Smoothly Differentiable and Efficiently Vectorizable Contact Manifold Generation

Onur Beker; Andreas René Geist; Anselm Paulus; Nico Gürtler; Ji Shi; Sylvain Calinon; Georg Martius

cs.ROFeb 23, 2026

Smoothly Differentiable and Efficiently Vectorizable Contact Manifold Generation

Authors: Onur Beker, Andreas René Geist, Anselm Paulus, Nico Gürtler, Ji Shi, Sylvain Calinon, Georg Martius

Organizations: University of Tübingen · Idiap Research Institute · EPFL

Abstract

Simulating rigid-body dynamics with contact in a fast, massively vectorizable, and smoothly differentiable manner is highly desirable in robotics. An important bottleneck faced by existing differentiable simulation frameworks is contact manifold generation: representing the volume of intersection between two colliding geometries via a discrete set of properly distributed contact points. A major factor contributing to this bottleneck is that the related routines of commonly used robotics simulators were not designed with vectorization and differentiability as a primary concern, and thus rely on logic and control flow that hinder these goals. We instead propose a framework designed from the ground up with these goals in mind, by trying to strike a middle ground between: i) convex primitive based approaches used by common robotics simulators (efficient but not differentiable), and ii) mollified vertex-face and edge-edge unsigned distance-based approaches used by barrier methods (differentiable but inefficient). Concretely, we propose: i) a representative set of smooth analytical signed distance primitives to implement vertex-face collisions, and ii) a novel differentiable edge-edge collision routine that can provide signed distances and signed contact normals. The proposed framework is evaluated via a set of didactic experiments and benchmarked against the collision detection routine of the well-established Mujoco XLA framework, where we observe a significant speedup. Supplementary videos can be found at https://github.com/bekeronur/contax, where a reference implementation in JAX will also be made available at the conclusion of the review process.

Figures & tables

Fig. 1: We introduce a contact manifold generation framework that is smoothly differentiable, fast, and well-suited for vectorization. This is enabled by two contributions. The first is a pipeline for constructing signed distance fields (SDF) for arbitrary surfaces from a representative set of analytical primitives. These SDFs then serve two roles: (i) acting as a differentiable acceleration structure to efficiently select vertex–face (V–F) and edge–edge (E–E) contact candidates, and (ii) providing signed distances, normals, and activity indicators for V–F and E–E interactions in a way that retains curvature information (since by default, they happen between flat, planar facets). The second contribution is a method to make witness point computations for V–F and E–E pairs smoothly differentiable by solving the associated quadratic programs via an analytical active-set approach.

Fig. 2: Visualizations of how the proposed SDF primitives behave with increasing levels of smoothing.

Fig. 3: ( Left ) A didactic experiment illustrating how E-E witness points behave under the proposed smoothing scheme. There are two edges lying on the

xy

plane, edge-2 being fixed and extending in the

x

direction with

y=-1.2

, and edge-1 rotating around the

z

axis of the origin. The colored lines plot the positions traced by the witness point

\bm{e}_{1}^{\star}(\theta)=[\bm{e}_{1,x}^{\star},\bm{e}_{1,y}^{\star},0]

on edge-1 as it completes a half rotation

\theta:0\to\pi

. Without smoothing (top left),

\bm{e}_{1}^{\star}(\theta)

exhibits a jump as the system passes through the pathological parallel configuration

\theta=\pi/2

. The center and bottom plots visualize the smoothing contributions of L 2 regularization and employing the approximations from Section III . ( Right ) The gradients of

\bm{e}_{1}^{\star}(\theta)

for the three cases above. It can be seen that in the absence of any smoothing, the finite difference gradients are uninformative (e.g., very large near the pathological configuration), which is fixed by the proposed smoothing scheme.

Fig. 4: Creating E-E Contacts: The witness points $\bm{e}_{i}^{\star}$ , as well as the corresponding unsigned distances and normals, are obtained from the (smoothed) analytical active set method described in S. IV-D2 . The subsequent sign assignment for distances and normals in turn utilizes the SDFs by evaluating the dot product of the unsigned normal direction and the SDF gradient for the penetrated surface evaluated at each witness point.

Fig. 5: Contact manifold visualizations for box-box collisions. SDFs are obtained from SQ primitives (

\epsilon_{1}=\epsilon_{2}=0.1

). For illustration clarity, E-E contacts use the simplest cube mesh with

8

vertices,

12

edges, and

6

faces, whereas V-S contacts use a more densely meshed version of the same cube with

98

vertices. In both cases, only half of the contact points (belonging to the top box) are visualized, resulting in two manifolds: one with

12\times 12=144

E-E contacts and one with

98

V-S contacts. For both, two versions are shown: a version where normals are scaled with the activity indicator (to visualize which contacts are actually selected by the routine), and a version where they are unscaled. It can be seen that normals point in the right directions, and the correct points are selected.

Fig. 6: Plots showing the speed and vectorizability of the proposed active-set approach for witness point computation (Ours, and the non-smoothed version Ours-NS), in comparison with log-barrier smoothing approaches. The latter are implemented via two interior point solvers, primal-dual Newton iterations on the KKT system (IP), and L-BFGS iterations with a log barrier penalty (IP-L). It can be seen that the proposed routines are fast and efficiently vectorizable.

Fig. 7: Plots showing the speed and vectorizability of the proposed contact manifold generation routine, compared to the analogous routine of the well-established MJX simulator. The setup involves collisions between two non-convex armadillo geometries across 100 random configurations per-trial. Surfaces are represented with up to 18 SQs each, which is a controlled variable characterizing the effects of geometry complexity on the runtime. The second controlled variable is the vectorization batch size. Notice the logarithmic scale and the orders of magnitude improvement compared to MJX.

Fig. 8: A plot demonstrating how possible simplifications within the proposed contact manifold generation framework effect runtimes across different vectorization batch sizes. Ours denotes the proposed pipeline without any simplifications, Ours-NE denotes a setting where E-E contacts are turned off and only V-S contacts are considered, and Our-NE-S denotes a setting where only one sided V-S contacts are created.

Panel of Figure 9 — Fig. 9: Original high-resolution meshes, SQ primitive decompositions, as well as mesh and SDF representations (used by the proposed method) corresponding to the $4$ objects used in simulation videos. The SQ decomposition visualizations also involve an overlay of the original high-resolution mesh, to demonstrate the representation power of SQ primitives.

Fig. 10: Larger versions of plots from Fig.6 of the main paper.

Fig. 11: Larger versions of plots from Fig.7 of the main paper.

ID	File	Description
Folder: armadillo_armadillo_rollout/
A1	angle_1.mp4	Armadillo–Armadillo collision rollout (view angle 1). Shows contact manifold behavior on a non-convex mesh pair.
A2	angle_2.mp4	Same rollout as A1 from view angle 2.
A3	angle_3.mp4	Same rollout as A1 from view angle 3.
A4	angle_4.mp4	Same rollout as A1 from view angle 4.
A5	angle_1_E_E_contacts_off.mp4	Same setup as A1, but with E–E contacts disabled (V–S only). Demonstrates the “Ours-NE” simplification discussed in the supplementary material Fig. 8 .

TABLE I: Supplementary videos. Each scenario is shown from four camera angles, plus an ablation with E–E contacts disabled (V–S only), an E–E on/off comparison, and an interactive visualizer demonstration.

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Novel Algorithms for Smoothly Differentiable and Efficiently Vectorizable Contact Manifold Construction

Onur Beker, Andreas René Geist, Anselm Paulus, Georg Martius

University of Tübingen

Signed Distance Functions Contact-Rich Robotic Manipulation Differentiable Programming Differentiable Physics

Open paper AskPDF arXiv

Generating intelligent robot behavior in contact-rich settings is a research problem where zeroth-order methods currently prevail. Developing methods that make use of first/second order information about rigid-body dynamics in the presence of contact holds great promise in terms of increasing the solution speed and computational efficiency. The main bottleneck in this research direction is the difficulty in obtaining gradients and Hessians that are actually useful for numerical optimization, due to pathologies in all three steps of a common simulation pipeline: i) collision detection, ii) contact dynamics, iii) time integration. This abstract proposes a method that aims to address the collision detection part of the puzzle, via a novel pipeline designed from scratch with smooth (i.e. twice) differentiability and massive vectorizability on GPUs as the main priorities. This is in contrast to standard collision detection routines that are instead optimized for runtime on CPUs and minimal memory footprint, but do employ logic and control flow that hinder differentiability and vectorization. The proposed pipeline consists of the following contributions: i) highly expressive and compute efficient SDF representations, ii) differentiable broad-phase and narrow-phase routines that use these representations to generate vertex-SDF and edge-SDF contacts, iii) a differentiable routine for convex decomposition based contact blending.

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