A Robust Strain-Based Cosserat Rod Finite Element Formulation for Modeling Soft Robots

Soft robots are increasingly expected to operate in unstructured environments where compliance, environmental coupling, and embedded sensing fundamentally shape their behavior. However, the Cosserat rod models commonly used in soft robotics originate from two largely independent traditions: robotics-driven formulations optimized for control or intuitive simplicity, and mechanics-driven formulations optimized for structural generality.Considered separately, neither tradition fulfills the requirements for sensing, fabrication, environmental interaction, and numerical reliability of modern soft robotic systems. Drawing on our modeling and design experience with soft robotic systems, we identified a set of design challenges that guided the development of a suitable Cosserat rod formulation. To address these challenges, we propose a Cosserat rod model expressed directly in strain coordinates, matching the quantities measured by embedded sensors and eliminating the need for global pose tracking. A finite-element discretization supports localized tactile forces and environmental interactions, while a globally singularity-free quaternion representation enables full floating-base capability. We demonstrate that a Petrov–Galerkin projection, in contrast to the existing Bubnov–Galerkin formulation in soft robotics, decouples the choice of velocity coordinates and virtual displacements from the configuration parametrization. This enables the dynamics to be expressed in terms of sensor-aligned velocity variables, such as nodal absolute velocities from reliable existing sensors, rather than parametrization-dependent strain rates. This alignment simplifies estimator and controller design and also yields a constant mass matrix as a computational benefit. In addition, the formulation provides a unified mechanism for embedding hardware-intended strain constraints directly through intrinsic Lagrange-multiplier enforcement, ensuring that the model evolves strictly within the robot’s physically realizable strain manifold. The resulting finite-element formulation is total Lagrangian, objective, locking-free, globally singularity-free, and path-independent. Robust convergence and a significant reduction in nonlinear solver iterations (98.4% and 99.8% relative to the second most efficient comparison formulation) are demonstrated on well-known benchmark examples. By integrating and extending strengths from both robotics-oriented and mechanics-oriented approaches, this framework offers a practical, robust, sensor-aligned, and computationally efficient Cosserat rod model suited for soft robots operating beyond controlled laboratory environments.