An LMI approach to robust multimodel controller design for a four-link lightweight manipulator arm

AN LMI APPROACH TO ROBUST MUL TTMODEL CONTROLLER DE ...

14th World Congress ofTFAC

B-ld-Ol-l

Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China

AN LMI APPROACH TO ROBUST MULTIMODEL CONTROLLER DESIGN FOR A FOUR-LINK LIGHTWEIGHT MANIPULATOR ARM

J. Uhlig, H. Werner, N. Fabritz, J. Kita, H. Unbehauen

Control Engineering Laboratory (ESR) Faculty ofElectrical Engineering and lriformation Technology Ruhr-University Bochum 44780 Bochum, Germany E-mail: [email protected] [email protected] [email protected] [email protected] [email protected] FAX: #49234 7094-101 FTP: ftp.esr.ruhr-uni-bochum.de WWW: htlp://esr.ruhr-uni-bochum.de

Abstract: An alternative to a complex nonlinear state feedback controller design for robot manipulators is to use a linear state feedback controller based on linearized manipulator models. For a given range of operation, the nonlinear dynamic model of the manipulator is linearized in a number of operating points (OP). New linear matrics inequalities (LMI)-methods can be used to design a linear state feedback controller which achieves good performance simultaneously in all operating points. This approach is applied to a laboratory four-link flexible manipulator. Linear models have been identified experimentally in suitable operating points, and the performance of a robust state feedback controller is illustrated by experimental results and comparison with a pole placement controller. Copyright © 1999 IFAC Keywords: robotic manipulators, robust control, state feedback

I. INTRODUCTION An alternative to the expensive cqnstruction of multilink rigid manipulators is given by light-weight manipulator arms which, however, have the disadvantage of high elasticity. Position control of multi-link lightweight manipulator arms is no more possible with the conventionally applied control approaches for rigid manipulators but needs highly sophisticated control schemes. Several theoretical

and experimental results have been presented in literature for the control of single-link and two-link flexible manipulators (Amerongen, 1991), (Bayo, 1989), (Khorrami, 1993), (Oakley, 1990), (Yurkovich, 1990b). The construction of these plants does not allow to handle large p aylo ads. Multi-link rigid and flexible manipulators are highly non linear systems, which would need also a rather complex nonlinear controller design. An alternative is to use a linear state feedback controller based on a Iinearized

488

Copyright 1999 IF AC

ISBN: 008 0432484

14th World Congress oflFAC

AN LMl APPROACH TO ROBUST MUL TlMODEL CONTROLLER DE .. .

manipulator model. Oakley and Cannon (1990) as well as Y oshikawa et al. (1990) proposed to control the endpoint position by LQ controllers. Amerongen et al. (1991) proposed to control a two-link flexible manipulator with a pole placement state space technique. A serious problem of linear state feedback for manipulators lies in the simplifing assumptions. Controller design is based on a linearized model for one operating point. Using variable pay loads or more than two manipulator links leads to very large model errors and thus to poor control performance. An alternative to cope with the variable operating conditions would be the application of an adaptive or robust controller design which would provide acceptable tracking behaviour or disturbance rejection in the whole range of operation. Hilhorst (1992) proposed a multimodel approach, Yurkovich et al. (1990) a gain-scheduling scheme. In this paper, a robust multi-model controller is presented for the control of a four-link leightweight manipulator arm. Recent advances in application of convex programming techniques to control problems have made a multimodel approach in state space to robust control an interesting alternative to frequency domain methods. Given a family of linear models, there are by now standard software tools available to search for state feedback gains which place the closed loop poles for each model simultaneously in a desired region in the complex plane, and if such gains exist one can search for one which satisfies a given HiH" condition (Gahinet et at., 1995). In this paper, application of such a multi-model approach to the four-link manipulator is presented. For comparison, an additional pole placement controller has been designed based on the step response of a linear model. We show real-time results for both controllers. The outline of this paper is as follows. The plant is described in section 2. In section 3 follows a description of the linear identification experiments. Linear models are used in section 4 to

design a multimodel robust controller using LMI methods. The comparison between the robust mu1timodel controller and the direct pole placement controller is presented in section 5. Conclusions are drawn in section 6.

2. THE FOUR-LINK LIGHTWEIGHT MANIPULATOR ARM A four-link flexible manipulator was built at the

Control Engineering Laboratory at the RuhrUniversity Bochum (Fabritz, 1997). DC-motors with Harmonic Drive gears are used as actuators. The length of each link, which consists of aluminium with hollow-profile, is 60 cm. At the end effector of the manipulator different payloads can be fixed. Fig. 1 shows the principle plant structure. The working area is a semi-circular plane. The joint angles tpj, j = 1, ... ,4 are measured by optical encoders and used for feedback. Measured values are collected in a real-time measurement system. An emergency shutdown system is integrated, so that the manipulator will be stopped if an exception handling is deteced. A transputer network with seven transputers (TSOO) and a digital signal processor (C 40) and a PC 486 DX2/66 as a host computer serves as control and operation device, respectively. The sampling rate is 15 ms. For operation and visualisation a specially developed grafical software package is implemented on a PC.

3. IDENTIFICATION OF LINEAR MODELS Tn order to apply a linear controller to the nonlinear system, linearized models are required. Linear discrete-time state space models for the manipulator of the form

Fig. I: Principle plant structure, topview

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ISBN: 0 08 043248 4

14th World Congress oflFAC

AN LMl APPROACH TO ROBUST MUL TlMODEL CONTROLLER DE ...

x(k + 1) = A;x(k) + B;(k)u(k), i

= 1, ...,4

(1)

y(k) = Cix(k)

(2)

were estimated; the sampling time was taken as r-= 15 ms. The matrices Ab Bi and C; describe the system in

addition a second order model for each link was identified independently and these models were combined to a decentralized block diagonal representation

best results when the links were modelled by second order models. So the state vector is x(k) = [lPJ.(k) lPJ.(k-l) CP2(k) 1I'2(k-l)]. (3)

A=

u3(k)

(4)

u4(k)f.

Likewise, the controlled variables are y(k)

= Cx(k) = [lPI(k) fP2(k) 'P3(k) fP4(k)f .(5)

To test the multimodel controller design, a test trajectory (Fig. 2) was defined, and four operating points, shown in Tab. 1, were chosen to cover the range of operation. Tab. I: Joint angles for the operating points lP:t(k)

CP2 (k)

CP3 (k)

OPl

0.0 0

0.0 0

0.0

OP2

14.6° 51.5 0 -14.6

-59.2° -87.3° 59.2

OP3 OP4

fP4(k) 0.0° 58.6° 86.3° -58.6

0

-57.4° -84.6° 57.4

For these four operating points, multivariable models of the form (1),(2) for the links were estimated. In

0

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0

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J

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l_

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Jlde<.3(k))' 10

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ink 4 _ _,..,.,.,..--_

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:'----,.U-t.=----r-·-o

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ink 2 00,-____- ,______, -____,

0 0 , - - - - - - , - - - - - - , - - -- -__,

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Here the coupling between links is interpreted as disturbance. Identification was carried out by using the ARX-function of the System Identification Toolbox from Matlab (Ljung 1991). The results show that the dynamic behavior is very different in different operating points. Because the plant in operating point 3 is difficult to identify, we show the model validation for this case. The validation is done in open loop to be sure that the model error is not reduced by the controller. Figure 2 shows the measured angles (ym....iCk) and the simulated values for decentralized- and MIMO-models Cvd«.t
IIrk 1

\--~-·-·~r\ -+--~-t~ --1..-----~-----J

0 0

(6)

,o,---_ _~---,___--__, o

0 0 0

The actuator variables are the voltages applied to the motors between the links; they are stacked into a control vector u2{k)

0

0

0 0

0

-ano -ai ,1.1 0 0

operating point i. Identification experiments showed

u(k) = [ul(k)

1

0

5

10

15

time[s]

Fig. 2; Model validation in an open loop simulation for the SI SO and the MIMO model for operating point 3

490

Copyright 1999 IF AC

ISBN: 0 08 043248 4

14th World Congress oflFAC

AN LMT APPROACH TO ROBUST MULTTMODEL CONTROLLER DE...

the horizontal axis of 0,3 and a half-length of the vertical axis of 0, 15. With weight matrices

4. CONTROLLER DESIGN This section describes the design of an observerbased state feedback controller for the manipulator arm. State estimation is done by using an identity observer with observer gain F 0 and with observer equation x(k + I) = AoX(k) + Bo"Ck) + Fo(YCk)- jr(k») (7) y(k) = Cox(k) . (8)

the resulting controller matrix F is

The control law is u(k) = Fx(k) + VW(k)

0 0 0.45 -028 0 F= [ 0 0 0.45 -0.30 0 0 o 0 0 0 -0.13 0.18

(9)

= f(x T (k)Qx(k) + UT (k)Ru(k»)

° 01 0

0

o

°

0

10

(10)

(11)

So the closed-loop eigenvalues are constrained to be in a desired pole region Dc simultaneously for all operating points. It was not possible to find a controller for the MTMO-modells by using the LMIToolbox if the poleregion Dc was chosen smaller then the unit ciree!. For the SISO-modells Dc was chosen as ellipsoid with centre at 0.6, a half-length of

Tab. 2: Effects of mismatch between observer model and ~lant on closed Operating point 2

Operating point 3

ZS

Zr

ZS

Zr

0.014 0.015 0.040 0.040 0.179 0.179 0.188 0.188 0.392 0.400 0 .527 0.768 0.769 0.795 0.834 + iO,065 0.834 - iO.065

- 0.115 0.117 - 0.173 0.007 0.007 - 0.019 - 0.019

0.014 0.015 0.040 0.040 0.179 0 .179 0.188 0.188 0.530 0.651 - 0.049i 0.651 + 0.049i 0.686 - 0.106i 0.686 +0.106i 0.705 0.837 - O.072i 0.837 +O.072i

O.Qll O.Qll 0.100 0.100 0.032 0.032 - 0.039 - 0.039 0.542 0.716 0.716 0.771 0.771 0,823 0.940 0.940

- 0.317i + 0.317i - 0.348i +0.348i 00411 +O.305i 00411 - 0.305i 0.625 0.710 +0.085i 0.710 - O.085i 0.758 0.840 0.947 - 0.087i 0,947 +0.087i

(12)

0

0 ]

0

0

0 0' o 0 0 0 0 0 -0.16 0.30 (13) Due to the special construction of the manipulator ann, the quocient between torque and power for link 3 and 4 are much higher compared to the ones obtained for link I and 2, This quocient is mainly responsible for the selection of the weight matrix R during the design. In order to avoid possible overshoots, the entries in R corresponding to link 3 and 4 are selected having much higher values. The problem with using a state observer is that for multimodel problems the seperation principle does not hold. Aditional, it is difficult to fmd an observer gain matrix which satisfy the condition to place the observer poles for all operation points in the typically small observer pole region. Here, the problem is solved in a heuristic way. An observer was designed for each operating point. The LMl Control Toolbox was used to find an observer gain which places the observer poles in an ellipsoid with centre at 0.1, a half-length of the horizontal axis of 0.1 and a half-length of the vertical axis of 0.05.

k=O

under the constraint eig(A; + B;F) c Dc, j = 1,2,3.

01

°

were x(k) is the estimated state, wCk) the command input and V a pre-filter. The LMI Control Toolbox for Matlab (Gahinet et. al. 1995) was used for simultaneous pole-region assignment with quadratic performance index, i,e. to find a state feedback gain F which minimizes the cost function J

o1

-IR[ 010000 Q-, -0

loo~

Eoles

Operating point 4 ZS

Zr

- 0.347i +0.347i - 0.382i +0.382i - 0.397i + 0.397i - 0.447i +0.447i

0.014 0.015 0.040 0.040 0.179 0.179 0.188 0.188 0.393

- 0.090i +0.090i - 0.099i +O.099i

00436

0.148 0.175 0.276 0.067 0.067 0.012 0.012 - 0.001 - 0.001 0.508 0.605 0.764 0 .809 0.816 0.951 0.951

- O.087i +O.087i

0.473 0.771 0.779 0.791 0,835 - 0.069i 0,835 + 0.069i

- 0.271i +0.271i - 0.280i + 0.280i - O.380i +0.380i

- 0.076i +0.076i

491

Copyright 1999 IFAC

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AN LMI APPROACH TO ROBUST MUL TTMODEL CONTROLLER DE ...

14th World Congress ofTFAC

The weights were chosen as

Qo

= I,Ro =1.

(14)

\

These observers were tested in real-time experiments. The observer designed for operating point 1 with the observer gain matrix 0.24 0.25

Fo =

[

o

0

0

0

0

0

0.26 0.26 0

0

° 0

o

o

o

o o

3.89 4.14

00000

o

) /

0.9

(15) shows the best results in all operating points. To obtain zero steady state error, a pre-·filter V was used given by

V=(co(I-Ao+BoFtIBot.

Fig. 3: Circle with the one-model controller

(16)

Table 2 shows the poles zs for which controller and observer were designed, and the closed-loop poles ZT resulting from a mismatch between plant and observer for the OP2, OP3 and OP4. The first eight rows of the columns zs show the observer poles. Because the observer is designed for the operating point 1, the poles of observer and plant are identical with the closed loop poles for OPl. The table indicates that, in OP2, OP3 and OP4, the mismatch between observer and plant leads to slower and less damped poles in comparison with the desired behavior.

5. EXPERIMENTAL RESULTS The performance of the controller described in the previous section was tested by mean of different realtime experiments. For comparison purposes, a "onemodel" pole placement controller has also been designed for the first operation point. This controller was designed in a way to achieve a response without overshoot. It is possible to operate the manipulator arm in three different ways: 1) The manipulator can be moved according to given motor angles, 2) it can be moved to a user defined endpoint with the Tool Centre Point (TCP) or 3) following a trajectory. In order to obtain a better visualization of the results, only these one concerning to the third case, i.e. following a trajectory will be shown in this paper. The multimodel controller shows a better behaviour for all kinds of movement than the direct designed pole placement controller. The goal was to follow different geometric figures (circle, triangle and square) in different areas of the working space. These areas do not include the operating points defmed in table 1. The choosen trajectories illustrate the worst-case and they are representative for the whole working area.

Fig. 4: Circle with the multimodel controller Figures 3 and 4 show the results of the one-model and the multimodel controller for a circle with a radius of 35 cm in the right halfplane of the working space. As one can see, the multimodel controller is able to follow the trajectory better. The behaviour of the controlled signal by using the multimodel and the other pole-placement controller for link 3 can be seen from the plot of the signals Yu.n(k) and yPp(k) in the figure 5. Because of the highest quocient between torque and power for link 3 this link is most difficult to control and most interesting to iIlustrate. The signal YLMl(k) tracks the reference signal better than ypp(k) because the multimodel controller was developed for all operating points simultaneously. Also in all other tested working areas good results have been achieved by the multimodel controller. ;reference signal and controlled variables of link. 3

~5r---------~--------~--------~ :"'.:;~:·····'·····'·'··'~·'··'·'·····'·---·······'f

-70 -15 ·80

... '........:-. . .:-. . . . . ~~,.

i..... . ,\---....... \.; ........... +_ .... --.-....... ..... \\ •.•.....•.•.... ,

y.,CkJ ;

\1.

C

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I

·90 -GIj

-100

Yu.uO:):

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l

....... l"'\:"'"

"'.,

-105 ... .1100~---~~--':-~~--~--L10----------'15

lime [5]

Fig. 5: Reference and controlled signals for the circle

492

Copyright 1999 IF AC

ISBN: 008 0432484

AN LMI APPROACH TO ROBUST MUL TTMODEL CONTROLLER DE ...

14th World Congress ofTFAC

REFERENCES Amerongen, J.van.; Kruise, L., Lohnberg, P., Tiemego, M.J.L. (1991). State-Feedback Controller for a Two-Degree of Freedom Flexible Robot Arm. 3. IFACIIFIPIIMACS Symposium on Robot Control, SYROCO 91, Vienna, Austria, 16.-18. Sep. 1991, pp.399404.

Fig. 6: Square with the one-model controller

Fig. 7: Square with the muItimodel controller Because the separation principle does not hold for a multimodel design, the resulting output feedback controller does not perform the desired dynamic response exactly. The same results can be seen in the figures 6 and 7, which show the results following a square with a length of a side of 50 cm.

6. CONCLUSION A multimodel robust controller design for a four-link

lightweight manipulator, based on LMI methods has

been presented. Linear models have been identified in four operating points. A robust controller has been designed for the four models simultaneously, however the observer is designed for one operating point only. Because the separation principle does not hold, the resulting closed-loop eigenvalues are different from the original design. Even though, realtime results showed performance superior to that achieved with a pole-placement controller designed only for one operating point. Further research activities are directed towards LMI methods for robust observer design as well as multirate techniques for simultaneous realization of state feedback gains (Wemer, 1998).

Bayo, E., Papadopoulos, P., Stubbe, J., Sema, M.A. (1989). Inverse Dynamics and Kinematics of Multi-Link Elastic Robots: An Iterative Domain Approach. The Frequency International Journal of Robotics Research, VoU, No.6, pp.49-62. Fabritz, N. (1997). Ein offenes Automatisierungssystem fUr einen mehrgliedrigen elastischen Manipulator. Fortschrittberichte VD! Reihe 8: MejJ-, Steuerungs- und Regelungstechnik Nr. 603, VDI-Verlag, Dlisseldorf, Germany. Gahinet, P., Nemirovski, A., Laub, A. J., Chilali, M. (1995). LMI Control Toolbox. The MathWorks, Natick, MA, USA. Hilhorst, R.A. (1992). Supervisory Control of ModeSwitch Processes. Dissertation, University Twente, Enschede, The Netherlandes Khorrami, F., Jain, S. (1993). Nonlinear Control with End-Point Accelerati.on Feedback for a TwoLink Flexible Manipulator: Experimental Results. Journal of Robotic Systems, Vol.10, No.4, pp.505-530. Ljung, L. (1991). System Identification Toolbox. The MathWorks, Natick, MA, USA. Oakley, C.M., Cannon, R.H. (1990): Anatomy of an Experimental Two-Link Flexible Manipulator under End-Point Control. Proceedings of the 29. Conference on Decision and Control ' Honolulu, Hawaii, VoU, pp.507-513. Wemer, H. (1998). Robust Multimodel Design Using Fast Output Sampling - An LMI Approach. To appear in Automatica. Yoshikawa, T., Murakami, H., Hosoda, K. (1990). Modeling and Control of a Three Degree of Freedom Manipulator with Two Flexible Links. Proceedings of the 29. Conference on Decision and Control, Honolulu, Huwai, Vol.4, pp.2532-2537. Yurkovich, S., Hillsley, K.L., Tzes, AP. (1990a): Identification and Control for a Manipulator with Two Flexible Links. Proceedings of the 29. Conference on Decision and Control, Honolulu, Hawaii, Vol.4, pp. 1995-2000. Yurkovich, S., Tzes, A.P., Lee, I., HilIsley, K.L. (1990b): Control and System Identification of a Two-Link Flexible Manipulator. Proc. ofthe 1990 IEEE Conference on Robotics and

Automation,

Cincinnati,

Ohio,

Vo1.3,

pp.1626-1631.

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ISBN: 008 0432484