Nonlinear Dyn (2010) 59: 61–72 DOI 10.1007/s11071-009-9520-1 ORIGINAL PAPER Nonlinear adaptive task space control for a 2-DOF redundantly actuated parallel manipulator Weiwei Shang · Shuang Cong Received: 21 February 2008 / Accepted: 27 April 2009 / Published online: 13 May 2009 © Springer Science+Business Media B.V. 2009 Abstract A nonlinear adaptive (NA) controller in the task space is developed for the trajectory tracking of a 2-DOF redundantly actuated parallel manipulator. The dynamic model with nonlinear friction is established in the task space for the parallel manipulator, and the linear parameterization expression of the dynamic model is formulated. Based on the dynamic model, a new control law including adaptive dynamics compen- sation, adaptive friction compensation and error elim- ination items is designed. After defining a quadratic performance index, the parameter update law is de- rived with the gradient descent algorithm. The stabil- ity of the parallel manipulator system is proved by the Lyapunov theorem, and the convergence of the track- ing error and the error rate is proved by the Barbalat’s lemma. The NA controller is implemented in the tra- jectory tracking experiments of an actual 2-DOF re- dundantly actuated parallel manipulator, and the ex- periment results are compared with the APD con- troller. Keywords Parallel manipulator · Redundant actuation · Coulomb + viscous friction · Task space · Nonlinear adaptive control W. Shang · S. Cong ( ) Department of Automation, University of Science and Technology of China, Hefei, 230027, People’s Republic of China e-mail: scong@ustc.edu.cn 1 Introduction Over the past two decades, parallel manipulators have attracted the attention of many researchers. Although a vast literature is dedicated to kinematics and dy- namics, studies on control strategies are relatively few. In literature, there are two types of basic control strategies for parallel manipulators: kinematic control strategies and dynamic control strategies. In the kine- matic control strategies, parallel manipulators are de- coupled into a group of single-axis systems, so they can be controlled by a group of individual controllers. Proportional-Derivative (PD) control [1, 2], nonlinear PD control [3, 4], and artificial-intelligence based con- trol [5, 6] belong to this type of control strategies. The nonlinear dynamics is not considered in these controllers, so the complex computation of dynam- ics can be avoided and the controller design can be simplified considerably. However, these controllers do not always produce high performance, and there is no guarantee of stability at the high speed. Unlike the kinematic control strategies, full dynamic model of parallel manipulator is taken into account in the dy- namic control strategies. So the nonlinear dynamics of the manipulators can be compensated and higher performance can be achieved with dynamic strategies. The traditional dynamic strategies are the augmented PD (APD) controller and compute-torque controller [7, 8]. Furthermore, to gain a deep insight of the dy- namics and the traditional dynamic strategies of par- allel manipulators, Liu and Li [9] developed a unified