Control of Robot Manipulators in Joint Space R. Kelly, V. Santibanez and A. Loria

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.1049 1050." Canudas C., Fixot N., 1991, Robot control via estimated state feedback ,IEEE Transactions on Automatic Control, Vol.36, No.12, December." Canudas C., Fixot N., �str�m K.J., 1992, Trajectory tracking in robotmanipulators via nonlinear estimated state feedback , IEEE Transactionson Robotics and Automation, Vol.8, No.1, February." Ailon A., Ortega R., 1993, An observer-based set-point controller for robotmanipulators with flexible joints , Systems and Control Letters, Vol.21,October, pp.329 335.The motion control problem for a time-varying trajectory qd(t) without ve-locity measurements, with a rigorous proof of global asymptotic stability ofthe origin of the closed-loop system, was first solved for one-degree-of-freedomrobots (including a term that is quadratic in the velocities) in" Lor�a A., 1996, Global tracking control of one degree of freedom Euler-Lagrange systems without velocity measurements , European Journal ofControl, Vol.2, No.2, June.This result was extended to the case of n-DOF robots in" Zergeroglu E., Dawson, D.M., Queiroz M.S.de, Krstić M., 2000, Onglobal output feedback tracking control of robot manipulators , in Proceed-ings of Conferenece on Decision and Control, Sydney, Australia, pp.50735078.The controller called here, P D with gravity compensation and charac-terized by Equations (13.2) (13.3) was independently proposed in" Kelly R., 1993, A simple set point robot controller by using only positionmeasurements , 12th IFAC World Congress, Vol.6, Sydney, Australia,July, pp.173 176." Berghuis H., Nijmeijer H., 1993, Global regulation of robots using onlyposition measurements , Systems and Control Letters, Vol.21, October,pp.289 293.Problems 309The controller called here, P D with desired gravity compensation andcharacterized by Equations (13.16) (13.17) was independently proposed inthe latter two references; the formal proof of global asymptotic stability waspresented in the second.Problems1.Consider the following variant of the controller P D with gravity com-pensation1:�� = Kpq + Kv� + g(q)�� = -Ax - ABq�� = x + Bqwhere Kp, Kv " IRn�n are diagonal positive definite matrices, A =diag{ai} and B = diag{bi} with ai, bi real strictly positive numbers.Assume that the desired joint position qd " IRn is constant.a) Obtain the closed-loop equation expressed in terms of the state vectorT� �xT qT qT.b) Verify that the vector�# �# �# �#x 0�# �# �# �#�q = 0 " IR3n�q 0is the unique equilibrium of the closed-loop equation.c) Show that the origin of the closed-loop equation is a stable equilibriumpoint.Hint: Use the following Lyapunov function candidate2:1 1� � � � � �V (x, q, q) = qTM(q)q + qTKpq2 21� �+ (x + Bq)T KvB-1 (x + Bq).22.Consider the model of robots with elastic joints (3.27) and (3.28),1This controller was analyzed in Berghuis H., Nijmeijer H., 1993, Global regulationof robots using only position measurements , Systems and Control Letters, Vol.21, October, pp.289 293.2By virtue of La Salle s Theorem it may also be proved that the origin is globallyasymptotically stable.310 P D Control� � �M(q)q + C(q, q)q + g(q) +K(q - �) =0�J� - K(q - �) =�.It is assumed that only the vector of positions � of motors axes, but not�the velocities vector �, is measured.We require that q(t) �! qd, where qdis constant.A variant of the P D control with desired gravity compensation is3�� = Kp� - Kv� + g(qd)� = -Ax - AB�� = x + B�where�� = qd - � + K-1g(qd)�q = qd - qand Kp, Kv, A, B " IRn�n are diagonal positive definite matrices.a) Verify that the closed-loop equation in terms of the state vectorTT�T �� T qT � qT � may be written as�# �#�# �#�-A� + AB��# �#�# �#�# �#�# �#�# �#�-q�q�# �#�# �#�# �#�# �#�# �#�#d�# �#=�#�� # �#-��# �#�# �#�# �#dt�# �#�# �#�# �#�# �# -1��# �#� � ��q�# �#�#M(q) -K(� - q) +g(qd) - C(q, q)q - g(q) �#�# �#�# �#� � ��J-1 Kp� - Kv(� - B�) +K(� - q)�where � = x + B qd + K-1g(qd).b) Verify that the origin is an equilibrium of the closed-loop equation.c) Show that if �min{Kp} >kg and �min{K} >kg, then the origin is astable equilibrium point.Hint: Use the following Lyapunov function and La Salle s Theorem2.7.3This controller was proposed and analyzed in Kelly R., Ortega R., Ailon A.,Loria A., 1994, Global regulation of flexible joint robots using approximate dif-ferentiation , IEEE Transactions on Automatic Control, Vol.39, No.6, June, pp.1222 1224.Problems 3111� � � �� V (q, �, q, �) =V1(q, �, q, �) + qTKq + V2(q)2T1� �+ � - B� KvB-1 � - B�2where1 1 T 1� � � � �T �� V1(q, �, q, �) = qT M(q)q + � J� + � Kp�2 2 21�T � �T�+ � K� - � Kq2� � �V2(q) =U(qd - q) -U(qd) + qT g(qd)and verify thatT� � � � �� V (q, �, q, �) =- � - B� KvB-1A � - B�.3.Use La Salle s Theorem 2.7 to show global asymptotic stability of theorigin of the closed-loop equations corresponding to the P D controllerwith gravity compensation, i.e.Equation (13.4).4.Use La Salle s Theorem 2.7 to show global asymptotic stability of theorigin of the closed-loop equations corresponding to the P D controllerwith desired gravity compensation, i.e.Equation (13.18).14Introduction to Adaptive Robot ControlUp to this chapter we have studied several control techniques which achievethe objective of position and motion control of manipulators.The standingimplicit assumptions in the preceding chapters are that:" The model is accurately known, i.e.either all the nonlinearities involvedare known or they are negligible." The constant physical parameters such as link inertias, masses, lengths tothe centers of mass and even the masses of the diverse objects which maybe handled by the end-effector of the robot, are accurately known.Obviously, while these considerations allow one to prove certain stabilityand convergence properties for the controllers studied in previous chapters,they must be taken with care.In robot control practice, either of these assump-tions or both, may not hold.For instance, we may be neglecting considerablejoint elasticity, friction or, even if we think we know accurately the massesand inertias of the robot, we cannot estimate the mass of the objects carriedby the end-effector, which depend on the task accomplished.Two general techniques in control theory and practice deal with thesephenomena, respectively: robust control and adaptive control.Roughly, thefirst aims at controlling, with a small error, a class of robot manipulatorswith the same robust controller.That is, given a robot manipulator model,one designs a control law which achieves the motion control objective, with asmall error, for the given model but to which is added a known nonlinearity.Adaptive control is a design approach tailored for high performance ap-plications in control systems with uncertainty in the parameters.That is,uncertainty in the dynamic system is assumed to be characterized by a setof unknown constant parameters.However, the design of adaptive controllersrequires the precise knowledge of the structure of the system being controlled.314 14 Introduction to Adaptive Robot ControlCertainly one may consider other variants such as adaptive control forsystems with time-varying parameters, or robust adaptive control for systemswith structural and parameter uncertainty.In this and the following chapters we concentrate specifically on adaptivecontrol of robot manipulators with constant parameters and for which weassume that we have no structural uncertainties [ Pobierz całość w formacie PDF ]

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