Technical note Nonlinear computed torque control for a high-speed planar parallel manipulator Weiwei Shang * , Shuang Cong Department of Automation, University of Science and Technology of China, Hefei, Anhui 230027, PR China article info Article history: Received 28 September 2008 Accepted 7 April 2009 Keywords: Parallel manipulator Trajectory tracking Nonlinear PD Nonlinear computed torque abstract A new computed torque (CT)-type controller termed nonlinear CT (NCT) controller is developed and applied to a high-speed planar parallel manipulator. The NCT controller is designed by replacing the linear PD in the conventional CT controller with the nonlinear PD (NPD) algorithm. The stability of the parallel manipulator system with the NCT controller is proven using the Lyapunov theorem, and the pro- posed controller is further proven to guarantee asymptotic convergence to zero of both tracking error and error rate. The superiority of the proposed NCT controller is verified through the trajectory tracking experiments of an actual high-speed planar parallel manipulator, and the experiment results are com- pared with the CT controller. Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved. 1. Introduction Comparing with the serial ones, the parallel manipulators have potential advantages in terms of high stiffness, accuracy, and speed [1]. Especially the high accuracy and speed performances make the parallel manipulators widely applied to the following fields, like the pick-and-place operation in food, medicine, electronic industry and so on. At present, the key issues are the ways to meet the de- mand of high accuracy in moving process under the condition of high speed. In order to realize the high speed and accuracy motion, it is very important to design efficient controllers for parallel manipulators. In literatures, there are two basic control strategies for parallel manipulators [2]: kinematic control strategies and dynamic con- trol strategies. In the kinematic control strategies, parallel manip- ulators are decoupled into a group of single axis control systems, so they can be controlled by a group of individual controllers. Propor- tional-derivative (PD) control [3,4], nonlinear PD (NPD) control [5,6], and fuzzy control [7] belong to this type of control strategies. These controllers do not always produce high control performance, and there is no guarantee of stability at the high speed. Unlike the kinematic control strategies, full dynamic model of parallel manip- ulators is taken into account in the dynamic control strategies. So the nonlinear dynamics of parallel manipulator can be compen- sated and higher performance can be achieved with the dynamic strategies. The traditional dynamic strategies are the augmented PD (APD) controller and the compute-torque (CT) controller [8– 10]. In the APD controller, the control law is the combination of the PD control term and the feed-forward dynamic compensation term calculated with the desired velocity and acceleration signals. If the accurate dynamic model is established, the APD controller can achieve satisfied tracking performances. However, the feed- forward term cannot restrain the disturbance, thus the tracking accuracy will be affected when the disturbance exists. In the traditional CT controller, the control law is the combina- tion of the PD control term and the feedback dynamic compensa- tion term calculated with the actual velocity and desired acceleration signals. That it is to say, the CT controller is a PD con- troller plus a feedback inner loop. Thus, the CT controller can get better trajectory tracking and disturbance rejection ability. How- ever, the CT controller owns two main drawbacks. Firstly, the dy- namic compensation is calculated based on the dynamic model with the fixed dynamic parameters, but the parameters are vari- able during the trajectory tracking. Thus the dynamic compensa- tion in the CT controller cannot get good compensation performance. In order to realize better dynamic compensation, adaptive controller is designed for the parallel manipulators [11]. And the external disturbance is usually decreased by the adaptive robust controllers [12,13]. Secondly, in the CT controller, the linear PD with the proportional and derivative constant is used to elimi- nate the tracking error. For the presence of nonlinear factors such as modeling error and friction in the dynamic model of the parallel manipulators, sometimes the CT controllers will not get satisfied tracking accuracy. Thus, some new methods are used to tune the PD gains of the CT controller. In [14,15], the intelligent methods are used to optimize the PD gains in CT controller, and the trajec- tory tracking accuracy of the manipulators are improved. However, it is difficult to implement this type of controllers in practice for the complex structure and enormous calculation. Fortunately, the NPD control has good suppression of the uncertain factors, and its control performances are superior to the conventional PD 0957-4158/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2009.04.002 * Corresponding author. Tel.: +86 551 36071489; fax: +86 551 3603244. E-mail address: wwshang@ustc.edu.cn (W. Shang). Mechatronics 19 (2009) 987–992 Contents lists available at ScienceDirect Mechatronics journal homepage: www.elsevier.com/locate/mechatronics