rest-to-rest maneuvers. The gimbal angle excursions reach a maximum

of 39 degrees with a maximum attitude rate of almost 5.4 degrees/s.

0 10 20 30−0.5

0

0.5Control Torque

Nm

0 10 20 30−1

0

1

2

3

4

5

6Attitude Rate

degr

ees/

s

0 10 20 30−40

−20

0

20

40Gimbal Angles

degr

ees

time (s)0 10 20 30

−15

−10

−5

0

5

10

15Gimbal Rates

degr

ees/

s

time (s)

Figure 4.3: Simulation results of the Quaternion Feedback Control Logic in one axis

Figure 4.4 and Figure 4.5 display the simulation results for a simultaneous 30o pitch, 20o

roll and 5o yaw maneuver. The total settling time is a little bit longer since the rotation

around the different axes influence each other. The corresponding quaternions for the

maneuver are also shown.


0 5 10 150

0.05

0.1

0.15

0.2q1

0 5 10 150

0.1

0.2

0.3q2

0 5 10 15−8

−6

−4

−2

0x 10

−3 q3

time (s)0 5 10 15

−10

0

10

20

30

40Euler Angles

degr

ees

time (s)

Figure 4.4: Simulation results of the Quaternion Feedback Control Logic in three axis

0 5 10 15−0.5

0

0.5Roll Torque

Nm

0 5 10 15−0.5

0

0.5Pitch Torque

Nm

time (s)

0 5 10 15−0.04

−0.02

0

0.02

0.04Yaw Torque

Nm

time (s)

Figure 4.5: Simulation results of the Quaternion Feedback Control Logic in three axis

Chapter 5

Control Demand for Imaging

Sequence

5.1 Imaging Sequence Objective

The use of CMG’s provide a satellite with more agility and thus enabling it to do rapid

pointing maneuvers. Pointing an entire satellite with larger moment of inertia instead of

sweeping the imaging system from side to side, allows the imaging system to achieve higher

definition due to smaller noise sensitivity and improves the resolution for it’s images. The

objective of this chapter is to design a control mode for the imaging phase making use

of the Moving Demand approach. This approach implement consecutive slew maneuvers

via a time varying attitude and rate demand. The advantage of this approach is that the

desired 3-axis attitude and rate are known at all times which ensures better tracking of

the attitude command.

In [7], the imaging sequence shown consist of 4 slew maneuvers. Starting from nadir

pointing, the satellite will slew to the attitude qS1, shown in Figure 5.1. The second slew

is from qS1 to qMID. Imaging starts during slew 2 after a specific constant slew rate is

reached. This rate is continued into slew 3 until the imaging period stops. After imaging

the satellite will slew until qS3. The last slew is back to nadir pointing under 3-axis

control.

A few inputs for the Moving Demand calculation is needed to dictate the target attitude

and rate to be achieved at a certain target time. The inputs for the demand derivation is

as follows:

• The time tMID at the mid-point of the imaging slew.

• The target attitude quaternion qMID at tMID.

46

Chapter 5 — Control Demand for Imaging Sequence 47

• The required orbit referenced body rate vector, ωoIMG, during imaging.

• The lead-in angle up to the mid-point of the imaging period, ρS2,TOT .

Nadir Pointing

Along Track

Start of Imaging

End of Imaging

Across Track

qS3

qS1

qMID

Figure 5.1: Image Sequence Schematic

5.2 Detailed Slew Sequence

The detailed image sequence consists of 4 slews determined by the Moving Demand. These

slews are defined in detail in Table 5.1.

It is assumed that the settling times ∆tSET12 and ∆tSET34 are taken as constants derived

from the satellite’s characteristics. The value of ∆tPM is calculated as a fraction of

ρS2,TOT .

A detail derivation of the stages in the time varying demand, derived from the input

information, are given in the following sections.


Maneuver Attitude Description Comment

Slew1 Slew from nadir pointing Slew performed as eigenaxis

to target attitude qS1. rotation.

Slew time ∆tS1.

Total slew angle ρS1,TOT .

aS1 - Demanded angular acceleration

for first half of slew, −aS1 for

second half of slew.

Settle at qS1. Settling time ∆tSET12.

Slew 2 Slew from qS1 to qMID. Slew performed as eigenaxis

rotation.

Slew time ∆tS2.


Demanded angular acceleration of

aS2 until rate reaches ωoIMG,

then constant rate.

qMID reached at time tMID.

Slew 3 Continuation of Slew 2 Slew performed as eigenaxis

from qMID to qS3. rotation at rate ωoIMG for time

∆tPM .

Deceleration to rest with demanded

angular acceleration −aS3.

Slew time ∆tS3.

Settle at qS3. Settling time ∆tSET34.

Slew 4 Slew from attitude qS3 Slew performed as eigenaxis

to nadir pointing. rotation.

Slew time ∆tS4.


aS4 - Demanded angular acceleration

for first half of slew, −aS4 for

second half of slew.

Table 5.1: Mission Sequence of Attitude Demand


5.3 Attitude Demand Derivation

This section describes the derivation of the attitude and rate demands for the slews of

the Moving Demand.

5.3.1 Slew 2

The slewing axis for Slew 2, λSLEW2, can be derived from the required imaging rate since

they are in the same direction.

λSLEW2 =ωo

IMG

|ωoIMG|

(5.1)

The quaternion describing the transformation between the initial and final attitudes of

Slew 2, qS2,TOT , can be derived from the slewing axis and angle for Slew 2 as follows:

qS2,TOT =

λSLEW2(1) sin(12ρS2,TOT )



cos(12ρS2,TOT )

(5.2)

Given the final attitude of Slew 2, qMID, the initial demanded attitude which is also the

final attitude for Slew 1, qS1, can be derived as:

qS1 =

qS2,TOT (4) −qS2,TOT (3) qS2,TOT (2) −qS2,TOT (1)

qS2,TOT (3) qS2,TOT (4) −qS2,TOT (1) −qS2,TOT (2)

−qS2,TOT (2) qS2,TOT (1) qS2,TOT (4) −qS2,TOT (3)

qS2,TOT (1) qS2,TOT (2) qS2,TOT (3) qS2,TOT (4)

qMID (5.3)

The total time for Slew 2, ∆tS2, can be derived from the total slew angle, ρS2,TOT , and

the initial angular acceleration, aS2, as follows:

∆tS2 =ρS2,TOT

|ωoIMG|

+|ωo

IMG|2aS2

(5.4)

where the angular acceleration, aS2, is a constant value depending on the satellites char-

acteristics such as moment of inertia and maximum torque from the CMG’s. The start

time for Slew 2, tST2, is then given by:

tST2 = tMID − ∆tS2 (5.5)


The previous equations of Section 5.3.1 derived the start time and attitude for Slew 2.

This can be used to derive the rotation angle and rate as a functions of time, based on

an initial angular acceleration aS2 and constant imaging rate as follows:

ρS2(t) =

0 (t < tST2

12aS2(t− tST2)

2

(tST2 ≤ t < tST2 +

|ωoIMG|aS2

)

|ωoIMG|

(t− tST2 − |ωo

IMG|2aS2

) (tST2 +

|ωoIMG|aS2

≤ t < tMID

)

(5.6)

ρS2(t) =

0 (t < tST2

aS2(t− tST2)(tST2 ≤ t < tST2 +

|ωoIMG|aS2

)

|ωoIMG|

(tST2 +

|ωoIMG|aS2

≤ t < tMID

)

(5.7)

Now the demanded quaternion qD(t) and demanded orbit-referenced body rate ωoD(t) for

the period tST2 ≤ t < tMID can be determined as follows:

qSLEW2(t) =

λSLEW2(1) sin(12ρS2(t))



cos(12ρS2(t))

(5.8)

qD(t) =

qSLEW2(4) qSLEW2(3) −qSLEW2(2) qSLEW2(1)

−qSLEW2(3) qSLEW2(4) qSLEW2(1) qSLEW2(2)

qSLEW2(2) −qSLEW2(1) qSLEW2(4) qSLEW2(3)

−qSLEW2(1) −qSLEW2(2) −qSLEW2(3) qSLEW2(4)

qS1 (5.9)

ωoD(t) = ρS2(t).λSLEW2 (5.10)

5.3.2 Slew 1

Slew 1 is performed as an eigenaxis rotation with the starting attitude nadir pointing and

the final attitude at qS1. The rotation axis and angle can then be derived as:

ρS1,TOT = 2 arccos(qS1(4)) (5.11)

λSLEW1 =[qS1(1) qS1(2) qS1(3)]T

sin(ρS1,TOT/2)(5.12)


A constant demanded angular acceleration, aS1, is performed for the first half of Slew 1,

followed by a constant demanded angular deceleration, −aS1, for the second half. The

total time used for Slew 1, ∆tS1, is derived from the total slew angle as follows:

∆tS1 = 2

√ρS1,TOT

aS1(5.13)

and the start time for Slew 1, tST1, is given by:

tST1 = tMID − ∆tS2 − ∆tSET12 − ∆tS1 (5.14)



an angular acceleration parameter aS1 as follows:

ρS1(t) =

0 (t < tST1)12aS1(t− tST1)

2(tST1 ≤ t < tST1 + ∆tS1

2

)ρS1,3(t)

(tST1 + ∆tS1

2≤ t < tST1 + ∆tS1

)ρS1,TOT (tST1 + ∆tS1 ≤ t < tST1 + ∆tS1 + ∆tSET12)

(5.15)

ρS1(t) =

0 (t < tST1)

aS1(t− tST1)(tST1 ≤ t < tST1 + ∆tS1

2

)ρS1,3(t)

(tST1 + ∆tS1

2≤ t < tST1 + ∆tS1

)0 (tST1 + ∆tS1 ≤ t < tST1 + ∆tS1 + ∆tSET12)

(5.16)

with

ρS1,3(t) = 2√ρS1,TOTaS1(t− tST1) − aS1(t− tST1)

2

2− ρS1,TOT

ρS1,3(t) = aS1∆tS1

2− aS1(t− tST1 − ∆tS1

2)

where the time interval ∆tSET12, between the end of Slew 1 and the start of Slew 2,

serve as a settling period for the rotation angle . Now the demanded quaternion qD(t)

and demanded orbit-referenced body rate ωoD(t) for the period tST1 ≤ t < tST2 can be

determined as follows:

qD(t) =




cos(12ρS1(t))

(5.17)



5.3.3 Slew 3

Slew 3 is a continuation of Slew 2 with the rotation axis identical to that for Slew 2. It is

performed as an eigenaxis rotation starting at the midpoint of the image defined by the

time tMID and attitude qMID. The duration of Slew 3, ∆tS3, is given by:

∆tS3 = ∆tPM +|ωo

IMG|aS3

(5.19)

where ∆tPM is the time after the image midpoint where braking commences and the

second term is the time it takes for a constant angular deceleration of aS3 until the rate

is zero. The total rotation angle for Slew 3 is derived as follows:

ρS3,TOT = |ωoIMG|

(∆tPM +

|ωoIMG|aS3

)− |ωo

IMG|22aS3

(5.20)




ρS3(t) =

0 (t < tST3)

|ωoIMG| (t− tST3) (tST3 ≤ t < tST3 + ∆tPM)

ρS3,3(t) (tST3 + ∆tPM ≤ t < tST3 + ∆tS3)

ρS3,TOT (tST3 + ∆tS3 ≤ t < tST3 + ∆tS3 + ∆tSET34)

(5.21)

ρS3(t) =

0 (t < tST3)

|ωoIMG| (tST3 ≤ t < tST3 + ∆tPM)

ρS3,3(t) (tST3 + ∆tPM ≤ t < tST3 + ∆tS3)

0 (tST3 + ∆tS3 ≤ t < tST3 + ∆tS3 + ∆tSET34)

(5.22)

with

ρS3,3(t) = |ωoIMG| (t− tST3) − 0.5aS3(t− tST3 − ∆tPM)2

ρS3,3(t) = |ωoIMG| − aS3(t− tST3 − ∆tPM)


the period tST3 ≤ t < tST3 + ∆tS3 + ∆tSET34 can be determined as follows:

qSLEW3(t) =




cos(12ρS3(t))

(5.23)


qD(t) =





qMID (5.24)


5.3.4 Slew 4

The start time of Slew 4, tST4, is the same as the end time of Slew 3 with the settling

interval ∆tSET34 included and is given by:

tST4 = tST3 + ∆tS3 + ∆tSET34 (5.26)

Since the rotation axis for Slew 3 is the same as for Slew 2, the quaternion describing the

transformation between the initial and final attitudes of Slew 3, qS3,TOT , is given by:

qS3,TOT =




cos(12ρS3,TOT )

(5.27)

The initial attitude of Slew 4 is identical to the final attitude for Slew 3, qS3, and the

initial attitude of Slew 3 is known to be qMID. Thus qS3 can be derived as:

qS3 =

qS3,TOT (4) qS3,TOT (3) −qS3,TOT (2) qS3,TOT (1)

−qS3,TOT (3) qS3,TOT (4) qS3,TOT (1) qS3,TOT (2)

qS3,TOT (2) −qS3,TOT (1) qS3,TOT (4) qS3,TOT (3)

−qS3,TOT (1) −qS3,TOT (2) −qS3,TOT (3) qS3,TOT (4)

qMID (5.28)

Since the final attitude for Slew 4 is nadir pointing, the rotation axis and angle for Slew

4 are derived as:

ρS4,TOT = 2 arccos(qS3(4)) (5.29)

λSLEW4 =−[qS3(1) qS3(2) qS3(3)]T

sin(ρS4,TOT/2)(5.30)


A constant demanded angular acceleration, aS4, is performed for the first half of Slew 4,

followed by a constant demanded angular deceleration, −aS4, for the second half. The

total time used for Slew 4, ∆tS4, is derived from the total slew angle as follows:

∆tS4 = 2

√ρS4,TOT

aS4(5.31)




ρS4(t) =

0 (t < tST4)12aS4(t− tST4)

2(tST4 ≤ t < tST4 + 1

2∆tS4

)ρS4,3(t)

(tST4 + 1

2∆tS4 ≤ t < tST4 + ∆tS4

)ρS4,TOT (tST4 + ∆tS4 ≤ t < tEND)

(5.32)

ρS4(t) =

0 (t < tST4)

aS4(t− tST4)(tST4 ≤ t < tST4 + 1

2∆tS4

)ρS4,3(t)

(tST4 + 1

2∆tS4 ≤ t < tST4 + ∆tS4

)0 (tST4 + ∆tS4 ≤ t < tEND)

(5.33)

with

ρS4,3(t) = 2√ρS4,TOTaS4(t− tST4) − aS4(t− tST4)

2

2− ρS4,TOT

ρS4,3(t) = aS4∆tS4

2− aS4(t− tST4 − ∆tS4

2)


the period tST4 ≤ t < tEND can be determined as follows:

qSLEW4(t) =

λSLEW4(1) sin(ρS4(t)2

)


)


)

cos(ρS4(t)2

)

(5.34)

qD(t) =





qS3 (5.35)




An approach was presented in Section 5 to derive the demanded attitude quaternion and

orbit reference body rate for the entire imaging sequence. This is based on input values

for the defined image mid-point time, tMID, the target attitude quaternion at the mid-

point, given as qMID, the required imaging body rate vector (orbit referenced) denoted

by ωoIMG, and the size of the lead-in angle up to the mid-point of the imaging period,

ρS2,TOT .

The simulation is based on the following parameter selection:

• tMID = 65 seconds after the start of slew 1.

• qMID = [0.2156 0.01886 − 0.08509 0.9726]T (0o pitch, 25o roll, −10o yaw).

• ωoIMG = [0 − 0.6 0]T degrees/s.

• ρS2,TOT = 20o.

The settling times at the end of slew 1 and slew 3 are assumed to be ∆tSET12 = ∆tSET34 =

5 seconds. The demanded angular accelerations for each slew are chosen as aS1 = aS2 =

aS3 = aS4 = 0.005 rad/s2 and the time ∆tPM = 0.5ρS2/ |ωoIMG|. This simulation uses

the three-axis quaternion feedback controller from Section 4.2.3. A faster settling time

is chosen for this simulation, since the commanded angles are very small. A 5% settling

time of ts = 0.1 seconds is used with a damping ratio of ζ = 0.9. The undamped natural

frequency ωn = 3/ζts = 33.33 rad/s and the integral time constant T = 10/ζωn = 0.333

seconds. The controller gains are calculated as:

k = ω2n + 2ζωn/T = 1291

c = 2ζωn + 1/T = 63

The generated quaternion demand and the orbit-referenced body rate demands are shown

in Figure 5.2 and Figure 5.3 respectively. The settling periods at the end of Slew 1 and

Slew 3 are clearly visible in both figures. In Figure 5.3 the body rates of the satellite

for the different slew maneuvers can be seen. During Slew 1 and Slew 3 the body rate

ramps to a maximum value and then reduces linearly to 0. This behaviour is expected

since both slews are performed with an angular acceleration for the first half of the slew,

followed by an angular deceleration.


Slew 2 starts with an acceleration phase until the required imaging rate is obtained and

then continues at a constant rate. This rate is maintained during Slew 3 until the end of

the imaging period, where after the rate reduces at a constant deceleration to 0.

0 20 40 60 80 100 120−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25Quaternion Demand

time (s)

q1q2q3

Slew 1 &SettlingTime

Slew 2


Slew4

ImagingPeriod

Figure 5.2: Quaternion Demand for Imaging Sequence

The commanded torque for the imaging sequence and the corresponding gimbal rate is

shown in Figure 5.4 and Figure 5.6 respectively. The maximum torque needed is 0.12 Nm

and the maximum gimbal rate is 2.2 deg/s. The errors between the commanded Euler

angles and the measured Euler angles are also depicted. These angle errors are very small,

especially during imaging (Slew 2 and Slew 3) when accuracy is most important. The

maximum error during imaging is less than 0.001 degrees.

From these simulation results one can see that the maneuvers are quite accurate when

using the Moving Demand approach since the reference angle and rate demand are con-

tinuously known.


0 20 40 60 80 100 120−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5Orbit Referenced Rate Demand

time (s)

wxwywz


Slew 2


Slew 4

ImagingPeriod

Figure 5.3: Body Rate Demand for Imaging Sequence

0 20 40 60 80 100 120

−0.1

−0.05

0

0.05

0.1

0.15Torque Command

Nm

time (s)

Tx

Ty

Tz

Imaging Period

Figure 5.4: Simulation Result for Imaging Sequence - Torque Command


0 20 40 60 80 100 120−20

−15

−10

−5

0

5

10

15

20

25

30Euler Angles

Deg

rees

time (s)

PitchRollYaw

Imaging Period

Figure 5.5: Simulation Result for Imaging Sequence - Euler Angles

0 20 40 60 80 100 120−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5Gimbal Rate

Deg

rees

/s

time (s)

Delta dotx

Delta doty

Delta dotz

Imaging Period

Figure 5.6: Simulation Result for Imaging Sequence - Gimbal Rate


0 20 40 60 80 100 120−6

−4

−2

0

2

4

6x 10

−3 Euler Angle Error

degr

ees

time (s)

Pitch ErrorRoll ErrorYaw Error

Imaging Period

Figure 5.7: Simulation Result for Imaging Sequence - Euler Angle Error

Chapter 6

Measurements and Results

The Control Moment Gyro module designed in Chapter 3 is tested on an air bearing table

and the measurements are compared to the simulation results of Chapters 2, 4 and 5 to

verify the functionality of the CMG design. Communications between the CMG module

and the PC is done via an RF link to minimize external forces. The CMG module with

battery, RF link and rate gyro is mounted on a cart with three carbon nozzles which are

used to levitate the system. This whole setup will be referred to as the CMG system. The

air bearing table, levitating cart and CMG setup are further described in Appendix C.

6.1 Calibration

6.1.1 Gimbal Accuracy

The step size of the gimbal is measured by means of a laser pointer. The laser pointer

was attached to the gimbal, pointing horizontally onto a wall. After a rotation of 100

steps, the distance covered on the wall was measured. The typical geometry are shown

in Figure 6.1.

1805 mm

60 mm

θ

Wall

Laser

Pointer

Figure 6.1: Measuring the gimbal’s step size

60

Chapter 6 — Measurements and Results 61

The angle, θ, which represents 100 steps is calculated as:

θ = arctan60

1805= 1.904o

Thus the measured size of one gimbal step = 0.01904o. From Section 3.3.2, the rotational

resolution of the gimbal is:

Motor step size

Total gear ratio=

15

786= 0.01908o

The measured gimbal step angle of 0.01904o is very close to the calculated value of

0.01908o.

6.1.2 Moment of Inertia Calculation

The aim of the first test on the air bearing table is to calculate the moment of inertia

of the cart and CMG system. This is done by using the CMG as a reaction wheel with

the spin-axis of the momentum wheel perpendicular to the ground level. The momentum

wheel is accelerated from zero and the angular acceleration of the cart is measured. The

moment of inertia of the ideal cart and CMG system Is is given from the conservation of

angular momentum as:

Is =Iwω

θ

with Iw = 0.0015 kgm2 the moment of inertia of the momentum wheel, ω the angular

acceleration of the momentum wheel and θ the angular acceleration of the cart. The

results of five consecutive tests are shown in Table 6.1. The measurements shown in the

table are the average for the first few seconds of each maneuver when the torque is still

at maximum.

ω/100 (o/s2) θ (o/s2) Is (kgm2)

9.9 2.85 0.521

9.8 2.85 0.5158

9.9 2.80 0.5304

10.5 2.96 0.5321

9.88 2.65 0.559

Table 6.1: Results from Moment of Inertia Tests


The average moment of inertia calculated from the measurements for the cart and CMG

system is Is = 0.535 kgm2. Figure 6.2 display the angular acceleration of the momentum

wheel and the cart, as well as the angular rotation and angular rate of the cart.

0 5 10 150

50

100

150

200

250

300

350Gyro Angular Rotation

Ang

le (

degr

ees)

0 5 10 150

5

10

15

20

25

30Gyro Rate

degr

ees/

s

time (s)

0 5 10 15

−2

0

2

4

6

8

10

12Gyro Acceleration & Wheelacc/100

degr

ees/

s2

time (s)

Gyro AccWheel Acc/100

Figure 6.2: Angular acceleration of Momentum Wheel and Cart (Gyro)

6.1.3 Glass Surface Test

Since the CMG setup consists of only one CMG, the resulting torque output of the CMG is

around two axes. The spin-axis of the momentum wheel is parallel to the ground plane and

the gimbal is rotated around an axis also parallel to the ground plane, but perpendicular

to the spin-axis as shown in Figure 6.3. For small gimbal angles, the output torque will

be large about an axis perpendicular to the ground plane. This will cause the cart to

rotate. The torque generated about the second axis will be very small and will have no

effect since the cart can only rotate around one axis.

The gimbal is rotated through a specific angle and then back to zero again. Repeating

this process will be referred to as ‘mirror’. The CMG is ‘mirrored’ from +15 degrees to

an inclination angle of -15 degrees with a gimbal rate of 500 steps/s (9.54 o/s). A wheel

speed of ω = 2000 rpm is used for the tests.


Spin-axis

Gimbal-axisGround plane

Figure 6.3: Diagram of the CMG setup

When a CMG is gimballed to a certain inclination angle and back to zero again, the

angular rate of the cart should reach a maximum at the maximum gimbal inclination angle

and return to zero when the gimbal is back at zero degrees, as described in Section 2.2.1.

In Figure 6.4 the gimbal inclination angle is plotted with the angular rate of the cart

(Gyro).

0 10 20 300

50

100

150

200

250


Ang

le (

degr

ees)

0 10 20 30

−10

0

10

20

30Gyro Rate & Gimbal Angle

degr

ees/

s &

deg

rees

0 10 20 30−10

−5

0

5

10Gyro Acceleration

degr

ees/

s2

time (s)0 10 20 30

−10

−5

0

5

10Gimbal Rate

degr

ees/

s

time (s)

Gyro RateGimb Angl

Figure 6.4: Measured results of ’Mirror’ Test

As shown in the measured results, the angular rate of the cart does not always return

to zero when the gimbal angle is zero. This is as a result of the unevenness of the glass


surface and a little bit of friction. Since it is not possible to achieve a 100% even glass

surface by adjusting the legs of the table, all the measurements done on the air bearing

table will have a small error.

6.1.4 Constant Angular Rate Test

It is important to know the magnitudes of the disturbances due to the glass surface of

the air bearing table. Once it is determined, the accuracy of the results from the tests

on the air bearing table can more easily be explained. The disturbances are measured by

looking at the effect it has when the cart is rotating at a constant rate.

0 10 20 30 400

100

200

300

400

500


degr

ees

0 10 20 30 400

5

10

15

20Gyro Rate

degr

ees/

s

time (s)

0 10 20 30 40−2

0

2

4

6

8Gyro Acceleration

degr

ees/

s2

time (s)

Figure 6.5: Measurements for constant angular rate input

The CMG is gimballed to an inclination angle of 30 degrees at a rate of 500 steps/s. In an

ideal case this would result in the cart moving at a constant angular rate. The measured

values are shown in Figure 6.5. Looking at the angular rate and acceleration of the cart,

one can clearly see the effect friction and an uneven glass surface has on the cart. The

change in angular rate can be divided into two parts. The one is a linear decrease in the

angular rate due to friction, and the other one is a periodic disturbance due to the uneven

surface.


The average deceleration of the cart due to friction is measured as θ = −0.586 o/s2. Using

Equation 2.3, the disturbance friction torque is calculated as:

Tavg = Isθ = 5.47 mNm

where the moment of inertia of the cart Is = 0.535 kgm2. The unevenness of the glass

surface causes a sinusoidal disturbance in the acceleration of the cart, with an amplitude

of between 0.5 and 1 o/s2 which results in a maximum torque of approximately 4.7 to 9.3

mNm.

6.2 CMG Tests

6.2.1 Rest-to-Rest Slew

The CMG is gimballed from +15 degrees to an inclination angle of -15 degrees, and back

to +15 degrees again, at a gimbal rate of 500 steps/s (9.54 o/s). This results in the cart

rotating almost 55 degrees with a maximum angular rate measured as 17.4 degrees/s.

The average angular acceleration of the cart is measured as 5.77 degrees/s2. The shape of

the results displayed in Figure 6.6 are very similar to the shape of the simulation results

from Section 2.3.

From Equation 2.6 the CMG torque is calculated as:

T = h× δ = 0.052315 Nm

with the angular momentum h = 209.42 × 0.0015 × cos δ Nms and the gimbal rate δ =

0.1665 rad/s. Since the gimbal angle δ is small, cos δ ≈ 1. Using this with Equation 2.3

gives:

θ =T

Is=

0.052315

0.535= 0.09779 rad/s2 (5.61 o/s2)

This calculated value of the angular acceleration (5.61 o/s2) is close to the average mea-

sured value of 5.77 o/s2. Using Equation 2.5, the rotation of the cart can be calculated.

Since the cart accelerates for the first half of the slew, and decelerates for the second

half, the rotation of the cart is calculated for the first half and then multiplied by two to

obtain the total slew rotation angle. The time, thalf , for half of the slew is obtained from

Figure 6.6 as thalf = 3.15 seconds. The rotation angle is then calculated as:

θT = 2 × θhalf =T

Ist2half = 0.9703 rad (55.6 o)


0 2 4 60

10

20

30

40

50


Ang

le (

degr

ees)

0 2 4 6−20

−10

0

10

20

30Gyro Rate & Gimbal Angle

degr

ees/

s &

deg

rees

0 2 4 6−8

−6

−4

−2

0

2

4

6Gyro Acceleration

degr

ees/

s2

time (s)0 2 4 6

−10

−5

0

5

10Gimbal Rate

degr

ees/

s

time (s)

Gyro RateGimb Angl

Figure 6.6: Measured results of the Rest-to-Rest Slew

The calculated value of the rotation angle (55.6 o) is very similar to the measured value

(≈ 55 o) from the test. The calculated values of the angular acceleration and rotation

angle proof the validity of the measurements.

6.2.2 Moving Demand Test

The Moving Demand simulation done in Section 5 is used to obtain an array of gimbal

angle rotations which will cause the cart to do a slew maneuver similar to the pitch

rotation of the satellite in the moving demand simulation of Section 5.4. The moment

of inertia of the CMG system as measured in Section 6.1.2 is used with the following

parameters:

• tMID = 25 seconds after the start of slew 1.

• qMID = [0 0 0 1]T (0o pitch, 0o roll, 0o yaw).

• ωoIMG = [0 − 6 0]T degrees/s.

• ρS2,TOT = 80o.


The angular rotation and rate obtained from a mathematical simulation are displayed

together with the measured results in Figure 6.7.

0 10 20 30 40−40

−20

0

20

40

60Angular Rotation

degr

ees

0 10 20 30 40−5

0

5

10

15Angular Rate

degr

ees/

s

0 10 20 30 40−60

−40

−20

0

20

40


degr

ees

time (s)0 10 20 30 40

−10

−5

0

5

10

15Gyro Rate

degr

ees/

s

time (s)

Figure 6.7: Measured results of the Moving Demand

From the Measured Angular Rate figure, the effect of friction and the uneven glass surface

is clearly visible, especially when the cart is suppose to move at a constant angular

rate. The friction causes a constant deceleration of the cart. Keeping the effects of the

disturbances in mind, the measured results are quite similar to the simulations and clearly

illustrates the pitch rotation of the Moving Demand.

Chapter 7

Conclusions

This thesis showed the design of a Single Gimbal Control Moment Gyro actuator which

can be used in small satellite applications. The CMG was tested on an air bearing table

to evaluate the working of the controller and the CMG. A mathematical model of a small

satellite with an actuator configuration containing six CMG’s and an attitude controller

were derived to simulate the rotation of the satellite when using CMG’s. These simulations

were compared to practical tests done with a single CMG prototype on the air bearing

table to verify the performance of the CMG.

7.1 Results Obtained

The following results were obtained in this thesis:

• The BLDC motor was successfully controlled with inner analogue current and outer

discrete speed control loops to within 0.6 rpm of the commanded wheel speed.

• The gimbal has a rotational resolution of 0.0191 degrees.

• A mathematical model of a satellite was developed according to the parameters

given in Section 1.1 and was used to simulate the behaviour of the satellite with

CMG’s. It was shown in the simulations of Section 2.3 that a 30 degree rotational

maneuver is possible within 10 seconds.

• A three-axis quaternion feedback control law under slew rate constraint was de-

rived for the satellite’s attitude control. This was successfully implemented in the

mathematical model of the satellite to demonstrate a 30 degree rotation within the

permitted 10 seconds with less than 0.2% overshoot.

68

Chapter 7 — Conclusions 69

• A control mode known as the Moving Demand approach, used during imaging, was

also introduced into the mathematical model. This approach successfully imple-

mented consecutive slew maneuvers via a time varying attitude and rate demand to

ensure better tracking of the attitude command. Tracking errors were kept below

0.005 degrees during simulations.

• A single CMG prototype was designed and built according to the specifications of

Section 1.1 and tested on an air bearing table. The results of the tests correspond

to the mathematical simulations and the theory of Chapter 2, 4 and 5.

7.2 Additional Work

The following is a list of additional work that can still be done:

• The steering logic of the CMG’s can be extended to a pseudoinverse steering logic.

This would mean only four CMG’s are required to gimbal at different gimbal rates,

giving the controller full 3-axis freedom, but also making it more vulnerable to

singularities.

• The prototype CMG can be used as a VSCMG and a control law for an integrated

power and attitude control system can be developed.

• All the electronics used for controlling the CMG can be combined on one pcb.

This would dramatically downsize the area needed and thus a more compact CMG

‘package’ can be designed.

• A complete CMG product with all the electronics included can be developed, de-

signed to specific specifications. This product has to be as compact and light as

possible so it can be used on a small satellite.

• The surface of the air bearing table needs to be leveled more accurately to conduct

tests with reduced disturbances. From the tests done in Chapter 6, one can see the

unevenness of the glass surface has a noticable influence on the measurements. Re-

search in developing a smooth surface which is 100% level, would thus tremendously

increase the quality of the measurements.

Bibliography

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[2] AHMED, J., COPPOLA, V., and BERNSTEIN, D., “Adaptive Asymptotic

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of Guidance, Control, and Dynamics, 1998, Vol. 21, No. 5, No. 5, pp. 684–691.

[3] ARSAPE. Low voltage driver for 2 phase stepper motors, May 2000.

[4] CYGNAL INTEGRATED PRODUTS, INC. C8051F040/1/2/3 Mixed-Sygnal ISP

FLASH MCU Family , August 2002.

[5] DE LA MORINAIS, G. C., SALENC, C., and PRIVAT, M., “Mini CMG

Development for Future European Agile Satellite.” 5th International ESA

Conference on Spacecraft Guidance Navigation and Control Systems, Frascati

(Rome), October 2002.

[6] DEFENDINI, A., FAUCHEUX, P., GUAY, P., MORAND, J., and HEIMEL, H.,

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October 2002.

[7] DUNGATE, D. G. et al., “Topsat Imaging Mode ADCS Design.” 5th International

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[8] FORD, K. A. and HALL, C. D., “Singular Direction Avoidance Steering for

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[9] HEIBERG, C., BAILEY, D., and WIE, B., “Precision Spacecraft Pointing Using

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BIBLIOGRAPHY 71

[10] LAPPAS, V. J., STEYN, W. H., and UNDERWOOD, C., “Attitude Control for

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[11] LITEF GMBH. uFORS User Manual , May 2001.

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[13] RADIOMETRIX LTD. TX2 & RX2 Data Sheet , February 2002.

[14] SCHAUB, H. and JUNKINS, J. L., “Singularity Avoidance Using Null Motion and

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[16] STEYN, W. H., “Reaction Wheel Electronics Design.” Private communication,

January 2004.

[17] TSIOTRAS, P. and SHEN, H., “Satellite Attitude Control and Power Tracking

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[18] UNITRODE CORPORATION. UC2625 - Brushless DC Motor Controller ,

November 1999.

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Appendix A

CMG Machine Drawing

73

APPENDIX A - CMG MACHINE DRAWING 74

Figure A.1: Drawing of CMG stand with Gimbal

Appendix B

Schematics of Brushless DC Motor

Electronics

75

APPENDIX B - SCHEMATICS OF BRUSHLESS DC MOTOR ELECTRONICS 76

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NL

6C

AN

H7

RS

8

U3

PC

A82C

250?

TX

D1

GN

D2

VC

C3

RX

D4

Vre

f5

CA

NL

6C

AN

H7

RS

8

U4

PC

A82C

250?

+5

V

+5

V

CA

NS

EL

CA

NH

_A

CA

NL

_A

CA

NH

_B

CA

NL

_B

nC

AN

SE

L

CA

NR

X

CA

NT

X

CA

NR

X_A

CA

NR

X_B

CA

N tra

nceiv

ers

and m

ultip

lexer

R3

120

R2

120

Note

: O

nly

the

last

node

should

hav

e a

term

inat

ion r

esis

tor.

RE

2R

O1

DE

3A

6

B7

DI

4

VC

C8

GN

D5

U1

LT

C1480

AD

CS

_S

YN

C_H

AD

CS

_S

YN

C_L

+3

.3V

AD

CS

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YN

C

RS

485 d

river

for

AD

CS

_S

YN

C

L1

EX

XC

3216

A+

3.3

V+

3.3

V

Analo

gue G

round P

lane

Dig

ital G

round P

lane T

P4

AG

ND

TP

5G

ND

R13

1M

C22

1nFC

hassis

Gro

und P

lane

Dig

ital G

round P

lane

C18

1uF

C6

0.1

uF

R10

0E

+5

V

TP

6+

5V

C24

10uF

, 25V

C2

4.7

uF

, 20V

C7

0.1

uF

+5

V

TP

7+

3.3

V

R14

0.2

7R

, 1%

X2

DC

FIL

TE

R

R12

0E

RE

G5_P

WR

_S

UP

PL

Y

RE

G5_P

WR

_R

ET

UR

N

TM

ST

CK

TD

IT

DO

+3

.3V

A+

3.3

V+

3.3

V

VDD24

VDD41

VDD57

AV+3

DGND25

DGND40

DGND56

AV+6

AGND4

AGND5

TM

S5

8

TC

K5

9

TD

I6

0

TD

O6

1

/RS

T6

2

XT

AL

11

7

XT

AL

21

8

MO

NE

N1

9

VR

EF

7

VR

EF

A8

AIN

0.0

9

AIN

0.1

10

AIN

0.2

11

AIN

0.3

12

HV

CA

P1

3

HV

RE

F1

4

HV

AIN

+1

5

HV

AIN

-1

6

CA

NT

X2

CA

NR

X1

DA

C0

64

DA

C1

63

P0

.05

5

P0

.15

4

P0

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3

P0

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2

P0

.45

1

P0

.55

0

P0

.64

9

P0

.74

8

P1

.02

9

P1

.12

8

P1

.22

7

P1

.32

6

P1

.42

3

P1

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2

P1

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1

P1

.72

0

P2

.03

7

P2

.13

6

P2

.23

5

P2

.33

4

P2

.43

3

P2

.53

2

P2

.63

1

P2

.73

0

P3

.04

7

P3

.14

6

P3

.24

5

P3

.34

4

P3

.44

3

P3

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2

P3

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9

P3

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8

U5

C8051F

041

R4

100k

TX

D0

RX

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SD

AS

CL

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TP

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DC

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C

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TA

L2

C5

0.1

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DA

C B

yp

ass C

ap

s

C17

<tb

d>

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16M

Hz

C3

22pF

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R9

10k

SD

A

R8

10k

SC

L

SD

AS

CL

+3

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CA

NT

XC

AN

RX

+5V

_se

nse

+5V

_se

nse

+5V

AV

RE

F

R1

120

Pow

er

filter

and c

urr

ent m

easure

ment re

sis

tor

Initia

l desig

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3/2

003

Pow

er

Regula

tor

Decouplin

g c

apacitors

C9

0.1

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C12

0.1

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C13

0.1

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C15

0.1

uF

C16

0.1

uF

+5

V

A+

3.3

V

C11

0.1

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C10

0.1

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+3

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OU

T1

SE

NS

E2

GN

D3

BY

P4

SH

DN

5

GN

D6

GN

D7

IN8

U6

LT

1763-3

.3

C25

0.0

1uF

+5

V

+5

V

X1

DC

FIL

TE

R

R11

0E

UN

RE

G_P

WR

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UP

PL

Y

UN

RE

G_P

WR

_R

ET

UR

NV

BU

S_

GN

D

Pow

er

filter

AIN

1A

IN2

TX

D1

RX

D1

P1

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1.1

TX

D1

RX

D1

AIN

3

C21

1nF

C14

0.1

uF

C1

4.7

uF

, 20V

XC

FS

econ

d D

raft

(Add

apte

d fo

r R

W d

esig

n)12

-11-

2003

4-O

ct-2

004

Prin

t

ES

L04-1

00000-2

13-0

1-0

.0A

Ori

gin

ally

the

Gen

eric

Node:

AD

CS

_S

EL

+3

.3V

C54

0.1

uF

123

U2A

SN

74H

C00N

GN

D7

VC

C1

4

U2E

SN

74H

C00N

89 1

0

U2C

SN

74H

C00N

11

12

13

U2D

SN

74H

C00N

4 56

U2B

SN

74H

C00N

Pow

er

Addit

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100nF

outp

ut

cap. at

gro

undin

g c

ircu

it.

Gen

eric

CA

N N

od

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CA

NL

_A

CA

NH

_A

CA

N_S

HIE

LD

_A

AD

CS

_S

YN

C_H

AD

CS

_S

YN

C_L

CA

N_S

HIE

LD

_B

J3J2

J1

AD

CS

_S

YN

C_S

HIE

LD

CA

NL

_A

CA

NH

_A

CA

NH

_B

CA

NL

_B

CA

NH

_B

CA

NL

_B

CA

N

CA

NH

_B

CA

NL

_B

AD

CS

_S

YN

C_

SH

IEL

D

CA

N_

SH

IEL

D_

BA

DC

S_

SY

NC

_L

CA

NL

_B

AD

CS

_S

YN

C_

H

CA

NH

_B

CA

NH

_A

CA

NL

_A

CA

N_

SH

IEL

D_

A

CA

NL

_A

CA

NH

_A

UN

RE

G_P

WR

_S

UP

PL

Y

UN

RE

G_P

WR

_R

ET

UR

NR

EG

5_P

WR

_S

UP

PL

Y

RE

G5_P

WR

_R

ET

UR

NU

NR

EG

_P

WR

_S

UP

PL

Y

UN

RE

G_P

WR

_R

ET

UR

N

RE

G5_P

WR

_S

UP

PL

Y

RE

G5_P

WR

_R

ET

UR

N

UN

RE

G_

PW

R_

SU

PP

LY

UN

RE

G_

PW

R_

SU

PP

LY

UN

RE

G_

PW

R_

RE

TU

RN

UN

RE

G_

PW

R_

RE

TU

RN

RE

G5

_P

WR

_R

ET

UR

N

RE

G5

_P

WR

_R

ET

UR

N

RE

G5

_P

WR

_S

UP

PL

Y

RE

G5

_P

WR

_S

UP

PL

Y

R6

10k

R5

10k

R7

10k

+3

.3V

TM

S

TD

IT

CK

TD

O

TX

D0

RX

D0


12

34

ABCD

43

21

D C B A

SH

TO

F6

3V

ER

SIO

N:

0.0

A

DW

G.N

O.:

ES

L0

3-1

04

02

0-2

13

-03

-0.0

A

TIT

LE

:

Bru

sh

less D

C M

oto

r E

lectr

on

ics

CO

PY

RIG

HT

RE

SE

RV

ED

FIL

E:

BLD

Cdri

verc

ontr

olle

r.S

ch

RE

VIS

ION

DE

TA

ILS

VE

RS

ION

C/N

OT

ED

RW

NC

HK

AP

PR

S.E

NG

DA

TEESL

ISE

NS

E2

5

E/A

IN

(+

)1

VR

EF

2

ISE

NS

E3

ISE

NS

E1

4

DIR

6

SP

EE

D-I

N7

H1

8

H2

9

H3

10

RC

-BR

AK

E21

QU

AD

SE

L22

0V

-CO

AS

T23

SS

TA

RT

24

RC

-OS

C25

PW

M IN

26

E/A

OU

T27

E/A

IN

(-)

28

PD

C12

PD

B13

PD

A14

GND15

PU

C16

PU

B17

PU

A18

VCC19 T

AC

H-O

UT

20

PWR VCC11

U8

UC

2625

+C

42

4.7

uF

, 35V

C35

100nF

VD

D6

INA

2

INB

4

GN

D3

OU

TA

7

OU

TB

5

U9

MA

X4427E

SA

VD

D6

INA

2

INB

4

GN

D3

OU

TA

7

OU

TB

5

U10

MA

X4427E

SA

+12V

+12V

+12V

C43

4n7

C34

100nF

C23

1nF

CO

AS

TD

IRC

MD

QU

AD

SE

L

+5V

C44

10nF

+12V

TP

16

TP

17

TP

20

TP

14

TP

15

TP

19

TP

29

PD

AT

P30

PD

BT

P31

PD

C

TP

13

TP

18

PD

AP

DB

PD

C

PD

AP

DB

PD

C

PU

A

PU

BP

UC

+12V

+12V

+12V

HA

LL3

HA

LL2

HA

LL1

R37

150k

+5V

ISE

NS

E1

ISE

NS

E2

DIR

CM

DP

WM

_D

IR

4 56

U11B

HC

86

TP

32

SP

EE

D_P

PR

TP

35

DIR

CM

D

1 23

14 7V+

V-

U11A

HC

86

+5V

UC

2625

MA

X4427E

SA

's

SP

EE

D_P

PR

PW

M_D

IR

E/A

_O

UT

R32

68k, 1%

R30

10k

R31

20k

+5V

RE

F

SP

EE

D_O

UT

SP

EE

D_P

PR

R36

0E

PG

ND

Ref

er a

lso t

o g

ener

ic_node.

Sch

C46

220pF

+5V

C45

220pF

TP

37

CH

AN

NE

L_A

TP

36

CH

AN

NE

L_B

+5V

TP

38

SP

EE

D_D

IR

VC

C14

GN

D7

D2

CLK

3

nS4

nR1

Q5

nQ

6

U12A

74H

CT

74

SP

EE

D_D

IRS

PE

ED

_D

IR

CH

AN

NE

L_A

CH

AN

NE

L_B

VC

C14

GN

D7

D12

CLK

11

nS10

nR13

Q9

nQ

8

U12B

74H

CT

74

+5V

C47

100nF

+5V

74H

C74

CH

AN

NE

L_A

CH

AN

NE

L_B

C32

100nF

+C

40

4.7

uF

, 35V

+12V

C33

100nF

+C

41

4.7

uF

, 35V

+12V

C36

100nF

+5V

74H

C86

XC

FS

econ

d D

raft

12-1

1-20

03

4-O

ct-2

004

Prin

t

CH

AN

NE

L_A

CH

AN

NE

L_B

R33A

10k

R33B

10k

R33C

10k

R34A

10k

R34B

10k

R34C

10k

+12V

PG

ND

+12V

PG

ND

+5V

+5V

GN

DG

ND

EN

CO

DE

R

QU

AD

SE

LP

WM

_Q

UA

DS

EL

TP

33

QU

AD

SE

L

9

10

8U

11C

HC

86

PW

M_Q

UA

DS

EL

CO

AS

TP

WM

_C

OA

ST

TP

34

CO

AS

T

12

13

11

U11D

HC

86

PW

M_C

OA

ST

CH

AN

NE

L_A

CH

AN

NE

L_B


12

34

ABCD

43

21

D C B A

SH

TO

F6

4V

ER

SIO

N:

0.0

A

DW

G.N

O.:

ES

L0

3-1

04

02

0-2

13

-04

-0.0

A

TIT

LE

:

Bru

sh

less D

C M

oto

r E

lectr

on

ics

CO

PY

RIG

HT

RE

SE

RV

ED

FIL

E:

BLC

Dm

osfe

ts.S

ch

RE

VIS

ION

DE

TA

ILS

VE

RS

ION

C/N

OT

ED

RW

NC

HK

AP

PR

S.E

NG

DA

TEESL

+12V

C38

100nF

+C

53

22uF

, 25V

PG

ND

C37

100nF

PG

ND

ISE

NS

E1

ISE

NS

E2

PH

AS

E C

PH

AS

E B

PH

AS

E A

PG

ND

PG

ND

PG

ND

TP

24

TP

25

TP

26

TP

28

TP

27

TP

40

+12V

TP

39

PG

ND

R44

240

R45

240

R42

0.2

7R

43

0.2

7

PD

A

PD

B

PD

C

PU

A

PU

B

PU

C

+12V

ISE

NS

E1

ISE

NS

E2

PH

AS

E A

PH

AS

E B

PH

AS

E C

HA

LL

A

HA

LL

B

HA

LL

C

+5V

GN

D

C52

2n2

C51

2n2

C50

2n2

TP

23

TP

22

TP

21

HA

LL3

HA

LL2

HA

LL1

HA

LL

1H

AL

L2

HA

LL

3

G

SD

2

7,8

1

Q3A

IRF

7319

GS D

4

5,6

3Q

2B

IRF

7319

GS D

4

5,6

3Q

3B

IRF

7319

G

SD

2

7,8

1

Q2A

IRF

7319

GS D

4

5,6

3Q

1B

IRF

7319

G

SD

2

7,8

1

Q1A

IRF

7319

XC

FS

econ

d D

raft

12-1

1-20

03

4-O

ct-2

004

Prin

t

R35A

10k

R35B

10k

R35C

10k

C49A

10nF

C49B

10nF

C49C

10nF

C48A

10nF

C48B

10nF

C48C

10nF

R40A

100k

R40B

100k

R40C

100k

R38C

10

R38B

10

R38A

10

R39A

10

R39B

10

R39C

10

R41D

1k

R41C

1k

R41B

1k

+12V

PG

ND

+12V

PG

ND

+5V

+5V

GN

D

D8

30B

Q100

D9

30B

Q100

D10

30B

Q100

D3

30B

Q100

D5

30B

Q100

D7

30B

Q100D

230B

Q100

D6

30B

Q100

D4

30B

Q100

HA

LLA

+5V

HA

LLB

PH

AS

E A

HA

LLC

PH

AS

E B

GN

DP

HA

SE

C


12

34

ABCD

43

21

D C B A

SH

TO

F6

5V

ER

SIO

N:

0.0

A

DW

G.N

O.:

ES

L0

3-1

04

02

0-2

13

-05

-0.0

A

TIT

LE

:

Bru

sh

less D

C M

oto

r E

lectr

on

ics

CO

PY

RIG

HT

RE

SE

RV

ED

FIL

E:

BLD

Ccurr

ents

ense.S

ch

RE

VIS

ION

DE

TA

ILS

VE

RS

ION

C/N

OT

ED

RW

NC

HK

AP

PR

S.E

NG

DA

TEESL

567

48

U7B

LM

358A

48

2 31

U7A

LM

358A

R26

12k, 1%

R24

39k

R23 0

R21 0

R28

2M

2C

19

1uF

R25

39k

R27

12k, 1%

R29

2M

2 C20

1uF

E/A

_O

UT

R18

100k, 1%

R16

10k

R17

10k

R19

100k, 1%

R20

10k

C31

100nF

C26

100nF

C29

100nF

ISE

T

TP

8IS

EN

SE

1T

P9

ISE

NS

E2

TP

10

I_M

ON

TP

11

ISE

T

TP

12

E/A

_O

UT

I_M

ON

ISE

NS

E2

ISE

NS

E1

+5V

+5V

ISE

T

ISE

T_M

ON

E/A

_O

UT

LM

358A

C27

100nF

+5V

RE

F

2.5

V R

EF

+5V

RE

F

ISE

T_M

ON

I_M

ON

XC

FS

econ

d D

raft

12-1

1-20

03

17-N

ov-2

004

Prin

t

+5V

+5V

AG

ND

AG

ND

NC

1

IN2

NC

3

GN

D4

NC

5O

UT

6N

C7

NC

8D

1

MA

X6166

C39

100nF

TP

41

2.5

V R

EF


12

34

ABCD

43

21

D C B A

SH

TO

F6

6V

ER

SIO

N:

0.0

A

DW

G.N

O.:

ES

L0

3-1

04

02

0-2

13

-06

-0.0

A

TIT

LE

:

Bru

sh

less D

C M

oto

r E

lectr

on

ics

CO

PY

RIG

HT

RE

SE

RV

ED

FIL

E:

BLC

Ddri

verI

ntr

fs.S

ch

RE

VIS

ION

DE

TA

ILS

VE

RS

ION

C/N

OT

ED

RW

NC

HK

AP

PR

S.E

NG

DA

TEESL

XC

FS

econ

d D

raft

12-1

1-20

03

4-O

ct-2

004

Prin

t

WP

24 W

P28

WP

25 W

P29

WP

26 W

P30

WP

27 W

P31

BL

DC

MO

TO

R

HA

LLA

+5V

HA

LLB

PH

AS

E A

HA

LLC

PH

AS

E B

GN

DP

HA

SE

C

Pow

er

CA

N

WP

1W

P9

WP

2W

P10

WP

3W

P11

WP

4W

P12

WP

5W

P13

WP

6W

P14

WP

7W

P15

WP

8

CA

NH

_B

CA

NL_B

AD

CS

_S

YN

C_S

HIE

LD

CA

N_S

HIE

LD

_B

AD

CS

_S

YN

C_L

CA

NL_B

AD

CS

_S

YN

C_H

CA

NH

_B

CA

NH

_A

CA

NL_A

CA

N_S

HIE

LD

_A

CA

NL_A

CA

NH

_A

WP

23

WP

19W

P22

WP

18W

P21

WP

17W

P20

WP

16

UN

RE

G_P

WR

_S

UP

PLY

UN

RE

G_P

WR

_S

UP

PLY

UN

RE

G_P

WR

_R

ET

UR

N

UN

RE

G_P

WR

_R

ET

UR

N

RE

G5_P

WR

_R

ET

UR

N

RE

G5_P

WR

_R

ET

UR

N

RE

G5_P

WR

_S

UP

PLY

RE

G5_P

WR

_S

UP

PLY

12

34

56

78

910

P2

JTA

G

+3.3

V

TM

ST

DI

TC

KT

DO

JT

AG

TM

ST

DI

TC

KT

DO

TX

D0

RX

D0

1 2 3

P4

UA

RT

0

TX

D0

Ser

ial

Inte

rface

RX

D0

+3.3

V+

3.3

VG

ND

GN

D

CH

AN

NE

L_A

CH

AN

NE

L_B

WP

32

WP

34 W

P33

WP

35

CH

AN

NE

L_A

+5V

CH

AN

NE

L_B

+5V

+5V

Op

tica

l In

crem

enta

l E

nco

der

TX

D0

RX

D0

1 2 3

P8

UA

RT

1

TX

D1

RX

D1

Appendix C

CMG Setup for Practical Tests

The practical tests of the CMG were done on an air bearing table, which consists of a table

with a glass surface. The CMG with electronics are placed on an aluminium frame which

has three carbon nozzles and a small gas canister. The canister is filled with nitrogen to

a pressure of about 15 MPa. This is slowly released through the carbon nozzles, which

then lifts the frame from the glass surface, leaving it almost frictionless and very sensitive

to any disturbance. For this reason the glass surface has to be as level as possible and

dustfree. The aluminium frame with gas canister is shown in Figure C.1.

Figure C.1: Picture of aluminium frame with gas canister and nozzles

82

APPENDIX C - CMG SETUP FOR PRACTICAL TESTS 83

The air bearing table has a metal frame with 15 legs of which the height are seperately

adjustable. The legs can be extended or shortened by adjusting a nut. A cable is attached

to the frame next to each leg. This is used to pull the frame down and keep it in place

once the height of the table has been adjusted. A 35mm thick Superwood surface is placed

on top of the metal frame. This is covered by a layer of felt to protect the glass surface.

The glass surface is then placed on top of the felt. A picture of the table is shown in

Figure C.2.

Figure C.2: Picture of the table with glass surface

A miniature fibre optic rate sensor [11] was used in the CMG setup to measure the

angular rotation of the CMG system. The sensor transmits an angle increment to the

microcontroller each time it receives a trigger pulse. One angle increment has a resolution

of 0.00024414 degrees. The angle increment is transmitted as RS422 by the sensor and has

to be converted to RS232 before the microcontroller can receive the data. The measured

data is then transmitted from the microcontroller to a PC via an RF link to minimize the

external disturbances.

A 12V battery was used to provide power to the BLDC motor and RF link. An LM317

adjustable regulator was used to adjust the 12V supply to 5V for the rate sensor, stepper

motor and all the electronics. A diagram of the setup is shown in Figure C.3 and a picture

in Figure C.4.

APPENDIX C - CMG SETUP FOR PRACTICAL TESTS 84

Stepper

MotorRate

Gyro

RF Link

BLDC

Motor

12V

Power

Supply

CMG

Electronics

PC

RF Link

Figure C.3: Diagram of the CMG setup

Figure C.4: Picture of the CMG setup

Appendix D

CMG Interface Program

The program, CMGprobe, is used as interface between the user and the Control Moment

Gyro. The CMG is fully controllable from this program and has functions such as load

and save. A picture of the control panel is shown in Figure D.1. The control panel consists

mainly of the following four parts: stepper motor control, BLDCM control, data sampling

and data display.

D.1 Stepper Motor Control

The stepper motor control panel is used to control the gimbal maneuvers. The rotation

angle of the gimbal is controlled by the Gimble Rotation window. A maximum angle

increment of 90 degrees is allowed to protect the gimbal. The gimbal rate can be modified

in the Steps/Sec window. A minimum of 10 steps/sec and a maximum of 500 steps/sec is

allowed. The direction of the gimbal rotation is set by the Clockwise bit. The Mirror bit

is used when the gimbal needs to rotate by an angle increment and back. This process

repeats until it is stopped. When the gimbal needs to rotate by a certain amount of

stepper motor steps, the Steps window is used. The total angle deviation of the gimbal

is displayed in the Gimbal Angle window.

The definition of the buttons are as follows:

• Start - When the Start button is pressed, the gimbal will rotate by the angle specified

in the Gimbal Rotation window at a rate of Steps/Sec and in the direction selected

by the Clockwise bit.

• Stop - The Stop button will immediately stop the gimbal.

• Step - Pressing the Step button will cause the gimbal to step the amount of steps

85

APPENDIX D - CMG INTERFACE PROGRAM 86

specified in the Steps window at a rate of Steps/Sec in the selected direction.

• Go to Zero - This button cause the gimbal to rotate back to the zero degree gimbal

angle position.

• Zero Angle - This button resets the Gimbal Angle to zero.

• Start Sending - When this button is pressed, gimbal excursion angles are read in

from the file specified in the window. This is used to implement a ’Moving Demand’.

Each 100 ms the new excursion angle will be sent to the microcontroller and the

needed gimbal rate is calculated from the difference between two following angles

excursions.

D.2 BLDCM Control

The speed of the BLDC motor is controlled with the BLDCM Control panel. The current

loop of the BLDCM control can be accessed directly from this panel for current control.

A definition of the buttons are given below:

• Output Current Dem - The current loop command is directly controllable with this

button. The input range is from -4095 to 4095 where a negative value means rotation

in the anti-clockwise direction.

• Switch RW On/Off - Power to the momentum wheel is switched on and off, de-

pending on the RW On bit.

• Send Wheel Speed - A new speed command for the momentum wheel is sent when

the Send Wheel Speed button is pressed. The speed range for the momentum wheel

is -5000 to 5000 rpm with an accuracy of 0.6 rpm.

• Send Controller Gains - The controller gains of the speed loop can be changed by

sending new gain values. The default values are: K1 = 91 and K2 = 74.

D.3 Data Sampling

The telemetry received from the microcontroller can be sampled and stored in a file. This

is done by entering the path in the Files Directory window and a name for the file in the

File Name window. When the Start Sampling button is pressed, the telemetry received

will be written to the specified file. Sampling will stop when the Stop Sampling button

is pressed.


D.4 Data Display

The received telemetry is displayed in the Data Received window. It includes the mo-

mentum wheel speed, gimbal angle excursion, package number and the gyro angle. The

correct comport for the serial link can also be set on this panel.


Figure D.1: Picture of the CMGprobe control panel

Appendix E

Matlab Simulation Design and

S-function Code

89

APPENDIX E - MATLAB SIMULATION DESIGN AND S-FUNTION CODE 90

Uc

Pitch

Roll

Yaw

wheel speed

gim

b a

ngle

Torq

ue c

om

m

CM

G H

_H

do

t

Eu

ler

An

gle

s

Qu

ate

rnio

ns

W_

Sa

t

Sate

llite

Contr

olD

erivation

S-F

unction

Co

mm

an

d T

orq

ue

W_

Sa

t

CM

G d

ata

Gim

b r

ate

s

Req g

imb r

ate

s

Req G

imb r

ate

-K-

R2D

8

-K-

R2D

2

-K-

R2D

1Q

uate

rnio

ns

Quate

rnio

n d

em

and

Pa

ram

ete

rs

Input P

ara

mete

rs

In1

gim

b

wh

ee

l sp

ee

d

Gim

b a

ngle

s

Eule

r A

ngle

s

em

Qu

ate

rnio

ns

Qu

ate

rn c

om

m

W_

sa

t

Co

mm

an

d T

orq

ue

Contr

ol B

lock

Body R

ate

dem

and

3se

t G

imb

ra

tes

H_

Hd

ot

To

tal

da

ta

3 S

ets

of V

SC

MG

s

Figure E.1: Picture of Matlab Simulation


function [sys,x0,str,ts] = ControlDerivation(t,x,u,flag)

switch flag, case 0, [sys,x0,str,ts]=mdlInitializeSizes; case 3, sys=mdlOutputs(t,x,u); case 1, 2, 4, 9, % Unused flags sys=[]; otherwise error(['Unhandled flag = ',num2str(flag)]);end% end sfuntmpl

%=========================================================================% mdlInitializeSizes% Return sizes, initial conditions, and sample times for the S-function.%=========================================================================%function [sys,x0,str,ts]=mdlInitializeSizessizes = simsizes;sizes.NumContStates = 0;sizes.NumDiscStates = 0;sizes.NumOutputs = 7;sizes.NumInputs = 9;sizes.DirFeedthrough = 1;sizes.NumSampleTimes = 1; % at least one sample time is needed

sys = simsizes(sizes);x0 = [];str = [];ts = [0 0];% end mdlInitializeSizes

%=========================================================================% mdlOutputs% Return the block outputs.%=========================================================================function sys=mdlOutputs(t,x,u)

%Constantsas1 = 0.005; % Torque/Is =versnelling (0.1/20)= 0.005as2 = as1;as3 = as1;as4 = as1;deltaT12 = 5; % Setttling time after slew 1 (seconds)deltaT34 = deltaT12;

%Inputtmid = u(1);ps2 = u(2)*pi/180;wimage = [u(3) u(4) u(5)]'*pi/180;qmid = [u(6) u(7) u(8) u(9)]';

%Derivations%Slew2size_w = sqrt(wimage(1)^2+wimage(2)^2+wimage(3)^2);n_slew2 = wimage/size_w; %Eq5-1qs1s2 = [n_slew2(1)*sin(ps2/2) n_slew2(2)*sin(ps2/2) n_slew2(3)*sin(ps2/2) cos(ps2/2)]'; %Eq5-2qs1 = [qs1s2(4) -qs1s2(3) qs1s2(2) -qs1s2(1); qs1s2(3) qs1s2(4) -qs1s2(1)


-qs1s2(2); -qs1s2(2) qs1s2(1) qs1s2(4) -qs1s2(3); qs1s2(1) qs1s2(2) qs1s2(3) qs1s2(4)] * qmid; %Eq5-3Dts2 = ps2/size_w + 0.5*size_w/as2; %Eq5-4ts2start = tmid-Dts2; %Eq5-5%Slew1ps1 = 2*acos(qs1(4)); %Eq5-11n_slew1 = [qs1(1) qs1(2) qs1(3)]'/sin(ps1/2); %Eq5-12Dts1 = 2*sqrt(ps1/as1); %Eq5-13ts1start = tmid - Dts2 - deltaT12 - Dts1; %Eq5-14%Slew3Dtpostmid = 0.5*ps2/size_w; %choose postmid half the distance of slew2Dts3 = Dtpostmid + size_w/as3; %Eq5-19ps3 = size_w*(Dtpostmid + size_w/as3) -0.5*size_w^2/as3; %Eq5-20%Slew4ts4start = tmid + Dts3 + deltaT34; %Eq5-26 ts3start = tmidqs2s3 = [n_slew2(1)*sin(ps3/2) n_slew2(2)*sin(ps3/2) n_slew2(3)*sin(ps3/2) cos(ps3/2)]'; %Eq5-27qs3 = [qs2s3(4) qs2s3(3) -qs2s3(2) qs2s3(1); -qs2s3(3) qs2s3(4) qs2s3(1) qs2s3(2); qs2s3(2) -qs2s3(1) qs2s3(4) qs2s3(3); -qs2s3(1) -qs2s3(2) -qs2s3(3) qs2s3(4)]*qmid; %Eq5-28ps4 = 2*acos(qs3(4)); %Eq5-29n_slew4 = -[qs3(1) qs3(2) qs3(3)]'/sin(ps4/2); %Eq5-30Dts4 = 2*sqrt(ps4/as4); %Eq5-31

if ((t >= ts1start) && (t < ts2start)) %Slew1 if (t < (ts1start + 0.5*Dts1)) Ps1t = 0.5*as1*(t - ts1start)^2; %Eq5-15 Pds1t = as1*(t - ts1start); %Eq5-16 elseif ((t >= (ts1start + 0.5*Dts1)) && (t < ts1start + Dts1)) Ps1t = 2*sqrt(ps1*as1)*(t - ts1start) - 0.5*as1*(t - ts1start)^2 -ps1; %Eq5-15 Pds1t = 0.5*as1*Dts1 - as1*(t - ts1start - 0.5*Dts1); %Eq5-16 elseif (t >= ts1start + Dts1) Ps1t = ps1; %Eq5-15 Pds1t = 0; %Eq-16 end; qDt = [n_slew1(1)*sin(Ps1t/2) n_slew1(2)*sin(Ps1t/2) n_slew1(3)*sin(Ps1t/2) cos(Ps1t/2)]'; %Eq5-17 wDt = Pds1t*n_slew1; %Eq5-18

elseif ((t >= ts2start) && (t < tmid)) %Slew2 if (t < (ts2start + size_w/as2)) Ps2t = 0.5*as2*(t - ts2start)^2; %Eq5-6 Pds2t = as2*(t - ts2start); %Eq5-7 else Ps2t = size_w*(t - ts2start - 0.5*size_w/as2); %Eq5-6 Pds2t = size_w; %Eq5-7 end qslew2t = [n_slew2(1)*sin(Ps2t/2) n_slew2(2)*sin(Ps2t/2) n_slew2(3)*sin(Ps2t/2) cos(Ps2t/2)]'; %Eq5-8 qDt = [qslew2t(4) qslew2t(3) -qslew2t(2) qslew2t(1); -qslew2t(3) qslew2t(4) slew2t(1) qslew2t(2); qslew2t(2) -qslew2t(1) qslew2t(4) qslew2t(3); -qslew2t(1) -qslew2t(2) -qslew2t(3) qslew2t(4)] * qs1; %Eq5-9 wDt = Pds2t*n_slew2; %Eq5-10

elseif ((t >= tmid) && (t < ts4start)) %Slew3 if (t < (tmid + Dtpostmid)) Ps3t = size_w*(t - tmid); %Eq5-21 Pds3t = size_w; %Eq5-22


elseif ((t >= (tmid + Dtpostmid)) && (t < (tmid + Dts3))) Ps3t = size_w*(t - tmid) -0.5*as3*(t - tmid -Dtpostmid)^2; %Eq5-21 Pds3t = size_w - as3*(t - tmid -Dtpostmid); %Eq5-22 elseif (t > (tmid + Dts3)) Ps3t = ps3; %Eq5-21 Pds3t = 0; %Eq5-22 end qslew3t = [n_slew2(1)*sin(Ps3t/2) n_slew2(2)*sin(Ps3t/2) n_slew2(3)*sin(Ps3t/2) cos(Ps3t/2)]'; %Eq5-23 qDt = [qslew3t(4) qslew3t(3) -qslew3t(2) qslew3t(1); -qslew3t(3) qslew3t(4) qslew3t(1) qslew3t(2); qslew3t(2) -qslew3t(1) qslew3t(4) qslew3t(3); -qslew3t(1) -qslew3t(2) -qslew3t(3) qslew3t(4)] * qmid; %Eq5-24 wDt = Pds3t*n_slew2; %Eq5-25

elseif (t >= ts4start) %Slew4 if (t < (ts4start + 0.5*Dts4)) Ps4t = 0.5*as4*(t - ts4start)^2; %Eq5-32 Pds4t = as4*(t - ts4start); %Eq5-33 elseif ((t >= (ts4start + 0.5*Dts4)) && (t < (ts4start + Dts4))) Ps4t = 2*sqrt(ps4*as4)*(t - ts4start) - 0.5*as4*(t - ts4start)^2 -ps4; %Eq5-32 Pds4t = 0.5*as4*Dts4 - as4*(t - ts4start - 0.5*Dts4); %Eq5-33 elseif (t > (ts4start + Dts4)) Ps4t = ps4; %Eq5-32 Pds4t = 0; %Eq5-33 end qslew4t = [n_slew4(1)*sin(Ps4t/2) n_slew4(2)*sin(Ps4t/2) n_slew4(3)*sin(Ps4t/2) cos(Ps4t/2)]'; %Eq5-34 qDt = [qslew4t(4) qslew4t(3) -qslew4t(2) qslew4t(1); -qslew4t(3) qslew4t(4) qslew4t(1) qslew4t(2); qslew4t(2) -qslew4t(1) qslew4t(4) qslew4t(3); -qslew4t(1) -qslew4t(2) -qslew4t(3) qslew4t(4)] * qs3; %Eq5-35 wDt = Pds4t*n_slew4; %Eq5-36

else qDt = [0 0 0 1]'; wDt = [0 0 0]';end

sys(1) = qDt(1);sys(2) = qDt(2);sys(3) = qDt(3);sys(4) = qDt(4);sys(5) = wDt(1);sys(6) = wDt(2);sys(7) = wDt(3);% end mdlOutputs

Appendix F

RF Link

The RF link consist of TX2 and RX2 data link modules which are miniature UHF FM

radio transmitters and receivers. The modules are designed by Radiometrix and the

product numbers are TX2-433-40-5V (transmitter module) and RX2-433-40-5V (receiver

module).

TX2 Transmitter

The TX2 transmitter module is a two stage, SAW controlled FM transmitter operating

between 2V and 6V and is available in 433.92 MHz. The TX2 module delivers nominally

+9dBm from a 5V supply at 12mA. The module measures 32 x 12 x 3.8 mm. Figure F.1

displays the circuit diagram of the transmitter module.

TXD

Vcc

1 2 3 4 5

100 nF 1k

15k

VP0106

Switch

RadiometrixTX2 UHF Transmitter

Antenna

Figure F.1: TX2 Circuit

The TXD input will accept serial digital data (0V to 5V) at a maximum rate of 40 kbps.

The Switch input turns the transmitter off when in a high level state (5V) and turns the

transmitter on when in a low level state (0V). A helix antenna is used as described below.

The supply voltage, Vcc, is 5V and should have a voltage ripple of less than 0.1Vp-p.

94

APPENDIX F - RF LINK 95

RX2 Receiver

The RX2 receiver module is a double conversion FM superhet receiver capable of handling

data rates of up to 40kbps. The SIL style RX2 receiver measures 48 x 17.5 x 4.5 mm. It

will operate from a supply of 3V to 6V and draws 14mA when receiving. It has a fast-

acting carrier detect and a power-up enable time of less than 1ms. This allows effective

duty cycle power saving and a -107 dBm sensitivity. The circuit diagram of the receiver

module is shown in Figure F.2.

RXD

Vcc

1 2 3 4 5

47k

2N3906

RadiometrixRX2 UHF Receiver

6 7

NAND Gates

Antenna

Figure F.2: RX2 Circuit

A helix antenna is used as described below. The Carrier Detect (pin 3) drives an ex-

ternal PNP transistor to obtain a logic level carrier detect signal. This signal and the

digital data output signal (pin 7) is fed to a quad 2-input NAND schmitt trigger package

(CD74HCT132) to obtain the RXD output signal. This RXD signal is always in a high

level state (5V) when the Carrier Detect is in a low level state. Thus will data only be

received when a carrier cygnal is detected.

Voltage Regulator Design

An LM317 3-terminal adjustable regulator is used to give a constant voltage of 5V to the

Radiometrix modules. A few features of the LM317 are:

• Guaranteed 1% output voltage tolerance.

• Guaranteed maximum 0.01%/V line regulation.

• 80 dB ripple rejection.

The LM317 circuit is shown in Figure F.3. Tantalum capacitors are used for the 10 µF

capacitors. The input voltage to the regulator can vary between 7V and 15V. If it is more


ADJ

Vin V0

V (7-15V)in

1 uF

82010 uF

27010 uF

Vcc (5V)

LM317T

Figure F.3: Voltage Regulator Circuit

than 15V, a heatsink will be needed.

Antenna Design

A Helical antenna is used with the following parameters:

• 0.5 mm enameled copper wire.

• close wound on 3.2 mm diameter former.

• 433 MHz = 24 turns.

RF Link Unit

The RF link consists of two units; one unit at the PC side and one unit in the CMG

system. Each unit has a receiver module, transmitter module and a voltage regulator. A

picture of the unit is shown in Figure F.4. When data is not transmitted, the Switch input

should be in a high level state to ensure that the transmitter is off. Since the receiver

module is next to the transmitter, all the data transmitted will be received by the receiver

on the same module. For this reason it is convenient to make use of package ID’s.

The receiver module has a data slicer which uses the average received cygnal to determine

if the input is high or low. If the received cygnal is low or high for a too long period, the

average will hit the rail of the data slicer and then the data received will be incorrect.

To overcome this problem, every second byte which is transmitted should be the inverse

of the previous byte. The average voltage will then stay in the middle. Each time the

transmitter is switched on, the byte 10101010 should be sent repeatedly for the first 5 ms

to ensure the average of the data slicer is in the region of 2.5V.


Figure F.4: Picture of the RF Link Unit

Appendix G

Datasheets

G.1 Brushless DC Motor

G.2 Stepper Motor

G.3 Stepper Motor Gearhead

G.4 Stepper Motor Controller

98

APPENDIX G - DATASHEETS 99

1234

56789

10111213

1415161718

1920

21

2223

24

252627

282930

UNRP2 max.

max.

noIoMHCoCv

knkEkMkI

n/ ML

mJ

max.

Rth 1 / Rth 2

w1 / w2

=

ne max.

Me max.

Ie max.

Volt

W%

rpmAmNmmNmmNm/rpm

rpm/VmV/rpmmNm/AA/mNm

rpm/mNmµHmsgcm2

.103rad/s2

K/Ws

°C

NNN

mmmm

g

rpmmNmA

3056 ... B

21,5 mNm

12 24 36 481,6 7,0 13,7 24,548 49 49 4973 73 74 74

8 790 8 200 8 840 8 7400,168 0,075 0,056 0,04295 93 99 1000,91 0,91 0,91 0,911,4 .10-4 1,4 .10-4 1,4 .10-4 1,4 .10-4

750 350 251 1861,334 2,861 3,981 5,37412,74 27,32 38,02 51,320,078 0,037 0,026 0,019

94 90 91 89160 720 1 400 2 52013 13 13 1213,6 13,6 13,6 13,670 68 73 73

3,3 / 9,419 / 1 034

– 30 ... +125

72 / 5118 / 1262

0,0150

190

28 000 28 000 28 000 28 00020,7 21,4 21,2 21,51,94 0,93 0,66 0,50

3056 K 012 B 024 B 036 B 048 B

5,0 10,0 25,020,015,0

5 000

10 000

15 000

20 000

25 000

30 000

M UmNmI

n UrpmI

ne max. = 28 000 rpm

n = 22 000 rpm

Me max. = 21,5 mNm

00

49 Watt

30/1, 38/1, 38/2

5500, 5540

Series

Brushless DC-ServomotorsElectronic Commutation

Nominal voltageTerminal resistance, phase-phaseOutput power 1)

Efficiency

No-load speedNo-load current (with shaft ø 4,0 mm)Stall torqueFriction torque, staticFriction torque, dynamic

Speed constantBack-EMF constantTorque constantCurrent constant

Slope of n-M curveTerminal inductance, phase-phaseMechanical time constantRotor inertiaAngular acceleration

Thermal resistanceThermal time constant

Operating temperature range

Shaft bearingsShaft load max.:– radial at 3000/20000 rpm (7,4 mm from mounting flange)– axial at 3000/20000 rpm (axial push-on only)– axial at standstill (axial push-on only)Shaft play:– radial– axial

Housing materialWeightDirection of rotation

Speed up to 2)

Torque up to 1) 2)

Current up to 1) 2)

1) at 22 000 rpm2) thermal resistance Rth 2 by 55% reduced

ball bearings, preloaded

aluminium, black anodized

electronically reversible

Recommended values

Recommended area for continuous operation

For combination withGearheads:

Encoders:

Drive Electronics:refer to “Combination Chart”, pages 14-15


3056 K ... B

3056 K ... B - K312

N

S

AABBCC+5VGND

ø30 ±0,1 ø13 -0,005 0

A

ø4+0,003-0,002

ø0,050,02

19

45

M2 5

12,6max.15

1,4

14

56

±0,3

±0,3

4x90

5

A

7

M1,6

3x120

20,9

3ø4

+0,003-0,002

9,6 57,8±0,3

K1000:

K1155:

Cable and connection information

Function ColourHall sensor greenPhase brownHall sensor bluePhase orangeHall sensor greyPhase yellowLogical supply redLogical black Coil winding 3 x 120°

Connection

with rear end shaft

deep

CableSingle wires, material PTFELength 300 mm ± 15 mm3 conductors, AWG 205 conductors, AWG 26

deep

Options

Motors in autoclavable version.

Motors for operationwith Motion Controller MCBL 2805.


AM 1524V 3 V 6 V 12 V 24 A 0,25

7

4x M1,6 x 1,5

60° 60°

ø10

ø14,5 ø15 ø1,5

ø6

7,5

1

16,5

2,54

4 BA

1

4x ø0,780°

2,54

108,5

10

1 AB

4

±0,0

5

-0,070

-0,020

-0,008-0,004

1,2

ø14,5

16,5

ø6-0,020

4,3

1 2,4

DIN 58400

x=+0,35Z=9 m=0,2

ø2,38 -0,0150

ø6 -0,020

8,1

1

DIN 58400Z=21 m=0,2

ø4,64 -0,08-0,04

ø13-0,030

5,15

1,75

DIN 58400

x=+0,0496Z=15 m=0,3

ø5,248 -0,113-0,078

±0,15,15

1,75

DIN 58400

x=+0,2156Z=9 m=0,3

ø3,587-0,113-0,078

±0,1

ø13-0,030

(mNm)

0

1

3

10000

0,1

0,2

(W)

(Step/s)400(rpm)

2

0,3

2000800

30001200

40001600

0,5

1

1,5

(mNm)

0

1

3

5000 100000

(W)

(Step/s)2000 4000(rpm)15000

6000

2

200008000

1 2 3 4Phase A + – – +

Phase B + + – –

CWCCW

1 3 6 12 24 V DC2 10 35 150 590 12,5 Ω3 4,25 15 65 239 5,5 mH4 0,28 0,15 0,075 0,037 0,25 A5 3,1 6 12 24 3,56 6 mNm7 10 mNm8 0,9 mNm9 37 °C/W

10 130 °C11 –40 ... +70 °C12 220 s13 1514 ± 1015 45 ·10-9 kgm2

1617

0,5 6,0 N0,5 3,0 N

1815 12 µm150 ~0 µm

19 12 g20 200 V21 120 Hz22 0,4 ms

Series

Front face view

Mechanical Power (W)

Torque Start/Stop (mNm)

Torque Slew rate (mNm)

Voltage mode (V) Current mode (A)PowerTorquePowerTorque

Torque/Speed curves measured with a load inertia of 10 ·10-9 kgm2

Speed Speed

Stepper MotorsTwo phases, 24 steps per revolution For combination with:

Gearheads: 15A, 15/5, 15/8, 16A, 16/7Encoders: AE 23B8Drive Electronics: AD VL M, AD VM M, AD CM M

Front shaft Solder tag PCB

Round PCB

AM 15242. Rear shaft

Pignon Afor Gearheads15A

for Gearhead16/7

for Gearheads15/5, 15/8, 16A

Pignon Bfor Gearheads15A

CWCCW

A+ –1 2

B

+

–

3

4

Front face view

Rotor

Voltage mode Current modeNominal voltage UN

Phase resistance (at 20°C)Phase inductance (1kHz)Nominal current per phase (both phases ON)Back-EMF amplitude V/k step/sHolding torque 1) (at nominal current in both phases)Holding torque 1) (at twice the nominal current)Residual and friction torqueThermal resistance winding-ambient airWinding temperature tolerated, max.Ambient temperature rangeThermal time constantStep angle (full step) degreeAngular accuracy 2) % of full stepRotor inertiaShaft bearings sintered bronze sleeves ball bearings, preloadedShaft load, max.: (standard) (optional)– radial (3 mm from bearing)– axialShaft play, max.:– radial (0,2N)– axial (0,2N)WeightIsolation test voltageResonance frequencyElectrical time constant

1) with bipolar driver2) 2 phases ON, balanced phase current


AM 1524L2 L1 M max. M max.

g mm mm mNm mNm %22 :1 21 29,9 33,8 60 150 ≠ 6141 :1 21 29,9 33,8 60 150 ≠ 6176 :1 24 32,0 35,9 100 300 = 51

141 :1 24 32,0 35,9 100 150 = 51262 :1 26 34,1 38,0 100 300 ≠ 43485 :1 26 34,1 38,0 100 150 ≠ 43900 :1 28 36,2 40,1 100 300 = 37

1 670 :1 28 36,2 40,1 100 150 = 37

15/8

0,1 Nm

15/8

15/8

5000 rpm0°

≤ 25 N≤ 5 N 2)

≤ 5 N 2)

≤ 0,02 mm= 0 mm 2)

– 30 … + 100 °C

L1

ø15

2

ø16

10,92

M2

2-56UNC

ø14,5 ø16

2,8

ø7

ø3

6,7 4,3

11,9

12,7

14,2

1,5

L2

2x

2x

-0,1 -0,043-0,016

-0,02 0

-0,015 0

-0,012-0,006

±0,2

±0,3

±0,3

±0,3

±0,3

±0,3

±0,5

+0,2(1524)3

3

Series

Spur GearheadsFor combination with:Stepper motor: AM 1524

Housing material metalGeartrain material all steelRecommended max. input speed for:– continuous operationBacklash, (preloaded) 1)

Bearings on output shaft preloaded ball bearingsShaft load, max.:– radial (6,5 mm from mounting face)– axialShaft press fit force, max.Shaft play (on bearing output):– radial– axialOperating temperature range

reduction ratio(nominal)

weightwithoutmotor

withoutmotor

length output torqueefficiencydirection

of rotation(reversible)

continuousoperation

intermittentoperation

Specifications

Zero Backlash

Orientation with respect to motorterminal circuit board is not defined

deep

deep

1) These gearheads are available preloaded tozero backlash only with motors mounted.

2) Limited by the preloaded ball bearings.A higher axial load negates the preload.

withmotor


AD VL M

53,5

22*76,3

1

4

1

12

5214,2 14,2*

4x M2 x 5,3

ARSAPESwitzerland

10,1

5*

ACC

DEC

RUN

STOP

FMA

XFM

IN

48

12 (M1)

*(M2, M3)

0

500

1000

1 2 30

1500

4

2000

3 3 3 V DC14 14 14 V DC

14 16 16 mA400 400 400 mA

0 ... 0,6 0 ... 0,6 0 ... 0,6 V DC1,6 ... 14 1,6 ... 14 1,6 ... 14 V DC

– 10 10– 2 000 2 000

0 ... +70 0 ... +70 0 ... +70 °C22 30 34 g

VL M1 VL M2 VL M3Series

Drive ElectronicsLow Voltage For combination with:

Stepper motor: AM 0820, AM 1020, AM 1524

Motor connector

Commandconnector

GND

Power supply (V+)

Switch

+5V CC GND

DIRF/H

• AD VL M1 basic drive is composed of a translator (full step and half step mode)and a power stage which is in this case in voltage mode.

• AD VL M2 contains the basic drive AD VL M1 and a pulse generator delivering variableclock frequency.

• AD VL M3 contains the basic drive AD VL M1 and a pulse generator with generation of ramps.This circuit can provide a velocity profile to start and stop the stepper motorwith acceleration and deceleration ramps.

The drivers type AD VL M are designed todrive the two phase stepper motors type AM ...3 types of drivers are available:

Velocity profile example

* only valid for the AD VL M2 and AD VL M3

Power supply voltage:- min.- max.

Power supply currentOutput current, max. (for each phase)

Logic input level:- low- high

Direction of rotation CW / CCW CW / CCW CW / CCWStep mode full step / half step full step / half step full step / half step

Step frequency:- min. full step/s- max. full step/s

Operating temperature rangeWeight

General description / Features / Command connector functions

Scale reduced

OPO

Speed(rpm)

Time (s)

4x2,7 (only for M1)

1 I OPO > full step mode, one phase ON (wave)2 I F/H > half step mode; default or low logic level = full step mode, 2 phases ON3 I DIR > ccw ; default or low logic level = cw4 I CLK > external clock input, active on the positive edge of the clock pulse5 I RUN > starts the clock generator *6 I STOP > stops the clock generator *7 I Inhibit > disables the current in both coils of the motor8 O Busy > low level as long as the clock is active *9 – GND > ground potentional: 0 Volt

10 O VCC > +5V output11 – GND > ground potentional: 0 Volt12 I VCO > external voltage input (Voltage Controlled Oscillator) *

Pin I /O Function: active on high logic level

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