EquiMus: Energy-Equivalent Dynamic Modeling and Simulation of Musculoskeletal Robots Driven by Linear Elastic Actuators

TL;DR

EquiMus addresses the challenge of simulating and controlling rigid–soft musculoskeletal robots with distributed mass and loop closures by introducing an energy-equivalent lumped-mass discretization that maps elastic actuator dynamics onto discrete rigid bodies while preserving energy and work. Implemented in MuJoCo, the framework supports loop closures, real-time performance, and seamless integration with model-based control and reinforcement learning. Validation on a pneumatic robotic leg shows close sim-to-real agreement, with superior fidelity compared to native MuJoCo actuators, enabling PID auto-tuning, MPC, and RL-based control. The work provides a scalable, practical tool for designing and controlling hybrid soft-rigid systems, while outlining future extensions to capture actuator nonlinearities and richer constitutive behaviors.

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. To 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.

Figures (8)

• Figure 1: Overview and motivation of the proposed energy-equivalent modeling framework EquiMus.
• Figure 2: Schematic diagram of the soft actuator and its lumped mass distribution. The actuator is subjected to gravity, driving forces $F$, elastic forces, viscous resistance $c\dot{l}$ at endpoints, and constraint forces $F_\mathrm{c}$ from the rigid skeletons. $\delta\vec{r}$ denotes the virtual displacement. Since $F_\mathrm{c}$ is internal to the system, it is excluded from the Lagrangian dynamics. Driving forces and viscous forces are defined as positive as shown in figure.
• Figure 3: MJCF hierarchical structure of the EquiMus model, showing body–joint–geom relationships and key attributes. The “...” node denotes the remaining rigid skeleton structure, omitted here for clarity. Dashed arrows indicate <equality> constraints, including joint equality and body connection.
• Figure 4: Comparison between simulation and physical implementation of the robotic leg system. (A) MuJoCo-based energy-equivalent model of the robotic leg, illustrating the complete structure and the corresponding connectivity graph. Black nodes represent rigid links; black solid lines indicate tree joints; black dashed lines denote loop-closure joints (loop joints). Two blue lines represent pneumatic tubing. For clarity, the MAA module is omitted, and the focus is on the BAA structure. (B) Physical experimental setup, including the coordinate frame of the bionic robotic leg with pneumatic artificial muscles and the OptiTrack motion capture system. A simplified force analysis diagram is also included to aid understanding of the joint loading conditions.
• Figure 5: Verification of dynamic equivalence in simulation. The stance phase (A) and swing phase (B) are tested. Joint trajectories from simulation and theoretical models show strong agreement, demonstrating the validity of the proposed formulation.