J-PARSE: Jacobian-based Projection Algorithm for Resolving Singularities Effectively in Inverse Kinematic Control of Serial Manipulators

TL;DR

J-PARSE introduces a Jacobian-based projection framework to enable stable, first-order inverse kinematics control near singularities in serial manipulators. By constructing a Safety Jacobian to preserve non-singular mobility, a Projection Jacobian to isolate non-singular directions, and Singular Projections to modulate motion along singular directions, the method yields continuous, stable joint velocities even at or beyond the workspace boundary. The approach provides local Lyapunov stability guarantees in continuous time and practical discrete-time guidelines, with extensive results on planar and 6-DOF manipulators, including teleoperation and visual servoing in SE(3). The work offers a tunable, intuitive mechanism to expand usable manipulator workspace, with strong implications for teleoperation, servoing, and data-driven policy learning under singular configurations.

Abstract

J-PARSE is an algorithm for smooth first-order inverse kinematic control of a serial manipulator near kinematic singularities. The commanded end-effector velocity is interpreted component-wise, according to the available mobility in each dimension of the task space. First, a substitute "Safety" Jacobian matrix is created, keeping the aspect ratio of the manipulability ellipsoid above a threshold value. The desired motion is then projected onto non-singular and singular directions, and the latter projection scaled down by a factor informed by the threshold value. A right-inverse of the non-singular Safety Jacobian is applied to the modified command. In the absence of joint limits and collisions, this ensures safe transition into and out of low-rank configurations, guaranteeing asymptotic stability for reaching target poses within the workspace, and stability for those outside. Velocity control with J-PARSE is benchmarked against approaches from the literature, and shows high accuracy in reaching and leaving singular target poses. By expanding the available workspace of manipulators, the algorithm finds applications in teleoperation, servoing, and learning. Videos and code are available at https://jparse-manip.github.io/.

Figures (19)

• Figure 1: By appropriately modifying the commanded velocity along singular directions, the manipulator is able to move both into and out of singular configurations. A safety Jacobian matrix $\boldsymbol{J}_s$ is created by setting a minimum aspect ratio for the manipulability ellipsoid, preventing it from losing dimension and making it possible to invert $\boldsymbol{J}_s$ to obtain a feasible set of joint speeds from the modified command.
• Figure 2: Summary of the real-time J-PARSE algorithm.
• Figure 3: Simulation results for the planar $2$-R manipulator, comparing algorithms listed in Table \ref{['tab:2r']}.
• Figure 4: The UFactory X-Arm is used for simulated velocity control. Methods are compared using two waypoint sequences, each containing points within and beyond the reachable space of the manipulator.
• Figure 5: Comparison of J-PARSE vs PseudoInverse.