多体系统的线性时间变分积分器
A Linear-Time Variational Integrator for Multibody Systems
摘要：
提出了一种多体系统的变分积分方法。变分积分器将多体系统的运动方程转化为离散的欧拉-拉格朗日（DEL）方程，将前向积分转化为DEL方程的求根问题。变分积分器已经被证明在保持系统的基本性质（例如动量和能量）方面比许多常用的数值积分器更鲁棒和准确。然而，现有技术的算法受到$O(n^3)$复杂性的影响，这对于在广义坐标中具有大量自由度$n$的铰接多体系统是禁止的。我们的主要贡献是推导出一个递归算法，评估DEL方程在$O(n)$，规模以及复杂的多体系统，如人形机器人。受递归Newton-Euler算法的启发，我们的关键见解是为每个物体而不是整个系统单独制定DEL方程。此外，我们引入了一种新的拟牛顿法，利用基于脉冲的动力学算法，这也是$O(n)$，以避免昂贵的雅可比反演求解DEL方程。我们证明了可扩展性和效率，以及可扩展性完整的约束，通过几个案例研究。
英文摘要：
We present an efficient variational integrator for multibody systems. Variational integrators reformulate the equations of motion for multibody systems as discrete Euler-Lagrange (DEL) equations, transforming forward integration into a root-finding problem for the DEL equations. Variational integrators have been shown to be more robust and accurate in preserving fundamental properties of systems, such as momentum and energy, than many frequently used numerical integrators. However, state-of-the-art algorithms suffer from $O(n^3)$ complexity, which is prohibitive for articulated multibody systems with a large number of degrees of freedom, $n$, in generalized coordinates. Our key contribution is to derive a recursive algorithm that evaluates DEL equations in $O(n)$, which scales up well for complex multibody systems such as humanoid robots. Inspired by recursive Newton-Euler algorithm, our key insight is to formulate DEL equation individually for each body rather than for the entire system. Furthermore, we introduce a new quasi-Newton method that exploits the impulse-based dynamics algorithm, which is also $O(n)$, to avoid the expensive Jacobian inversion in solving DEL equations. We demonstrate scalability and efficiency, as well as extensibility to holonomic constraints through several case studies.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
