A virtual-variable-length method for robust inverse kinematics of multi-segment continuum robots

Abstract

This paper proposes a new, robust method to solve the inverse kinematics (IK) of multi-segment continuum manipulators. Conventional Jacobian-based solvers, especially when initialized from neutral/rest configurations, often exhibit slow convergence and, in certain conditions, may fail to converge (deadlock). The Virtual-Variable-Length (VVL) method proposed here introduces fictitious variations of segments' length during the solution iteration, conferring virtual axial degrees of freedom that alleviate adverse behaviors and constraints, thus enabling or accelerating convergence. Comprehensive numerical experiments were conducted to compare the VVL method against benchmark Jacobian-based and Damped Least Square IK solvers. Across more than $1.8\times 10^6$ randomized trials covering manipulators with two to seven segments, the proposed approach achieved up to a 20$\%$ increase in convergence success rate over the benchmark and a 40-80$\%$ reduction in average iteration count under equivalent accuracy thresholds ($10^{-4}-10^{-8}$). While deadlocks are not restricted to workspace boundaries and may occur at arbitrary poses, our empirical study identifies boundary-proximal configurations as a frequent cause of failed convergence and the VVL method mitigates such occurrences over a statistical sample of test cases.

Figures (6)

• Figure B1: Schematic of Constant Curvature segment, as in Allen2020.
• Figure B2: Illustration of convergence progression to desired configuration (a) identified with a green color, from two distinct initial guesses (b) and (c) when using a traditional Jacobian method. In (b) a solution is obtained after 81 iterations, as demonstrated by the final configuration overlapping with the reference state. In (c) a solution is not reached due to a deadlock configuration, shown in red in (d). In (e), solution via the proposed VVL method is reached within 30 iterations; notice the initial variable segments' length which aid in the convergence process.
• Figure B3: Workspace of the continuum manipulator (blue) and failed IK solutions (red) obtained with Jacobian method (a)-(f) and DLS method (g)-(l). Subfigures (a) and (g) correspond to the two-segment manipulator, (b) and (h) to a three-segment manipulator, (c) and (i) to a four-segment manipulator, (d) and (j) to a five-segment manipulator, (e) and (k) to a six-segment manipulator, and (f) and (l) to a seven-segment manipulator. Subfigures (a)–(f) are obtained using the benchmark Jacobian method described in Sec. \ref{['subsec: Inverse Kinematics of CC Model with Screw Theory']}, while (g)–(l) are obtained using the DLS method described in Sec. \ref{['subsec: The DLS Method']}.
• Figure B4: Workspace of the continuum manipulator, in blue, and failed IK solutions in red obtained with the VVL method (sec.\ref{['subsec: The Augment Jacobian Method']}), respectively for the case of a manipulator made of (a) two segments, (b) three segments, (c) four segments, (d) five segments, (e) six segments and (f) seven segments.
• Figure C1: Convergence success rate with tolerances of 10$^{-4}$, 10$^{-6}$, 10$^{-8}$ for manipulators with 2, in (a), 3, in (b), and 5, in (c), segments respectively for the Jacobian method (indicated with the $o$ symbol in the legend), the VVL method (indicated by the $\text{v}$ symbol), and the DLS method (indicated by the $\text{d}$ symbol).