Constrained Optimization of Multi-Degree-of-Freedom Mechanisms for Near-Time-Optimal Motions

This paper presents a method to design multi-degree-of-freedom mechanisms for near-time optimal motions. The design objective is to select system parameters, such as link lengths and actuator sizes, that will minimize the optimal motion time of the mechanism along a given path. The exact time-optimization problem is approximated by a simpler procedure that maximizes the acceleration near the end points. Representing the directions of maximum acceleration with the acceleration lines, and the reachability constraints as explicit functions of the design parameters, we transform the constrained optimization to a simpler curve-fitting procedure. This problem is formulated analytically, permitting the use of efficient gradient-based optimizations instead of the zero order optimization that is otherwise required. It is shown that with the appropriate choice of variables, the reachability constraints for planar mechanisms are linear in the design parameters. Consequently, the reachability of the entire path can be guaranteed by satisfying the reachability of only two extreme points along the path. This greatly simplifies the optimization problem since it reduces the dimensionality of the constraints and it permits the use of efficient projection methods. Examples for optimizing the dimensions of a five-bar planar mechanism demonstrate close correlation between the approximate and the exact solutions and better computational efficiency of the constrained optimization over previous penalty-based methods.