The geometrical structure of constraints leads to the important conclusion that components of velocities and accelerations in the normal space of the constraint hypersurface are totally determined by the constraints. Orthogonal base vectors of the constraint’s normal and tangent spaces can be obtained by the QR decomposition with column permutation. A numerical method of closed-loop multibody systems with singular configurations is presented, in which the traditional hypothesis on independence of constraints is abandoned. Instead of correcting constraint violations at the end of each integration step, positions and velocities are modified to satisfy their constraint equations before they are used to form equations of motion. Such an approach can collaborate with any standard ODE solver. In order to systematically generate constraint equations and obtain the corresponding joint’s reaction forces, rotational and translational constraints of a closed-loop are standardized as part of six explicit equations, and a clear relationship between Lagrange multipliers and joint’s reaction forces is derived based on the principle of virtual power equivalence. The proposed method can dynamically identify independent constraints and modify the equations of motion accordingly. The numerical examples validated its effectiveness.

约束的几何结构得出一个重要结论，即约束超曲面法向空间中的速度和加速度分量完全由约束决定。约束的法线和切线空间的正交基向量可以通过列置换的 QR 分解获得。提出了一种奇异构型闭环多体系统的数值方法，摒弃了传统的约束独立性假设。不是在每个积分步骤结束时纠正约束违规，而是修改位置和速度以满足它们的约束方程，然后再用于形成运动方程。这种方法可以与任何标准 ODE 求解器协作。为了系统地生成约束方程并获得相应的关节反作用力，将闭环的旋转和平移约束标准化为六个显式方程的一部分，并根据该原理推导出拉格朗日乘子与关节反作用力之间的明确关系虚功率等价。所提出的方法可以动态识别独立约束并相应地修改运动方程。数值例子验证了其有效性。所提出的方法可以动态识别独立约束并相应地修改运动方程。数值例子验证了其有效性。所提出的方法可以动态识别独立约束并相应地修改运动方程。数值例子验证了其有效性。