Application of Nonlinear PD Learning Control to a Closed Loop Manipulator P. R. Ouyang W. J. Zhang Department of Aerospace Engineering Department of Mechanical Engineering Ryerson University University of Saskatchewan Toronto, Ontario, M5K 2B3 Canada Saskatoon, Saskatchewan, S7N 5A9 Canada pouyang@ryerson.ca wjz485@usask.ca Abstract— In this paper, A new control method called nonlinear PD learning control (NPD-LC) is proposed and applied for the trajectory tracking control of a closed loop manipulator. The proposed control algorithm is a combination of a nonlinear PD feedback control and a directly iterative learning (feedforward) control. Consequently, this new control method possesses both adaptive and learning properties. One of the features of the NPD- LC controller is that the learning process is directly based on the previous torque profile of a repetitive trajectory tracking. It is proved that the asymptotic convergence for both tracking positions and tracking velocities is guaranteed based on the NPD- LC controller. Experimental studies are conducted for a closed loop manipulator under different operation conditions. It is demonstrated that the NPD-LC control method can achieve a fast convergence speed. Index Terms—Nonlinear PD control, iterative learning control, closed loop manipulator, trajectory tracking. I. INTRODUCTION Systems such as robot manipulators, which track given reference trajectories and operate in a repetitive mode, are very common in industrial processes. Trajectory tracking with proportional-derivative (PD) or proportional-integral- derivative (PID) control [1] may not lead to high accuracy tracking performance, especially at high speed processes. To address this issue, many researchers studied nonlinear PD (NPD) control and demonstrated NPD control is better than PD control in terms of increasing damping, reducing rise time, and improving tracking accuracy [2-4]. Since the control gains of NPD control can be adjusted on-line as functions of tracking errors, NPD control is superior to PD control. In recent years, many adaptive control techniques [5-7], which can accommodate changing environments and are insensitive to modeling errors, have been reported in the robotics literature as effective alternatives to PD/PID control. Most of these control methods are based on techniques that use the regression matrices to make the system dynamics linear with respect to the unknown system parameters. In parallel with adaptive control techniques, many substantial studies have been carried out on iterative learning control (ILC) [8-12] for the purpose of improving system performance. Arimoto [8] defined ILC as a class of control algorithms that achieve an asymptotic zero tracking error by an iterative process. In such a process, the same tracking task is repeatedly performed by the system, starting always from the same initial conditions. Emulating human learning, ILC uses knowledge obtained from the previous trial to modify the control input for the current trial so that a better performance can be achieved from iteration to iteration. Following the same idea used in the ILC method, we apply the learning strategy to NPD control in an iterative mode and develop a new control law, called nonlinear PD learning control (NPD-LC) law. The main purpose of this paper is to study the application of NPD-LC to a closed loop manipulator. This paper is organized as follows. In Section II, the NPD-LC control technique is proposed, and the similarities and differences between NPD-LC and ILC are analyzed. The convergence analysis is provided in Section .. Experimental studies to verify the effectiveness of the proposed control scheme are described in Section IV. Conclusions are given in Section V. II. NPD-LC ALGORITHM DESIGN A. Dynamic Model For a robot manipulator with n joints driven by n actuators, where each joint consists of a prismatic or rotational joint, it is known that the dynamic equation can be expressed by [1] (())() (() ( )) ( ) ( ( )) () Dqt qt C q t ,q t qt Gqt Tt       (1) where 1 2 () [ () () ( )] T n n qt qt q t q t    is the joint position vector, () n qt   is the joint velocity vector, and () n qt   is the joint acceleration vector. ( ( )) nxn Dqt  is the inertia matrix, (() ( )) ( ) n C q t ,q t qt    denotes a vector containing the Coriolis, centrifugal and other coupling terms, ( ( )) n Gqt  is the gravitational and other disturbance vector, and () n Tt  is the input torque vector. It is known [1] that () 2 ( ) Dq C q,q    is a skew-symmetric matrix. If the desired joint trajectory is 3 () d q t C  for [0, ] f t t  , and () () () d et q t qt   , equation (1) can be linearized along the desired trajectory ( ( ), ( ), () d d d q t q t q t   ) as follows 1 ()() [ () ( )] ( ) ()() (,,,) () Dtet Ct Ct et Ftet neeet Tt          (2) 978-1-4244-2495-5/08/$25.00 © 2008 IEEE. 102 Proceedings of the 2008 IEEE/ASME International Conference on Advanced Intelligent Mechatronics July 2 - 5, 2008, Xi'an, China Authorized licensed use limited to: Ryerson University Library. Downloaded on October 7, 2009 at 11:49 from IEEE Xplore. Restrictions apply.