Geometric Stiffness for Real-time Constrained Multibody Dynamics

Abstract

Figure 1: A heavy vehicle attached to a nearly inextensible cable is dropped from the end of a crane. The simulation involves very large mass ratios, a heterogeneous collection of joints, and remains stable at a time step of 1/60 s. Cable dynamics are well preserved by our adaptive damping method. AbstractThis paper focuses on the stable and efficient simulation of articulated rigid body systems for real-time applications. Specifically, we focus on the use of geometric stiffness, which can dramatically increase simulation stability. We examine several numerical problems with the inclusion of geometric stiffness in the equations of motion, as proposed by previous work, and address these issues by introducing a novel method for efficiently building the linear system. This offers improved tractability and numerical efficiency. Furthermore, geometric stiffness tends to significantly dissipate kinetic energy. We propose an adaptive damping scheme, inspired by the geometric stiffness, that uses a stability criterion based on the numerical integrator to determine the amount of non-constitutive damping required to stabilize the simulation. With this approach, not only is the dynamical behavior better preserved, but the simulation remains stable for mass ratios of 1,000,000-to-1 at time steps up to 0.1 s. We present a number of challenging scenarios to demonstrate that our method improves efficiency, and that it increases stability by orders of magnitude compared to previous work.

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

“…and the algorithm is continued in the standard way described in the paper. It should be noted that there are some parallels between this notion and the results obtained by the other authors [2,40]. The new terms added here are similar to the concept of geometric stiffness defined for real-time simulation of complex multibody systems.…”

“…(12) and (13) at the k + 1 (next) time-instant. After scaling the resulting equations by t 2 4 , we get…”

Primal/Dual Descent Methods for Dynamics

“…Servin et al [SLM06] formulated a linear finite element method (FEM) as a set of compliant constraints. This approach was made robust by the inclusion of geometric stiffness terms that include second order constraint information [TNGF15,ATK17]. Goldenthal et al [GHF * 07] proposed a fast constraint projection method using direct solvers, while Position-based Dynamics employs iterative, local, nonlinear constraint projections on the same constraint-based formulation [MHHR07,Sta09].…”

Non-smooth Newton Methods for Deformable Multi-body Dynamics

“…They use temporal coherence of Lagrange multipliers to build the system Jacobian and compute a Cholesky decomposition, followed by a projected-Gauss Seidel solve for contact. For smaller problems our approach is compatible with direct solvers, however we avoid the requirement of dense matrix decompositions by using a diagonal geometric stiffness approximation inspired by the work of Andrews et al [2017], this improves stability and allows us to apply iterative methods.…”