Optimal Inverse Kinematic Solutions for Redundant Manipulators by Using Analytical Methods to Minimize Position and Velocity Measures

Two methods are presented to obtain optimal inverse kinematic solutions for redundant manipulators, according to two different performance criteria stipulated in the position and velocity levels. Both methods are analytical throughout except their final stages, which involve the numerical solution of a simplified minimization problem in a position-level case and the numerical integration of a set of differential equations derived optimally in a velocity-level case. Owing to the analytical nature of the methods, the multiple and singular configurations of the manipulator of concern can be identified readily and studied in detail. The methods are applicable for both serial and parallel redundant manipulators. However, they are demonstrated here for a humanoid manipulator with seven revolute joints. In the demonstrations, the first performance criterion is stipulated in the position level as the minimization of the potential energy. In that case, the optimal inverse kinematic solution is first obtained in the position level for a specified position of the hand. Then, it is compatibly extended to the velocity level for a specified motion of the hand. In the main analytical part of the solution, six of the joint variables are expressed in terms of the selected seventh one. Then, the optimal value of the selected joint variable is determined numerically by a simple one dimensional scanning. The second performance criterion is stipulated in the velocity level as the minimization of the kinetic energy. In that case, the optimal inverse kinematic solution is first obtained in the velocity level and then extended to the position level by integration. The main analytical part of the solution provides an optimally determined set of nonlinear differential equations. These differential equations are then integrated numerically in order to obtain the corresponding solution in the position level. However, the corrections needed to eliminate the numerical integration errors are still obtained analytically. The distinct optimal behaviors of the manipulator according to the mentioned criteria are also illustrated and compared for a duration, in which the hand moves in the same specified way.