# A Smooth Analytical Formulation of Collision Detection and Rigid Body Dynamics With Contact

arXiv: https://arxiv.org/abs/2503.11736 · HTML: https://mlsanity.com/paper/2503.11736/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Mar 2025.

## Built on by (most cited first, 3 of 3)

- [Differentiable Simulation of Hard Contacts with Soft Gradients for Learning and Control](https://mlsanity.com/paper/2506.14186/) (2025)
- [SoftJAX & SoftTorch: Empowering Automatic Differentiation Libraries with Informative Gradients](https://mlsanity.com/paper/2603.08824/) (2026)
- [Learning Physics-Guided Residual Dynamics for Deformable Object Simulation](https://mlsanity.com/paper/2607.13451/) (2026)

## Abstract (arXiv)

Generating intelligent robot behavior in contact-rich settings is a research problem where zeroth-order methods currently prevail. A major contributor to the success of such methods is their robustness in the face of non-smooth and discontinuous optimization landscapes that are characteristic of contact interactions, yet zeroth-order methods remain computationally inefficient. It is therefore desirable to develop methods for perception, planning and control in contact-rich settings that can achieve further efficiency by making use of first and second order information (i.e., gradients and Hessians). To facilitate this, we present a joint formulation of collision detection and contact modelling which, compared to existing differentiable simulation approaches, provides the following benefits: i) it results in forward and inverse dynamics that are entirely analytical (i.e. do not require solving optimization or root-finding problems with iterative methods) and smooth (i.e. twice differentiable), ii) it supports arbitrary collision geometries without needing a convex decomposition, and iii) its runtime is independent of the number of contacts. Through simulation experiments, we demonstrate the validity of the proposed formulation as a "physics for inference" that can facilitate future development of efficient methods to generate intelligent contact-rich behavior.