An End-to-End Framework for Modelling Pneumatic Soft Robots Based on Differentiable Finite Element Methods

This article presents a novel SOFA based finite element method for the soft body modeling and the corresponding dynamic simulation and control of a pneumatic morphing soft quadrotor. The proposed modeling preserves the physical interpretability and control structure of traditional quadrotor dynamics, while capturing the complex, time-varying behavior of pneumatically actuated soft arms. In SOFA, the soft pneumatically actuated arms are discretized as a tetrahedral mesh following an elastic material law that produces internal forces adequate to the real dynamic behavior of the body. Pneumatic actuation governed by both periodic and error-based control signals is applied within the internal cavities to analyze the morphing capability. Finally, a proportional-integral controller is proposed to study the controlled dynamic behavior and morphing capabilities of the pneumatic arm, wherein the pneumatic actuation to the soft arm is controlled to achieve the desired target position. The simulation results show the effectiveness of the proposed novel modeling framework and the related controller design.

Soft robots and tactile sensors have demonstrated great potential in delicate manipulation tasks. Soft pneumatic robots enable safe contact through compliance, and vision-based tactile sensors offer high-resolution touch perception. However, learning tactile manipulation with compliant robots has been challenging, bottlenecked by the lack of efficient simulation. Existing simulators typically model them in isolation, and exhibit large calibration gaps that are difficult to overcome efficiently. We present PneuTac, a unified framework for tactile-feedback manipulation with soft pneumatic robots. We leverage the material point method (MPM) for modelling the dynamics of the soft robot and the deformable tactile membrane, and 3D Gaussian splatting (3DGS) for rendering. Real-to-sim modelling is done with a simple vision-based method, to then train action and perception networks for efficient simulation with surrogate models. We use the framework to drive a tactile-guided pipeline to collect demonstrations in simulation. Through experiments on a custom-designed pneumatic soft finger with a tactile sensing tip, together with additional cross-device evaluations, we show that PneuTac is capable of accurately modelling soft robots with tactile sensors, and that policies trained with simulation-augmented demonstrations outperform baselines trained on the same real data on three real-world contact-rich compliant manipulation tasks, making it a practical framework for tactile manipulation on compliant hardware.

Reduced-order models based on Cosserat-rod theory are now well established, and modeling theory is no longer the primary bottleneck in soft-robot control. Their implementations, however, do not support the differentiable, GPU-parallel, and control-oriented workflows that underpin advanced rigid-robotics applications. Here, we fill this gap with SoRoMoX (Soft Robot Models in JAX), a fully numerical, JIT-compilable Python/JAX framework. SoRoMoX implements articulated, Piecewise Constant Strain, and Variable Strain models through a unified, control-ready interface that provides inertia matrices, gravitational and elastic forces, Jacobians, and their derivatives. To our knowledge, it is the first rod/strain-based soft-robot modeling framework that runs directly on GPUs, with Warp kernels accelerating parallel continuum-model execution, and is end-to-end differentiable with respect to states, inputs, and parameters. Sequential CPU rollouts are up to 27.0 times faster than SoRoSim, while GPU-parallel GVS rollouts increase throughput by up to 679.7 times. This performance enables workflows that were previously impractical or impossible: static-equilibrium system identification with 66% lower marker RMSE; residual-force learning with a further 64% reduction; computed-torque tracking with RMSE reduced by a factor of approximately 500 relative to model-free PD; control-gain optimization with up to 98% lower loss than untuned gains; safety-constrained control using high-order control barrier functions to keep the peak contact force within a prescribed 5 N bound, compared with 33.5 N without the safety constraint; and reinforcement-learning policy training up to 7 times faster than a CPU PyElastica discrete-rod baseline through massively parallel rollouts.