The dynamic equations of motion of the constrained multibody mechanical system are mixed differential-algebraic equations (DAEs). The numerical solution of the DAE systems solved using ordinary-differential equation (ODE) solvers may suffer from constraint drift phenomenon. To solve this problem, Baumgarte proposed a constraint stabilization method in which a position and velocity terms were added in the second derivative of the constraint equation. Baumgarte’s method is a proportional-derivative (PD) type controller design. In this paper, an controller is included to form a proportional-integral-derivative (PID) controller so that the steady state error of the numerical integration can be reduced. Stability analysis methods in the digital control theory are used to find out the correct choice of the coefficients for the PID controller.
A PID Type Constraint Stabilization Method for Numerical Integration of Multibody Systems
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Lin, S., and Chen, M. (April 14, 2011). "A PID Type Constraint Stabilization Method for Numerical Integration of Multibody Systems." ASME. J. Comput. Nonlinear Dynam. October 2011; 6(4): 044501. https://doi.org/10.1115/1.4002688
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