Energy-Optimal Collision-Free Motion Planning for Multi-Axis Motion Systems: An Alternating Quadratic Programming Approach

This work investigates energy-optimal motion planning for a class of multi-axis motion systems where the system dynamics are linear time-invariant and decoupled in each axis. Solving the problem in a reliable and efficient manner remains challenging owing to the presence of various constraints on control and state, non-convexity in the cost function, and obstacles. This work shows how the cost function can be convexified by taking into account the system dynamics, while decomposing decision variables to obtain a convex representation of collision avoidance constraints. With the convexified cost function and constraints, the original problem is decomposed into two quadratic programming (QP) problems. An alternating quadratic programming (AQP) algorithm is proposed to solve both QP problems alternatingly and iteratively until convergence. Requiring an initial feasible trajectory as a guess, AQP necessarily converges to an energy-efficient solution which is homotopic to the initial guess. Under certain circumstances, AQP is guaranteed to produce a local optimum. Simulation demonstrates that AQP is computationally efficient and reliable while claiming comparable energy saving as the Mixed-Integer QP approach.