Lie Group Implicit Kinematics for Redundant Parallel Manipulators: Left-Trivialized Extended Jacobians and Gradient-Based Online Redundancy Flows for Singularity Avoidance
Yifei Liu, Kefei Wen
AI summary
Problem
Kinematically redundant parallel manipulators lack effective methods to preserve singularity margins and maintain kinematic feasibility along prescribed end-effector paths.
Approach
The authors model kinematics as a Lie group implicit constraint, compute left-trivialized extended Jacobians via automatic differentiation, and drive redundancy variables along a gradient-based flow to avoid singularities along fixed end-effector paths.
Key results
- Left-trivialized extended Jacobians computed via automatic differentiation without closed-form derivations
- Gradient-based redundancy flow maintains well-conditioned Jacobians along dense-coverage orientation trajectories
- Damped-Newton step generator enables soft real-time planning at approximately 2 kHz on laptop hardware
- Validated on a (6+3)-DoF Stewart platform and an isotropic Spherical–Revolute platform
Why it matters
This framework provides a mechanism-agnostic, differentiable approach to singularity avoidance that enables real-time deployment of redundant parallel robots in industrial and interactive applications.
Abstract
We present a Lie group implicit formulation for kinematically redundant parallel manipulators that yields left- trivialized extended Jacobians for the extended task variable x = (g, ρ) ∈SE(3) × R. On top of this model we design a gradient-based redundancy flow on the redundancy manifold that empirically maintains a positive manipulability margin along prescribed SE(3) trajectories. The framework uses right- multiplicative state updates, remains compatible with automatic differentiation, and avoids mechanism-specific analytic Jaco- bians; it works with either direct inverse kinematics or a numeric solver. A specialization to SO(2)3 provides computation-friendly first- and second-order steps. We validate the approach on two representative mechanisms: a (6+3)-degree-of-freedom (DoF) Stewart platform and a Spherical–Revolute platform. Across dense-coverage orientation trajectories and interactive gamepad commands, the extended Jacobian remained well conditioned while the redundancy planner ran at approximately 2 kHz in software-in-the-loop on a laptop-class CPU. The method integrates cleanly with existing kinematic stacks and is suitable for real-time deployment.