Parallel O(log n) algorithms for open- and closed-chain rigid multibody systems based on a new mass matrix factorization technique

In this paper, parallel O(log n) algorithms for computation of rigid multibody dynamics are developed. These parallel algorithms are derived by parallelization of new O(n) algorithms for the problem. The underlying feature of these O(n) algorithms is a drastically different strategy for decomposition of interbody force which leads to a new factorization of the mass matrix (M). Specifically, it is shown that a factorization of the inverse of the mass matrix in the form of the Schur Complement is derived as M(exp -1) = C - B(exp *)A(exp -1)B, wherein matrices C, A, and B are block tridiagonal matrices. The new O(n) algorithm is then derived as a recursive implementation of this factorization of M(exp -1). For the closed-chain systems, similar factorizations and O(n) algorithms for computation of Operational Space Mass Matrix lambda and its inverse lambda(exp -1) are also derived. It is shown that these O(n) algorithms are strictly parallel, that is, they are less efficient than other algorithms for serial computation of the problem. But, to our knowledge, they are the only known algorithms that can be parallelized and that lead to both time- and processor-optimal parallel algorithms for the problem, i.e., parallel O(log n) algorithms with O(n) processors. The developed parallel algorithms, in addition to their theoretical significance, are also practical from an implementation point of view due to their simple architectural requirements

Diferencijalno-geometrijsko modeliranje i dinamička simulacija diskretnih mehaničkih sustava s kinematičkim vezama

A formulation for the kinematics of multibody systems is presented, which uses Lie group concepts. With line coordinates the kinematics is parameterized in terms of the screw coordinates of the joints. Thereupon, the Lagrangian motion equations are derived, and explicit expressions are given for the objects therein. It is shown how the kinematics and thus the motion equations can be expressed without the introduction of body-fixed reference frames. This admits the processing of CAD data, which refers to a single (world) frame. For constrained multibody systems, the Lagrangian motion equations are projected to the constraint manifold, which yields the equations of Woronetz. The mathematical models for numerical integration routines of MBS are surveyed and constraint gradient projective method for stabilization of constraint violation is presented.U radu je prikazano matematičko modeliranje kinematike diskretnih mehaničkih sustava s kinematičkim vezama pomoću Lievih grupa. Kinematika sustava parametarizirana je koristeći vijčane koordinate zglobova kinematičkog lanca. Nastavljajući se na takav kinematički model, Lagrangeove dinamičke jednadžbe gibanja sustava izvedene su u nastavku rada. Koristeći takav pristup, pokazano je kako se kinematički model, a također i dinamičke jednadžbe gibanja mehaničkog sustava, mogu izvesti bez upotrebe lokalnih koordinatnih sustava vezanih za pojedina tijela kinematičkog lanca. Takvo matematičko modeliranje omogućava izravnu upotrebu CAD podataka koji se, u pravilu, izražavaju u jedinstvenom koordinantnom sustavu. U slučaju dodatnih kinematičkih ograničenja narinutih na sustav, jednadžbe gibanja izvedene su projiciranjem Lagrangeovih jednadžbi na višestrukost ograničenja, čime se model izražava u obliku jednadžbi Woronetza. U radu su, nadalje, prikazane formulacije matematičkih modela koji se koriste kao podloga numeričkih algoritama za vremensko integriranje jednadžbi dinamike, a također je, uz izrađeni numerički primjer, opisana i metoda stabilizacije numeričkih rješenja na višestrukosti ograničenja

EFFICIENT PARALLEL COMPUTER SIMULATION OF THE MOTION BEHAVIORS OF CLOSED-LOOP MULTIBODY SYSTEMS

ABSTRACT This paper presents an efficient parallelizable algorithm for the computer-aided simulation and numerical analysis of motion behaviors of multibody systems with closed-loops. The method is based on cutting certain user-defined system interbody joints so that a system of independent multibody subchains is formed. These subchains interact with one another through associated unknown constraint forces c f at the cut joints. The increased parallelism is obtainable through cutting joints and the explicit determination of associated constraint forces combined with a sequential O(n) method. Consequently, the sequential O(n) procedure is carried out within each subchain to form and solve the equations of motion while parallel strategies are performed between the subchains to form and solve constraint equations concurrently. For multibody systems with closed-loops, joint separations play both a role of creation of parallelism for computing load distribution and a role of opening a closed-loop for use of the O(n) algorithm. Joint separation strategies provide the flexibility for use of the algorithm so that it can easily accommodate the available number of processors while maintaining high efficiency. The algorithm gives the best performance for the application scenarios for n>>1 and n>>m, where n and m are number of degree of freedom and number of constraints of a multibody system with closed-loops respectively. The algorithm can be applied to both distributed-memory parallel computing systems and shared-memory parallel computing systems. INTRODUCTION In practice, dynamical systems can be modeled as multibody systems. Examples of such systems include, but are not limited to, spacecraft, robotic systems, mechanisms in machinery, automotive applications, land vehicles, underwater vehicles, ships, aircraft, the human body, molecular chain in biotechnology fields, and even quantum dynamical system at nano scale

Chapter 35: Free Simulation Software and Library

International audienceWith the advent of powerful computation technologies and efficient algorithms , simulators became an important tool in most engineering areas. The field of humanoid robotics is no exception; there have been numerous simulation tools developed over the last two decades to foster research and development activities. With this in mind, this chapter is written to introduce and discuss the current-day open source simulators that are actively used in the field. Using a developer-based feedback, we provide an outline regarding the specific features and capabilities of the open-source simulators, with a special emphasis on how they correspond to recent research trends in humanoid robotics. The discussion is centered around the contemporary requirements in humanoid simulation technologies with regards to future of the field

Proceedings of the Fifth NASA/NSF/DOD Workshop on Aerospace Computational Control

The Fifth Annual Workshop on Aerospace Computational Control was one in a series of workshops sponsored by NASA, NSF, and the DOD. The purpose of these workshops is to address computational issues in the analysis, design, and testing of flexible multibody control systems for aerospace applications. The intention in holding these workshops is to bring together users, researchers, and developers of computational tools in aerospace systems (spacecraft, space robotics, aerospace transportation vehicles, etc.) for the purpose of exchanging ideas on the state of the art in computational tools and techniques