Robust control of flexible manipulators via μ-synthesis

*Corresponding author. Tel.: #965-481-1188-x5884; fax: #965-484-7131.

E-mail address: [email protected] (M. Karkoub).

Control Engineering Practice 8 (2000) 725}734

Robust control of #exible manipulators via k-synthesis

Mansour Karkoub!,*, Gary Balas", Kumar Tamma#, Max Donath#

!Mechanical and Industrial Engineering Department, Kuwait University, Kuwait"Aerospace Engineering and Mechanics Department, University of Minnesota, MN, USA

#Mechanical Engineering Department, University of Minnesota, MN, USA

Received 31 March 1999; accepted 5 October 1999

Abstract

An experimental #exible arm serves as testbed to investigate the e$cacy of the k-synthesis design technique in the control of #exiblemanipulators. A linearized model of the testbed is derived for control design. Discrepancies and errors between the linearized modeland the physical system are accounted for in the control design via uncertainty models. These uncertainties include: unmodeledhigh-frequency dynamics, errors in natural frequencies and damping levels and actuator and sensor errors. Colocated andnoncolocated controllers are designed using k-synthesis. It is observed, theoretically and experimentally, that the k-synthesis designtechnique is a viable control tool for tip tracking with #exible manipulators. ( 2000 Elsevier Science ¸td. All rights reserved.

Keywords: Robust control; k-Synthesis; Flexible manipulators

1. Introduction

Two driving requirements for high-performance robotmanipulators are high-speed and accurate end-pointtracking. Existing manipulators are massive and bulkywhich usually translates into high-power consumptionand slow speeds. To increase speed, arm inertia is re-duced which can result in increased #exibility of themanipulator and degradation of end-point tracking per-formance. One approach to improve performance is tofeedback end-point positions or accelerations resulting ina noncolocated control design problem. A drawback ofthis approach is that there is usually no actuatorcolocated with the sensor at the tip. This leads to a non-colocated control problem whose closed-loop stabilitycan be sensitive to modeling errors or uncertainty(Cannon & Schmitz, 1984; Park & Asada, 1994).

Recent papers have discussed a variety of control strat-egies for the control of #exible mechanical systems.Kwon and Book (1994) proposed a time-domain inversedynamics approach for noncolocated control of a single-link arm. This work extends the method of feedforwardcompensation based on inverse dynamics to non-colocated control. Experimental results of accurate joint

tracking response are presented. The potential draw-backs of this method include the restriction that thetorque command pro"le may not excite the unmodeledhigh-frequency modes, and the lack of robustness tomodeling errors and sensor noise. Kang and Yang (1994)propose synthesis of noncolocated controllers for #exiblemechanical systems using time delays. Again the issue ofrobustness to changes in the plant dynamics and timedelay are not speci"cally addressed. An alternativecontrol design is proposed by Park and Asada (1994) inwhich the torque actuator is relocated closer to the tip.Overall this approach may be limited in its real-worldapplication. Several other authors have attempted eitherto modify the structure of the #exible arm, see Ghazaviand Gordaninejad (1995), or feedback shear strains tocontrol the structural #exibility vibrations, see Luo andGuo (1997). Although these methods achieve good vibra-tion suppression, none of the controllers address the issueof robustness to modeling errors or uncertainties. Todate, these control techniques work well on rigid or fairlyrigid manipulators whose low-frequency modes are bothwell known and do not directly a!ect the performancespeci"cation.

Colocated pointing and disturbance-rejection controlalgorithms for #exible manipulators insensitive topayload dynamics were developed by Cannon andSchmitz (1984). Though robust to payload dynamics,their end-point tracking performance was poor.

0967-0661/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 9 6 7 - 0 6 6 1 ( 0 0 ) 0 0 0 0 6 - X

Fig. 1. Photograph of the Human Motion Laboratory testbed.

High-performance, end-point tracking control usingnoncolocated control was achieved by Cannon andSchmitz (1984) but required accurate knowledge of thepayload. Burdakov (1998) devised a compensatory strat-egy for parametric uncertainties.

The approach taken in this paper, similar to that ofCannon and Schmitz, is to design a high-performance,noncolocated end-point tracking controller that is robustto an unknown tip mass. Errors in the system, modelerrors and uncertainty in the payload are directly ac-counted for in the control design process. The structuredsingular value (k) synthesis technique, `D}Ka iteration, isused to design the end-point tracking controller (Balas,Doyle, Glover, Packard & Smith, 1991; Packard, Doyle& Balas, 1993). The bene"t of k-synthesis is that it allowsthe direct inclusion of modeling errors or uncertainties,measurement and control inaccuracies and performancerequirements into a common control problem formula-tion. This technique is well established and has beenapplied extensively in the area of #exible structurecontrol using colocated and noncolocated actuators andsensors with great success (Balas & Doyle, 1994; Smith,Chu & Fanson, 1994).

An experimental, single-link #exible arm testbed at theUniversity of Minnesota is used to represent a typical#exible manipulator. A linearized plant model, of a non-linear, dynamic model of the manipulator, is developedinitially. Discrepancies between the linear model and thephysical system are included in the k-synthesis frame-work via weighting functions to develop robust,high-performance controllers. Errors and uncertaintiesaccounted for in the control design procedure include:unmodeled high-frequency dynamics, uncertainty innatural frequencies and damping levels and actuator andsensor errors.

The following is an outline of the paper. Section 2 de-scribes the experimental testbed, the performance objec-tives and the linearized model of the #exible arm. InSection 3, the D}K iteration technique for control designis presented. The selection of performance and uncertain-ty weighting functions are discussed. Two controllers aresynthesized for the #exible arm. The initial controller isdesigned with an aggressive performance and robustnessobjective. Due to excessive overshoot in the initial design,the slew rate tracking response is reduced in the seconddesign. Theoretical and experimental result for both de-signs are presented. Section 3.3 investigates the robust-ness of the controllers to tip mass variations, hub-anglecommands and time delays. Section 4 summarizes thedi!erence in performance between all three controllers.

2. Experimental 6exible manipulator

An experimental #exible arm located in the HumanMotion Laboratory at the University of Minnesota is

used to represent a typical #exible manipulator in thispaper, see Fig. 1. The testbed consists of a controller,a servo board, a servo drive, a servo motor, and analuminum beam. The characteristics of the servo motorand the aluminum beam are given in Tables 1 and 2.

A nonlinear dynamical model was derived for the arm(see Fig. 2) in Karkoub (1994) based on Timoshenkobeam theory and the assumed modes method. This non-linear model is linearized around the stationary operat-ing point to obtain a linear model of the arm. Transferfunctions from torque input to hub-angle and tip de#ec-tion are computed using the data listed in Table 1. The"rst two bending modes of the beam at 23 and 179 rad/sare included in the linear model. The "rst and secondbending modes are assumed to have damping values of0.88 and 1.41%, respectively. A control design model forthe testbed which includes all the hardware componentsis subsequently obtained. The derived linear modelneglects arm dynamics above the second bending mode.In addition, the dynamics of the electronic equipment arenot included in the linear model.

The focus of this paper is on the selection of perfor-mance and robustness weights in the k framework to

726 M. Karkoub et al. / Control Engineering Practice 8 (2000) 725}734

Table 1Material and geometrical properties of the arm

Variable Value

I): Hub moment of inertia 2.3]10~4 kg m2

Mp: Tip mass 0.3333 kg

I: Beam moment of inertia 0.429 kg m2

¸: Beam length 0.974 mo: Material density 1.341 kg/mE: Modulus of elasticity 7.00]1010 N/m2

Table 2Servo motor characteristics

Variable Value

Im: Rotor moment of inertia 0.01384]10~4 kg m2

Kb: Back EMF constant 24.9 V/krpm

Ku : Tachometer sensitivity 7.0 V/krpmK

t: Motor torque sensitivity 0.24 Nm/amp

¸m: Motor inductance 1.8]10~3 mHR: Motor resistance 1.39 )¹m: Motor mechanical time constant 5.8]10~3 s¹e: Motor electrical time constant 1.29]10~3 s

Fig. 2. Schematic of the #exible arm manipulator.

achieve the closed-loop performance and robustness ob-jectives. The controllers are designed using a hub-anglesensor and tip accelerometer for feedback measurementsand a hub torque actuator for control. The followingare the closed-loop performance and robustnessrequirements:

f Steady-state, tip tracking error of no more than 2%.f Rise time of approximately 2 s.f Increase the damping of the "rst #exible mode by

a factor of "ve.f Insensitivity to changes in the second and higher#exible modes.

f Robustness to variations in the manipulator tip mass.

All of the control objectives are achieved. There isgood correlation between the theoretical simulations andthe experimental testbed results for the controllers. Thenext three sections provide a description of the testbedcomponents.

2.1. Experimental setup

The following hardware components are used for real-time control on the experimental testbed of the #exiblemanipulator (Fig. 3):

f A CIMROC II controller made by PaR Systems Inc.f A servo motor made by Kolmorgen Corp. model num-

ber TT-2042-A.f A harmonic drive made by Emhart Corp. with a gear

ratio of 180.f A tachometer whose shaft is connected to the motor

shaft and produces a voltage proportional to the speedof the rotor.

f An encoder made by Intel Corp. and outputs 16 digitalbits of information on the angular position of the arm.

f A Macintosh IIfx for data acquisition and real-timecontrol using a National Instruments data acquisitionboard, NB-MIO-16, and the 33 MHz DSP (digitalsignal processing) board.

f A multi-purpose accelerometer made by IC SensorsCorp. It has a sensitivity of 1.0 V/G.

f The arm is a thin aluminum strip 0.947 m long,0.0762 m wide, and 0.0064 m thick.

2.2. Theoretical and experimental transfer functions

The theoretical torque input to hub-angle and tip-acceleration transfer functions were derived using aBernoulli}Euler beam model (Kanoh, 1991; Karkoub,1994). The results are shown in Fig. 4. The "rst two modesare determined theoretically to be at 25 and 178 rad/s,respectively, which are within 2% of the experimentallymeasured values. Experimentally, the "rst mode wasfound to be at 24.5 rad/s and the second mode at181 rad/s. The theoretical linear model was updatedto incorporate the experimental natural frequencyinformation.

The testbed components must be included to obtainthe transfer function of the voltage input to the resolverand accelerometer voltage output (see Fig. 3). That is,models of the input signal "lter, servo drive, and servocard need to be included in the overall transfer function.The servo motor has a moment of inertia equal to J

mand

a viscous damping coe$cient b. The inductance andresistance of the motor are ¸ and R, respectively. Themotor has a back EMF constant equal to K

b. The result-

ant torque supplied by the motor is transmitted througha gear pair (gear ratio is 180). Inside the motor there isa tachometer for measuring the speed of the motor and

M. Karkoub et al. / Control Engineering Practice 8 (2000) 725}734 727

Fig. 3. Schematic diagram of the #exible manipulator system: servo motor and #exible arm.

Fig. 4. Experimental and theoretical bode plots of voltage input totip-acceleration transfer functions (theoretical (solid) and experimental(dotted)) and voltage input to hub-angle transfer functions (theoretical(plus) and experimental (dash}dotted)).

has a gain Ku . All these constants are listed in Table 1.The CIMROC control box has a velocity loop represent-ed by the gain G and a seventh-order, low-pass "lter (seeFig. 3). The voltage input to tip acceleration transferfunction, < to yK , and voltage input to hub-angle, < toh (see Fig. 3), are obtained and the results are shown inFig. 4. These new transfer functions are superimposed onthe experimental ones for comparison (see Fig. 4). Notethe excellent correlation between the linear theoreticalmodel and the experimental data.

The voltage to tip and hub transfer functions of themanipulator are obtained using a 0.5}50 Hz pseudo-random noise signal input to the servo drive motor. Theinput signal is chosen so that it excites as many modes aspossible. Other types of input signals, such as varyingsinusoids, were used to excite the #exible modes of therobot arm, the results were almost identical. The tipacceleration, resolver output voltage, and input signal arerecorded. The time domain data is transformed into thefrequency domain via a fast Fourier transform (FFT) and"ltered using a Hanning window. The frequency-domaindata is computed with 2048 FFT points and 50% over-lap. Matlab is used to process the data (The Mathworks,1992, Tse, Morse & Hinkle, 1978).

3. Control design

The k-synthesis technique is used to design controllersfor the #exible manipulator. The method used to performthe k-synthesis procedure is `D}K iterationa. D}K iter-ation is a two-step minimization process: the "rst step isa minimization of the H

=norm over all stabilizing con-

trollers K while the scaling matrix D is held "xed, and thesecond step is a minimization over a set of scalings,D while the controller K is held "xed (Balas et al., 1991;


Fig. 5. Interconnection control design diagram for the #exible arm manipulator.

Doyle, 1982). The controllers are designed using the D}Kiteration software in the `k-analysis and synthesisa tool-box (Balas et al., 1991). A more detailed review ofk-synthesis can be found in Balas et al. (1991), Doyle(1982) and Packard et al. (1993).

Controllers are designed for the #exible arm manipula-tor using hub-angle and tip acceleration feedback. Thek framework requires performance and robustness speci-"cations be described as weighting functions on systemtransfer functions. An interconnection structure isformed, consisting of the model uncertainties and per-formance objectives, such that the closed-loop k valueacross frequency is less than one when all of the objec-tives are satis"ed.

The performance objectives include: less than 2%steady-state tip tracking error, a 2 s rise time and afactor of 10 increase in the "rst mode damping of the arm.Two performance weighting functions are used in thek-framework to achieve these objectives. The rise timeand steady-state tracking performance objectives onthe hub response are de"ned in terms of the weight (seeFig. 5).

=)6"}1%3&

"901

100s#1

12s#1

.

This weight directly e!ects the slew rate of the manipula-tor by penalizing the di!erence between the commandedand sensed hub response. The DC gain of 90 re#ects thesteady-state error objective, less than 2% tracking erroror 1.1%. The roll o! frequency of =

!##}1%3&, 0.5 rad/s,

corresponds to the 2 s rise time speci"cation. Above50 rad/s, the tracking error is allowed to degrade bya factor of 1.5 from the open-loop response.

Increasing the "rst mode damping level by a factor of"ve is addressed with the =

5*1}1%3&performance weight

(see Fig. 5). The open-loop magnitude response of thetransfer function from the hub actuator voltage totip acceleration has a peak of 20, at approximately24.5 rad/s, as seen in Fig. 4. A factor of "ve increase in the"rst mode damping ratio would correspond to a reduc-tion of the 24.5 rad/s peak magnitude by a factor of "vegiven that the transfer function magnitude stays thesame. The control design objective is to achievea H

=norm less than one for all the weighted closed-loop

transfer functions, which would result in a k value lessthan one across frequency. Therefore selecting=

5*1}1%3&to

be a constant 0.25 would result in the weighted hubcommand to tip acceleration response having a peakmagnitude response at 24.5 rad/s of "ve. A closed-loopk value of the interconnection structure shown in Fig. 5


Fig. 6. Hub time response using controller d1, hub-angle and tip-acceleration feedback: command (dashed), experimental (dotted), theor-etical (solid).

of less than one would imply that the magnitude of theweighted transfer function from hub command to tipacceleration would be less than one. This corresponds toa factor of "ve increase in the damping value of the "rstbending mode at 24.5 rad/s.

It is important for the controllers to be insensitive tounmodeled #exible mode dynamics, neglected nonlin-earities and variations in the manipulator tip mass. Thereare two forms of model uncertainty included in the con-trol design formulation, additive and input multiplicativeuncertainty, to achieve these objectives. The additiveuncertainty weight, denoted by =

!$$*5*7%in Fig. 5, ac-

counts for high-frequency dynamics and nonlinearitiesneglected in the control design model. The magnitude ofthe=

!$$*5*7%weighting function is selected to encompass

the peaks of the second and higher modes of vibration ofthe manipulator. This ensures that these modes are gainstabilized by the controller, provided that the weightedH

=norm of the transfer function from w

2to z

2is less

than one. Achieving an H=

norm less than one alsoserves to limit the bandwidth of the system to 60 rad/sand enforces a roll-o! constraint on the closed-loopsystem. Since=

!$$*5*7%is greater than 0.08 above 60 rad/s,

it helps account for modeling errors due to nonlineare!ects not directly accounted for in the linear model. Theadditive uncertainty weight is modeled as

=!$$*5*7%

"

0.631s2#5.31s#22.6

s2#59.41s#2270.

A multiplicative input uncertainty model, =!#56!503

inFig. 5, is included in the control problem formulation toaccount for input modeling errors and imperfections inthe linearized actuator model. The actuator uncertaintyweight is selected to be

=!#56!503

"0.10150

s#11

500s#1

.

This weight implies that there is approximately 10%error in the input signal at lower frequency that rises to100% uncertainty at high-frequency.=

!#56!503weights the

transfer function from z1

to w1. As in the case of the

additive uncertainty weight, the closed-loop system willbe robust to 10% input error at low-frequency rising to100% error at high-frequency if the H

=norm of the

weighted transfer function =!#56!503

(I#PC)~1 is lessthan one where P represents the #exible arm model andC the controller. The multiplicative input uncertaintyweight helps limit the bandwidth of the system, whileincreasing the system robustness to changes in the plantmodel. Though the variations in the manipulator tipmass are not directly accounted for in the control designmodel, both the additive and multiplicative uncertaintyweights have been designed with robustness to variationsin the tip mass in mind. The magnitude of the multiplica-tive and additive weights at the "rst natural frequency of

the system, 24.5 rad/s, account for approximately 15%variation in its damping and resonance frequency.

Models of sensor noise associated with the hub-anglesensor and tip accelerometer,=

4%/403}/0*4%, are included in

the problem formulation since all of the feedback signalsare corrupted to some extent by noise. A constant noiseweight across frequency of 10~4 re#ects experimentalmeasurement of the sensor noise.

3.1. Control design I

A controller is designed for the #exible arm manipula-tor using hub-angle and tip-acceleration feedback. FiveD}K iterations were performed to synthesize the control-ler. The resulting 27 state controller was reduced to 10states using balanced-realization model-reductionand implemented on the theoretical model and experi-mental testbed. The time-response results are shown inFigs. 6 and 7.

The designed k-synthesis controller leads to very goodhub and tip responses. The rise time of the arm is 0.8 s,with a 20% overshoot and a tip settling time of 2 s. Notethat the experimental hub response has an overshootwhich does not correspond with the predicted response.A possible cause of the overshoot in the hub time re-sponse is the rapid slew-rate. A fast slew-rate may serveto accentuate nonlinear e!ects or time delays that wereunaccounted for in the control design model.

3.2. Control design II

The hub tracking performance speci"cation is relaxedleading to a slower response time (about a second slower)in this control design. The roll-o! frequency of the con-


Fig. 7. Tip-acceleration time-response controller d1, hub-angle andtip-acceleration feedback: experimental (solid), theoretical (dotted). Fig. 8. Hub-angle time response using controller d2, hub-angle and

tip-acceleration feedback: command (dashed), experimental (dotted),and theoretical (solid).

Fig. 9. Tip-acceleration time response using controller d2, hub-angleand tip-acceleration feedback: theoretical (solid), experimental (dotted).

troller is shifted from 24 to 16 rad/s by shifting thecross-over frequency of the additive uncertainty weightby 5 rad/s. The new additive uncertainty weight isgiven by

=!$$*5*7%

"

0.6632s2#1.31s#1.82

s2#22.82s#180.51.

The other weighting functions are the same as used in the"rst noncolocated design.

Five D}K iterations were performed to synthesize thesecond, 27 state noncolocated controller. This controlleris reduced to 10 states using balanced-realization modelreduction. The hub response speed of this controller isreduced in comparison to the previous controllers asshown in Figs. 8 and 9, respectively. Note that the theor-etically predicted responses and the experimental re-sponses match very well with zero steady-state trackingerror. The cost of eliminating the overshoot is a 1 s reduc-tion in the system step response. Note the large reductionin the relative tip acceleration as seen in Fig. 9. Themanipulator tip is practically at the same angularposition as the hub.

3.3. Robustness of the control designs

To investigate the robustness of the two controllers,a 217 g mass is added to the manipulator tip. The addi-tional mass increased the tip mass by 70% with little orno a!ect on the damping level. The additional massshifted the "rst mode from 25 to 21 rad/s and the secondmode from 178 to 171 rad/s. The theoretical and experi-mental closed-loop time responses of Controller I areshown in Figs. 10 and 11. The time responses with Con-troller II implemented were identical to the original timeresponses in Figs. 8 and 9. The variation in the #exible

modes of the system were accounted for indirectly in thek-synthesis problem formulation by the appropriateselection of input and additive uncertainty weights.It is apparent from this experiment that the additionalmass resulted in no signi"cant change in the behavior ofthe closed-loop system.

Hub-angle commands of 11 and 483 are input to themanipulator to observe the e!ect of nonlinearities on thedynamic response of the manipulator (see Figs. 6, 7, and12}15). Based on these experimental results the size of theinput command does not have a signi"cant in#uence onthe slew response of the #exible arm. Also, it appears thatthe overshoot exists in the hub-angle response even forsmall input commands. This implies that, although


Fig. 10. Hub-angle response of the #exible arm using Design I control-ler with modi"ed tip mass (tip mass"530 g): command (dashed),theoretical (solid), experimental (dotted).

Fig. 11. Tip-acceleration response of the #exible arm using DesignI controller with modi"ed tip mass (tip mass"530 g): experimental(dotted), theoretical (solid).

Fig. 12. Hub-angle response of the #exible arm to an 113 input com-mand using Design I controller: command (dashed), experimental (dot-ted), and theoretical (solid).

Fig. 13. Tip-acceleration response of the #exible arm to an 113 inputcommand using Design I controller: theoretical (solid), experimental(dotted).

nonlinearities from the unmodeled dynamics have amarginal e!ect on the response of the manipulator, theyare not the main contributors to the discrepancies be-tween the theoretical and experimental results. Theauthors believe that a phase di!erence between the theor-etical and experimental transfer function near the cross-over frequency, 24 rad/s, causes the observed overshoot.To verify this conjecture, an all-pass "lter is added tothe controller input to introduce a 0.26 s phase lag.The transfer function of the "lter is given by

¹(s)"1!0.13s

1#0.13s.

Simulation responses show an overshoot in the theoret-ical hub time response on the same order as seen experi-mentally. This leads the authors to believe that thediscrepancies between the experimental and the theoret-ical hub time responses are due to phase error caused byan inaccurate representation of one or several testbedcomponents.

4. Conclusions

The "rst controller was designed using hub-angleand tip-acceleration feedback. The experimental and


Fig. 14. Hub-angle response of the #exible arm to a 483 input com-mand using Design I controller: command (solid), experimental (dot-ted), and theoretical (solid).

Fig. 15. Tip-acceleration response of the #exible arm to a 483 inputcommand using Design I controller: theoretical (solid), experimental(dashed).

theoretical responses of the hub and tip-accelerationmatch very well except for the overshoot in the hubtime response. The tip accelerometer indirectly monitorsthe tip tracking objective and provides direct sensing ofthe lightly damped, #exible manipulator modes.

The second controller eliminated the overshoot fromthe hub-angle time response by reducing the controllerbandwidth from 24 to 16 rad/s. This led to a slowerresponse time though, the experimental and theoreticalresponses for the tip acceleration and hub-angle are inexcellent agreement (Figs. 8 and 9).

Tests were performed on the controllers to investigatetheir robustness to the size of the input command and the

tip mass. The response of the #exible manipulator systemis very similar for large and small input commands andvirtually unchanged in the presence of the additionaltip mass.

Overall, k-synthesis control design techniques weresuccessfully used in designing good tip tracking control-lers for a #exible manipulator experiment. This techniqueleads to controllers that are robust to unmodelled dy-namics, input and parameter uncertainty, and sensornoise. The robustness of the control designs were veri"edtheoretically and experimentally.

Acknowledgements

The authors wish to acknowledge the generous "nan-cial support from NASA Langley Control/Structure In-teraction Group (NAG-1-821), NSF (ECS-9110254), andthe University of Minnesota McKnight Land-GrantProfessorship Program. The authors would also liketo thank the Productivity Center at the Mechanical En-gineering Department.

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Robust control of flexible manipulators via μ-synthesis