A unified numerical solution framework for solving DAEs of multibody system dynamics with holonomic and nonholonomic constraints

Studying and proposing a unified numerical solution framework and adapting it to simulation software is important to address the increasing complexity of models and constraints in modern engineering. In this paper, based on the state-space method (SSM), a unified framework that can solve holonomic constraints, linear nonholonomic constraints, nonlinear nonholonomic constraints, and various combinations of these constraints is proposed. The nonholonomic constraints directly restrict the velocity coordinates, resulting in no corresponding position constraint equations. Therefore, the traditional SSM is insufficient to solve such differential algebraic equations (DAEs). In the proposed SSM, we directly integrate the ordinary differential equations derived from index-1 DAEs, ensure that the acceleration constraints are satisfied, and provide initial values for the dependent variables. Subsequently, position and velocity constraint equations are solved to update dependent variables, strictly ensuring the satisfaction of constraints at three levels. To avoid coordinate identification at each step and thereby improve computational efficiency, we also propose an SSM based on singular value decomposition (SSM-SVD). Finally, several examples are selected to compare the numerical simulation results with the theoretical solutions, and to compare the results for a rolling disk on a rough plane with ADAMS. The results show that the algorithm proposed in this paper has excellent generalization and promising applications in simulation software.