Analysis of Optimal Dynamic Manipulation for Robotic Manipulator Based on Pontryagin's Minimum Principle

A branch of dynamic manipulation is robotic throwing which recently plays a major role in the performance improvement of robotic manipulations. This paper aims at the optimal throwing using a robotic arm. Launching the captured object to a specified target by minimum control effort with the specified final time is the mission of the throwing robot in this study. Taking into account the indirect method based on the fundamental theorem of the calculus of variations, the optimal throwing problem is defined as the optimal control problem (OCP). By employing Pontryagin's Minimum Principle (PMP), the optimality conditions are extracted and then considering the throwing equation as a moving-end boundary condition (BC), the corresponding OCP converts to a two-point boundary value problem (BVP). By applying the moving-end BCs, the best releasing condition is automatically obtained. Next, a simple algorithm is presented to solve the resulting BVP which leads us to the multiple optimal solutions that have been sorted according to the defined cost function values. This multiplicity of solutions allows for the selection of the most practical optimal solution. The feasibility of the derived framework is validated by some simulations and experiments done for a fabricated throwing robot.