# Fast and Feature-Complete Differentiable Physics for Articulated Rigid Bodies with Contact

arXiv: https://arxiv.org/abs/2103.16021 · HTML: https://mlsanity.com/paper/2103.16021/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 43 references that are in the map.
- First posted on arXiv 30 Mar 2021.

## Builds on (most cited first, 16 of 16)

- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [DiffTaichi: Differentiable Programming for Physical Simulation](https://mlsanity.com/paper/1910.00935/) (2019)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [A Compositional Object-Based Approach to Learning Physical Dynamics](https://mlsanity.com/paper/1612.00341/) (2016)
- [A Differentiable Physics Engine for Deep Learning in Robotics](https://mlsanity.com/paper/1611.01652/) (2016)
- [ChainQueen: A Real-Time Differentiable Physical Simulator for Soft Robotics](https://mlsanity.com/paper/1810.01054/) (2018)
- [Flexible Neural Representation for Physics Prediction](https://mlsanity.com/paper/1806.08047/) (2018)
- [Scalable Differentiable Physics for Learning and Control](https://mlsanity.com/paper/2007.02168/) (2020)
- [Theano: A Python framework for fast computation of mathematical expressions](https://mlsanity.com/paper/1605.02688/) (2016)
- [SPNets: Differentiable Fluid Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1806.06094/) (2018)
- [Interactive Differentiable Simulation](https://mlsanity.com/paper/1905.10706/) (2019)
- [Learning to Slide Unknown Objects with Differentiable Physics Simulations](https://mlsanity.com/paper/2005.05456/) (2020)
- [Augmenting Differentiable Simulators with Neural Networks to Close the Sim2Real Gap](https://mlsanity.com/paper/2007.06045/) (2020)
- [Automatic Differentiation and Continuous Sensitivity Analysis of Rigid Body Dynamics](https://mlsanity.com/paper/2001.08539/) (2020)
- [Constrained Sampling-based Trajectory Optimization using Stochastic Approximation](https://mlsanity.com/paper/1911.04621/) (2019)

## Built on by (most cited first, 30 of 52)

- [Brax -- A Differentiable Physics Engine for Large Scale Rigid Body Simulation](https://mlsanity.com/paper/2106.13281/) (2021)
- [Do Differentiable Simulators Give Better Policy Gradients?](https://mlsanity.com/paper/2202.00817/) (2022)
- [Gradients are Not All You Need](https://mlsanity.com/paper/2111.05803/) (2021)
- [Human Trajectory Prediction via Neural Social Physics](https://mlsanity.com/paper/2207.10435/) (2022)
- [NeuPhysics: Editable Neural Geometry and Physics from Monocular Videos](https://mlsanity.com/paper/2210.12352/) (2022)
- [Differentiable Physics Simulation of Dynamics-Augmented Neural Objects](https://mlsanity.com/paper/2210.09420/) (2022)
- [DiffSkill: Skill Abstraction from Differentiable Physics for Deformable Object Manipulations with Tools](https://mlsanity.com/paper/2203.17275/) (2022)
- [Differentiable Simulation of Soft Multi-body Systems](https://mlsanity.com/paper/2205.01758/) (2022)
- [Differentiable Dynamics for Articulated 3d Human Motion Reconstruction](https://mlsanity.com/paper/2205.12256/) (2022)
- [Learning Quadruped Locomotion Using Differentiable Simulation](https://mlsanity.com/paper/2403.14864/) (2024)
- [A Survey on Physics Informed Reinforcement Learning: Review and Open Problems](https://mlsanity.com/paper/2309.01909/) (2023)
- [DexDeform: Dexterous Deformable Object Manipulation with Human Demonstrations and Differentiable Physics](https://mlsanity.com/paper/2304.03223/) (2023)
- [SAM-RL: Sensing-Aware Model-Based Reinforcement Learning via Differentiable Physics-Based Simulation and Rendering](https://mlsanity.com/paper/2210.15185/) (2022)
- [Combining Physics and Deep Learning to learn Continuous-Time Dynamics Models](https://mlsanity.com/paper/2110.01894/) (2021)
- [SAGCI-System: Towards Sample-Efficient, Generalizable, Compositional, and Incremental Robot Learning](https://mlsanity.com/paper/2111.14693/) (2021)
- [Contact Points Discovery for Soft-Body Manipulations with Differentiable Physics](https://mlsanity.com/paper/2205.02835/) (2022)
- [A Survey on Reinforcement Learning Methods in Character Animation](https://mlsanity.com/paper/2203.04735/) (2022)
- [Learning Tool Morphology for Contact-Rich Manipulation Tasks with Differentiable Simulation](https://mlsanity.com/paper/2211.02201/) (2022)
- [SoftMAC: Differentiable Soft Body Simulation with Forecast-based Contact Model and Two-way Coupling with Articulated Rigid Bodies and Clothes](https://mlsanity.com/paper/2312.03297/) (2023)
- [DeepSimHO: Stable Pose Estimation for Hand-Object Interaction via Physics Simulation](https://mlsanity.com/paper/2310.07206/) (2023)
- [Differentiable Physics Simulations with Contacts: Do They Have Correct Gradients w.r.t. Position, Velocity and Control?](https://mlsanity.com/paper/2207.05060/) (2022)
- [Jade: A Differentiable Physics Engine for Articulated Rigid Bodies with Intersection-Free Frictional Contact](https://mlsanity.com/paper/2309.04710/) (2023)
- [Differentiable Visual Computing for Inverse Problems and Machine Learning](https://mlsanity.com/paper/2312.04574/) (2023)
- [DiffVL: Scaling Up Soft Body Manipulation using Vision-Language Driven Differentiable Physics](https://mlsanity.com/paper/2312.06408/) (2023)
- [Improving Gradient Computation for Differentiable Physics Simulation with Contacts](https://mlsanity.com/paper/2305.00092/) (2023)
- [Human Motion Prediction under Unexpected Perturbation](https://mlsanity.com/paper/2403.15891/) (2024)
- [Learning Object Properties Using Robot Proprioception via Differentiable Robot-Object Interaction](https://mlsanity.com/paper/2410.03920/) (2024)
- [Differentiable Analog Quantum Computing for Optimization and Control](https://mlsanity.com/paper/2210.15812/) (2022)
- [From 2D to 3D Cognition: A Brief Survey of General World Models](https://mlsanity.com/paper/2506.20134/) (2025)
- [Embodied Co-Design for Rapidly Evolving Agents: Taxonomy, Frontiers, and Challenges](https://mlsanity.com/paper/2512.04770/) (2025)

## Abstract (arXiv)

We present a fast and feature-complete differentiable physics engine, Nimble (nimblephysics.org), that supports Lagrangian dynamics and hard contact constraints for articulated rigid body simulation. Our differentiable physics engine offers a complete set of features that are typically only available in non-differentiable physics simulators commonly used by robotics applications. We solve contact constraints precisely using linear complementarity problems (LCPs). We present efficient and novel analytical gradients through the LCP formulation of inelastic contact that exploit the sparsity of the LCP solution. We support complex contact geometry, and gradients approximating continuous-time elastic collision. We also introduce a novel method to compute complementarity-aware gradients that help downstream optimization tasks avoid stalling in saddle points. We show that an implementation of this combination in an existing physics engine (DART) is capable of a 87x single-core speedup over finite-differencing in computing analytical Jacobians for a single timestep, while preserving all the expressiveness of original DART.