1

Smart Materials & Structures

Accurate Torque Control of a Bi-directional Magneto-

rheological Actuator Considering Hysteresis and Friction

Effects

Phuong-Bac Nguyen and Seung-Bok Choi

Smart Structures and Systems Laboratory, Department of Mechanical Engineering, Inha

University, Incheon 402-751, Korea

Corresponding Author

Phone: +82-32-860-7319

Fax: +82-32-1716

Email: seungbok@inha.ac.kr

Homepage: http://www.ssslab.com

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Abstract

This paper presents a novel type of magneto-rheological (MR) actuator called bi-directional magneto-

rheological (BMR) actuator and accurate torque control results considering both hysteresis and friction

compensation. The induced torque of this actuator varies from negative to positive values. As a result, it

can works as either brake or clutch depending on the scheme of current input. In our work, the

configuration of the actuator as well as its driving system is presented first. Subsequently, a congruency

hysteresis based (CBH) model to take account the effect of the hysteresis is proposed. After that, a

compensator based on this model is developed. In addition, the effect of dry friction, which exists

inherently with MR actuators in general, is also considered. In order to assess the effectiveness of the

hysteresis compensator, several experiments on modeling and control of the actuator with different

waveforms are carried out.

Keywords : Magnetorheological (MR) Fluid, MR Brake, MR Clutch, Bi-Directional Actuator, Hysteresis

Compensator, Friction Effect, Torque Control

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1. Introduction

Magneto-rheological (MR) fluid is a functional material whose solidification responses to an application

of a magnetic field. With the appearance of this field, the MR particles, which are distributed randomly at

initial, work as dipole magnets and line-up in such a way that the total potential energy is minimized.

Consequently, a chain-like structure or solidification state of MR fluid is formed. Nowadays, products

featuring MR fluid can easily be found in enormous fields in industry. The reason is that MR fluid

possesses some salient performances such as fast response to input excitation, large range of change of

rheological characteristic and reversibility. Moreover, its price is reasonable due to the facility of

commercialized MR products.

Conventional hydraulic devices are integrated in the passive control system to absorb vibratory energy or

load transmission path of the disturbing vibration. This control type is just suitable in a narrow band of

operation frequency. It has significant limitations in systems where broadband disturbances of highly

uncertainty nature are encountered [1-4]. An outstanding advantage of MR fluid versus conventional

hydraulics is that its rheology can change with the stimulus of a magnetic field. Therefore, it can

substitute them in many applications such as vibration absorbers, dampers, mounts, valves, etc., to

broaden the band of operating frequency of the system. In addition, since brakes and clutches featuring

MR fluid can generate a varying torque, they hold a good candidate as actuators in precise control

systems [5-7]. In general, actuators featuring MR fluids have been used mostly as dissipating or

transmitting devices separately. In detail, in MR brakes, the induced torque is resistant to the external

excitation [5, 8]. Otherwise, in MR clutches, the propulsive torque is generated to drive a system [9, 10].

In our best knowledge, there is no MR actuator that can generate resistant and propulsive torques

simultaneously.

Our objective in this research work is to propose such a novel device which can produce both resistant

and propulsive torques simultaneously. In our work, the actuator consists of two rotors counter to each

other, two coils to generate two magnetic fields independently. In order to transmit the power to the

rotors, a motor and a corresponding gear system to generate two opposite rotations are implemented.

When a coil is excited, the actuator works as a clutch. In reverse sense, when another is excited, it works

as a brake. Consequently, the generated torque can vary from negative to positive value depending on the

excitation scheme of the current input of two coils. In detail, it can generate propulsive, braking and zero

torques. Among these torques, the zero one is almost impossible in current MR actuators due to the

unavoidable effect of dry friction in sealing [11, 12]. Owing to this significant advantage, it is possible to

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apply this actuator in haptic systems for medical surgery, where resistant, propulsive and zero force might

exist to react the motion of the manipulator when it manoeuvers in a patient’s body [13-15] .

The paper is organized as follows. Section 2 provides an introduction to the BMR actuator both in

configuration and principle of working. Due to the effect of inherent hysteresis of the steel component,

the actuator response is highly nonlinear. Consequently, a model considering hysteresis for the actuator is

developed and its corresponding compensator is presented in section 3. Several experimental results are

demonstrated and discussed in section 4. At last, concluding remarks are given in section 5.

2. Configuration of BMR Actuator

Conventional MR brakes or clutches consists of one rotor inside a casing and MR fluid between them.

When the rotor is driven, shear friction between their contact surfaces is generated to resist the relative

motion of the casing and rotor. The amount of the torque due to the shear friction can be changed due to

the variation of the solidification of the MR fluid. The proposed BMR actuator is different from the

conventional one in that there are two rotors in the opposite rotations arranged inside an outer casing as

shown in figure 1. The casing might be either fixed or rotatable. However, the speed of the rotors is set to

be much faster than the casing so that the casing’s speed can be considered to be insignificant. The

distance between the rotors and the casing is a gap filled with MR fluid. The magnetic fields through MR

fluid parts around the rotors are created separately by two copper coils implemented in the actuator. In

addition, in order for two magnetic fields not to interfere to each other, a non-magnetic element is placed

between two parts of the casing. Since two rotors rotate counter to each other, two generated frictions are

in opposite directions. They are driven by a motor via a special gear system. This gear system has one

input and two outputs. Or in other words, it is fed by the motor and transmits to two shafts with opposite

rotations as shown in figure 2.

The shear friction between the casing and the rotors forms the friction torque components. As two

independent current sources are applied to the coils, the magnetic fields are generated in two separated

zones as shown in figure 3. The solidification of the MR fluid at the gap between the rotors and outer

casing occurs promptly. Consequently, there exist two induced friction torque components. The resultant

torque of the BMR actuator is the subtraction of these two components whose direction depends on the

dominance of one of these components. Its form can be expressed as follows:

1 2T T T (1)

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where, 1T and 2T are two induced torque components generated from the rotors 1 and 2. The magnitude

of the resultant torque is the difference of these components due to their opposite directions;

1 2T T T (2)

Based on the geometry of the BMR actuator, the expression for these components can be given in the

following forms.

20

2

2

2

22 , 1,22

RR

S

D

RDai e

b

i iT dD

z ir dr

Where, ai and ei are the shear stresses acting on the MR fluid at the surfaces of rotors ( 1,2)i i and

the faces of the casing whose values can be mathematically expressed by Bingham’s model as follows:

ai y aiK (3)

ei y eiK (4)

where, K is called the consistency; y is the yield stress of MR fluid at the surfaces of rotors and casing

respectively whose magnitude depends on the magnetic intensity. In equation (3), i is the shear rate of

MR fluid on the faces of the rotors whose values can be determined as follows:

, 1,22

R i

f

c

ai iD

t

(5)

, 1,2i c

e

f

it

ri

(6)

where, 1 2, and c are the angular velocities of the rotors 1, 2 and casing.

With the assumption that casing’s angular velocity is insignificant compared to those of the rotors and

substituting equations (3)-(6) to equation (2), the torque components can be obtained in the following

form:

20

2

2

2

22 , 1,22

RR

S

D

RDy

b

i y

Dd rT z dr i

(7)

6

In reality, in order to increase the effectiveness and avoid the energy loss, just only one coil is excited and

the other is de-excited. Consequently, the resultant torque can be expressed as follows with the existence

of dry friction.

1

2

if coil 1 is excited

if coil 2 is excited

f

f

T TT

T T

where, fT is dry friction torque due to sealing which is obtained via experiment.

3. Inverse Compensation of BMR Actuator

It is noted that the magnitude of the yield stress at the different locations in the MR fluid depends on the

magnetic field applied to it. As well known the magnetic is generated as a current source applied to the

copper coils. In summary, the value of the yield stress at the different locations depends on the magnitude

of the current source. In order to obtain this value, several analytical and finite element analysis

approaches have been proposed. However, all these approaches have neglected the hysteresis effect that is

inherent with steel components of the MR device. In this section, an approach to modeling the MR

actuator considering the hysteresis based on experimental data is proposed. In our work, a congruency

based hysteresis (CBH) model, which is applied successfully in modeling a piezoelectric actuator [16,

17], is adopted. In order to keep the consistency of the paper, the CBH model and its corresponding

compensator are reviewed.

3.1 CBH model for BMR actuator

The model is developed based on the assumption that the hysteresis has the congruency property. This

property means that induced minor hysteresis loops of output torque corresponding to the same

consecutive maximum and minimum peaks of input currents are congruent with each other. The model is

derived with two cases of input currents: the increase and the decrease. It is remarkable that the output

torque of the actuator varies from negative to positive value corresponding to the scheme of input currents

applied to two coils. At one time, just only one coil is excited: either coil 1 or 2. Therefore, for facility, a

both-side-range, nominal input current, whose negative (positive) value is equivalent to the excitation one

to coil 1 (2), is presented and considered as an input theoretically. The real input currents applied to two

coils equivalent to this nominal one can be expressed symbolically as follows:

2

1 if 0

if 0c

c n

n

I I I

I I I

(8)

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where, 1 2, ccI I are input currents applied to coil 1, 2, respectively; I is nominal current. At instance, in

the case of monotonic increase of the input nominal current as shown in figure 4, the response of the

corresponding output torque versus the input current is hysteretic as shown in figure 5. In the figure,

0 ,as 1

as and 2

as ; 0 ,ds 1

ds stand for the main and the first-order descending segments, respectively. As

demonstrated in figure 5, the output torque at the instantaneousness [ ]t n can be expressed as follows:

min

2 2( ) ( )ex

aT I T I T (9)

where min

2( )exT I is the torque at the minimum input min

2I obtained in the past; 2

aT is the difference

between the torque at the instantaneousness [ ]t n and min

2t when the response follows in the second-order

segment 2

as . The term 2

aT can be determined thank to the congruency property as follows. The

congruency can be taken advantage that with a higher-order minor hysteresis curve, there always exists a

first-order segment that is congruent with it. This segment departs from the main descending 0

ds as

demonstrated in figure 6 and symbolized min

2( , )aS I I . Consequently, 2

aT is the difference of two values

corresponding to the input currents min

2[ ],nI t I on this first-order segment given by

min min min

2 2 2 2( , [ ]) ( , )a a a

nT S I I t S I I (10)

Substituting equation (10) into equation (9) yields the detail expression of the output torque as follows:

min min min min

2 2 2 2( [ ]) ( ) ( , [ ]) ( , ).a a

n ex nT I t T I S I I t S I I (11)

The generalized form of the output torque while considering the m th order ascending segment can be

induced in the following form

min min min min( [ ]) ( ) ( , [ ]) ( , ).n ex

a a

m m m mnT I t T I S I I t S I I (12)

Consider the monotonic decrease of input excitation case with input excitation and corresponding output

torque as shown in figures 7 and 8. In these figures, 0

as is the main ascending; 1

ds and 2

ds stand

respectively for the first and second-order descending segments. The output torque at the

instantaneousness [ ]t n is expressed as follows:

max

2 2( [ ]) ( )ex

d

nT I t T I T (13)

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where 2

dT is the difference between the torque at the instantaneousness [ ]t n and max

2t when the

response follows on the segment 2

ds . This term can also be obtained indirectly based on this first-order

curve as follows:

max max max

2 2 2 2( , ) ( , [ ]).d d

n

dT S I I S I I t (14)

By substituting equation (14) into equation (13), the expression of the output torque is expressed in detail

in the following form.

max max max max

2 2 2 2( [ ]) ( ) ( , [ ]) ( , ).d d

n ex nT I t T I S I I t S I I (15)

Similarly, the generalized form of the output displacement while considering the m th order descending

segment can be expressed as

max max max max( [ ]) ( ) ( , [ ]) ( , ).n ex

d d

m m m mnT I t T I S I I t S I I (16)

In summary, equation (12) and equation (16) constitute the approach to modeling the BMR actuator

considering hysteresis in the cases of input monotonic increase and monotonic decrease of the nominal

current, respectively. In order to employ the expressions for hysteresis (12) and (16), two data sets of the

discretized first-order ascending and descending curves ( , )a

iS I I , ( , )d

iS I I must be determined in

advance. If the segments min( , )a

mS I I and max( , )d

mS I I in equations (12) and (16) are not coincident with

the discretized ones in the datasets, the interpolation method, which was proposed in [17], can be adopted

to obtain the values of these segments.

3.2 Inversed CBH model for BMR actuator

The output torque of a BMR actuator versus the input nominal current can be simply symbolized in

following form.

( [ ]) [ ( [ ])]T t n H I t n (17)

In the reverse sense, the necessary applied nominal current to obtain a desired torque can be expressed as

1( [ ]) [ ( [ ])]I t n H T t n (18)

where 1[ ]H

is the inverse hysteretic operator. Consequently, high precise control can be achieved if

there exists an inverse hysteretic operator 1[ ]H

such that the composition of [ ]H and 1[ ]H

drives

the actuator to follow the desired torque. The inverse model for the compensator in this work is assumed

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to be the hysteretic operator that possesses the same properties as that of the CBH model. It implies that

this model also owns the wiped-out, loop closing between the consecutive control points and congruency,

from which the CBH model is developed. Therefore, similarly, with a torque considered as the input

applied to an inverse hysteresis model, the necessary driving current can be expressed by modifying

equations (12) and (16) slightly as in the following forms:

min min min min( [ ]) ( ) ( , [ ]) ( , ),d

a a

m m m mn ex nI T t I T I T T t I T T (19)

max max max max( [ ]) ( ) ( , [ ]) ( , ).d

d d

m m m mn ex nI T t I T I T T t I T T (20)

In the above equations, the terms min( , [ ])a

m nI T T t and min min( , )a

m mI T T stand for the values of the segment

min( , )a

mI T T at desired [ ]nT t and min

mT respectively; max( , [ ])d

m nI T T t and max max( , )d

m mI T T are the values

of the segment max( , )d

mI T T at [ ]nT t and max

mT respectively. Similarly, the interpolation method in [17]

can be adopted to obtain the segments min( , )a

mI T T and max( , )d

mI T T in the case they are not coincident

with the database sets of first order ascending and descending current segments.

4. Experimental Results and Discussions

A setup for the experiment is shown in figure 9. The apparatus consists of the BMR actuator with its

driving system, a data acquisition board SCB68 and a PXI controller of National Instrument corporation.

In addition, a 6-DOF force sensor Nano 17 is used to measure the output signal. It is remarkable that the

longitudinal force is measured instead of the torque via a lever as shown in figure 10. Subsequently, the

output torque can be induced easily. Moreover, because the control signals from the PXI system are

voltages 1 2,u u while the coils need the current sources, several current voltage-current converters are

utilized. These converters have a converting coefficient 0.125. It means that 1V of control voltage is

converted to 0.125A of current. As mentioned above, the actuating torque of the BMR actuator comes

from the effects of the magnetic field-induced and dry friction shear stresses. The contributor due to dry

friction exists because of the imperfect of manufacturing and assembly as well as sealing. Figure 11

shows a performance of the BMR actuator at zero current before and after motor running. Because there

is no magnetic field, the torque is completely occurred from the effect of dry friction. It is not constant but

varying periodically in a revolution of the rotors of the actuator. In order to obtain the magnetic field

induced torque, the dry friction component should be identified and excluded from the measured torque of

the BMR actuator. To identify the friction torque, an incremental encoder system including a disk and

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three photodiode is inserted at the linking position between driving system and the actuator as shown in

figure 10. Thank to this encoder system, the variation of the friction torque in one revolution can be

identified successfully. The form of the friction torque is proposed as follows:

2 3 4 71 5 6sin( ) sin(2 ) sin(3 )fT f f f f f f f (21)

Where, is the angular position of the rotors; 1 7,...f f are the coefficients which are identified via

experimental data and optimization. To obtain these coefficients, a minimization of the following cost

function is carried out.

2

1

( ) ( )ex

N

k

fJ T k T k

(22)

where, ( )exT k is the measured torque value at zero current at the time k . Figure 12 shows the identified

friction torque and corresponding experimental data. It is observable that the identified friction torque

agrees well with experimental data. In order to assess the effectiveness of excluding the effect of the

friction torque from the performance of the actuator’s torque, an experiment with applied multi-step of

input current is carried out. Figure 13 shows the performance of the actuator with and without excluding

the effect of friction. It is realized that the signal becomes smoother. Moreover, the friction torque

generated in the actuator is almost independent with the stimulus of the input current. After that, in order

to model and control the BMR actuator, two datasets of first order curves ( , )a

iI T T and ( , )d

iI T T need to

be obtained in advance. The procedure to identify them was given in detail in [17]. First, two

experimental data of first-order ascending and descending torque curves after excluding the effect of

friction are obtained as shown in figures 14 and 15. From these data, two dataset of first-order ascending

and descending current curves are achieved using curve fitting method and shown in figures 16 and 17.

In order to evaluate the effectiveness of the proposed model and corresponding compensator, several

experiments are carried out. First of all, an experiment for compensation for the friction is carried out. In

this experiment, the hysteresis effect is not taken account. Consequently, relationship between the current

and the torque is assumed to be linear. Figure 18 shows a triangle waveform of a nominal input current. It

is noteworthy to remind that the actual currents to two coils are equivalent to the positive and negative

values of the nominal one, respectively. Figure 19 shows two responses in the cases of with and without

friction compensation. It is observable that the response with friction compensation is smoother, the

fluctuation amplitude is smaller. Or in other word, its quality becomes better that that without considering

the effect of friction. It is also noted that the effect of hysteresis is significant in both responses. It is

reasonable because the effect of hysteresis is not considered in this experiment.

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The accuracy of the hysteresis model and its compensation is analyzed in an open-loop tracking control

experiment. In this experiment, a triangular waveform is applied as a reference to the hysteresis

compensator. Based on the formulations shown in section 2, the compensator produces a proper control

signal to BMR actuator via a voltage-current converter. The block diagram for the open-loop control in

this experiment is shown in figure 20. Figure 21 shows the shape of the reference as well as the real

response. It is realized from the result shown in that the actual response matches well with reference.

Figure 22 shows that the hysteresis effect is almost eliminated although the fluctuation of the response is

significant because the effect of friction is not taken account in this experiment.

At last, an open-loop torque tracking control experiment considering both hysteresis and friction effects is

carried out with a random multi-peak reference. The block diagram for the control is shown in figure 23.

As shown in the figure, the control voltage is a summation of two compensations from hysteresis and

friction effects. The performances of the open-loop control of the BMR brake demonstrated in figure 24

show that the quality of the response with both friction and hysteresis compensations is better in

comparison with that without considering the friction effect. Figure 25 shows clearly that the error of the

response with both compensations is smoother. Moreover, it is smoother compared to that without friction

compensation.

5. Conclusions

In this work, a novel kind of MR actuators called BMR actuator was presented and accurate control

responses of the BMR were experimentally investigated. The BMR actuator consists of two rotor rotating

counter to each other. With such a configuration, it can work as either a clutch or brake. Or in other

words, it can generate either a repulsive or resistance torque. In order to transmit successfully two counter

rotations to the rotors of the BMR actuator, a motor and corresponding driving gear system was also

presented. After that, an approach to model the BMR actuator considering the effect of hysteresis was

proposed. In our work, the CBH model, which was applied successfully in modeling and control of a

piezoelectric actuator, was adopted. Subsequently, from the hysteresis model, a compensator was

developed. In addition, because it is necessary to use seals to prevent leakage of MR fluid in the BMR

actuator, the existence of the dry friction is unavoidable. More seriously, due to the imperfect of the

manufacturing process, the generated friction torque is not constant and tends to fluctuate periodically.

Consequently, in order to obtain accurately the reference datasets for the CBH model, this friction effect

must be identified and excluded. In our work, a sinusoidal form of friction torque was proposed. Its

parameters were obtained via an identification process with the objective to minimize a least square cost

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function. In order to evaluate the performance of the hysteresis model and its compensator, several

experiments were undertaken. The results obtained from these experiments revealed that the CBH model

and its corresponding compensator can be applied successfully to modeling a BMR actuator. Moreover,

the proposed sinusoidal model of friction torque is enough efficient to catch well the friction effect. As

well known, due to wide range of generated torque (from negative to positive), the BMR actuator has a

good potential as a multi-function device and can be integrated in medical haptic systems where sensing

force from a patient body might be a resistance or repulsion. It is finally remarked that as a second step of

this research work the proposed BMR will be utilized to make a medical haptic device which can be

integrated with a slave robot system for minimally invasive surgery.

Acknowledgement

This work was supported by Inha University Research Grant and the National Research Foundation of

Korea (NRF) grant funded by the Korea Government (MEST) (No. 2012-0005613).

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References

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[2]Lee H S and Choi S B 2000 Control and response characteristics of a magneto-rheological fluid

damper for passenger vehicles J Intel Mat Syst Str 11 80-7

[3]Niwa N, Kobori T, Takahashi M, Midorikawa H, Kurata N and Mizuno T 2000 Dynamic loading test

and simulation analysis of full-scale semi-active hydraulic damper for structural control Earthquake Eng

Struc 29 789-812

[4]Kurata N, Kobori T, Takahashi M, Ishibashi T, Niwa N, Tagami J and Midorikawa H 2000 Forced

vibration test of a building with semi-active damper system Earthquake Eng Struc 29 629-45

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rheological fluid brake J Mater Process Tech 129 559-62

[6]Nguyen Q H, Choi S B, Lee Y S and Han M S 2009 An analytical method for optimal design of MR

valve structures Smart Mater Struct 18

[7] Stanway R 2004 Smart fluids: current and future developments Mater Sci Tech-Lond 20 931-9

[8]Li W H and Du H 2003 Design and experimental evaluation of a magnetorheological brake Int J Adv

Manuf Tech 21 508-15

[9]Lee U, Kim D, Hur N and Jeon D 1999 Design analysis and experimental evaluation of an MR fluid

clutch J Intel Mat Syst Str 10 701-7

[10]Saito T and Ikeda H 2007 Development of normally closed type of magnetorheological clutch and its

application to safe torque control system of human-collaborative robot J Intel Mat Syst Str 18 1181-5

[11]Liu B, Li W H, Kosasih P B and Zhang X Z 2006 Development of an MR-brake-based haptic device

Smart Mater Struct 15 1960-6

[12]Blake J and Gurocak H B 2009 Haptic Glove With MR Brakes for Virtual Reality Ieee-Asme T Mech

14 606-15

[13]Ohnishi K, Shimono T and Natori K 2009 Haptics for medical applications Artificial Life and

Robotics 13 383-9

14

[14]Rosen J, Hannaford B and Satava R M 2011 Surgical Robotics: Systems, Applications, and Visions:

Springer)

[15]Okamura A M 2009 Haptic feedback in robot-assisted minimally invasive surgery Curr Opin Urol 19

102-7

[16]Nguyen P B and Choi S B 2012 Compensator design for hysteresis of a stacked PZT actuator using a

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using the congruency property Smart Mater Struct 20

15

Captions for Tables and Figures

Figure 1 Configuration of the BMR actuator

Figure 2 BMR actuator and its driving gear system

Figure 3 Magnetic field in the BMR actuator

Figure 4 Monotonic increase of the input nominal current

Figure 5 Response of the output torque corresponding to the monotonic increase of the input

nominal current

Figure 6 Demonstration of the first-order ascending curve congruent with the higher order one

Figure 7 Monotonic decrease of the input nominal current

Figure 8 Response of the output torque corresponding to the monotonic decrease of the input

nominal current

Figure 9 Apparatus for the experimental setup

Figure 10 Close View of the BMR cctuator

Figure 11 Performance of the BMR actuator at the off-state

Figure 12 Experimental and corresponding identified torque at off-state

Figure 13 Responses of the BMR actuator corresponding to the multi-step input nominal current

Figure 14 Experimental data of first-order ascending torque curve

Figure 15 Experimental data of first-order descending torque curve

Figure 16 Dataset of first-order ascending current curves obtained using curve fitting

Figure 17 Dataset of first-order descending current curves obtained using curve fitting

Figure 18 Triangular waveform of the nominal input current

Figure 19 Response of the BMR actuator corresponding to the triangular waveform of the nominal

input current

Figure 20 Block diagram for open-loop control considering hysteresis

Figure 21 Torque responses of the BMR actuator with triangular waveform of reference

Figure 22 Hysteretic relationship between the torque and reference in the second experiment

16

Figure 23 Block diagram for open-loop control considering both hysteresis and friction effects.

Figure 24 Performance of the proposed compensator with a random multi-peak reference

Figure 25 Error of the proposed compensator in comparison with the reference

17

Figure 1 Configuration of the BMR actuator

Coil Holder

MR Fluid

Shaft 1

Shaft 2

Casing’s

Shaft

Rotor 1 Rotor 2

Coil 1 Coil 2

Casing

Non-Magnetic Partition

18

Figure 2 BMR actuator and its driving gear system

19

Figure 3 Magnetic field in the BMR actuator

Coil 1 Coil 2

Magnetic Field

20

Figure 4 Monotonic increase of the input nominal current

21

Figure 5 Response of the output torque corresponding to the monotonic increase of the input nominal

current

22

Figure 6 Demonstration of the first-order ascending curve congruent with the higher order one

23

Figure 7 Monotonic decrease of the input nominal current

24

Figure 8 Response of the output torque corresponding to the monotonic decrease of the input nominal

current

25

Figure 9 Apparatus for the experimental setup

Voltage-Current

Converter

BMR Actuator

Angular

Position

Control

Voltage

Control

Current

PXI system

26

Figure 10 Close view of the BMR actuator

Lever

Rotary Disk

Photodiode

Force Sensor

Nano 17

(a) Front View

(b) Side View

27

Figure 11 Performance of the BMR actuator at the off-state

0 5 10 15 20 25 30 35 40

-20

0

20

40

60

80

100

To

rqu

e (

N.m

m)

Time (sec)

Motor Off Motor On

28

0 45 90 135 180 225 270 315 360

-20

-10

0

10

20

30

40

50

Experimental Torque

Identified TorqueT

orq

ue (

N.m

m)

Angular Position (deg)

Figure 12 Experimental and corresponding identified torque at off-state

29

Figure 13 Responses of the BMR actuator corresponding to the multi-step input nominal current

0 200 400 600 800 1000 1200

-600

-400

-200

0

200

400

600

Torq

ue (

N.m

m)

Time (sec)

Without Friction Elimination

With Friction Elimination

30

-10 -8 -6 -4 -2 0 2 4 6 8 10

-600

-400

-200

0

200

400

600

To

rque

(N

.mm

)

Current (0.125A)

Figure 14 Experimental data of first-order ascending torque curve

31

-10 -8 -6 -4 -2 0 2 4 6 8 10

-600

-400

-200

0

200

400

600

To

rque

(N

.mm

)

Current (0.125A)

Figure 15 Experimental data of first-order descending torque curve

32

-600 -400 -200 0 200 400 600

-10

-8

-6

-4

-2

0

2

4

6

8

10

Cu

rre

nt

(0.1

25A

)

Torque (N.mm)

Figure 16 Dataset of first-order ascending current curves obtained using curve fitting

33

-600 -400 -200 0 200 400 600

-10

-8

-6

-4

-2

0

2

4

6

8

10

Cu

rre

nt

(0.1

25A

)

Torque (N.mm)

Figure 17 Dataset of first-order descending current curves obtained using curve fitting

34

0 50 100 150 200

-10

-5

0

5

10

Inpu

t N

om

inal C

urr

ent

(0.1

25A

)

Data (sec)

Figure 18 Triangular waveform of the nominal input current

35

Figure 19 Response of the BMR actuator corresponding to the triangular waveform of the nominal input

current

-10 -8 -6 -4 -2 0 2 4 6 8 10

-600

-400

-200

0

200

400

600

To

rque

(N

.mm

)

Current (0.125A)

Without Friction Compensation

With Friction Compensation

36

Figure 20 Block diagram for open-loop control considering hysteresis

1 2,I I

Currents

T 1 2,u u

Voltages

dT

Desired

Torque

Actual Torque

Hysteresis

Compensator

1[ ]H

K

Voltage-Current

Converter

BMR Actuator

37

0 50 100 150 200

-600

-400

-200

0

200

400

600

To

rque

(N

.mm

)

Time (sec)

Real Response

Reference

Figure 21 Torque responses of the BMR actuator with triangular waveform of reference

38

-400 -200 0 200 400

-600

-400

-200

0

200

400

600

Re

sp

on

se

(N

.mm

)

Reference (N.mm)

Real Response

Reference

Figure 22 Hysteretic relationship between the torque and reference in the second experiment

39

Figure 23 Block diagram for open-loop control considering both hysteresis and friction effects.

1 2,I I

Currents

Angular

Position

T 1 2,u u

Voltages

dT

Desired

Torque

Actual Torque

Hysteresis

Compensator1[ ]H

K

Voltage-Current

Converter

BMR Actuator

Encoder Friction

Compensator

40

0 50 100 150 200

-600

-400

-200

0

200

400

600

T

orq

ue

(N

.mm

)

Time (sec)

With Friction Compensation

Without Friction Compensation

Reference

Figure 24 Performance of the proposed compensator with a random multi-peak reference

41

0 50 100 150 200

-60

-40

-20

0

20

40

60

Err

or

(N.m

m)

Time (sec)

Without Friction Compensation

With Friction Compensation

Figure 25 Error of the proposed compensator in comparison with the reference