By integrating these derivatives into the model, the researchers reported computational speed-ups of up to 1000 times across six different robotic platforms, enabling faster simulations for design, control, and optimization.
The advance addresses a major challenge in soft robotics: accurately simulating systems that combine rigid components with continuously deformable structures.
Bridging the Gap Between Rigid and Soft Robot Simulation
Modern robot simulation tools rely heavily on rigid-body algorithms with gradient information. These methods allow highly optimized simulations and are widely used in advanced robotic control systems.
However, extending these techniques to soft robots has proven difficult. Unlike rigid machines, soft robotic structures deform continuously, which makes their dynamics more complex to compute.
Many soft robots, such as continuum manipulators and flexible robotic arms, are often modeled as slender rods using Cosserat theory. The geometric variable strain (GVS) model builds on this idea by providing a unified mathematical framework capable of representing both rigid and soft components within the same system.
In this approach, soft bodies are described through strain fields, while rigid joints are represented as equivalent constant-strain Cosserat rods. This allows hybrid soft-rigid robots to be simulated using a consistent strain-based formulation.
Although previous studies had introduced differentiable Cosserat dynamics, an explicit analytical Jacobian matrix for the discretized GVS model had not yet been developed. The new work addresses this limitation.
Analytical Derivatives Improve Simulation Efficiency
To overcome this challenge, the researchers derived analytical derivatives for the GVS model and implemented them in the SoRoSim toolbox for MATLAB.
These derivatives allow solvers to compute the Jacobian matrix directly rather than approximating it numerically. In conventional simulations, numerical techniques such as finite differences are often used to estimate derivatives. While effective, these approximations can be computationally expensive and slower to evaluate.
By contrast, analytical derivatives provide exact expressions for these quantities, significantly improving computational efficiency.
The research team extended the Recursive Newton–Euler Algorithm (RNEA) to compute derivatives of inverse dynamics with respect to generalized coordinates, velocities, and accelerations. This recursive formulation enables efficient calculation while maintaining the accuracy of the dynamic model.
To validate the method, the team applied it to a cable-driven soft manipulator with 24 degrees of freedom. Simulations using analytical derivatives closely matched results obtained using traditional approaches, with tip position differences of less than one millimeter.
Despite the near-identical results, the computational performance improved substantially. Dynamic simulations ran more than four times faster, Jacobian calculations were eight times faster, and static simulations also showed roughly eightfold improvements.
Extending the Method to Hybrid Robotic Systems
The researchers then extended the framework to more complex hybrid robotic systems containing rigid joints, rigid links, soft links, branched structures, and closed-chain mechanisms.
To handle these configurations, they established a set of modeling rules that treat each robotic link as a combination of a rigid joint and a body within the strain-based representation. Recursive transformations allow the simulation to account for interactions between different branches of the robot’s kinematic structure.
The framework also supports key constraint types often found in robotic systems.
For rigid joints controlled through specified joint coordinates rather than forces, the dynamics are formulated as differential-algebraic equations. Closed-chain mechanisms introduce additional constraints that are modeled using Lagrange multipliers, with Baumgarte stabilization used to maintain numerical stability during simulations.
These capabilities were tested across several robotic platforms. A hybrid serial robot combining rigid joints and a soft link achieved threefold improvements in dynamic simulations and eightfold improvements in static analysis.
A more complex fin-ray robotic finger, which includes multiple soft links and closed-chain joints, demonstrated the method’s ability to handle highly constrained systems where conventional solvers can struggle.
The most dramatic case involved a hybrid parallel robot with three soft pillars. In that system, the analytical derivative approach reduced simulation time from more than an hour to just 2.46 seconds, representing an approximately 1800-fold speed increase.
Supporting Advanced Robotics Applications
The framework also incorporates external forces such as contact interactions, hydrodynamic loads, and applied wrenches. Simulations using Hertz contact models and underwater vehicle dynamics both showed significant computational improvements when analytical derivatives were applied.
By integrating the method into the SoRoSim MATLAB toolbox, the researchers have made differentiable simulation tools more accessible for soft robotics research.
Faster simulations could support advanced workflows such as trajectory optimization, model-predictive control, reinforcement learning, and automated design optimization.
Conclusion
The study demonstrates that introducing analytical derivatives into the GVS model can dramatically accelerate simulations of hybrid soft-rigid robots while maintaining high accuracy.
By extending derivative-based algorithms from rigid robotics to soft and hybrid systems, the researchers established a framework capable of efficiently simulating complex robotic structures.
With the implementation now available in the SoRoSim toolbox, the work provides a foundation for faster modeling and improved control strategies in the growing field of soft robotics.
Journal Reference
Mathew, A. T., Boyer, F., Lebastard, V., & Renda, F. (2025). Analytical derivatives of strain-based dynamic model for hybrid soft-rigid robots. The International Journal of Robotics Research, 45(1), 128–158. DOI:10.1177/02783649251346209. https://journals.sagepub.com/doi/full/10.1177/02783649251346209
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