Algorithms that use derivatives of governing equations have accelerated rigid robot simulations and improved their accuracy, enabling the modeling of complex, real-world capabilities. However, extending these methods to soft and hybrid soft-rigid robots is significantly more challenging due to the complexities in modeling continuous deformations inherent in soft bodies. A considerable number of soft robots and the deformable links of hybrid robots can be effectively modeled as slender rods. The Geometric Variable Strain (GVS) model, which employs the screw theory and the strain parameterization of the Cosserat rod, extends the rod theory to model hybrid soft-rigid robots within the same mathematical framework. Using the Recursive Newton-Euler Algorithm, we developed the analytical derivatives of the governing equations of the GVS model. These derivatives facilitate the implicit integration of dynamics and provide the analytical Jacobian of the statics residue, ensuring fast and accurate computations. We applied these derivatives to the mechanical simulations of six common robotic systems: a soft cable-driven manipulator, a hybrid serial robot, a fin-ray finger, a hybrid parallel robot, a contact scenario, and an underwater hybrid mobile robot. Simulation results demonstrate substantial improvements in computational efficiency, with speed-ups of up to three orders of magnitude. We validate the model by comparing simulations done with and without analytical derivatives. Beyond static and dynamic simulations, the techniques discussed in this paper hold the potential to revolutionize the analysis, control, and optimization of hybrid robotic systems for real-world applications.

翻译：利用控制方程导数的算法已加速了刚性机器人仿真并提高了其精度，使得对复杂现实世界能力的建模成为可能。然而，将这些方法扩展到软体和混合软硬机器人上则显著更具挑战性，这源于软体固有的连续变形建模的复杂性。大量软体机器人及混合机器人的可变形连杆可有效地建模为细长杆。几何可变应变（GVS）模型采用旋量理论和Cosserat杆的应变参数化，扩展了杆理论，可在同一数学框架内对混合软硬机器人进行建模。利用递归牛顿-欧拉算法，我们推导出了GVS模型控制方程的解析导数。这些导数促进了动力学的隐式积分，并提供了静力学残差的解析雅可比矩阵，从而确保了快速且准确的计算。我们将这些导数应用于六种常见机器人系统的力学仿真：软体缆驱机械臂、混合串联机器人、鳍条手指、混合并联机器人、接触场景以及水下混合移动机器人。仿真结果表明计算效率得到显著提升，加速比高达三个数量级。我们通过比较使用与不使用解析导数的仿真结果来验证该模型。除了静态和动态仿真，本文讨论的技术还有望彻底改变混合机器人系统在实际应用中的分析、控制与优化。