# Dojo: A Differentiable Physics Engine for Robotics

arXiv: https://arxiv.org/abs/2203.00806 · HTML: https://mlsanity.com/paper/2203.00806/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Mar 2022.

## Built on by (most cited first, 26 of 26)

- [Learning Quadruped Locomotion Using Differentiable Simulation](https://mlsanity.com/paper/2403.14864/) (2024)
- [Stabilizing Reinforcement Learning in Differentiable Multiphysics Simulation](https://mlsanity.com/paper/2412.12089/) (2024)
- [A Survey of Robotic Navigation and Manipulation with Physics Simulators in the Era of Embodied AI](https://mlsanity.com/paper/2505.01458/) (2025)
- [DeepSimHO: Stable Pose Estimation for Hand-Object Interaction via Physics Simulation](https://mlsanity.com/paper/2310.07206/) (2023)
- [Planning-Guided Diffusion Policy Learning for Generalizable Contact-Rich Bimanual Manipulation](https://mlsanity.com/paper/2412.02676/) (2024)
- [ContactGaussian-WM: Learning Physics-Grounded World Model from Videos](https://mlsanity.com/paper/2602.11021/) (2026)
- [One-Shot Real-to-Sim via End-to-End Differentiable Simulation and Rendering](https://mlsanity.com/paper/2412.00259/) (2024)
- [Implicit Bias of Policy Gradient in Linear Quadratic Control: Extrapolation to Unseen Initial States](https://mlsanity.com/paper/2402.07875/) (2024)
- [Differentiable Discrete Event Simulation for Queuing Network Control](https://mlsanity.com/paper/2409.03740/) (2024)
- [From 2D to 3D Cognition: A Brief Survey of General World Models](https://mlsanity.com/paper/2506.20134/) (2025)
- [PhysMind: From Video to Executable Worlds for Training-Free Physical Reasoning](https://mlsanity.com/paper/2608.04575/) (2026)
- [Safe Reinforcement Learning using Finite-Horizon Gradient-based Estimation](https://mlsanity.com/paper/2412.11138/) (2024)
- [SAD-Flower: Flow Matching for Safe, Admissible, and Dynamically Consistent Planning](https://mlsanity.com/paper/2511.05355/) (2025)
- [Is Linear Feedback on Smoothed Dynamics Sufficient for Stabilizing Contact-Rich Plans?](https://mlsanity.com/paper/2411.06542/) (2024)
- [React to Surprises: Stable-by-Design Neural Feedback Control and the Youla-REN](https://mlsanity.com/paper/2506.01226/) (2025)
- [Emergence of Physical Intelligence via Controllable Information Production](https://mlsanity.com/paper/2601.22449/) (2026)
- [NeuralActuator: Neural Actuation Modeling for Robot Dynamics and External Force Perception](https://mlsanity.com/paper/2607.11734/) (2026)
- [ABPT: Amended Backpropagation through Time with Partially Differentiable Rewards](https://mlsanity.com/paper/2501.14513/) (2025)
- [Approximately Optimal Global Planning for Contact-Rich SE(2) Manipulation on a Graph of Reachable Sets](https://mlsanity.com/paper/2601.10827/) (2026)
- [VisFly-Lab: Unified Differentiable Framework for First-Order Reinforcement Learning of Quadrotor Control](https://mlsanity.com/paper/2603.21123/) (2026)
- [JODA: Composable Joint Dynamics for Articulated Objects](https://mlsanity.com/paper/2605.09954/) (2026)
- [Integrating Mechanistic and Data-Driven Models for Neurological Disorders through Differentiable Programming](https://mlsanity.com/paper/2606.06094/) (2026)
- [URDF Synthesis from RGB-D Sequences via Differentiable Joint Inference and Energy-Consistent Verification](https://mlsanity.com/paper/2606.18861/) (2026)
- [Ostrich: Taking Large Strides Through Stiff Contact in Differentiable Dynamics](https://mlsanity.com/paper/2609.08800/) (2026)
- [Asking the World: Generalist Physical Reasoning through Agentic World Modeling and Probing](https://mlsanity.com/paper/2609.39135/) (2026)
- [E2DRL: End-to-End Differentiable Reinforcement Learning for 3D Obstacle Avoidance in Quadruped Robots](https://mlsanity.com/paper/2610.19627/) (2026)

## Abstract (arXiv)

We present Dojo, a differentiable physics engine for robotics that prioritizes stable simulation, accurate contact physics, and differentiability with respect to states, actions, and system parameters. Dojo models hard contact and friction with a nonlinear complementarity problem with second-order cone constraints. We introduce a custom primal-dual interior-point method to solve the second order cone program for stable forward simulation over a broad range of sample rates. We obtain smooth gradient approximations with this solver through the implicit function theorem, giving gradients that are useful for downstream trajectory optimization, policy optimization, and system identification applications. Specifically, we propose to use the central path parameter threshold in the interior point solver as a user-tunable design parameter. A high value gives a smooth approximation to contact dynamics with smooth gradients for optimization and learning, while a low value gives precise simulation rollouts with hard contact. We demonstrate Dojo's differentiability in trajectory optimization, policy learning, and system identification examples. We also benchmark Dojo against MuJoCo, PyBullet, Drake, and Brax on a variety of robot models, and study the stability and simulation quality over a range of sample frequencies and accuracy tolerances. Finally, we evaluate the sim-to-real gap in hardware experiments with a Ufactory xArm 6 robot. Dojo is an open source project implemented in Julia with Python bindings, with code available at https://github.com/dojo-sim/Dojo.jl.