# Differentiable Physics Simulations with Contacts: Do They Have Correct Gradients w.r.t. Position, Velocity and Control?

arXiv: https://arxiv.org/abs/2207.05060 · HTML: https://mlsanity.com/paper/2207.05060/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 52 references that are in the map.
- First posted on arXiv 8 Jul 2022.

## Builds on (most cited first, 27 of 27)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Brax -- A Differentiable Physics Engine for Large Scale Rigid Body Simulation](https://mlsanity.com/paper/2106.13281/) (2021)
- [DiffTaichi: Differentiable Programming for Physical Simulation](https://mlsanity.com/paper/1910.00935/) (2019)
- [A Differentiable Physics Engine for Deep Learning in Robotics](https://mlsanity.com/paper/1611.01652/) (2016)
- [Accelerated Policy Learning with Parallel Differentiable Simulation](https://mlsanity.com/paper/2204.07137/) (2022)
- [Do Differentiable Simulators Give Better Policy Gradients?](https://mlsanity.com/paper/2202.00817/) (2022)
- [NeuralSim: Augmenting Differentiable Simulators with Neural Networks](https://mlsanity.com/paper/2011.04217/) (2020)
- [Scalable Differentiable Physics for Learning and Control](https://mlsanity.com/paper/2007.02168/) (2020)
- [DiffCloth: Differentiable Cloth Simulation with Dry Frictional Contact](https://mlsanity.com/paper/2106.05306/) (2021)
- [DiSECt: A Differentiable Simulation Engine for Autonomous Robotic Cutting](https://mlsanity.com/paper/2105.12244/) (2021)
- [DiffPD: Differentiable Projective Dynamics](https://mlsanity.com/paper/2101.05917/) (2021)
- [An End-to-End Differentiable Framework for Contact-Aware Robot Design](https://mlsanity.com/paper/2107.07501/) (2021)
- [Fast and Feature-Complete Differentiable Physics for Articulated Rigid Bodies with Contact](https://mlsanity.com/paper/2103.16021/) (2021)
- [Efficient Differentiable Simulation of Articulated Bodies](https://mlsanity.com/paper/2109.07719/) (2021)
- [Differentiable Simulation of Soft Multi-body Systems](https://mlsanity.com/paper/2205.01758/) (2022)
- [Extending Lagrangian and Hamiltonian Neural Networks with Differentiable Contact Models](https://mlsanity.com/paper/2102.06794/) (2021)
- [Encoding Physical Constraints in Differentiable Newton-Euler Algorithm](https://mlsanity.com/paper/2001.08861/) (2020)
- [Learning Physical Constraints with Neural Projections](https://mlsanity.com/paper/2006.12745/) (2020)
- [ADD: Analytically Differentiable Dynamics for Multi-Body Systems with Frictional Contact](https://mlsanity.com/paper/2007.00987/) (2020)
- [DiffSDFSim: Differentiable Rigid-Body Dynamics With Implicit Shapes](https://mlsanity.com/paper/2111.15318/) (2021)
- [Learning to Slide Unknown Objects with Differentiable Physics Simulations](https://mlsanity.com/paper/2005.05456/) (2020)
- [Automatic Differentiation of Rigid Body Dynamics for Optimal Control and Estimation](https://mlsanity.com/paper/1709.03799/) (2017)
- [Augmenting Differentiable Simulators with Neural Networks to Close the Sim2Real Gap](https://mlsanity.com/paper/2007.06045/) (2020)
- [Differentiable Implicit Soft-Body Physics](https://mlsanity.com/paper/2102.05791/) (2021)
- [Solving Optimal Control Problems of Rigid-Body Dynamics with Collisions Using the Hybrid Minimum Principle](https://mlsanity.com/paper/2205.08622/) (2022)

## Built on by (most cited first, 7 of 7)

- [DiffMimic: Efficient Motion Mimicking with Differentiable Physics](https://mlsanity.com/paper/2304.03274/) (2023)
- [Improving Gradient Computation for Differentiable Physics Simulation with Contacts](https://mlsanity.com/paper/2305.00092/) (2023)
- [STAMP: Differentiable Task and Motion Planning via Stein Variational Gradient Descent](https://mlsanity.com/paper/2310.01775/) (2023)
- [VisualPatchWorld: Code World Models as Latent Structured Representations for Planning](https://mlsanity.com/paper/2607.25236/) (2026)
- [Dimensional Peeking for Low-Variance Gradients in Zeroth-Order Discrete Optimization via Simulation](https://mlsanity.com/paper/2602.00075/) (2026)
- [Where-to-Learn: Analytical Policy Gradient Directed Exploration for On-Policy Robotic Reinforcement Learning](https://mlsanity.com/paper/2603.27317/) (2026)
- [Amortising Trajectory Optimisation for Residual MPC via Implicit Contact Differentiation](https://mlsanity.com/paper/2607.24959/) (2026)

## Abstract (arXiv)

In recent years, an increasing amount of work has focused on differentiable physics simulation and has produced a set of open source projects such as Tiny Differentiable Simulator, Nimble Physics, diffTaichi, Brax, Warp, Dojo and DiffCoSim. By making physics simulations end-to-end differentiable, we can perform gradient-based optimization and learning tasks. A majority of differentiable simulators consider collisions and contacts between objects, but they use different contact models for differentiability. In this paper, we overview four kinds of differentiable contact formulations - linear complementarity problems (LCP), convex optimization models, compliant models and position-based dynamics (PBD). We analyze and compare the gradients calculated by these models and show that the gradients are not always correct. We also demonstrate their ability to learn an optimal control strategy by comparing the learned strategies with the optimal strategy in an analytical form. The codebase to reproduce the experiment results is available at https://github.com/DesmondZhong/diff_sim_grads.