Force tracking control of a flexible gripper featuring shape memory alloy actuators

Mechatronics 11 (2001) 677±690

Force tracking control of a ¯exible gripper featuring shape memory alloy actuators S.B. Choi *, Y.M. Han, J.H. Kim, C.C. Cheong Smart Structures and Systems Laboratory, Department of Mechanical Engineering, Inha University, 253 Yong Hyun-Dong, Nam-Gu, Incheon 402-751, South Korea Received 14 August 1999; accepted 5 April 2000

Abstract This paper presents a robust force tracking control of a ¯exible gripper using shape memory alloy (SMA) actuators. The governing equation of a partial di€erential form for the ¯exible gripper is derived by employing HamiltonÕs principle, and a state-space control model is obtained by retaining a ®nite number of vibration modes. In the formulation of the control model, time constant of the SMA actuator is treated as uncertain parameter. This is adopted due to the fact that the SMA actuator has di€erent time constant at the heating and cooling stage, respectively. The H1 -controller is then synthesized by treating the uncertain parameter as coprime factor. The speci®cations of stability margin and steady state error to achieve robust performance are imposed, and the ®rst-order reference model is augmented to avoid excessive overshoot. After analyzing the robust stability of the system using the singular value plot, the controller is experimentally realized and force tracking control responses for step and sinusoidal force trajectories are presented in time domain to demonstrate the e€ectiveness of the proposed methodology. Ó 2001 Published by Elsevier Science Ltd. Keywords: Shape memory alloy; Flexible gripper; Force control; H1 controller; Robust control

1. Introduction Recently, considerable attention is being given to the force control of small-sized or miniaturized ¯exible grippers. One of the tasks of these types of ¯exible grippers includes the assembly of tiny work pieces in the semiconductor industry and the sample collection for microscopic observations in the biochemical laboratory. In

*

Corresponding author. Tel.: +82-32-860-7319; fax: +82-32-868-1716. E-mail address: [email protected] (S.B. Choi).

0957-4158/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd. PII: S 0 9 5 7 - 4 1 5 8 ( 0 0 ) 0 0 0 3 4 - 9

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general, during the execution of the required task the ¯exible grippers are easily subjected to various system uncertainties including parameter variations and external disturbances. The robust control performance of the ¯exible gripper system to overcome the uncertainties can be hardly satis®ed by employing conventional actuating mechanisms such as electrical motors. However, the emergence of small and lightweight actuators associated with smart materials enables us to achieve the robust force control characteristics of the ¯exible gripper. So far, the smart materials include electro-rheological ¯uids [1], piezoelectric materials [2,3], optical ®bers [4], and shape memory alloys (SMAs) [5±9]. The SMA actuators, considered in this study, produce relatively large control forces compared to other actuating materials. In addition, design simplicity for control mechanism, high possibility of miniaturization, and low power consumption are salient properties of the SMA actuators. Accordingly, this class of actuators is primarily employed to vibration and acoustic control of ¯exible structures [5±7]. Recent researches on the smart systems associated with the SMA actuators also include temperature fuze, micro-robot and various medical devices such as blood clot ®lter [8]. It is well recognized that the most signi®cant problem to develop successful abovementioned mechanisms or systems is slow response time of the SMA actuator. This, of course, leads to the study on the improvement of the response time [9]. Especially, the SMA actuator has di€erent response characteristic at heating and cooling stages, respectively. Therefore, the actuator dynamics needs to be treated as an uncertain system, but with bounded deviation of time constant. In this work, we propose a small-sized ¯exible gripper using the SMA actuator, and undertake a force tracking control of the gripper by considering time constant as the uncertain parameter. Consequently, the main contribution of this paper is to show how H1 -controller can be satisfactorily employed for the robust force tracking control of the SMA-driven ¯exible gripper system subjected to actuator uncertainty. The e€ectiveness of the proposed control system is con®rmed by both simulation and experimental results. The partial di€erential equation governing the motion of the ¯exible gripper system is derived by employing HamiltonÕs principle. Upon retaining a ®nite number of vibration modes, a control model is obtained in Laplace domain. The overall transfer function of the system relating the input current to the output end-point gripping force is then established by adopting coprime factor uncertainty. A two degrees of freedom (TDF) control system based on the H1 technique is synthesized to guarantee the speci®cations of stability margin and steady state error. The eighth-order pre®lter and feedback compensator are then designed and experimentally realized. Force tracking results for step and sinusoidal trajectories are presented in time domain to demonstrate the e€ectiveness and robustness of the proposed control methodology. 2. Dynamic modeling Consider a ¯exible gripper system which has spring-type SMA actuators as shown in Fig. 1. The SMA actuators are installed on opposite sides of each cantilevered ®nger with a certain inclined angle (h). The end-point gripping force at coil spring is

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Fig. 1. The proposed ¯exible gripper with SMA actuators: (a) schematic con®guration; (b) photograph.

measured using the strain gage attached on the surface of the gripper root. It is seen from Fig. 1 that there are two beams in the proposed ¯exible gripper system. However, we formulate dynamic model for one beam under the assumptions that two beams are identical, and they are positioned symmetrically. Thus, from the Bernoulli±Euler beam theory the governing equation of motion for transverse de¯ection y…x; t† and associated boundary conditions are obtained as follows: EI

o4 y…x; t† o2 y…x; t† ~ ‡ q A ‡ k1 yd…x d ox4 ot2

L1 † ‡ Fa …t†d…x

y…x; t† jxˆ0 ˆ 0; oy…x; t† ˆ 0; ox xˆ0 o2 y…x; t† EI ˆ 0; ox2 xˆL2 o3 y…x; t† o2 y…x; t† EI ˆ M ‡ k2 y…x; t†jxˆL2 : ox3 xˆL2 ot2 xˆL2

L1 † ˆ 0;

…1†

…2†

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In the above, EI is the e€ective bending sti€ness, qd the mass density, A the crosssectional area, M the end-point mass, and Fa …t† is the e€ective actuating force of the SMA. The variables k~1 and k2 represent spring constant of the SMA actuator at operating temperature and spring constant of the coil spring at the end-point, respectively. It is noted that the spring constant k~1 is modelled as linear spring since the operating displacement range of the SMA spring is very small. By introducing the ith modal coordinate qi …t† and mode shape /i …x† of the whole system, a decoupled ordinary di€erential equation for each mode of the proposed gripper system is derived as follows [3]:  qi …t† ‡ 2fi xi q_ i …t† ‡ x2i qi …t† ˆ

Fa …t†/i …L1 † ; Ii

i ˆ 1; . . . ; 1;

…3†

where xi and fi is the natural frequency and the damping ratio of the ith mode, respectively, and Ii is the generalized mass given by Z L2 Ii ˆ qd A/2i …x† dx ‡ M/2i …L2 †: …4† 0

It is noted that in the control system model given by Eq. (3) viscous damping term (2fi xi q_ i ) has been added to each decoupled modal equation in an heuristic fashion. The inherent dynamic characteristics of the SMA actuator Fa …t† should be ®rst identi®ed to achieve successful implementation. Fig. 2 presents the measured step response of the SMA actuator employed in this work. By assuming that the force resulting from thermal expansion is negligible compared to the transformation force, the actuator dynamics can be expressed by:

Fig. 2. The dynamic characteristics of the SMA actuator.

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dFa …t† ‡ Fa …t† ˆ ka i…t†; dt …5† dFa …t† ‡ Fa …t† ˆ ka i…t†; sc dt where i…t† is the input current and ka is the input in¯uence coecient. The variables sh and sc represent time constants at the heating and cooling stages, respectively. We clearly see from Fig. 2 that the time constant sh at the heating stage is much di€erent from the time constant sc at the cooling stage. It is distilled from Fig. 2 that sh ˆ 1:2 s and sh ˆ 3:52 s. This was obtained by de®ning time constant as a time to reach 63.2% of the steady state value. This causes the control system to be divided into two parts, which is undesirable in the experimental realization. Therefore, we adopt the average value s0 ˆ …sh ‡ sc †=2 as a nominal time constant and treat the di€erence from the average value as an uncertain parameter as follows: sh

Ds ˆ max…jsh

s0 j; jsc

s0 j†:

…6†

Thus, the actuator dynamics of the SMA is given by Fa …s† ka ˆ : i…s† s…s0 ‡ Ds† ‡ 1

…7†

Now, the overall transfer function between the input current (i…t†) and the output end-point gripping force (Fg …t†) is obtained as follows: P1 Fg …s† ka fk / …L †/ …L †=I g Piˆ1 22 i 2 i 1 2i : ˆ GD …s† ˆ …8† i…s† s …s0 ‡ Ds† ‡ 1 1 iˆ1 fs ‡ 2fi xi s ‡ xi g 3. Controller design 3.1. General formulation The loop shaping design procedure (LSDP) based on H1 robust stabilization was proposed by MaFarlame and Glover [10]. Limbeer et al. [11] extended the LSDP by introducing a TDF method to guarantee both robust stability and closed-loop responses to be followed a speci®ed target response model. In order to formulate the TDF control algorithm, the nominal plant given by Eq. (8) with Ds ˆ 0 needs to be described by normalized left coprime factor as follows: Pn ka fk2 /i …L2 †/i …L1 †=Ii g ~ 1 N~ : Piˆ1 G…s† ˆ …9† ˆM n 2 2 s s0 ‡ 1 iˆ1 fs ‡ 2fi xi s ‡ xi g It is noted that the control model (9) is truncated from the distributed system model (8) by retaining a ®nite number (n) of ¯exible modes. The number of the ¯exible modes to be controlled is determined from the investigation of the system responses before and after employing the controller proposed in this section. Through the computer simulation, the open-loop responses of the system are observed in time domain by investigating the response e€ect of each ¯exible mode, and the closed-loop responses due to residual modes are also observed by

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investigating the excitation magnitude, which is an indicator of the control spillover e€ect. In this work, dynamic response of the ®rst two modes was observed to be dominant over higher residual modes. The control e€ect due to the residual modes (we tested up to the ®fth mode) has been observed to be small. Now, the uncertain plant of Eq. (8) also needs to be described in the same manner as follows: Pn ka fk2 /i …L2 †/i …L1 †=Ii g ~ 1 N~D Piˆ1 GD …s† ˆ ˆM n D 2 2 s…s0 ‡ Ds† ‡ 1 iˆ1 fs ‡ 2fi xi s ‡ xi g ~ ˆ …M

DM † 1 …N~ ‡ DN †;

…10†

where DM and DN represent coprime factor uncertainty of the time constant satis‡ fying DM , DN 2 H1 and jDM ; DN j 6 c 1 . The variable c is target norm bound for H1 optimization. The TDF H1 -control scheme proposed in this work is shown in Fig. 3. The control objective is to design the pre®lter K1 and the feedback compensator K2 so that the robust stability and favorable force tracking performance can be achieved. In the block-diagram, W is weighting function on the input u, which needs to be determined so that the shaped plant GW meets desired loop shape. M0 is desired target model to ensure robust model matching property, q is scaling factor, and S is scaling matrix. From the block-diagram shown in Fig. 3, we can establish the following generalized H1 frame work:

:

Fig. 3. The con®guration of TDF H1 -control system.

…11†

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Now, using commercial software package, robust control toolbox for MATLAB, the stabilizing pre®lter K1 and the feedback compensator K2 can be designed to satisfy the following inequality:

;

…12†

~ 1 N~S † represents the shaped plant. where GS …ˆ M S 3.2. Application to the proposed system In this study, we consider the ®rst and second ¯exible modes in the design of the H1 -controller. As mentioned earlier, the number of ¯exible modes is determined from the investigation of the system response before and after employing the controller via computer simulation. For the proposed gripper system, it is evaluated that the ®rst two modes are dominant ones over higher residual modes. To identify system parameters such as xi and fi , frequency responses of the system, whose properties are listed in Table 1, are obtained. The measured values are as follows: x1 ˆ 21:1 Hz, x2 ˆ 132:2 Hz, f1 ˆ 0:0504, f2 ˆ 0:0124, U1 …L1 † ˆ 0:986, U1 …L2 † ˆ 2:0, U2 …L1 † ˆ 1:068, U2 …L2 † ˆ 2:002. In addition, the parameters of the SMA actuator are identi®ed as follows: ka ˆ 0:284, s0 ˆ 2:36, h ˆ 22:6°, M ˆ 5:5 g, k~1 ˆ 22:46 N/m, k2 ˆ 9:83 N/m. It is desirable to robustly stabilize the plant subjected to the variation of time constant and also favorable force tracking performance needs to be accomplished with the following speci®cations: (i) The robust stability should be guaranteed in the presence of ‹40% variation of time constant; (ii) The tracking error should be less than 2% in the steady state phase. To achieve these objectives, the loop shaping is ®rst carried out. Fig. 4 presents singular value plots of the unshaped and shaped plants. It is clearly observed that the gain of the system is very low before shaping. This physically implies that the nominal open-loop control system has a very low

Table 1 Dimensional and mechanical properties of the ¯exible gripper and SMA actuator Flexible ®nger

SMA spring

Material

Glass epoxy

Material

Nitinol

YoungÕs modulus Length Thickness Density Width

6.3 GPa 80 mm 0.6 mm 1865 kg/m3 6 mm

Coil spacing Coil turns Wire diameter Coil diameter Transition temperature

0.66 mm 15 0.36 mm 3.81 mm 38°C

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Fig. 4. Singular value plots of loop gain and shaped plant.

control bandwidth. Therefore, the following weighting function is cascaded with the plant to boost the low frequency gain. W ˆ

15  103 : s ‡ 13

…13†

The pole is introduced to increase high frequency role o€ rate, which guarantees robust stability of the proposed system. It is also noted from Fig. 4 that the singular value plot of the shaped plant favorably agrees with that of the loop gain of TDF controller. This implies that the loop shaping is satisfactorily performed. The desired target model M0 (refer to Fig. 3) needs to be determined by considering the response time of the control system. In this work, the target model is chosen by M0 ˆ

1 : 0:005s ‡ 1

…14†

This is a ®rst-order model, which is fast enough to meet time response speci®cation with small overshoot. As the last step, by choosing the scaling factor q ˆ 1:6 and c ˆ 3:4, we obtain the following controllers from Eq. (12): SK1 ˆ

FN …s† ; FD …s†

K2 ˆ

CN …s† ; CD …s†

where FN …s† ˆ 3:917  108 s6 ‡ 2:237  1010 s5 ‡ 2:78  1014 s4 ‡ 1:028  1016 s3 ‡ 1:277  1019 s2 ‡ 2:928  1020 s ‡ 2:298  1021 ;

…15†

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FD …s† ˆ s8 ‡ 963:9 s7 ‡ 9:269  105 s6 ‡ 6:842  108 s5 ‡ 1:74  1011 s4 ‡ 3:817  1013 s3 ‡ 6:206  1015 s2 ‡ 2:726  1017 s ‡ 2:574  1018 ; CN …s† ˆ

2:14  107 s6 7:834  1017 s2

5:345  109 s5

1:539  1013 s4

1:496  1020 s

3:485  1015 s3

2:187  1021 ;

CD …s† ˆ s8 ‡ 963:9 s7 ‡ 9:269  105 s6 ‡ 6:842  108 s5 ‡ 1:74  1011 s4 ‡ 3:817  1013 s3 ‡ 6:206  1015 s2 ‡ 2:726  1017 s ‡ 2:574  1018 : Fig. 5 presents the multiplicative stability margin (T) obtained from the designed pre®lter K1 and feedback compensator K2 . From this, the guaranteed gain margin (GM) and phase margin (PM) are computed using the following formulas: 1 1 6 guaranteed GM 6 1 ‡ ; kT k1 kT k1   1 guaranteed PM ˆ 2 sin 1 : 2kT k1

1

…16†

The above formulas represent the ranges of pure gain and phase before the guarantee of the stability is lost. For the proposed control system, the guaranteed GM is calculated by 2 to 0 (mag.), while the guaranteed PM by 66:67°.

Fig. 5. The multiplicative stability margin.

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4. Results and discussions An experimental apparatus for force tracking control of the proposed ¯exible gripper is shown in Fig. 6. The gripper system is set up in the thermal insulating chamber and the operating temperature is controlled to be maintained by 12  1°C. The end-point gripping force is measured by a strain gage. This is achieved by calibrating the strain magnitude (voltage signal) to the gripping force. The gripping force is ampli®ed through the strain gage conditioner and fed back into the microprocessor through the analog/digital (A/D) converter. Depending on reference input and output force, the H1 -controller calculates control input current, and the control input current is supplied to the SMA actuators via the D/A converter and voltage/ current converter. The MetraByteÕs DAS-20 I/O board has 12-bit resolution for both the D/A and A/D conversion, and the sampling rate for the controller implementation is chosen by 1 kHz. Fig. 7 presents simulated and measured force control responses for smooth step desired trajectory. It is clearly observed that the tracking performance is favorable, and the steady state tracking error is within ‹2% with proper control input current. The tracking response for the step trajectory, which has a period of 5 s is shown in Fig. 8. We see from the gripping force history that the actual force well settles down to the desired force of 0.01 N without exhibiting overshoot. And we can clearly see that there exists an excellent agreement between the simulation and experimental results. This advocates the validity of the proposed dynamic model as well as control logic. It is remarked that if we use the smallest time constant (rise time constant) in the computer simulation the control response from the simulation will be much better than one obtained from the experimental work. Fig. 9 presents force tracking responses for the sinusoidal trajectory which has a frequency of 0.25 Hz. It is distilled

Fig. 6. Experimental apparatus for force tracking control.

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Fig. 7. Force regulating control responses: (a) simulated; (b) measured.

Fig. 8. Force tracking control response for step trajectory: (a) simulated; (b) measured.

from the gripping force history that the actual trajectory tracks the desired one within 0.9 s after activating the controllers. It is also known from the input current history that the magnitude of the current in the measured one is higher than that in the simulated one. This is caused by relatively slow cooling speed, which directly represents the reacting force of the SMA actuator. It is well known that the response force of the SMA actuator is relatively slow compared with other smart material actuators. To evaluate the tracking capability of the proposed ¯exible gripper in terms of the response characteristic, a sinusoidal

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Fig. 9. Force tracking control response for sinusoidal trajectory: (a) simulated; (b) measured.

Fig. 10. Force tracking control response for sinusoidal trajectory with time-varying frequency: (a) simulated; (b) measured.

trajectory which has a time-varying frequency is adopted as a desired one, and control results are presented in Fig. 10. The frequency varies from 0.08 to 0.68 Hz. It is obvious from the tracking error history that the tracking performance is deteriorated as the frequency increases. This directly indicates the applicable limitation of

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the SMA actuator. Speci®cally, the control bandwidth of the proposed system is about 0.48 Hz. It is remarked that the control bandwidth of the system depends on not only the dynamic characteristic of the SMA actuator but also the system parameters and control hardware/software utilized for experimental implementation.

5. Conclusion A robust force tracking control of a ¯exible gripper using SMA actuators was presented in this paper. Following the construction of a control system model which contains the uncertain time constant, the TDF H1 -controllers were formulated on the basis of the speci®cations for robust stability and steady state error. The controllers were then experimentally realized and the tracking control responses for several types of desired tip-force trajectories such as step and sinusoidal functions were evaluated. Favorable tracking control performances were achieved in terms of tracking accuracy and control robustness. In addition, the agreement between experiment and simulation results is excellent. The results achieved in this study can be used as fundamental and useful guidelines for constructing of various ¯exible gripper mechanisms operated by SMA actuators. It is ®nally remarked that the test for tracking durability and the study on the improvement of tracking response need to be undertaken in the future.

Acknowledgements This work was partially supported by the Brain Korea 21 Project. This ®nancial support is gratefully acknowledged.

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[9] Russel RA, Gorbet RB. Improving the response of SMA actuators. In: Proceedings of the IEEE Conference on Robotics and Automation, 1995, p. 2299±304. [10] MaFarlame DC, Glover K. A loop shaping design procedure using H1 synthesis. IEEE Trans Autom Control 1992;37(60):759±69. [11] Limbeer DJ, Kasenally EM, Perkins JD. On the design of robust two degrees of freedom controllers. Automatica 1993;29(1):157±68.