Encountering singularities of a serial robot along continuous paths at high precision

Abstract

A new variable step (VS) method rapidly calculates continuous kinematic paths that encounter singularities of a serial robot. Mitigating adverse effects of such encounters by applying small deviations to the intended path achieves high precision. The method is motivated by a simplified geometric model representing the arm extension, overhead and wrist singularities found in a wrist-separable 6-axis robot. A variable step length limits the kinematic closure error in a high-order combined predictor-corrector. Squaring the determinant of the robot Jacobian matrix gives its Taylor series favorable convergence. The resulting estimate of the distance to the nearest singularity affecting convergence of other kinematic variables bounds the step length, avoiding jumps between branches of path bifurcations associated with robot singularities. Finally, a 2nd order bivariate expansion of the squared determinant targets a minimum determinant magnitude by controlling deviation from an intended path. This process obviates the need for matrix regularization giving unintended switches between solution branches. The VS method demonstrates the ability to deviate from an intended path through dead center of any of the above singularities in isolation, pairwise and three-fold combination.