Numerically stable solution to the 6R problem of Inverse Kinematics

Abstract

In this paper, a stable and accurate algorithm to compute all solutions of the inverse kinematics problem of a 6 revolute manipulator chain is presented. A system of equations is constructed based on the fundamental closure conditions, leading to a closed algebraic system of 20 equations involving 16 quantities, composed of trigonometric functions of five among the six unknown joint angles. Two among these five are stably eliminated using singular value decomposition (SVD) avoiding the need to consider special cases. The resulting system of equations involving three unknowns is solved by conversion to a generalized eigenvalue problem. The remaining three unknown angles are obtained using the previously computed pseudoinverse. In this formulation we exploit the inherently complex form of the system reducing it to 10 complex equations in 9 quantities, which substantially accelerates the SVD computation. The method's robustness is demonstrated through a comparison to current methods and several examples including known problematic cases where some axis or link lengths vanish, or some joint angles are 180 degrees, as well as cases where multiple eigenvalues arise.

Ring kinematics‐informed conformation space exploration

“…R6B6 can generate cycles without requirement for bonds with dependent dihedrals to form coterminal pairs. 36,37 However, currently the R6B6 source code is not publicly available.…”

“…Significant progress in using IK to generate cyclic molecules was achieved by Coutsias et al [35][36][37] Due to his efforts, there have been implemented two ring closure algorithms for sampling molecular conformers: TLC (Triaxial loop closure) 35 and R6B6 (6 rotors/6 bar). 36,37 To the best of our knowledge, these two programs exhaust the list of existing IK ring closure codes for chemical purposes.…”

“…Significant progress in using IK to generate cyclic molecules was achieved by Coutsias et al [35][36][37] Due to his efforts, there have been implemented two ring closure algorithms for sampling molecular conformers: TLC (Triaxial loop closure) 35 and R6B6 (6 rotors/6 bar). 36,37 To the best of our knowledge, these two programs exhaust the list of existing IK ring closure codes for chemical purposes. In this subsection, we will discuss how ring conformations can be generated using these algorithms, however, we are not going to dive into their mathematical details because these were thoroughly explained in the original articles.…”

“…Reviews on molecular representations for ML 1 and physics-informed ML models 45 ; mathematical details of TLC and R6B6 algorithms. [35][36][37]…”

Fruit Mapping with Shape Completion for Autonomous Crop Monitoring

“…Typical numerical approaches use Jacobian methods [14], [15] to iteratively find the inverse kinematics solution for a desired pose around the current robot configuration. Inverse or pseudo-inverse Jacobian, or singular value decomposition (SVD) [16] computations are expensive and especially prone to getting stuck in local minima. Recent approaches like TracIK [17] and BioIK [18] have mitigated these problems to an extent using random restarts with nonlinear optimization, and memetic algorithms that combine gradient-based optimization with genetic and particle swarm optimization, respectively.…”