Differentiable Material Point Method for the Control of Deformable Objects

Controlling the deformation of flexible objects is challenging due to their non-linear dynamics and high-dimensional configuration space. This work presents a differentiable Material Point Method (MPM) simulator targeted at control applications. We exploit the differentiability of the simulator to optimize a control trajectory in an active damping problem for a hyperelastic rope. The simulator effectively minimizes the kinetic energy of the rope around 2$\times$ faster than a baseline MPPI method and to a 20% lower energy level, while using about 3% of the computation time.

💡 Research Summary

The paper introduces a differentiable Material Point Method (MPM) simulator specifically designed for optimal control of deformable objects and demonstrates its effectiveness on an active‑damping task involving a hyperelastic rope. Traditional control of flexible bodies is hampered by nonlinear partial differential equations and an effectively infinite‑dimensional state space, which makes analytical solutions infeasible and discretization‑based methods either limited to simple geometries or computationally prohibitive. Recent works on differentiable simulation have shown promise, yet many either sacrifice physical realism (e.g., position‑based dynamics) or are restricted to one‑dimensional structures (e.g., discrete elastic rods). Moreover, existing differentiable MPM implementations prioritize visual fidelity over energy conservation, leading to excessive numerical dissipation at coarse resolutions.

To overcome these limitations, the authors reformulate the standard FLIP‑MPM algorithm into a first‑order ordinary differential equation for the particle state (S_t = {x_t, v_t, F_t}). By rearranging the two Euler integration steps into a single derivative computation, they obtain (\dot S_t = \tilde f(S_t, t)), which can be integrated with any explicit or implicit scheme. They choose a fourth‑order Runge‑Kutta (RK4) integrator, which dramatically reduces numerical dissipation and preserves mechanical energy even on low‑resolution grids. Control inputs (u_t) are injected either as body forces or as boundary conditions, and the cost function is defined as the time‑varying kinetic energy of the rope, i.e., (c_t(S_t, u_t)).

The differentiable implementation relies on the JAX library for automatic differentiation (AD) and GPU acceleration. The simulation horizon is divided into inner batches of length (N); each batch computes particle derivatives, integrates the dynamics, and accumulates the partial cost. After each batch, JAX provides the gradient of the accumulated cost with respect to the control variables, enabling gradient‑based optimization. The Adam optimizer is employed to adapt learning rates and ensure stable convergence. This nested loop structure balances memory consumption and the need to capture long‑term dynamic effects of the controls.

For benchmarking, the authors compare their gradient‑based approach with Model Predictive Path Integral (MPPI) control, a sampling‑based method that evaluates many noisy control trajectories and selects a weighted average of the best. MPPI does not exploit gradient information and therefore requires many samples to approximate the optimal policy.

Experimental results on the active‑damping scenario show that the differentiable MPM method converges roughly twice as fast as MPPI and achieves a final kinetic energy that is 20 % lower. In terms of computational effort, the proposed method uses only about 3 % of the total runtime required by MPPI, representing a 97 % reduction. These gains are achieved while using a coarse particle/grid resolution, thanks to the energy‑preserving integration scheme.

The paper concludes that a differentiable, energy‑conserving MPM simulator can serve as a powerful tool for dynamic control of high‑dimensional deformable systems. The authors highlight the potential for extending the framework to more complex geometries, heterogeneous material models, and multi‑object manipulation, as well as its applicability to real‑time robotic manipulation, virtual‑reality physics engines, and material parameter identification.