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<p>Forward dynamics simulation of constrained multibody systems results in a system of differential–algebraic equations (DAEs), requiring effective stabilization of constraints. This study investigates the influence of numerical integrators on the performance of the Baumgarte method in stabilizing nonholonomic constraints. Using digital control theory, stability regions for Baumgarte coefficients are derived for holonomic and nonholonomic constraints across various numerical methods, including Euler, Runge–Kutta, Adams–Bashforth, and Bézier integrators. These stability regions provide practical guidelines for selecting appropriate Baumgarte coefficients based on the integrator type and time step size for holonomic and nonholonomic constraints. The stability regions are validated through four numerical examples involving a rolling disc, a three-wheeled mobile robot, a mobile manipulator, and a six-wheeled rover with rocker-bogie suspension. These examples demonstrate the necessity of considering the type of constraints present in the system when selecting Baumgarte coefficients. Simulation of these nonholonomic systems shows that the presence of nonholonomic constraints reduces the stability region compared to a purely holonomic system. In the final example, a six-wheeled rover performing an obstacle-climbing maneuver is simulated to illustrate the advantages of numerical methods with larger stability regions for solving complex systems with nonholonomic and unilateral constraints. These methods allow for the use of larger Baumgarte coefficients while maintaining stability, resulting in a more attractive constraint manifold, reduced constraint violation, and improved response accuracy. Furthermore, in this example, the Bézier integrator enables larger time steps while maintaining acceptable constraint violation levels compared to the Adams–Bashforth method, thereby achieving a threefold improvement in computational efficiency. This improvement enhances the feasibility of real-time simulation for multibody dynamic systems.</p>

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Impact of numerical integrators on the stabilization of holonomic and nonholonomic constraints in the Baumgarte method

  • 期刊论文

  • 发表日期：

    2026年2月8日

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  • Alireza Ebrahimi,
  • Ali Azimi,
  • Mohammad M. Aghdam

摘要

Forward dynamics simulation of constrained multibody systems results in a system of differential–algebraic equations (DAEs), requiring effective stabilization of constraints. This study investigates the influence of numerical integrators on the performance of the Baumgarte method in stabilizing nonholonomic constraints. Using digital control theory, stability regions for Baumgarte coefficients are derived for holonomic and nonholonomic constraints across various numerical methods, including Euler, Runge–Kutta, Adams–Bashforth, and Bézier integrators. These stability regions provide practical guidelines for selecting appropriate Baumgarte coefficients based on the integrator type and time step size for holonomic and nonholonomic constraints. The stability regions are validated through four numerical examples involving a rolling disc, a three-wheeled mobile robot, a mobile manipulator, and a six-wheeled rover with rocker-bogie suspension. These examples demonstrate the necessity of considering the type of constraints present in the system when selecting Baumgarte coefficients. Simulation of these nonholonomic systems shows that the presence of nonholonomic constraints reduces the stability region compared to a purely holonomic system. In the final example, a six-wheeled rover performing an obstacle-climbing maneuver is simulated to illustrate the advantages of numerical methods with larger stability regions for solving complex systems with nonholonomic and unilateral constraints. These methods allow for the use of larger Baumgarte coefficients while maintaining stability, resulting in a more attractive constraint manifold, reduced constraint violation, and improved response accuracy. Furthermore, in this example, the Bézier integrator enables larger time steps while maintaining acceptable constraint violation levels compared to the Adams–Bashforth method, thereby achieving a threefold improvement in computational efficiency. This improvement enhances the feasibility of real-time simulation for multibody dynamic systems.