Adaptive Jacobian tracking control of rigid-link electrically driven robots based on visual task-space information☆
Introduction
Actuator dynamics represent an important component of a robotic system and overlooking them may cause detrimental effects on motion performance (Eppinger and Seering, 1987, Good et al., 1985, Tarn et al., 1991). In addition, actuator dynamics are often uncertain, due e.g. to calibration errors, or to parameter variation from overheating and changes in environment temperature. Recently, this research topic has received considerable attention and several control schemes for rigid-link robots have been developed to address it, with the actuator dynamics explicitly included in the controller design (Bridges et al., 1993, Burg et al., 1996, Colbaugh and Glass, 1995, Dawson et al., 1992, Ishii et al., 1999, Mahmoud, 1993, Oya et al., 2004, Su and Stepanenko, 1996, Su and Stepanenko, 1998, Yuan, 1995).
The aforementioned control schemes are all designed in joint space. For most robot applications, however, the desired position or path is specified in task space, such as Cartesian space or camera image space. To design controllers in joint space for such control tasks, inverse kinematics transformation should be carried out to get the desired joint-space position or trajectory. In order to avoid the problem of solving inverse kinematics, Takegaki and Arimoto (1981) derived a task-space controller for set-point regulation in Cartesian space using a transposed Jacobian matrix, and many other task-space control schemes have been proposed (Kelly, 1999, Kelly et al., 2000, Khatib, 1987, Lewis et al., 1993, Miyazaki and Masutani, 1990). These task-space control schemes require knowledge of the Jacobian matrix from joint space to task space and hence exact kinematic parameters. However, no physical parameters can be derived exactly, and furthermore assessing the overall kinematics is difficult when the robot picks up objects of various lengths with unknown orientations or grasping points. Similarly, in visual servoing, the extrinsic and intrinsic parameters of the camera system are apt to changes and uncertainties. In the presence of kinematics uncertainties, the controllers (Bridges et al., 1993, Burg et al., 1996, Colbaugh and Glass, 1995, Dawson et al., 1992, Ishii et al., 1999, Kelly, 1999; Kelly et al., 2000, Lewis et al., 1993, Mahmoud, 1993, Miyazaki and Masutani, 1990, Oya et al., 2004, Su and Stepanenko, 1996, Su and Stepanenko, 1998, Takegaki and Arimoto, 1981, Yuan, 1995) may give degraded performance or even become unstable.
To deal with the problem of uncertain kinematics, several task-space control methods with uncertain kinematics from joint space to task space have been proposed (Cheah et al., 2004, Cheah et al., 2006; Cheah et al., 2003, Dixon, 2004). However, in these results the actuator dynamics was not considered. Very recently, Liu and Cheah (2005) proposed a task-space regulation control scheme for rigid-link electrically driven (RLED) robots with uncertain kinematics. The control scheme proposed incorporated the actuator dynamics into the controller synthesis and constitutes the first result that can deal with all the uncertainties in robot kinematics, manipulator dynamics and actuator dynamics at the same time. The control scheme proposed therein is confined to regulation or setpoint control and hence the trajectory tracking control of robots with uncertain kinematics and actuator dynamics remains an open problem.
In this paper, we propose a task-space adaptive Jacobian control scheme for trajectory tracking of RLED robots using visual information as feedback signal. The controller synthesis is separated into two steps and adaptive observers are constructed to avoid the use of accelerations in the control voltage design. The proposed control scheme does not need accurate information either about actuator dynamics, or about robot kinematics and dynamics. Stability of the closed-loop system is established using Lyapunov analysis. Simulation results are presented to show the effectiveness of the proposed control scheme.
Section snippets
Problem formulation
In most applications of robot manipulators, a desired path for the end-effector is specified in task space. Let be a task-space vector defined by (Cheah et al., 1999, Lewis et al., 1993)where is the vector of generalized joint coordinates, is generally a nonlinear transformation describing the relation between joint space and task space. The task-space velocity is related to joint-space velocity aswhere is the Jacobian matrix from joint
Adaptive Jacobian tracking control of RLED robots
In this section, we propose a task-space adaptive control scheme for the tracking control problem of RLED robots. First, based on the second-order manipulator subsystem dynamics (4), a desired armature current signal is designed to ensure that the task-space tracking errors converge even in the presence of uncertainties in kinematics and motor torque constant matrix . Then, based on the actuator subsystem dynamics (5), a backstepping procedure is used to design a voltage control input u to
Simulation study
In this section, we present the simulation study results for the control scheme proposed in this paper. The simulation is based on a two-link RLED robot holding an uncertain object as shown in Fig. 1. In this simulation, the “fixed in workspace” camera configuration is adopted and the camera is placed perpendicular to the operating plane (Cartesian coordinates) of the robot. The offsets between the X and Y axes of the Cartesian and camera image coordinates are set to zero.
The manipulator
Conclusion
In this paper, an adaptive Jacobian control scheme is proposed for the tracking control of RLED robots with kinematic uncertainty using visual feedback information. This control method represents the first tracking control result in the literature for robot manipulators with uncertainties both in actuator dynamics and in robot kinematics and dynamics. The control scheme is developed based on a backstepping procedure and the overall closed-loop system is shown to be asymptotically stable via
Chao Liu was born in Shandong, China in 1978. He received B.E. degree in Control Science and Technology from Shandong University, China in 2000. From 2000–2001, he worked as a research engineer in Perspective Technology Development Ltd., China. Since 2002, he has been with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore to pursue his Ph.D. degree. He is a member of IEEE and an associate member of Sigma Xi. His research interests include nonlinear
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Chao Liu was born in Shandong, China in 1978. He received B.E. degree in Control Science and Technology from Shandong University, China in 2000. From 2000–2001, he worked as a research engineer in Perspective Technology Development Ltd., China. Since 2002, he has been with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore to pursue his Ph.D. degree. He is a member of IEEE and an associate member of Sigma Xi. His research interests include nonlinear systems control and robotics.
Chien Chern Cheah was born in Singapore. He received B.E. degree in Electrical Engineering from National University of Singapore in 1990, M.E. and Ph.D. degrees in Electrical Engineering, both from Nanyang Technological University, Singapore, in 1993 and 1996, respectively.
From 1990 to 1991, he worked as a design engineer in Chartered Electronics Industries, Singapore. He was a research fellow in the Department of Robotics, Ritsumeikan University, Japan from 1996 to 1998. He joined the School of Electrical and Electronic Engineering, Nanyang Technological University as an assistant professor in 1998. Since 2003, he has been an associate professor in Nanyang Technological University. In November 2002, he received the oversea attachment fellowship from the Agency for Science, Technology and Research (A*STAR), Singapore to visit the Nonlinear Systems laboratory, Massachusetts Institute of Technology. He is the program chair of the International Conference on Control, Automation, Robotics and Vision 2006.
Jean-Jacques Slotine was born in Paris in 1959, and received his Ph.D. from the Massachusetts Institute of Technology in 1983. He is currently Professor of Mechanical Engineering and Information Sciences, Professor of Brain and Cognitive Sciences, and Director of the Nonlinear Systems Laboratory at MIT. Prof. Slotine is the co-author of the textbooks “Robot Analysis and Control” (Wiley, 1986) and “Applied Nonlinear Control” (Prentice-Hall, 1991). He was a member of the French National Science Council from 1997 to 2002.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Yong-Yan Cao under the direction of Editor Mituhiko Araki.