The paper deals with robust motion control of robotic

systems with unknown friction parameters and payload mass. The parameters of the robot arm were considered known with a given precision. To solve the control of the robot with unknown payload mass and friction parameters, sliding mode control algorithm was proposed combined with robust parameter adaptation techniques. Using Lyapunov method it was shown that the resulting controller achieves a guaranteed final tracking accuracy. Simulation results are

presented to illustrate the effectiveness and achievable

control performance of the proposed scheme

Lantos, Béla

Márton, Lőrinc

English

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Sliding Mode Robot Control with Friction and Payload EstimationPaper:Sliding Mode Robot Control with Friction and PayloadEstimationLo˝rinc Ma´rton , and Be´la Lantos   Dept. of Electrical Engineering, Sapientia Hungarian University of Transylvania540053 Tg. Mures, P-ta. Trandafirilor 61, RomaniaE-mail: martonl@ms.sapientia.ro  Dept. of Control Engineering and Information Technology, Budapest University of Technology and EconomicsH-1117 Budapest, Magyar Tudo´sok ko¨ru´tja 2, HungaryE-mail: lantos@iit.bme.hu[Received May 5, 2004; accepted June 16, 2004]The paper deals with robust motion control of roboticsystems with unknown friction parameters and pay-load mass. The parameters of the robot arm wereconsidered known with a given precision. To solvethe control of the robot with unknown payload massand friction parameters, sliding mode control algo-rithm was proposed combined with robust parameteradaptation techniques. Using Lyapunov method it wasshown that the resulting controller achieves a guaran-teed final tracking accuracy. Simulation results arepresented to illustrate the effectiveness and achievablecontrol performance of the proposed scheme.Keywords: robot control, sliding control, adaptive com-pensation, Lyapunov stability1. Introduction1.1. MotivationNowadays the robotic applications require increasedtransient performances and path tracking proprieties. Toachieve these requirements the introduction of the mathe-matical model in the control algorithm is necessary. Thereare analytical methods to determine the exact mathe-matical model of a robot, for example using the Euler-Lagrange method [1–3]. The parameters of the model(masses, inertias, length of the arms) are often catalogdata which are given by the manufacturer of the roboticsystem [4, 5]. However especially in the case of the in-ertias the value of the parameters cannot be determinedexactly, they are known with a given precision. This un-certainty should also be taken into consideration in thecontrol algorithm.The friction phenomena which should be considered inevery mechanical system can be described with modelswhose parameters are time varying, depending externalfactors. These parameters cannot be determined a-priori.At the other hand there is a tendency in robotic industry tobuild robots with lightweight arms to avoid unnecessaryenergy consumption. For this reason in the robot modelthe mass of the payload cannot be neglected related to themasses of the arms. The mass of the payload in manyapplication is unknown, varies according to the specifictask of the robot. These considerations suggest that thefriction parameters and the mass of the payload should beestimated on-line. If only the friction forces and the pay-load are unknown in the model it is unnecessary to useadaptive control algorithms which estimate all the param-eters of the robot.1.2. Previous WorksIn the past years the robust adaptive control methods [6]became a wide spread method to control not only the lin-ear but also the nonlinear systems with unknown parame-ters. Robust adaptive control of robotic manipulators hasalso been intensively studied in the control community.Early results can be found in the work of Slotine andLi [7] who proposed the introduction of a simple switch-ing term in the adaptive robot control algorithm to handlemodelling errors.The introduction of a neural network in the adaptivecontrol law to handle uncertainties due to payload uncer-tainty was proposed by Leahy et al. [8]A decentralized robust adaptive control algorithm fortrajectory tracking was proposed by Fu [9]. The controllercan be implemented in a decentralized way, i.e., a sub-controller is independently and locally equipped for eachjoint servo loop.A robustified adaptation law (switching σ modifica-tion) combined with sliding control was proposed for sin-gle arm robotic systems by Chen and Papavassilopoulos[10] to increase the robustness of the control system andto improve the tracking error performance.Adaptive disturbance attenuating controller was pro-posed by Tomei for robotic manipulators without [11] andwith [12] friction compensation. It was considered that allthe disturbances are bounded and if the disturbance van-ishes the controller achieves asymptotic tracking.Artega [13] proposes an adaptive scheme designed inconjunction with a linear velocity observer. Boundednessof the estimated parameters, tracking error and state esti-mation are guaranteed.Vol.8 No.5, 2004 Journal of Advanced Computational Intelligence 553and Intelligent InformaticsMa´rton, L., and Lantos, B.In many works it is considered that the model of thecontrolled robot is completely unknown. In this case inthe feedforward term of the controller neural or fuzzymodels [14] are introduced to compensate the effect ofnonlinearities and model uncertainties. The parameters ofthese models are tuned on-line.Lewis et al. [15] combine the neural control techniquewith linear optimal control theory to control rigid manip-ulators with completely unknown models.An adaptive neural-network based tracking controlwith guaranteed H∞ performances was proposed forrobotic systems with plant uncertainties and external dis-turbances by Chang and Cheng [16].Barambones and Etxebarria [17] proposes a controllerfor generic manipulator with unknown parameters andsliding-mode control which robustifies the design andcompensates the neural approximation errors.Many works deal with the control of square nonlinearMulti Input Multi Output (MIMO) systems with poorlyunderstood dynamics. The developed controllers couldalso be applied for robot control. Direct and indirectfuzzy adaptive control schemes are proposed by Ordonezand Passino [18]. The robustness of the adaptation is in-creased with parameter projection algorithms [19, 20]In the work of Chang [21] fuzzy control methods arecombined with H∞ control theory to design robust outputtracking controller for nonlinear MIMO systems.Nonlinear robust adaptive control algorithm for MIMOsystems with unknown nonlinearities using switchingfunctions and RBF type neural networks was used by Xuand Ioannou [22].1.3. Robot ModellingThe mathematical model of an open chain, rigid, n de-gree of freedom robot is given by:H qq¨C q  q˙q˙D q  τ hF q˙ . . . . (1)where q is the joint position vector of the robot and thevector τ represents the control signal realized by currentcontrolled DC servos. The following notations is used:H q is the generalized inertia matrix, C q  q˙ shows theeffect of the centripetal and Coriolis forces, D q is thegravity effect, hF q˙ represents the effect of the frictionforce which acts on the joints of the robot.The following properties of the robot model are wellknown [1]:1 The matrix H q is positive definite2 The matrix ˙H q   2C q  q˙ is skew-symmetric,namely for any x  Rn we have:xT   ˙H q 2C q  q˙x  0 . . . . . . . . (2)1.4. AssumptionsTo develop a control law with friction and payload es-timation in the presence of modelling errors firstly let usconsider the following assumptions:Assumption Ia (Payload)Let us assume that the dimensions of the payload canbe neglected compared with the dimensions of the roboticarm so its length and inertia is neglected. In this case theterms of the robotic model can be written in the followingform:H q  HR qmHL q . . . . (3)C q  q˙q˙D q  CR q  q˙q˙DR qm CL q  q˙q˙DL q . (4)where m is the mass of the payload. HR, CR and DR areterms of the robotic arm model without payload. HL, CLand DL are known terms which can be determined fromthe robot model and do not contain the payload and fric-tion parameters. The analysis of these terms are presentedin Appendix A.Assumption Ib (Payload)The mass of the payload is bounded, namely 0  m mM, where mM is the known upper bound of m.Assumption IIa (Modelling errors)The modelling errors for the robotic arm are boundedwith known bounds, i.e.HR q  HRC q∆HR q . . . . . . . . . (5)CR q  q˙q˙DR qCRC q  q˙q˙DRC q∆CR q  q˙q˙∆DR q . (6)where HRC, CRC and DRC are approximately known termsof the robot model that are used in control algorithm and∆HR, ∆CR, ∆DR is the matrix and the vectors which con-tains the modelling errors. Each element of these termsare bounded: ∆HRi j   ∆HRi jM    ∆CRi   ∆CRiM ∆DRi   ∆DRiM . . . . . . . . . . . . (7)where  ∆HRi jM,  ∆CRiM,  ∆DRiM are known upperbounds.Assumption IIb (Modelling errors)The unmodelled dynamics and external disturbancescan be incorporated in a vector d which appears as an ad-ditive term in the robot model. Each element of this vectoris bounded with known bounds:di  diM . . . . . . . . . . . . . . (8)Assumption III (Friction Modelling)The friction force acts separately on each joint of therobot. It can be written in a linearly parameterized form:hFi q˙i  µ q˙iθ TFPiξ FPi q˙i 1 µ q˙iθTFNiξFNi q˙i. . . . . . . . . . . . . . . . . (9)where µ q˙i  1 if q˙i  0 and 0 otherwise. The vec-tors θFi contains the unknown parameters of the frictionmodel and ξFi q˙i are known regressor vectors. This rela-tion means that the friction can be described by two sep-arate linearly parameterized models, one for the positive(P) velocity regime and the other for negative (N) velocityregime. The switching between these two models occursat the zero velocity. If we incorporate the switching func-554 Journal of Advanced Computational Intelligence Vol.8 No.5, 2004and Intelligent Informatics

Sliding mode robot control with friction and payload estimation

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