Non-smooth Newton Methods for Deformable Multi-body Dynamics

Abstract

Fig. 1. The Fetch robot picking up and transferring a tomato to a mechanical scale. The tomato is modeled using tetrahedral FEM, while the robot and working mechanical scale are modeled as rigid bodies connected by revolute and prismatic joints. Our method provides full two-way coupling that allows for stable grasping and force sensing on the gripper. The robot is controlled by a human operator in real-time. Model provided courtesy of Fetch Robotics, Inc.We present a framework for the simulation of rigid and deformable bodies in the presence of contact and friction. Our method is based on a non-smooth Newton iteration that solves the underlying nonlinear complementarity problems (NCPs) directly. This approach allows us to support nonlinear dynamics models, including hyperelastic deformable bodies and articulated rigid mechanisms, coupled through a smooth isotropic friction model. The fixed-point nature of our method means it requires only the solution of a symmetric linear system as a building block. We propose a new complementarity preconditioner for NCP functions that improves convergence, and we develop an efficient GPU-based solver based on the conjugate residual (CR) method that is suitable for interactive simulations. We show how to improve robustness using a new geometric stiffness approximation and evaluate our method's performance on a number of robotics simulation scenarios, including dexterous manipulation and training using reinforcement learning.

The Matchstick Model for Anisotropic Friction Cones

“…Respectively, these frameworks use an LCP‐based formulation with direct solver, NCP‐based formulation with fully implicit Newton solver and iterative Gauss–Seidel type scheme [Erl17]. Figures – use Flex [MEM19], Figures – use PROX and Figures and use Vortex. This choice of solvers shows examples of semi‐implicit time‐integration and fully implicitly time‐integration, as well as global versus local coupled schemes, and fully coupled normal and friction forces too.…”

“…We will now outline how to use a parameterized F in a Newtontype framework using non-smooth functions, such as the Fischer-Burmeister function [MEM19]. Without loss of generality, assume that we have any type of complementary function, ψ(a, b) :…”

“…[BFA02], we demonstrate the compatibility of our Matchstick model using several solver types such as both semi‐implicit and fully implicit time‐stepping methods. These include a pivoting algorithm with direct solver, a nonlinear implicit Newton‐type solver [MEM19] and a PROX‐based iterative Gauss–Seidel scheme.…”

“…In contrast, Macklin et al . [MEM19] show similarity between r‐factors and preconditioners that create self‐dual isotropic cones; however, adaptive r‐factors can adapt locally to the anisotropy and rescale cones just in the direction needed. For certain specific conditions, a cone description of the friction tensor model can be recovered at added computational cost as we prove in Appendix A, and with additional restrictions the maximum principle of dissipation can be fulfilled.…”

“…We will now outline how to use a parameterized in a Newton‐type framework using non‐smooth functions, such as the Fischer–Burmeister function [MEM19]. Without loss of generality, assume that we have any type of complementary function, such that Using an implicit limit surface models of the cone, we let be the corresponding implicit function of , Then, by the principle of maximal dissipation, we can write: where is an auxiliary scalar variable.…”