Constraint Violation Stabilization Using Gradient Feedback in Constrained Dynamics Simulation

Conventional holonomic or nonholonomic constraints are defined as geometric constraints in this paper. If the total energy of a dynamic system can be computed from the initial energy plus the time integral of the energy input rate due to external or internal forces, then the total energy can be artificially treated as a constraint. The violation of the total energy constraint due to numerical errors during simulation can be used as information to control these errors. When geometric constraint control is combined with energy constraint control, numerical simulation of a constrained dynamic system becomes more accurate. An energy constraint control based on the gradient feedback of the energy constraint violation leads to a new method to control both geometric and energy constraint violations, so-called constraint violation stabilization using gradient feedback. A new convenient and effective method to implement energy constraint control in numerical simulation is developed based on the geometric interpretation of the relation between constraints in the phase space. Several combinations of energy constraint control with either Baumgarte's constraint violation stabilization method or the new constraint violation stabilization using gradient feedback are also addressed. Finally, a new method for implementing constraint controls is developed by using the Euler method for integrating constraint control terms, even when higher-order integration methods are used for all other integrations.