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This paper introduces an AD-assisted implicit derivative method for differentiable simulation in contact-rich trajectory optimization, leveraging the Implicit Function Theorem (IFT) to enhance efficiency and memory usage. By avoiding the pitfalls of finite differences and traditional unrolled automatic differentiation, the proposed method maintains nearly constant memory usage while significantly improving performance in multi-contact scenarios. The results demonstrate a substantial increase in success rates for residual Model Predictive Control (MPC) across various robotic tasks, achieving up to 98 percentage points improvement over standard iterative Linear Quadratic Regulator (iLQR) methods.
Achieving up to 98% higher success rates in contact-rich robotic tasks by optimizing memory and computation through implicit differentiation.
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$\times$ growth for unrolled AD. IFT memory grows slower with active contacts and model dimension, using 20$\times$ less memory at 256 contacts and 6$\times$ 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.