Smooth Time-Optimal Path Tracking for Robot Manipulators With Kinematic Constraints

A general motion planning problem is complex and it is often decomposed into two hierarchical stages. The first stage tries to find a feasible path satisfying high-level constraints such as the obstacle avoidance. At the second stage, some optimal methods are used to assign a velocity profile to the path generated at the first stage. How to compute a time-optimal velocity profile under specific constraints such as the velocity, acceleration, jerk, torque and torque rate boundaries has been widely studied because of the request of the greater efficiency in the industry process. Many method uses torque-based control to track the given path as fast as possible. This kind of method has a significant drawback that the accelerate profile may be discontinuous, or at least non-smooth, which is difficult to be realized physically and causes great stress on the actuators and vibrations. To get smoother acceleration profiles and reduce the wear on the actuators, jerk needs constraining and additionally the velocity and acceleration boundaries are considered in the time-optimal problem. Actually, the time-optimal problems considering jerk limits are non-convex and it’s not easy to solve in the ordinary way. Some works solve the time-optimal problem considering piece-wise constant form for the control input. Then the problem can be transformed into a convex problem, but the obtained trajectory may be not smooth enough for control. To ensure high degree of smoothness in the motion profile, this paper try to regard the control input as a continuous function with some unknown parameters. Specifically, the control input is represented as the square root of a polynomial function with unknown coefficients. Based on the special form of the control input, the velocity and acceleration constraints can be reformulated into linear form and the jerk constraints can be transformed into the difference of convex form, moreover, the objective is convex, so that the time-optimal problem can be solved through sequential convex programming (SCP). In order to verify the proposed method, simulations have been carried out on a 7-DoF manipulator platform. Two strategies are conducted. The first strategy is to just constrain the velocity and acceleration of the robot joints when the robot moves along the given path, while the other strategy constrains the jerk additionally for the same path. The result shows that all the designed constraints can be satisfied, and the obtained velocity, acceleration and jerk profiles are very smooth. Besides, because the jerk constraints of each joint are not involved in the first strategy, the optimized jerk profile sometimes exceed the jerk boundaries, which indicates that the proposed method is effective. Although there are more constraints and the problem is solved iteratively, it just takes about several seconds to finish the optimization, which is acceptable in the offline case.