Numerical Stability and Convergence Analysis of Geometric Constraint Enforcement in Dynamic Simulation Systems

The geometric constraint enforcement using Lagrange Multipliers is one of the popular methods to control the behavior and trajectories of dynamically simulated entities. Therefore, effective and efficient enforcement and proper integration of the geometric constraints is the key to a successful constraint management. This paper describes the formulation and integration of geometric constraints in a dynamic simulation and provides a guideline to choose a proper constraint enforcement method. In addition, the numerical stability and convergence analysis of two explicit and implicit constraint enforcement techniques are reported with respect to the step sizes of differential equations. Experiments show that our new implicit constraint enforcement technique demonstrates a superior stability over large time steps and fast system response compared to the explicit Baumgarte method. The implicit constraint enforcement method uses the future time step to estimate the correct magnitude of the constraint forces and doesn’t require problem dependent stabilization parameters as a second order state feedback term.