The implicit inversion method for calculating the forward dynamics input Jacobian

IF 2.6 2区 工程技术 Q2 MECHANICS
Gabriel Krög, Hubert Gattringer, Andreas Müller
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Abstract

This paper presents the implicit inversion method (IIM), a recursive method to evaluate the Jacobian of the forward dynamics w.r.t. the system inputs, using intermediate results obtained from an O(n) forward dynamics algorithm. The resulting coefficient matrix, called the inertia-weighted input matrix (IWIM), can be used to significantly improve the performance of solving optimal control problems that take into account system dynamics for only the current time step. As the relationship between inputs and accelerations appears fixed within a time step, this matrix can be evaluated in the initialization step of the optimization. This means that the forward dynamics only needs to be solved once at the initialization of the optimization, rather than having to solve the equations in every iteration of the optimization. The method presented in this paper especially targets a case where the forward dynamics are calculated using an O(n) method and takes advantage of variables that are already known through the evaluation of that method. These quantities allow us to obtain the inertia-weighted input matrix without having to convert the system to its generalized coordinate form. Exploiting the shape of the resulting equation, it is even possible to avoid an explicit inversion of any matrices in the process. Finally, runtime comparisons between three different methods to calculate the IWIM are made for several examples.

Abstract Image

计算前向动力学输入雅各布的隐式反演法
本文介绍了隐式反演法 (IIM),这是一种利用 O(n) 前向动力学算法获得的中间结果来评估系统输入时前向动力学雅各比的递归方法。由此得到的系数矩阵被称为惯性加权输入矩阵(IWIM),可用于显著提高仅考虑当前时间步的系统动态的最优控制问题的求解性能。由于输入和加速度之间的关系在一个时间步长内是固定的,因此可以在优化的初始化步骤中对该矩阵进行评估。这意味着只需在优化初始化时求解一次前向动力学,而无需在优化的每次迭代中求解方程。本文介绍的方法特别针对使用 O(n) 方法计算前向动力学的情况,并利用了通过评估该方法已经知道的变量。利用这些变量,我们无需将系统转换为广义坐标形式,即可获得惯性加权输入矩阵。利用所得方程的形状,我们甚至可以避免在此过程中对任何矩阵进行明确的反演。最后,针对几个示例对计算 IWIM 的三种不同方法进行了运行时间比较。
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来源期刊
CiteScore
6.00
自引率
17.60%
发文量
46
审稿时长
12 months
期刊介绍: The journal Multibody System Dynamics treats theoretical and computational methods in rigid and flexible multibody systems, their application, and the experimental procedures used to validate the theoretical foundations. The research reported addresses computational and experimental aspects and their application to classical and emerging fields in science and technology. Both development and application aspects of multibody dynamics are relevant, in particular in the fields of control, optimization, real-time simulation, parallel computation, workspace and path planning, reliability, and durability. The journal also publishes articles covering application fields such as vehicle dynamics, aerospace technology, robotics and mechatronics, machine dynamics, crashworthiness, biomechanics, artificial intelligence, and system identification if they involve or contribute to the field of Multibody System Dynamics.
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