Multiple constrained rivaling actuators in the optimal control of miniaturized manipulators

Miniaturized robotic manipulators are a key elements in future high-precision telesurgery with artificial fingers and hands. A major challenge is the development of suitable drives for the micro-joints, because those small drives are either slow or can produce only low specific forces. A possible solution are joints driven by weak, but fast, and strong, but slow, actuators acting in parallel. This results in an optimal control problem of a multibody system subject to rivaling controls. Not only the control amplitude, but also the rate of change of the control is constrained for the respective actuator. In optimal control theory, derivatives of controls are forbidden by definition. The straightforward approach-defining the derivatives of the controls as the new control variables-results in a very sensitive differential-algebraic control problem with a complicated structure. A new approach is proposed: The original control problem is converted into a piecewise defined nonlinear multi-point boundary value problem for ordinary differential equations with nonconstant dimension. Based on the minimum principle, a strict and careful mathematical analysis of the coupling of the single parts of the boundary value problem leads to new interior point conditions at the junction points. As an example, three-dimensional energy optimal trajectories are calculated for a six-sectional branched manipulator. Stable, efficient and very accurate numerical solutions are obtained by the new procedure.