GLOBAL OPTIMAL FEEDBACK-LINEARIZING CONTROL OF ROBOT MANIPULATORS Abbas Chatraei and Vaclav Záda ABSTRACT The present paper offers a new optimal feedback-linearizing control scheme for robot manipulators. The method presented aims at solving a special form of the unconstrained optimal control problem (OCP) of robot manipulators globally using the results of the Lyaponov method and feedback-linearizing strategy and without using the calculus of variations (indirect method), direct methods, or the dynamic programming approach. Most of these methods and their sub-branches yield a local optimal solution for the considered OCP by satisfying some necessary conditions to find the stationary point of the considered cost functional. In addition, the proposed method can be used for both set-point regulating (point-to-point) tasks (e.g. pick-and-place operation or spot welding tasks) and trajectory tracking tasks such as painting or welding tasks. However, the proposed method can not support the physical constraints on robot manipulators and requires precise dynamics of the robot, as well. Instead, it can be used as an on-line optimal control algorithm which produces the optimal solution without performing any kind of optimization algorithms which require time to find the optimal solution. Key Words: Robot manipulator, optimal feedback control, calculus of variations. I. INTRODUCTION The optimal trajectory planning for robot manipulators has received great attention in recent decades with many approaches presented to solve the OCP of robot manipulators. In general, these approaches can be categorized in three branches: (i) indirect methods, (ii) dynamic programming methods, and (iii) direct methods. Early time-OCP of robot manipulators was solved by indirect methods. This problem was seriously considered by Bobrow in his Ph.D. dissertation [1]. In this work a so-called position-velocity phase-plane algorithm is introduced accord- ing to which the desired path is parameterized by an arc length parameter “s” and then the robot dynamics are repre- sented in terms of this parameter, resulting in a reduction in the robot dynamics to second order. Then, the resulting second order OCP is solved in the phase plane. This proposed method was followed by some other research, resulting in improvement to Bobrow’s first work [2, 3], for instance by considering the singularity problem that appeared in OCP [4, 5]. Some other studies added an energy term to the cost function [6], and also actuator dynamics [7]. The dynamic programming method was also employed to obtain the minimum-time optimal trajectories [8–10]. In [8], Bobrow’s method was used to solve the OCP of robot manipulators, but for computing the optimal controls, a dynamic programming algorithm was developed to derive the reduced set of second order differential equations in terms of path parameter; thus removing the problem of dimensionality in dynamic programming. Although the above two methods have been used suc- cessfully in many applications, they have been replaced by direct methods in recent years. The basic idea of this method, in the case of robot manipulators, is that the joint trajectories are approximated by a parametric function, such as spline functions, and then, using nonlinear programming, the optimal values of the parameters used in the approximat- ing function are achieved. Many researchers have presented different approaches to generate the optimal joint trajecto- ries. Among these works, polynomial cubic spline functions and B-splines have been used in many studies [11–16]. There is also another sub-part of the direct methods. These kinds of methods, which include single shooting, colloca- tion, and multiple shooting methods, use a piecewise con- stant function to parametrize the control inputs (robot joints torques and forces) of the system. These methods are con- sidered in [17, 18], in detail. There is also extensive use of model predictive control (MPC) to develop optimal feedback control systems for nonlinear systems like robots [19]. Another practical application of optimal control is to design a control system for a dielectrophoretic system, presented in [20]. Except research based on the dynamic programming approach, which yield a global optimal solution to the Manuscript received April 15, 2011; revised October 17, 2012; accepted September 2, 2012. Abbas Chatraei (corresponding author, e-mail: abbas.chatraei@gmail.com) is with Department of Electrical Engineering-Islamic Azad University, Najafabad Branch, Isfahan-Iran. Václav Záda is with Mechatronics Faculty of Technical University of Liberec, Czech Republic. Asian Journal of Control, Vol. 15, No. 4, pp. 1178–1187, July 2013 Published online 9 November 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.633 © 2012 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society