Trajectory Planning and Speed Control for a Two-Link Rigid Manipulator

Abstract

Contributed by the Mechanisms Committee for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 1999. Associate Editor: G. S. Chirikjian. Industrial manipulators are frequently required to perform different tasks and to carry different masses. Further, if a manipulator operates in space crowded with obstacles, it must follow a planned path very closely to avoid collisions, especially if other moving objects are present. To follow trajectory and the velocity profiles simultaneously is a difficult task. One option is to divide the problem into two parts: path planning and speed control. Usually, the desired trajectory is given as a sequence of knots (positions of the robot’s tip) in space Cartesian coordinates, where the velocity and the acceleration of the joints are subject to constraints. The control is performed at the joint level and it is desirable, therefore, to construct the trajectory at that level. The given knots are first transformed into two sets of joint displacements, and then piecewise approximation polynomials are used to fit these two sequences of joint displacements. Cubic polynomials are sufficiently smooth to provide for continuous motion 12. The function of approximation for the joint trajectory passes through the given knots and provides sufficient information (at the knots as well as at the intermediate points) for the controller. Path planning has been studied by a number of authors. A combined trajectory planning and adaptive control of a two-link rigid manipulator (TLRM) was presented in Fotouhi-C., Nikiforuk and Szyszkowski 3. An optimum path planning problem at the joint level using the cubic spline polynomial was given in Lin, Chang and Luh 4 and later in Xiangrong and Xiangfeng 5. A similar problem was repeated in Wang and Horng 6 where minimum-time path planning for robot manipulators using cubic B-spline functions was solved. The approach used in Wang and Horng 6 was similar to Lin et al. 4, but used the B-spline function rather than a spline function. A procedure to construct a robot joint trajectory using B-splines was given in Thompson and Patel 7. With the B-spline approach, it was claimed that the local modification of the path was possible for one or more joints without affecting the other joints. However, unlike the other two papers, a search for minimum-time was not given in Thompson and Patel 7. A two-part trajectory planning for a robot manipulator was proposed in Wu and Jou 8. A cubic spline function was used to plan the geometric trajectory as well as the speed. The problem was transformed then into solving an initial boundary value problem. Numerical results were given only for maintaining a constant speed along the geometric path. Two algorithms for fine-tuning B-spline motions were presented in Srinivasan and Ge 9. A path-smoothing algorithm and a speed-smoothing algorithm were used to keep the path highly smooth and to keep unintended speed variation to a minimum. The speed algorithm used a rational spline image curve to obtain a near constant kinetic energy. Robot path planning using the concept of trigonometric splines was discussed in Simon and Isik 10. It was stated that the trigonometric splines outperformed algebraic splines. In this paper, the piecewise cubic spline function is used to construct the joint trajectories with a given speed profile. The problem is divided into two parts: geometric trajectory planning and trajectory speed control which is the main contribution of this paper. The path planning is done at the joint level using cubic spline functions. The trajectory of the robot is specified by a sequence of knots (positions of the robot’s tip) in space Cartesian coordinates. These knots are then transformed into two sets of joint displacements. Linear scaling of the time variable is applied to accommodate the velocity and acceleration constraints. A new approach, which uses a nonlinear scaling of the time variable to fit the manipulator’s ti