Redundant Optimal Control of Manipulators along Specified Paths

The time-optimal motion of a four-link manipulator is investigated with its end-effector following a prescribed path in space. The dynamic equations together with the spatial restrictions yield a nonlinear differential-algebraic system of differential index 3. Additionally, the optimal control problem for this DAE system contains multiple restrictions and several interior points. A transformation to minimum coordinates is the most elegant way to eliminate severe mathematical problems arising from the algebraic constraints. The transformation results in a system of linear equations of motion. The remaining controls are the angular acceleration of the fourth joint and the acceleration of the end-effector along the prescribed trajectory, replacing the four actuator torques in the original formulation. The structure of the remaining controls is unknown and too difficult to be estimated in practical applications. Introducing eight highly nonlinear control/state constraints is another price to be paid for the much simpler structure of the overall problem. However, only the state variables are affected by the nonlinearity, whereas the control variables appear linearly in these constraints. The set of admissible controls turns out to be a convex subset of the IR

2

. The optimal control minimizes the Hamiltonian which is also linear in this case; at every moment the structure of the controls can be determined by techniques of linear programming; the results are valid for multi-link manipulators with an arbitrary number of joints. With this information, the complete optimal control problem is transferred into a highly nonlinear multi-point boundary value problem and solved numerically by the advanced multiple shooting method JANUS. By backward transformation the actuator torques and their switching behaviour are easily obtained.