A modified Lie group-based synthesis for parallel mechanisms: Addressing uncontrollable and immovable limb cases

The synthesis of parallel manipulators using Lie group theory represents kinematic pairs through displacement operators, allowing the deduction of displacement subgroups from velocity fields using Lie-algebraic structures. However, two key limitations exist: (1) the effect of uncontrollable passive degrees of freedom, which are not reflected in the mechanism’s true degrees of freedom, and (2) the impact of ineffective joints and immovable limbs, which hinders the effective application of existing Lie group-based synthesis methods. These oversights limit the applicability of Lie group-based synthesis for certain parallel manipulators, leading to incorrect results during the intersection operation in synthesis. Failing to detect this phenomenon at the synthesis stage may lead to wrong mechanism. This paper introduces an enhanced method that integrates passive degrees of freedom and the effects of immovable limbs into the synthesis process, providing a correct representation of the motion in previously overlooked mechanisms. This approach extends the applicability of Lie group-based synthesis to various manipulators, ensuring consistent results, including for kinematically redundant parallel manipulators with passive motion in their kinematotropic reconfigurable platforms.