{"title":"The implicit inversion method for calculating the forward dynamics input Jacobian","authors":"Gabriel Krög, Hubert Gattringer, Andreas Müller","doi":"10.1007/s11044-024-10010-8","DOIUrl":null,"url":null,"abstract":"<p>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.</p>","PeriodicalId":49792,"journal":{"name":"Multibody System Dynamics","volume":null,"pages":null},"PeriodicalIF":2.6000,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Multibody System Dynamics","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1007/s11044-024-10010-8","RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0
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.
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.