M-ABD: Scalable, Efficient, and Robust Multi-Affine-Body Dynamics

TL;DR

A novel framework that leverages the linear kinematic mapping of Affine Body Dynamics to isolate geometric nonlinearities of the system and provide a suite of specialized solvers tailored for diverse joint topologies, including chains, trees, closed loops, and irregular networks is introduced.

Abstract

Simulating large-scale articulated assemblies poses a significant challenge due to the numerical stiffness and geometric complexity of jointed structures. Conventional rigid body solvers struggle with the high nonlinearity induced by rotation parameterization. This difficulty becomes more pronounced for multiple two-way-coupled bodies. This paper introduces a novel framework that leverages the linear kinematic mapping of Affine Body Dynamics (ABD). As ABD targets near-rigid objects, the constitutive variations of different materials become negligible, which justifies a co-rotational approach to isolate geometric nonlinearities of the system. This insight enables the use of constant system matrices that can be pre-factorized throughout the simulation, even with fully implicit integration schemes. To manage the high DOF counts of large-scale systems, we map primal body coordinates onto a compact dual space defined by minimal joint degrees of freedom. By solving the resulting KKT systems, our method ensures exact constraint enforcement and physically accurate motion propagation. We provide a suite of specialized solvers tailored for diverse joint topologies, including chains, trees, closed loops, and irregular networks. Experimental results show that our approach achieves interactive rates for systems with hundreds of thousands of bodies on a single CPU core, while maintaining excellent stability at large time steps.

Figures (22)

• Figure 2: Control points and tetrahedron. We can map the generalized coordinate of an affine body to any non-degenerate tetrahedron or the control tetrahedron. Its four corners are named control points of the affine body. Concatenating their positions gives the CP coordinate.
• Figure 3: Rotational joints. The affine coordinate linearizes rotational constraints between affine bodies. By pre-parameterizing $\bm{q}$ to the CP coordinate $\bm{y}$, a ball joint becomes a point-point constraint; a hinge joint can be modeled as a 6-DOF edge-edge constraint; and a universal joint is the superposition of two hinge joints (with a virtual intermediate body). If we transfer the interconnected bodies into the local frame, the hinge joint can be enforced with 5 DOFs, and the universal joint only consumes 5 DOFs i.e., with the minimum DOFs needed.
• Figure 4: Prismatic joint. A prismatic joint filters out-of-axis movements. The last constraint keeps $\beta$ non-rotatable.
• Figure 5: Block-sparse dual matrix. The constraint gradient matrix $\nabla \widetilde{\bm{C}}$ is block-sparse, and the dual matrix is also block-sparse. An off-diagonal block is non-zero iff two joints are on the same affine body.
• Figure 6: Spinning box. We compare ABD (solid curves) against an implicit RBD baseline (dashed curves) for a cube with initial $\bm p_0=[100, 0, 0]^\top~{kg\,m/s}$ and $\bm L_0=[0, 100, 0]^\top~{kg\,m^2/s}$. Co-rotated ABD formulation closely matches the behavior of an implicit RBD.