On Solving the Dynamics of Constrained Rigid Multi-Body Systems with Kinematic Loops

TL;DR

This work tackles the problem of simulating highly constrained rigid-body systems with kinematic loops by formulating forward dynamics as a nonlinear complementarity problem and its dual NSOCP, then exploring constraint relaxation and proximal-optimization techniques. It provides a unified, maximal-coordinate CRBD framework, builds a comprehensive benchmarking suite, and systematically compares a broad class of solvers including PGS variants and ADMM-based methods. A key finding is that the De Saxcé correction substantially improves physical fidelity and convergence, and that ADMM-NCP offers robust performance on highly coupled systems, especially when paired with stabilization and softening augmentations. The results offer practical guidance for selecting and tuning solvers in robotics and Audio-Animatronics contexts, and highlight avenues for further speedups via preconditioning and GPU-ready implementations.

Abstract

This technical report provides an in-depth evaluation of both established and state-of-the-art methods for simulating constrained rigid multi-body systems with hard-contact dynamics, using formulations of Nonlinear Complementarity Problems (NCPs). We are particularly interest in examining the simulation of highly coupled mechanical systems with multitudes of closed-loop bilateral kinematic joint constraints in the presence of additional unilateral constraints such as joint limits and frictional contacts with restitutive impacts. This work thus presents an up-to-date literature survey of the relevant fields, as well as an in-depth description of the approaches used for the formulation and solving of the numerical time-integration problem in a maximal coordinate setting. More specifically, our focus lies on a version of the overall problem that decomposes it into the forward dynamics problem followed by a time-integration using the states of the bodies and the constraint reactions rendered by the former. We then proceed to elaborate on the formulations used to model frictional contact dynamics and define a set of solvers that are representative of those currently employed in the majority of the established physics engines. A key aspect of this work is the definition of a benchmarking framework that we propose as a means to both qualitatively and quantitatively evaluate the performance envelopes of the set of solvers on a diverse set of challenging simulation scenarios. We thus present an extensive set of experiments that aim at highlighting the absolute and relative performance of all solvers on particular problems of interest as well as aggravatingly over the complete set defined in the suite.

Figures (52)

• Figure 1: A visual depiction of the non-smooth initial-value problem (\ref{['eq:prelim:initial-value-problem']}) using the example of a rigid sphere rolling-off of one plane and then bouncing on another. When the sphere drops the first plane, the impact would result in a contact that is initially open, then closes momentarily and opens once again. Although the position and orientation of the sphere would remain continuous functions of time, the linear and angular velocities would change instantaneously in order to satisfy the contact constraints. This means that the velocities are in effect not continuous, and nor are the positions and orientations smooth functions of time. Therefore, the sphere's trajectory consists of the piece-wise smooth segments (violet lines) and atomic points $\bigcup_{i}{t_{i}}$ at the moments of impact (gray nodes). The former is determined by the smooth dynamics represented by the acceleration-level $\text{NCP}_{s,t}$ of (\ref{['eq:prelim:constrained-system-ncp']}), while the latter defined at each instance $t_{i}$ is determined by the impulsive $\text{NCP}_{ns,t_{i}}$ of (\ref{['eq:prelim:impulsive-constrained-system-ncp']}). Direct time-integration is intractable analytically thus necessitating the application of time-discretization to approximate the integral in (\ref{['eq:prelim:time-integration-problem']}).
• Figure 2: Free-body diagram of a simple constrained system with: (1) $n_{b}=2$ rigid bodies $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$, (2) $n_{j}=1$ joints $\mathcal{J}_{1}$ with revolute constraints, and (3) $n_{c}=2$ contacts $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ between the bodies and the ground.
• Figure 3: Binary collisions between bodies $A_{k}$ and $B_{k}$ for arbitrary contact with index $k$. In all configurations, the gap-function (\ref{['eq:gap-function-def']}) can be used to compute the signed distance $d_{c,k}\,\mathbf{n}_{c,k}$ (black arrows) of $B_{k}$ w.r.t. $A_{k}$, where $\mathbf{n}_{c,k}$ is the normal vector (blue arrows) w.r.t the plane that is tangent to the two contacting points of the bodies. Choosing $A_{k}$ as the reference body is a merely a matter of convention. When $d_{c,k} > 0$ (case (a)), then there is no interpenetration, and the distance is interpreted as the distance between the nearest points between the bodies. Conversely, when $d_{c,k} \leq 0$ (case (b)), interpenetration occurs, and $d_{c,k}$ becomes the penetration depth measured between the most interpenetrating points.
• Figure 4: The set-valued force laws along the normal and tangent directions. The red segments represent the sets of admissible values that the contact reactions $\lambda_{N}, \lambda_{T}$ can take while the respective contact velocities are zero. The blue segments represent the admissible constant values when non-zero velocities are present in the respective directions.
• Figure 5: The Coulomb friction cone $\mathcal{K}_{\mu}$ (a) and the relationships between the former to its dual cone $\mathcal{K}_{\mu}^{*}$ and polar cone $\mathcal{K}_{\mu}^{o}$ (b). As conjugates of $\mathcal{K}_{\mu}$, the dual and polar cones are those whose elements make obtuse angles with each $\boldsymbol{\lambda} \in \mathcal{K}_{\mu}$, i.e. $\mathbf{x} \in \mathcal{K}_{\mu}^{*} \,,\,\, \mathbf{x}^{T}\,\boldsymbol{\lambda} \geq 0$, and $\mathbf{x} \in \mathcal{K}_{\mu}^{o} \,,\,\, \mathbf{x}^{T}\,\boldsymbol{\lambda} \leq 0$.

Theorems & Definitions (11)

• Definition A.1
• Lemma A.1
• proof
• Remark
• Lemma A.2
• proof
• Corollary A.2.1
• Theorem B.1
• Remark
• proof