Differentiable Simulation of Hard Contacts with Soft Gradients
for Learning and Control
Anselm Paulus*, A. René Geist*, Pierre Schumacher, Vít Musil, Simon Rappenecker, Georg Martius
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Differentiable Simulation of Hard Contacts with Soft Gradients
ICLR 2026
Abstract
Abstract
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.
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.
Top: Gradient-based MPC using CFD
Combining CFD with MJX enables gradient-based MPC of a musculo-skeletal robot using only a single distance between ball and goal as cost.
What is DiffMJX?
What is DiffMJX?
For hard contact settings or small simulation stepsizes, the gradients provided by MuJoCo XLA are erroneous. These errors stem from discontinuities in the system dynamics introduced by contacts, which affect the accuracy of numerical integration.
DiffMJX integrates Diffrax - an library providing numerical differential equation solvers - atop MuJoCo XLA.
In turn, DiffMJX adds the following functionalities to MuJoCo XLA:
Tradeoff simulation / gradient accuracy for compute time via adaptive stepsize controllers during integration.
Tradeoff GPU memory consumption for compile / compute time by using advanced checkpointing methods.
Switch between different methods for computing adjoints for backpropagation.
In addition, we show how to smoothen collision detection via Softjax.
Left: Real-world cube parameter estimation
Estimation of cube parameters in MuJoCo via DiffMJX from real-world data of a cube being thrown onto a table. After training, the MuJoCo simulation accurately predicts the real-world trajectories. Despite contacts, the parameters are learned via run-of-the-mill gradient descent.
What is CFD?
What is CFD?
For using robot simulation in an optimization, we would like to have informative gradients. In particular, if two objects are not in contact, then the simulator gradients between the object states are zero. To obtain gradients between Contacts From a Distance (CFD), we extend MuJoCo's contact model to create small contact forces for positive signed-distances. By resorting to straight-through estimation, this softened contact model is only used when computing simulator gradients keeping the forward simulation untouched.
Top: Robot hovers due to applying large artifical contact forces in the simulation.
Left: Illustration of CFD
During backpropagation the gradients of the forward simulation are replaced by the gradients of the soft forward simulation.
Comparisons of sampling-based MPC to gradient-based MPC with CFD
Ball catch
Switch ball positions in-hand
Bionic tennis
cube_reorientation_plane_CFD.mp4Cube reorientation on plane
cube_reorientation_CFD.mp4Cube reorientation in-hand
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