Efficient parallel formulation for dynamics simulation of large articulated robotic systems

Abstract

This paper presents a recursive and parallel formulation for the dynamics simulation of large articulated robotic systems based on the Hamilton's canonical equations. 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 dynamics simulation based on such formulation. In this paper we consider open-loop kinematic chains that are connected by kinematic joints. Initially, the standard set of Hamilton's canonical equations are joined together with the constraint equations at the velocity level. The formulation allows to determine the system's joint velocities and impulsive constraint forces in a divide and conquer framework. This operation results in logarithmic numerical cost in parallel implementation. Subsequently, the time derivatives of the total joint momenta are evaluated at the constant expense. In case of sequential implementation, the entire algorithm exhibits linear computational cost. The proposed method is exact, non-iterative and does not require the direct calculation of the system's Hamiltonian nor its partial derivatives. Numerical test cases reveal negligible energy drift without the use of any additional constraint stabilization techniques. The results are compared against more standard acceleration based formulation and the preliminary outcome from real-life physical experiment.

FPGA acceleration of planar multibody dynamics simulations in the Hamiltonian–based divide–and–conquer framework

“…A class of Hamiltonian-based divide-and-conquer algorithms for multibody system dynamics that exhibit near-optimal, logarithmic computational complexity have been recently proposed in [25], [26], [27]. The methods can be applied both for open-and closed-loop MBS and represent a unique suite of algorithms that attempts to parallelize constrained Hamilton's equations practically.…”

Hamiltonian direct differentiation and adjoint approaches for multibody system sensitivity analysis

“…According to the literature, this approach often proves to be more efficient and numerically stable, when compared to the acceleration-based formulation, mainly due to the lowered differential index. [27][28][29][30] Now, let us augment the Lagrangian function with a term explicitly enforcing the kinematic velocity constraints (2):…”

“…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.…”

Joint–coordinate adjoint method for optimal control of multibody systems

“…Various formulations of the equations of motion (EOM) yield different adjoint systems, each characterized by different properties [7,8]. Recent works [9][10][11] have demonstrated that using constrained Hamilton's canonical equations, in which Lagrange multipliers enforce constraint equations at the velocity level, one can obtain more stable solutions for differential-algebraic equations (DAEs) as compared to acceleration-based counterparts [12,13]. This phenomenon is partially connected with the differential index reduction of the resultant Hamilton's equations.…”