# Amortising Trajectory Optimisation for Residual MPC via Implicit Contact Differentiation

arXiv: https://arxiv.org/abs/2607.24959 · HTML: https://mlsanity.com/paper/2607.24959/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 42 references that are in the map.
- First posted on arXiv 27 Jul 2026.

## Builds on (most cited first, 30 of 31)

- [End-to-End Training of Deep Visuomotor Policies](https://mlsanity.com/paper/1504.00702/) (2015)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [TD-MPC2: Scalable, Robust World Models for Continuous Control](https://mlsanity.com/paper/2310.16828/) (2023)
- [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)
- [Residual Reinforcement Learning for Robot Control](https://mlsanity.com/paper/1812.03201/) (2018)
- [DiffTaichi: Differentiable Programming for Physical Simulation](https://mlsanity.com/paper/1910.00935/) (2019)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Differentiable MPC for End-to-end Planning and Control](https://mlsanity.com/paper/1810.13400/) (2018)
- [Plan Online, Learn Offline: Efficient Learning and Exploration via Model-Based Control](https://mlsanity.com/paper/1811.01848/) (2018)
- [Accelerated Policy Learning with Parallel Differentiable Simulation](https://mlsanity.com/paper/2204.07137/) (2022)
- [Exploring Model-based Planning with Policy Networks](https://mlsanity.com/paper/1906.08649/) (2019)
- [Scalable Differentiable Physics for Learning and Control](https://mlsanity.com/paper/2007.02168/) (2020)
- [Fast and Feature-Complete Differentiable Physics for Articulated Rigid Bodies with Contact](https://mlsanity.com/paper/2103.16021/) (2021)
- [Crocoddyl: An Efficient and Versatile Framework for Multi-Contact Optimal Control](https://mlsanity.com/paper/1909.04947/) (2019)
- [MPC-Net: A First Principles Guided Policy Search](https://mlsanity.com/paper/1909.05197/) (2019)
- [Large Scale Model Predictive Control with Neural Networks and Primal Active Sets](https://mlsanity.com/paper/1910.10835/) (2019)
- [PLATO: Policy Learning using Adaptive Trajectory Optimization](https://mlsanity.com/paper/1603.00622/) (2016)
- [Differentiable Implicit Soft-Body Physics](https://mlsanity.com/paper/2102.05791/) (2021)
- [Dexterous Contact-Rich Manipulation via the Contact Trust Region](https://mlsanity.com/paper/2505.02291/) (2025)
- [Primal-Dual iLQR for GPU-Accelerated Learning and Control in Legged Robots](https://mlsanity.com/paper/2506.07823/) (2025)
- [Differentiable Physics Simulations with Contacts: Do They Have Correct Gradients w.r.t. Position, Velocity and Control?](https://mlsanity.com/paper/2207.05060/) (2022)
- [Contact Models in Robotics: a Comparative Analysis](https://mlsanity.com/paper/2304.06372/) (2023)
- [Contact-Implicit Model Predictive Control for Dexterous In-hand Manipulation: A Long-Horizon and Robust Approach](https://mlsanity.com/paper/2402.18897/) (2024)
- [Highly-Efficient Differentiable Simulation for Robotics](https://mlsanity.com/paper/2409.07107/) (2024)
- [Improving Gradient Computation for Differentiable Physics Simulation with Contacts](https://mlsanity.com/paper/2305.00092/) (2023)
- [Differentiable Simulation of Hard Contacts with Soft Gradients for Learning and Control](https://mlsanity.com/paper/2506.14186/) (2025)
- [Whole-Body Model-Predictive Control of Legged Robots with MuJoCo](https://mlsanity.com/paper/2503.04613/) (2025)
- [GATO: GPU-Accelerated and Batched Trajectory Optimization for Scalable Edge Model Predictive Control](https://mlsanity.com/paper/2510.07625/) (2025)
- [Contact-Implicit Trajectory Optimization with Hydroelastic Contact and iLQR](https://mlsanity.com/paper/2202.13986/) (2022)

## Abstract (arXiv)

Differentiable simulation can accelerate contact-rich trajectory optimisation by exposing local sensitivities of task outcomes to controls. Existing approaches either use finite differences, which are expensive and step-size sensitive; differentiate iterative contact solvers by unrolling automatic differentiation (AD), which stores a growing computation trace; or require intricate, solver-specific KKT sensitivity derivations. We introduce an AD-assisted implicit derivative for regularised smooth contacts and apply it to Mujoco MJX, based on the Implicit Function Theorem (IFT). The method differentiates the stationarity residual at the tolerance-converged solution, avoiding both solver unrolling and hand-assembled KKT systems. IFT keeps compiled temporary memory nearly constant with solver effort, changing by less than 4% from one to ten iterations versus 10.6× growth for unrolled AD. IFT memory grows slower with active contacts and model dimension, using 20× less memory at 256 contacts and 6× less at 16 contacts and 96 DoF. We further introduce optimiser distillation for residual MPC, amortising batched full-horizon iLQR into a policy that guides short-horizon residual iLQR. Across Finger, Franka, and Unitree, this raises six-step success by 28-98 percentage points over standard iLQR.