# Differentiable Simulation of Hard Contacts with Soft Gradients for Learning and Control

arXiv: https://arxiv.org/abs/2506.14186 · HTML: https://mlsanity.com/paper/2506.14186/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 78 references that are in the map.
- First posted on arXiv 17 Jun 2025.

## Builds on (most cited first, 30 of 38)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Learning Dexterous In-Hand Manipulation](https://mlsanity.com/paper/1808.00177/) (2018)
- [Learning Quadrupedal Locomotion over Challenging Terrain](https://mlsanity.com/paper/2010.11251/) (2020)
- [Brax -- A Differentiable Physics Engine for Large Scale Rigid Body Simulation](https://mlsanity.com/paper/2106.13281/) (2021)
- [Sim-to-Real: Learning Agile Locomotion For Quadruped Robots](https://mlsanity.com/paper/1804.10332/) (2018)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [DiffTaichi: Differentiable Programming for Physical Simulation](https://mlsanity.com/paper/1910.00935/) (2019)
- [Real-World Humanoid Locomotion with Reinforcement Learning](https://mlsanity.com/paper/2303.03381/) (2023)
- [Equinox: neural networks in JAX via callable PyTrees and filtered transformations](https://mlsanity.com/paper/2111.00254/) (2021)
- [Accelerated Policy Learning with Parallel Differentiable Simulation](https://mlsanity.com/paper/2204.07137/) (2022)
- [ChainQueen: A Real-Time Differentiable Physical Simulator for Soft Robotics](https://mlsanity.com/paper/1810.01054/) (2018)
- [Sample-efficient Cross-Entropy Method for Real-time Planning](https://mlsanity.com/paper/2008.06389/) (2020)
- [GPU-Accelerated Robotic Simulation for Distributed Reinforcement Learning](https://mlsanity.com/paper/1810.05762/) (2018)
- [NeuralSim: Augmenting Differentiable Simulators with Neural Networks](https://mlsanity.com/paper/2011.04217/) (2020)
- [MyoSuite -- A contact-rich simulation suite for musculoskeletal motor control](https://mlsanity.com/paper/2205.13600/) (2022)
- [Deep Regression on Manifolds: A 3D Rotation Case Study](https://mlsanity.com/paper/2103.16317/) (2021)
- [Global Planning for Contact-Rich Manipulation via Local Smoothing of Quasi-dynamic Contact Models](https://mlsanity.com/paper/2206.10787/) (2022)
- [Predictive Sampling: Real-time Behaviour Synthesis with MuJoCo](https://mlsanity.com/paper/2212.00541/) (2022)
- [Learning Agile Skills via Adversarial Imitation of Rough Partial Demonstrations](https://mlsanity.com/paper/2206.11693/) (2022)
- [Learning with 3D rotations, a hitchhiker's guide to SO(3)](https://mlsanity.com/paper/2404.11735/) (2024)
- [ContactNets: Learning Discontinuous Contact Dynamics with Smooth, Implicit Representations](https://mlsanity.com/paper/2009.11193/) (2020)
- [Simulation and Retargeting of Complex Multi-Character Interactions](https://mlsanity.com/paper/2305.20041/) (2023)
- [Learning Quadruped Locomotion Using Differentiable Simulation](https://mlsanity.com/paper/2403.14864/) (2024)
- [Backpropagation through Combinatorial Algorithms: Identity with Projection Works](https://mlsanity.com/paper/2205.15213/) (2022)
- [DEP-RL: Embodied Exploration for Reinforcement Learning in Overactuated and Musculoskeletal Systems](https://mlsanity.com/paper/2206.00484/) (2022)
- [A Review of Differentiable Simulators](https://mlsanity.com/paper/2407.05560/) (2024)
- [Bundled Gradients through Contact via Randomized Smoothing](https://mlsanity.com/paper/2109.05143/) (2021)
- [Adaptive Horizon Actor-Critic for Policy Learning in Contact-Rich Differentiable Simulation](https://mlsanity.com/paper/2405.17784/) (2024)
- [Sparsely Changing Latent States for Prediction and Planning in Partially Observable Domains](https://mlsanity.com/paper/2110.15949/) (2021)

## Built on by (most cited first, 5 of 5)

- [SoftJAX & SoftTorch: Empowering Automatic Differentiation Libraries with Informative Gradients](https://mlsanity.com/paper/2603.08824/) (2026)
- [NDPP-Grasp: Non-Differentiable Physical Plausibility Constraint-Guided Task-Oriented Dexterous Grasp Generation](https://mlsanity.com/paper/2606.02432/) (2026)
- [Amortising Trajectory Optimisation for Residual MPC via Implicit Contact Differentiation](https://mlsanity.com/paper/2607.24959/) (2026)
- [Ostrich: Taking Large Strides Through Stiff Contact in Differentiable Dynamics](https://mlsanity.com/paper/2609.08800/) (2026)
- [WorldAgent: Verification-Guided Agentic Physical World Construction](https://mlsanity.com/paper/2609.33208/) (2026)

## Abstract (arXiv)

Contact forces introduce discontinuities into robot dynamics that severely limit the use of simulators for gradient-based optimization. Penalty-based simulators such as MuJoCo, soften contact resolution to enable gradient computation. However, realistically simulating hard contacts requires stiff solver settings, which leads to incorrect simulator gradients when using automatic differentiation. Contrarily, using non-stiff settings strongly increases the sim-to-real gap. We analyze penalty-based simulators to pinpoint why gradients degrade under hard contacts. Building on these insights, we propose DiffMJX, which couples adaptive time integration with penalty-based simulation to substantially improve gradient accuracy. A second challenge is that contact gradients vanish when bodies separate. To address this, we introduce contacts from distance (CFD) which combines penalty-based simulation with straight-through estimation. By applying CFD exclusively in the backward pass, we obtain informative pre-contact gradients while retaining physical realism.