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 引用本文: 袁歆懿, 陈菊, 田强. SE(3)描述的多刚体系统约束违约问题研究. 力学学报, 2024, 56(8): 2351-2363.
Yuan Xinyi, Chen Ju, Tian Qiang. Research on constraints violation in dynamics of multibody systems based on SE(3). Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(8): 2351-2363.
 Citation: Yuan Xinyi, Chen Ju, Tian Qiang. Research on constraints violation in dynamics of multibody systems based on SE(3). Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(8): 2351-2363.

## RESEARCH ON CONSTRAINTS VIOLATION IN DYNAMICS OF MULTIBODY SYSTEMS BASED ON SE(3)

• 摘要: 多刚体系统动力学的指标-1微分代数方程组仅包含加速度级别约束方程, 未考虑位移和速度约束方程. 数值积分过程中位移和速度约束违约逐渐增加, 导致计算结果精度下降, 需引入约束稳定方法减小约束违约. 首先根据Hamilton变分原理, 建立了SE(3)描述的多刚体系统动力学模型. 其次给出4种约束稳定方法在SE(3)上形式: Baumgarte方法、罚函数方法、增广拉格朗日方法和约束违约增强稳定方法. 相较于传统多刚体系统动力学建模, 基于SE(3)的动力学建模可描述转动对平动的耦合效应, 能更精确地模拟系统的动力学行为, 可提升约束稳定方法的性能. 在SE(3)框架下, 将4种约束稳定方法与RKMK (Runge-Kutta Munthe-Kass)数值方法结合, 对含位移约束和速度约束的动力学方程进行求解. 最后通过两个动力学算例: 含球铰的空间双球摆算例以及含圆柱铰的曲柄滑块机构算例, 分别给出了4种约束稳定方法在SE(3)上对动力学系统的位移和速度约束违约的修正对比结果, 并分析了系统能量的变化图. 结果表明: SE(3)框架下, 4种约束稳定方法结合RKMK算法均表现出良好的保结构和保能量性质, 其中SE(3)上增广拉格朗日方法所得位移和速度约束违约较小, 总能量保持情况更好; SE(3)上Baumgarte方法能更好地平衡系统计算效率和精度.

Abstract: When simulating the dynamics of muti-rigid body systems, the Index-1 differential-algebraic equations (DAEs) only consider the acceleration constraint equations and ignore the position and velocity constraint equations. During the numerical integration procedure, the constraints violation at position and velocity levels increase, leading to unreliability of numerical results. Accordingly, constraint stabilization methods have to be introduced to reduce the violation of the constraints. In this work, the dynamic equations of muti-rigid body systems on SE(3) are offered based on the Hamilton’s principle. Then four constraint stabilization methods on SE(3) are introduced: the Baumgarte stabilization method, the penalty method, the augmented Lagrangian formulation and the constraint violation stabilization upgraded method. Compared with the traditional dynamic equations of muti-rigid body systems, the dynamic equations on SE(3) solved by integration scheme consider the coupling of rotations and transitions, leading more accurate configuration results and calculation of constraint error. The accurate calculation of constraint error improves the performance of the constraint stabilization methods on SE(3). The four kinds of constraint stabilization methods on SE(3) with dynamic equations of muti-rigid body systems are respectively simulated by RKMK (Runge-Kutta Munthe-Kass) integration scheme. Finally, two numerical examples, including a spatial double pendulum with spherical hinges and a crank slider mechanism with cylindrical hinges, are presented. The numerical results related to constraints violation at position and velocity levels, conservation of the total energy and simulation time are compared and analyzed. It is concluded that the four constraint stabilization methods on SE(3) are effective for preserving structure and energy conservation where RKMK is employed to simulate the dynamic equations of muti-rigid body systems. Compared with the other three constraint stabilization methods on SE(3), the augmented Lagrangian formulation method on SE(3) can provide better numerical accuracy and smaller constraints violation at both position and velocity levels. The Baumgarte stabilization method on SE(3) can balance the calculation efficiency and numerical accuracy well.

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