Abstract--Dynamic modeling and control are critical for unleashing soft robots' potential, yet remain challenging due to their complex constitutive behaviors and real-world operating conditions. Bio-inspired musculoskeletal robots, which integrate rigid skeletons with soft actuators, combine high load-bearing capacity with inherent flexibility. Although actuation dynamics have been studied through experimental methods and surrogate models, accurate and effective modeling and simulation remain a significant challenge, especially for large-scale hybrid rigid-soft robots with continuously distributed mass, kinematic loops, and diverse motion modes. T o address these challenges, we propose EquiMus, an energy-equivalent dynamic modeling framework and MuJoCo-based simulation for musculoskeletal rigid-soft hybrid robots with linear elastic actuators. The equivalence and effectiveness of the proposed approach are validated and examined through both simulations and real-world experiments on a bionic robotic leg. EquiMus further demonstrates its utility for downstream tasks, including controller design and learning-based control strategies.

Traditional robotic systems excel in high-precision and large-load operations, but achieving tasks that require robustness, dexterity, and flexibility necessitates high-precision sensors, high-precision structures, and advanced control algorithms. In situations where the absolute precision of sensing and control in each unit is not high, the human arm can effectively utilize its inherent structural characteristics, such as the serial and parallel hybrid kinematic structure and the rigid-flexible coupling dynamic characteristics, to achieve rapid, robust, safe, dexterous, and flexible operations through information processing in neural circuits [1-3]. Through the synergy of software and hardware, developing a neuromorphic intelligent robot system that embodies human-like structural characteristics and driving mechanisms holds significant inspirational and catalytic value for advancing novel high-performance robotic systems. However, simulating the musculoskeletal structure with physical devices poses significant challenges. Michael et al. [4] created the'Anthrob' robot, which is a reduced version of the human upper limb with 13 compliant muscles and four joints, However, the complexity of muscle units makes the extension of multi-muscle actuation challenging.

Muscles of the human body are composed of tiny actuators made up of myosin and actin filaments. They can exert force in various shapes such as curved or flat, under contact forces and deformations from the environment. On the other hand, muscles in musculoskeletal robots so far have faced challenges in generating force in such shapes and environments. To address this issue, we propose Patterned Structure Muscle (PSM), artificial muscles for musculoskeletal robots. PSM utilizes patterned structures with anisotropic characteristics, wire-driven mechanisms, and is made of flexible material Thermoplastic Polyurethane (TPU) using FDM 3D printing. This method enables the creation of various shapes of muscles, such as simple 1 degree-of-freedom (DOF) muscles, Multi-DOF wide area muscles, joint-covering muscles, and branched muscles. We created an upper arm structure using these muscles to demonstrate wide range of motion, lifting heavy objects, and movements through environmental contact. These experiments show that the proposed PSM is capable of operating in various shapes and environments, and is suitable for the muscles of musculoskeletal robots.