Regarding constrained mechanical systems, we are faced with index-3 differential-algebraic equation (DAE) systems. Direct discretization of the index-3 DAE systems only enforces the position constraints to be fulfilled at the integration-time points, but not the hidden constraints. In addition, order reduction effects are observed in the velocity variables and the Lagrange multipliers. In literature, different numerical techniques have been suggested to reduce the index of the system and to handle the numerical integration of constrained mechanical systems. This paper deals with an alternative concept, called collocated constraints approach. We present index-2 and index-1 formulations in combination with implicit Runge–Kutta methods. Compared with the direct discretization of the index-3 DAE system, the proposed method enforces also the constraints on velocity and—in case of the index-1 formulation—the constraints on acceleration level. The proposed method may very easily be implemented in standard Runge–Kutta solvers. Here, we only discuss mechanical systems. The presented approach can, however, also be applied for solving nonmechanical higher-index DAE systems.
Optimal control for stochastic singular integro-differential Takagi-Sugeno fuzzy system using ant colony programming