Lie Group Implicit Kinematics for Redundant Parallel Manipulators: Left-Trivialized Extended Jacobians and Gradient-Based Online Redundancy Flows for Singularity Avoidance

We present a Lie group implicit formulation for kinematically redundant parallel manipulators that yields left-trivialized extended Jacobians for the extended task variable$x=(g,\rho)\in \text{SE}(3)\times \mathcal {R}$. 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$\text{SE}(3)$trajectories. The framework uses right-multiplicative state updates, remains compatible with automatic differentiation, and avoids mechanism-specific analytic Jacobians; it works with either direct inverse kinematics or a numeric solver. A specialization to$\text{SO}(2)^{3}$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.