Enhanced Generalized Inverse Methods with Null-Space and Feedback Stabilization for Constrained Mechanical Systems

This study introduces two enhanced forms of the Udwadia–Kalaba Generalized Inverse Method (GIM) with built-in stabilization mechanisms, referred to as S1-GIM and S2-GIM, for mitigating drift in constrained dynamic systems. The classical GIM provides closed-form constrained accelerations but suffers from drift because it enforces constraints only at the acceleration level. The feedback-based S2-GIM enhances GIM by introducing proportional–derivative (PD) stabilization, while the pseudoinverse-based S1-GIM incorporates an additional null-space term, allowing dynamic shaping, such as damping or energy control, without violating constraints. Both formulations are compared with Baumgarte stabilization using a pendulum and a two-story shear building under holonomic and nonholonomic constraints. The results show that wherea Baumgarte stabilization demands careful tuning of feedback gains, while the original GIM is vulnerable to drift accumulation, both S1-GIM and S2-GIM suppress drift effectively, with S1-GIM offering greater flexibility through null-space shaping for applications requiring tailored dynamic control.