A parallel Hamiltonian formulation for forward dynamics of closed-loop multibody systems

Abstract

This paper presents a novel recursive divide-and-conquer formulation for the simulation of complex constrained multibody system dynamics based on Hamilton's canonical equations (HDCA). The systems under consideration are subjected to holonomic, independent constraints and may include serial chains, tree chains, or closed-loop topologies. Although Hamilton's canonical equations exhibit many advantageous features compared to their acceleration based counterparts, it appears that there is a lack of dedicated parallel algorithms for multi-rigid-body system dynamics based on the Hamiltonian formulation. The developed HDCA formulation leads to a two-stage procedure. In the first phase, the approach utilizes the divide and conquer scheme, i.e., a hierarchic assembly-disassembly process to traverse the multibody system topology in a binary tree manner. The purpose of this step is to evaluate the joint velocities and constraint force impulses. The process exhibits linear O(n) (n -number of bodies) and logarithmic O(log 2 n) numerical cost, in serial and parallel implementations, respectively. The time derivatives of the total momenta are directly evaluated in the second parallelizable step of the algorithm. Sample closed-loop test cases indicate very small constraint violation errors at the position and velocity level as well as marginal energy drift without any additional form of constraint stabilization techniques involved in the solution process. The results are comparatively set against more standard acceleration based Featherstone's DCA approach to indicate the performance of the HDCA algorithm.

Hamiltonian direct differentiation and adjoint approaches for multibody system sensitivity analysis

“…Moreover, Hamiltonian-based EOM have been successfully employed in the recursive divide and conquer framework. 25 The methods proposed in this article might be reformulated to take substantial benefits associated with parallelization. This issue represents current research tasks for the authors.…”

“…Since initial conditions of the dynamic analysis are prescribed, the appropriate variations at the time instant t = 0 are equal to zero. By inserting Equations (24), (25) into (23) and following the same path with expressions involving δq and δ̇, we come up with:…”

“…Multiple representations of equations of motion may be employed to model a MBS. Recent works of the authors [25][26][27] have demonstrated that by using a constrained Hamilton's canonical equations, in which Lagrange multipliers enforce constraint equations at the velocity level, one can obtain more stable solutions for DAEs as compared to acceleration-based counterparts. 28,29 This phenomenon is partially connected with differential index reduction of the resultant Hamilton's equations.…”

Index-3 divide-and-conquer algorithm for efficient multibody system dynamics simulations: theory and parallel implementation

“…The Articulated Body Algorithm (ABA) [15] was combined with the DCA in [6] to deliver significant speedups in computation times. Hamilton's canonical equations were used in [7,8] and showed good properties regarding the satisfaction of kinematic constraints. Augmented Lagrangian methods with configurationand velocity-level mass-orthogonal projections have also been employed [28]; the resulting algorithm has been proven to behave robustly during the simulation of mechanical systems with redundant constraints and singular configurations.…”

Parallel Computations and Co-Simulation in Universal Mechanism Software. Part I: Algorithms and Implementation

“…they require O(log N) operations in simulation of an articulated MBS with N rigid bodies on N processors. DCA, CFA and further developments of these algorithms [5][6][7][8][9][10] do not support implicit solver procedures for stiff MBS, which limits their application to simulation of rail vehicle dynamics.…”