Redundant parallel kinematic machines (RPKMs) offer higher dexterity, larger workspaces, and improved singularity avoidance compared with their minimally actuated counterparts. However, the presence of kinematic redundancy in closed-chain architectures introduces challenges in modeling, singularity characterization, and redundancy resolution. In particular, traditional approaches often rely on coordinate-dependent formulations or joint-space null-space methods that do not explicitly exploit the geometric structure of the configuration space. This thesis presents a unified Lie-group framework for the kinematic modeling and singularity-aware planning of RPKMs. The configuration space of an RPKM is modeled as a product Lie group, and the closed-chain constraints are expressed as Lie-group-valued implicit functions. Under a consistent right-perturbation convention, differential kinematics and Jacobian structures are derived directly on the configuration manifold, enabling singularity analysis in an extended task-space representation. Based on this geometric formulation, first-order and second-order redundancy flow methods are developed to optimize redundancy with respect to conditioning-related objectives. Unlike point-wise joint-space strategies, the proposed approach plans redundancy coherently along trajectories within the Lie algebra, allowing singularity margins to be maintained under dense excitation and continuous angular wrapping. The framework is validated through offline redundancy planning and teleoperation experiments on representative RPKMs, including Stewart-type platforms and a 3(∗SR) architecture. Results demonstrate consistently improved conditioning behavior relative to local damped least squares (DLS) strategies and confirm compatibility with a modular, differentiable software architecture implemented using JAX and MJX. These findings indicate that a unified Lie-group formulation provides a consistent and practically deployable foundation for redundancy-aware kinematic modeling and planning of RPKMs.
Yifei Liu (Thu,) studied this question.