Joint space trajectory planning for flexible manipulators

A trajectory planning approach for controlling flexible manipulators is proposed. It is demonstrated that choosing actual joint angles as the generalized rigid coordinates is the key to applying the proposed approach. From the observation of the special structure of the input matrix, the concepts of motion-induced vibration and inverse dynamics under a specified motion history of the joints are formed naturally. Based on the above concepts, trajectory planning in joint space is proposed by using the optimization technique to determine the motion of joints along a specified path in joint space or work space and for general point-to-point motion. The motion for each joint is assumed to be in a class consisting of a fifth-order polynomial and a finite terms of Fourier series. This parameterization of motion allows the optimal trajectory planning to be formulated as a standard nonlinear programming problem, which avoids the necessity of solving a two- point-boundary-value problem and using dynamic programming. Setting the accelerations to zero at the initial and the final times is used to obtain smoother motion to reduce the spillover energy into unmodeled high-frequency dynamics. A penalty term on vibration energy contained in the performance index is used to minimize the vibration of the system modeled by lower frequency only. The final simulation results show the effectiveness of the proposed approach and the advantage for proper trajectory planning.