Accelerated Reduced Gradient Algorithm for Solving the Inverse Kinematic Task of Redundant Open Kinematic Chains

Abstract

The solution of the differential inverse kinematic task of redundant open kinematic chains normally can be formulated as an optimization task under constraints in which the kinematic task means the constraint equations and the solution is made unambiguous by minimizing some cost function. This task only in the case of quadratic cost functions can be solved without numerical iteration. The ambiguity of the possible solutions can be utilized by “adding” certain elements of the null space of the Jacobian to this solution. If the cost function has more complex structure the more general numerical procedure of Lagrange’s Reduced Gradient Algorithm can be applied. Recently it was found that this procedure seriously can be accelerated if the computation of the Lagrange multipliers belonging to the individual constraint equations becomes unnecessary. In this case a single vector must be reduced with the components of another one. In the present paper this method is directly applied for the efficient solution of the inverse kinematic task of redundant robot arms. The method is exemplified in designing a particular kinematic structure for a robot arm for specified use.