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.
TL;DR
We proposed EquiMus, an energy-equivalent dynamics and simulation for the rigid-soft musculoskeletal robots with linear elastic actuators. The method captures dynamic mass redistribution, supports loop-closure constraints in MuJoCo, and remains real-time capable. Experiments on a pneumatic leg show close sim-to-real agreement and enable downstream usage in PID auto-tuning, model-based control, and reinforcement learning.
The robot dynamics are formulated using the vector form of the Lagrangian equation,
$$ (\frac{d}{dt} \frac{\partial }{\partial \dot{\mathbf{q}}} - \frac{\partial }{\partial \mathbf{q}}) (L_{\text{EA}}+L_{\text{other}}) = \mathbf{Q}_{\text{EA}} + \mathbf{Q}_{\text{other}} $$
where $L$, $\mathbf{q}$, and $\mathbf{Q}$ denote the Lagrangian, generalized coordinates, and generalized forces respectively. We decompose $L$ and $\mathbf{Q}$ into contributions from elastic actuators (EA) and rigid structures (other). $L_{\text{other}}$ and $\mathbf{Q}_{\text{other}}$ depend on $\mathbf{q}$, its derivative $\dot{\mathbf{q}}$, and external inputs. From an energy perspective, if the energy and forces of the elastic actuator can be discretized with rigid--body equivalents, the overall dynamics remain invariant, regardless of the specific configuration and type of soft actuators.
Figure: figure_lumped_mass_00
Shown in the figure above, the dynamic model of the linear elastic actuator (EA) can be equivalently represented by a discrete mass system. The method follows a 3-2-1 approach:
In MuJoCo, we construct the 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: MJCF hierarchical structure of the EquiMus model, showing body-joint-geom relationships and key attributes.
The "..." node denotes remaining rigid skeleton structure. Dashed arrows indicate
We validate EquiMus on a pneumatic musculoskeletal bionic robotic leg (TRO'25).
Figure: Comparison between simulation and physical implementation of the robotic leg system
Figure: 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
@article{zhu2025equimus,
title={EquiMus: Energy-Equivalent Dynamic Modeling and Simulation of Musculoskeletal Robots Driven by Linear Elastic Actuators},
author={Zhu, Yinglei and Dong, Xuguang and Wang, Qiyao and Shao, Qi and Xie, Fugui and Liu, Xinjun and Zhao, Huichan},
journal={IEEE Robotics and Automation Letters},
year={2025},
publisher={IEEE}
}