Accelerated Reduced Gradient Algorithm with Constraint Relaxation in Differential Inverse Kinematics

Bence Varga; Hazem Issa; Richárd Horváth; József Tar

doi:10.52846/stccj.2021.1.2.23

Authors

Bence Varga Óbuda University
Hazem Issa Óbuda University
Richárd Horváth Óbuda University https://orcid.org/0000-0002-4924-3244
József Tar Óbuda University https://orcid.org/0000-0002-5476-401X

DOI:

https://doi.org/10.52846/stccj.2021.1.2.23

Keywords:

differential inverse kinematic task, reduced gradient algorithm, Moore-Penrose pseudoinverse, redundant open kinematic chain, constraint relaxation

Abstract

The Moore-Penrose pseudoinverse-based solution of the differential inverse kinematic task of redundant robots corresponds to the result of a particular optimization underconstraints in which the implementation of Lagrange’s ReducedGradient Algorithm can be evaded simply by considering the zero partial derivatives of the ”Auxiliary Function” associated with this problem. This possibility arises because of the fact that the cost term is built up of quadratic functions of the variable of optimization while the constraint term is linear function of the same variables. Any modification in the cost and/or constraint structure makes it necessary the use of the numerical algorithm. Anyway, the penalty effect of the cost terms is always overridden by the hard constraints that makes practical problems in the vicinity of kinematic singularities where the possible solution still
exists but needs huge joint coordinate time-derivatives. While in the special case the pseudoinverse simply can be deformed, in
the more general one more sophisticated constraint relaxation can be applied. In this paper a formerly proposed accelerated
treatment of the constraint terms is further developed by the introduction of a simple constraint relaxation. Furthermore, the
numerical results of the algorithm are smoothed by a third order tracking strategy to obtain dynamically implementable solution.
The improved method’s operation is exemplified by computation results for a 7 degree of freedom open kinematic chain

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