The problem of generating an optimal feedrate trajectory has received a significant amount of attention in both the robotics and machining literature. The typical objective is to generate a minimum-time trajectory subject to constraints such as system limitations on actuator torques and accelerations. However, developing a computationally efficient solution to this problem while simultaneously guaranteeing optimality has proven challenging. The common constructive methods and optimal control approaches are computationally intensive. Heuristic methods have been proposed that reduce the computational burden but produce only near-optimal solutions with no guarantees. A two-pass feedrate optimization algorithm has been proposed previously in the literature by multiple researchers. However, no proof of optimality of the resulting solution has been provided. In this paper, the two-pass feedrate optimization algorithm is extended and generalized. The generalized algorithm maintains computationally efficiency, and supports the incorporation of a variety of state dependent constraints. By carefully arranging the local search steps, a globally optimal solution is achieved. Singularities, or critical points on the trajectory, which are difficult to deal with in optimal control approaches, are treated in a natural way in the generalized algorithm. A detailed proof is provided to show that the algorithm does generate a globally optimal solution under various types of constraints.
The Optimal Solution of a Non-Convex State-Dependent LQR Problem and Its Applications
