Pergamon Robotics & Computer-Integrate Manufacturing, Vol. 12, No. 1, pp. 29-39, 1996 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0736-5845/96 $15.00 + 0.00 073th5845(95)ouo26-7 l Paper THE PRACTICAL IMPLEMENTATION OF TIME-OPTIMAL CONTROL FOR ROBOTIC MANIPULATORS Z. SHILLER, H. CHANG and V. WONG Department of Mechanical, Aerospace and Nuclear Engineering, University of California Los Angeles, Los Angeles, CA 900954597, U.S.A. This paper presents experimental results for time-optimal control of robotic manipulators along specified paths. The implementation of time-optimal control represents several unique problems: (1) the control is generally discontinuous (bang-bang), (2) actuator dynamics are usually ignored in order to reduce the system order, and (3) the optimal control leaves no control authority to compensate for tracking errors caused by unmodeled dynamic and the delays introduced by the on-line feedback controller. To overcome these difficulties, we compensate for motor dynamics using a simplified friction model, and account for the dynamics of the feedback controller using trajectory preshaping. Implemented for the UCLA Direct Drive Arm, this is shown to drastically reduce the tracking errors compared to the errors obtained with no preshaping and no compensation for motor dynamics. The experimental results demonstrate the merit of time optimal control for reducing motion time as well as for increasing tracking accuracy. 1. INTRODUCTION Robotic applications can be divided into two major tasks: (1) point-to-point motions, as in parts handling and spot welding, and (2) specified path motions, as in laser cutting, arc welding, glue dispensing, and painting. The objective of point-to- point motions is to accurately reach the final point, whereas the objective of motions along specified paths is to accurately track the path at the specified speeds. This paper focuses on the latter, the specified path problem. For high productivity, it is desirable that the specified speeds be time-optimal so as to reduce motion time and thus minimize cycle times. Time- optimal motions constitute the highest speeds at which motion accuracy can be guaranteed. Attempt- ing to track the specified path at speeds higher than the time-optimal would require actuator torques that exceed the actuator limits. If the actuator limits represent the true actuator capabilities, then motions faster than the time-optimal would obviously result in tracking errors, or deviations from the specified path. Thus, accuracy comparable to the time-optimal motions using any velocity profile other than the time optimal can be achieved only at slower speeds and, hence, longer motion times. Efficient methods for optimizing motions along specified paths’*8P1*“1 have been developed for a rigid manipulator model, using the actuator torques/forces as the control inputs. It was shown that the time- optimal control saturates at least one actuator at all times, and that the actuator torques might be discontinuous at the switching points. This poses 29 difficult implementation issues since the discontin- uous control is impossible to realize because of the unmodeled motor dynamics, and the actuator saturation leaves no control authority to correct for the resulting tracking errors. Although research on time-optimal control of robotic manipulators dates back to the early 1970~,~ it has been demonstrated to date only in simulations. One exception is Ref. 3, which demon- strates a control algorithm that scales the time- optimal velocity profile to account for modeling errors. However, the tracking errors of the proposed controller were too large to consider the resulting motions of high accuracy, which leaves the practical benefits of time-optimal control unclear. In this paper, we discuss implementation issues and present experimental results for time-optimal control along specified paths. In particular, we evaluate the use of a simplified motor model and trajectory preshaping to compensate for the ignored actuator dynamics and the delays introduced by the feedback controller, and thus achieve high motion accuracy at high speeds. First, we demonstrate the significance of the unmodeled actuator/driver dynamics. Using open- loop control, it is shown that the dynamics of the brushless DC motor and static joint friction account for most of the tracking errors. An empirical linear, and hence invertible, model is then proposed to account for the motor/driver dynamics and joint friction. The empirical model, referred to as the viscous friction model, is used to compute the input to the motor driver that would produce the nominal