Robust trajectories following control of a 2-link robot manipulator via coordinate transformation for manufacturing applications Chieh-Li Chen, Tung-Chin Wu, Chao-Chung Peng n Department of Aeronautics and Astronautics, National Cheng Kung University, No. 1, University Road, Tainan 701, Taiwan article info Article history: Received 24 November 2007 Received in revised form 27 September 2010 Accepted 1 October 2010 Available online 4 December 2010 Keywords: Robot Manufacturing Tracking control Cross-coupled control Sliding mode abstract A common idea concerning trajectory control of robot manipulators is to tackle the motion of the end- effector. According to traditional trajectory designs, a prescribed profile in a work space is first decomposed into independent joint positions such that the success in a contouring task lies with good tracking capability of individual joints. To advance trajectory control precision without relying on high tracking performance, a contour control strategy for a robot manipulator is presented in this paper. Different from the traditional concept of trajectory control, a contour following control strategy is developed via a coordinate transformation scheme. The main advantage of the proposed control architecture is that the final contouring accuracy will not be degraded in case the tracking performance of the robot manipulator is not good enough. Moreover, using a concept of variable structure control theory, a smooth robust control algorithm is realized in the form of proportional control plus an integration term. The robustness of the control algorithm is also demonstrated. A number of experiments are conducted to demonstrate the advantage of the trajectories following control framework and validate the feasibility of the proposed controller. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction Robots are important for industrial automation and applica- tions. Similar to CNC machining, robotic systems have been applied to perform numerous tasks such as material assembling, welding, painting, manufacturing, etc. [1]. Regarding motion control of robot manipulators, separate operation of robot joints is an intuitive idea and has been widely investigated [2,3]. However, since the objective to be carried out is expressed in a work space (also known as a Cartesian space), the position of the end-effector is a paramount factor dominating the quality of the final product. Based on a given profile in a work space, there are two main approaches for trajectory control: one is to determine the desired joint space positions by solving with inverse kinematics, and the other is to directly deal with the dynamic model in the work space [4,5]. No matter which approach is used, superior tracking performance should be guaranteed to pursue a prescribed motion. In respect to both methods, if the tracking errors appearing in a joint space or a work space cannot be eliminated well, the end-effector would not be able to track the desired profile precisely. To put it another way, once good tracking ability cannot be accomplished, a synchronized motion fails and thereby gives rise to unsatisfactory machining results. From the trajectory control point of view, there are three major schemes, namely the tracking control, the point-to-point (PTP) positioning control and the contouring control. The tracking control approach is well known in the control field. As for PTP positioning control systems, the path and the corresponding velocity of the tool from one point to the other are not the main concerns. What is of interest is the accuracy of positioning the tool to the desired end target [6]. In contrast, the main control purpose of a contouring control system is to eliminate the distance from the end-effector position to a prescribed path. The definitions of positioning error and tracking error are well understood; however, the definition of contouring error is different from those of positioning and tracking errors. Instead of being time dependent, a contouring error is time independent with regard to real time command position and it is geometrically defined as the shortest distance from the end-effector position to a desired profile. However, it is a challenging issue to determine the contouring error in a straightforward manner and this is why to formulate the contour error model is not a facile work. Consider a particular circumstance when dealing with a con- touring control problem [7], as shown in Fig. 1. Suppose that A is the location of the end-effector and D is the corresponding command position at a time instant, providing the tracking error e A exists significantly, a resultant force, whose orientation is towards AD  ! , will be generated by controllers and will result in a mismatch path drifting away from the desired profile. Moreover, Fig. 1 also reveals that a good tracking performance is a sufficient, but not necessary Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/rcim Robotics and Computer-Integrated Manufacturing 0736-5845/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2010.10.004 n Corresponding author. Tel.: + 886 6 2757575x63634; fax: + 886 6 2389940. E-mail address: ccpeng24@gmail.com (C.-C. Peng). Robotics and Computer-Integrated Manufacturing 27 (2011) 569–580