We propose the formulation of forward and differentiable rigid-body dynamics using Lie-algebra rotation derivatives. In particular, we show how this approach can easily be applied to incremental-potential formulations of forward dymamics, and we introduce a novel definition of adjoints for differentiable dynamics. In contrast to other parameterizations of rotations (notably the popular rotation-vector parameterization), our approach leads to painlessly simple and compact derivatives, better conditioning, and higher runtime efficiency. We demonstrate our approach on fundamental rigid-body problems, but also on Cosserat rods as an example of multi-rigid-body dynamics.

Abstract The simulation of complex deformation problems often requires enrichment techniques that introduce local high‐resolution detail on a generally coarse discretization. The use cases include spatial or temporal refinement of the discretization, the simulation of composite materials with phenomena occurring at different scales, or even codimensional simulation. We present an efficient simulation enrichment method for both local refinement of the discretization and codimensional effects. We dub our method Voronoi filters, as it combines two key computational elements. One is the use of kinematic filters to constrain coarse and fine deformations, and thus provide enrichment functions that are complementary to the coarse deformation. The other one is the use of a centroidal Voronoi discretization for the design of the enrichment functions, which adds high‐resolution detail in a compact manner while preserving the rigid modes of coarse deformation. We demonstrate our method on simulation examples of composite materials, hybrid triangle‐based and yarn‐level simulation of cloth, or enrichment of flesh simulation with high‐resolution detail.

Abstract This paper presents a method to combine triangle and yarn models in cloth simulation, and hence leverage their best features. The majority of a garment uses a triangle‐based model, which reduces the overall computational and memory cost. Key areas of the garment use a yarn‐based model, which elicits rich effects such as structural nonlinearity and plasticity. To combine both models in a seamless and robust manner, we solve two major technical challenges. We propose an enriched kinematic representation that augments triangle‐based deformations with yarn‐level details. Naïve enrichment suffers from kinematic redundancy, but we devise an optimal kinematic filter that allows a smooth transition between triangle and yarn models. We also introduce a preconditioner that resolves the poor conditioning produced by the extremely different inertia of triangle and yarn nodes. This preconditioner deals effectively with rank deficiency introduced by the kinematic filter. We demonstrate that mixed yarns and triangles succeed to efficiently capture rich effects in garment fit and drape.

Graphical models jointly with non linear optimization have become the most popular approaches for solving SLAM and Bundle Adjustment problems: using a non linear least squares (NLSQs) description of the problem, these math tools serve to formalize the minimization of an error cost function that relates state variables through relative sensor observations. The simplest case just considers as state variables the locations of the sensor/robot in the environment deriving in a pose graph subproblem. In general, the cost function is based on the L2norm whose principal iterative solutions exploit the sparse connectivity of the corresponding Gaussian Markov Field (GMRF) or the Factor Graph, whose adjacency matrices are given by the fill-in of the Hessian and Jacobian of the cost function respectively. In this paper we propose a novel solution based on the L1norm as a back-end to the pose graph subproblem. In contrast to other NLSQs approaches, we formulate an iterative algorithm inspired directly on the Factor Graph structure to solve for the linearized residual ∥Ax - b∥1. Under the presence of spurious measurements the L1based solution can achieve similar results to the robust Huber norm. Indeed, our main interest in L1optimization is that it opens the door to the set of more robust non-convex Lpnorms where p ≤ 1. Since our approach depends on the minimization of a non differentiable function, we provide the theoretical insights to solve for the L1norm. Our optimization is based on a primal-dual formulation successfully applied for solving variational convex problems in computer vision. We show the effectiveness of the L1norm to produce both a robust initial seed and a final optimized solution on challenging and well known datasets widely used in other state of the art works.