On the Numerical Reliability of Differentiable Physics-Based Optimization for Robotic Material Manipulation

Three properties determine whether a differentiable simulator can drive gradient-based optimization through contact: simulation accuracy, gradient reliability, and per-iteration cost. Tape-based engines such as MJX and Newton Semi-Implicit require timesteps small enough to keep contacts numerically tractable, and their backpropagation memory grows linearly with the number of timesteps T. Surrogate models bound memory by approximating contact away, but the resulting gradients lose the geometry the optimization depends on. We present Ostrich, a GPU-accelerated rigid-body simulator that resolves hard contacts and friction with non-smooth Newton iteration at large timesteps (h ~ 0.1 s), and differentiates the converged residual via the implicit function theorem, reusing the forward Schur complement to compute the adjoint at O(1) memory per timestep. On real-robot trajectories over a pallet obstacle, Ostrich holds MuJoCo's sim-to-real accuracy up to a 50x larger timestep. Its gradients converge from random initializations where MJX descends slowly and Newton Semi-Implicit stalls; a warm iteration runs 211x faster than MJX's and 4.7x faster than Semi-Implicit's. On the same scene Ostrich differentiates 8,192 parallel worlds on a single 24 GB GPU, sustaining 29x checkpointed MJX's optimization throughput; without checkpointing both baselines exhaust memory at far fewer worlds. We close with a gradient-based trajectory optimization demonstration over triangle-mesh terrain across a 10 s horizon, a setting where prior engines either restrict to primitive geometry or face the convergence and memory limits shown above.

Differentiable simulation of soft bodies is a foundation for system identification, trajectory optimization, and Real2Sim transfer. Yet, existing methods such as the differentiable Projective Dynamics (DiffPD) struggle when faced with heterogeneous materials with extreme stiffness contrasts, hyperelasticity under large deformations, and contact-rich interactions, which are common scenarios in the real world. We present DiffPhD, a unified GPU-accelerated differentiable Projective Dynamics framework for heterogeneous materials that tackles these intertwined challenges simultaneously. Our key insight is a careful integration of: (i) stiffness-aware projective weights to embed heterogeneity into the global system; (ii) trust-region eigenvalue filtering lifted to the backward pass for stable hyperelastic gradients and a type-II Anderson Acceleration scheme with dual-gate convergence to stabilize forward iteration under large stiffness contrasts; and (iii) a unified GPU pipeline that reuses a single sparse factor across forward, backward, and contact computations, with stiffness-amplified Rayleigh damping folded into the same factor for heterogeneity-aware dissipation at zero recurring cost. DiffPhD achieves strict gradient accuracy while delivering up to an order-of-magnitude speedup over prior differentiable solvers on heterogeneous, hyperelastic, contact-rich benchmarks. Crucially, this speedup does not come at the cost of stability: DiffPhD remains convergent on stiffness contrasts up to 100x where prior PD solvers degrade. This unlocks end-to-end gradient-based optimization on regimes previously bottlenecked by either solver fragility or per-iteration cost -- shell--joint composite creatures, soft characters wielding stiff weapons, and soft-gripper robotic manipulation -- all handled within a single forward--backward pass.

Robotics simulators have improved significantly in computational speed and scalability, enabling them to generate years of simulated data for complex systems in minutes or hours. Despite these advances, efficiently and accurately computing simulation derivatives remains an open challenge. Addressing this would accelerate the convergence of reinforcement learning and trajectory optimization algorithms, particularly for contact-rich problems. This paper introduces a unifying framework for robotic simulation that accounts for all factors, including dynamics, collisions, and friction. The resulting algorithm computes analytical derivatives of the simulation by implicit differentiation, explicitly handling the intrinsic non-smoothness of the collision and frictional stages while exploiting the sparsity induced by the multi-body structure. Benchmark results demonstrate state-of-the-art performance, with timings ranging from $5\,μ$s for a 7-dof manipulator to $95\,μ$s for a 36-dof humanoid, an improvement of at least two orders of magnitude over alternative methods. Implemented in C++, the code will be open-sourced after the review process to support applications such as simulation-driven learning and real-time control.