Soft robots present significant modelling challenges due to their non-linearity, complex dynamics and potentially intricate geometries. These difficulties in accurate system identification and dynamics modelling limit their applications in precise robotics tasks. Prior modelling approaches typically suffer from trade-offs in accuracy, computational efficiency, or speed. The differentiable finite element method (FEM) offers a promising balance between these desiderata for soft robot modelling, and enables the use of gradients for efficient calibration and trajectory optimisation. In this paper, we propose an end-to-end differentiable FEM-based framework designed to streamline modelling and system identification for pneumatic soft robots, enabling the generation of reliable models for downstream tasks. The framework automates the conversion of computer-aided designs into voxelised tetrahedral viscoelastic FEM meshes that accurately represent the robot's geometry and actuation mechanism. By integrating easily acquired point cloud data with differentiable FEM, the framework achieves precise identification of material and dynamic actuation parameters using a minimal experimental setup. Additionally, the differentiable nature of the model facilitates trajectory optimisation by leveraging gradients from robot dynamics and contact interactions. We validate the framework by modelling a complex bellow-shaped pneumatic soft robot and demonstrate its efficacy in real-world motion planning tasks. Experimental results indicate high modelling accuracy, with a maximum positional error of less than 3 mm, and successful application in tasks such as path following and grasping.

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