Analytically Informed Inverse Kinematics Solution at Singularities

TL;DR

This work addresses the challenge of solving inverse kinematics near kinematic singularities where standard pseudoinverse-based methods can fail or converge poorly. It introduces Analytically Informed Inverse Kinematics (AI-IK), which first computes a regularizing perturbation from the tangent cone of the singular locus, moving the configuration from $q_0$ to a nearby regular state $q_0+x$ and then applies a standard iterative IK using a regularized Jacobian. The contribution combines analytic singular-motion analysis (tangent cones and Lie brackets) with a projector-based numerical regularization and an analytic calculation of the prolonged Jacobian, ensuring solvability even when EE motions are instantaneously infeasible. Experiments on a redundant 7-DOF Kuka LBR iiwa demonstrate robust convergence from singular configurations, highlighting AI-IK's potential to enable reliable, real-time IK in challenging singular regimes.

Abstract

Near kinematic singularities of a serial manipulator, the inverse kinematics (IK) problem becomes ill-conditioned, which poses computational problems for the numerical solution. Computational methods to tackle this issue are based on various forms of a pseudoinverse (PI) solution to the velocity IK problem. The damped least squares (DLS) method provides a robust solution with controllable convergence rate. However, at singularities, it may not even be possible to solve the IK problem using any PI solution when certain end-effector motions are prescribed. To overcome this problem, an analytically informed inverse kinematics (AI-IK) method is proposed. The key step of the method is an explicit description of the tangent aspect of singular motions (the analytic part) to deduce a perturbation that yields a regular configuration. The latter serves as start configuration for the iterative solution (the numeric part). Numerical results are reported for a 7-DOF Kuka iiwa.

Figures (5)

• Figure 1: a) 3R regional robot and b) 7-DOF Kuka iiwa 14 R820, both in a singular configuration.
• Figure 2: Convergence for 15 iterations. Notice that the error of the DPI solution stays constant. The PI and DPI solution starting with the analyticaly regularized configuration as well as the DPI starting with the random pertrbation converge.
• Figure 3: Convergence of the DPI for 20 different random perurbations (curves without markers), when a) $\lambda ^{2}=10^{-4}$ and b) $\lambda ^{2}=10^{-6}$. For comparison, the PI solution obtained with the analyticaly regularized start configuration is included.
• Figure 4: Convergence during 15 iterations when perform a general motion starting at the singularity.
• Figure 5: Convergence of the DPI for 20 different random perurbations (curves without markers), when $\lambda ^{2}=10^{-6}$. For comparison, the PI solution obtained with the analyticaly regularized start configuration, and the DPI solution with $\lambda ^{2}=10^{-6}$ are included.