Bridging the Gap: Simulating Softness in Robotic Systems

Author: Denis Avetisyan

A new framework accurately models the dynamics of robots combining rigid and flexible components, paving the way for more realistic control and learning.

EquiMus establishes an energy-equivalent modeling framework, offering a novel approach to understanding and potentially manipulating system dynamics through the fundamental principle of conserved energy.

This work introduces EquiMus, an energy-equivalent modeling and simulation framework for rigid-soft hybrid robots, enabling accurate dynamic simulations with loop closures for applications in reinforcement learning and control.

Despite advances in soft robotics, accurately modeling the dynamics of bio-inspired musculoskeletal robots—integrating rigid bodies with compliant actuators—remains a significant challenge, particularly for complex systems with kinematic loops. This paper introduces EquiMus: Energy-Equivalent Dynamic Modeling and Simulation of Musculoskeletal Robots Driven by Linear Elastic Actuators, a novel framework that leverages energy equivalence to achieve efficient and accurate dynamic simulation within the MuJoCo physics engine. By effectively capturing actuation dynamics and loop closure, EquiMus enables realistic simulations validated through both experiments and downstream control applications. Could this approach unlock more robust and adaptable control strategies for a new generation of hybrid robots?

Bridging the Divide: Simulating Hybrid Robot Bodies

Simulating robots integrating rigid and soft components presents a core challenge: fundamentally different modeling approaches are required. Traditional robotics focuses on rigid body dynamics, while accurately representing soft materials demands continuum mechanics. Combining these approaches efficiently and realistically remains a key obstacle. Existing simulation methods struggle to capture the interplay between these elements, introducing inaccuracies that propagate into control design and potentially destabilize robot behavior. Efficient and accurate simulation is crucial for developing adaptable robotic systems, enabling virtual prototyping and exploration of designs difficult to realize physically. The rules are there, waiting to be deciphered.

Verification in simulation confirms the dynamic equivalence of the stance and swing phases, as joint trajectories closely align between simulation results and theoretical models, validating the proposed formulation.

The ability to virtually refine these robots not only reduces development time and cost but unlocks designs beyond the reach of traditional methods.

EquiMus: Unifying Robot Dynamics with Energy Equivalence

EquiMus introduces an energy-equivalent modeling technique to unify the simulation of rigid and soft components. This approach bypasses traditional methods requiring distinct solvers, enabling a more cohesive and efficient simulation of complex robotic systems. By representing all components—rigid links and deformable muscles—in terms of energy, EquiMus facilitates seamless interaction and accurate dynamic behavior. This formulation eliminates complex interface models or penalty-based methods often used in hybrid simulations.

Analysis of a 3-DOF musculoskeletal robot demonstrates that joint-angle trajectories generated using the analytical SymPy method closely match those produced by the EquiMus implementation in MuJoCo, even with randomized morphologies and three highlighted muscles.

Leveraging the MuJoCo physics engine, EquiMus achieves efficient and scalable simulation with a speed of 141.9x real time and an average wall-clock step of 0.0071 ms. This performance is critical for real-time control and exploring large design spaces.

Validation Through Control: Sim-to-Real Performance with EquiMus

EquiMus integrates reinforcement learning algorithms, such as Proximal Policy Optimization (PPO), enabling robots to learn intricate locomotion and manipulation tasks directly from simulation or real-world interactions. Validation demonstrates significant improvements in control precision, achieving sim-to-real Root Mean Squared Errors (RMSEs) below 0.03253 radians for joint angle predictions, indicating robust transferability.

Quantitative analysis reveals high static equivalence, with RMSE values less than 0.001 radians for joint angle $\theta_1$ and less than 0.06 radians for $\theta_2$. Tracking triangular trajectories yields RMSEs of 0.092 rad ($\theta_1$) and 0.174 rad ($\theta_2$), demonstrating precise dynamic control capabilities.

Bio-Inspired Design: Unlocking Adaptability Through Simulation

EquiMus facilitates the creation and testing of articulated musculoskeletal robots, focusing on replicating the biomechanical efficiency and adaptability observed in living organisms. By modeling both rigid and compliant components within a single system, the framework enables the investigation of innovative actuator technologies, including pneumatic artificial muscles and other soft robotic elements. It supports comprehensive simulations, evaluating performance metrics like energy consumption, range of motion, and stability.

By establishing a robust link between simulation and physical implementation, EquiMus accelerates the development cycle for advanced robotic systems, promising more resilient and adaptable robots capable of navigating complex environments and performing intricate tasks. Innovation often arises from embracing the elegant disorder of nature.

Beyond the Horizon: Refinements and Future Extensions

Future robotic material simulation demands increased fidelity in modeling complex material behaviors. Integrating more sophisticated continuum mechanics models—like discrete material models and geometrical approximations—will enhance fidelity, particularly with highly deformable or fracturing materials. Computational cost remains a significant hurdle; leveraging surrogate models, such as Gaussian process regression or neural networks, can accelerate the design and optimization process.

Sensitivity analysis reveals the influence of various model parameters on overall system behavior.

Expanding the simulation framework to support loop closures and constraints will enable the modeling of more complex systems and tasks. Incorporating closed-loop control and feedback mechanisms will enhance realism and utility for robot design and validation.

The development of EquiMus directly embodies a philosophy of challenging established boundaries. The framework doesn’t simply accept existing simulation limitations; it actively seeks to bypass them by focusing on energy equivalence, effectively reverse-engineering the complexities of rigid-soft hybrid robot dynamics. As Grace Hopper famously stated, “It’s easier to ask forgiveness than it is to get permission.” This sentiment resonates with EquiMus’ approach; the researchers didn’t ask if accurate, energy-based modeling was possible – they built a system to demonstrate it, even if it meant circumventing traditional methods and tackling the difficult problem of loop closure within a simulation environment. This willingness to probe and question assumptions is central to both the spirit of innovation and the underlying principle of the framework.

Beyond Equilibrium: Charting Future Directions

The EquiMus framework, by embracing energy equivalence, sidesteps the traditional pitfalls of modeling rigid-soft hybrids. However, truly disruptive progress rarely stems from simply solving a problem; it arises from deliberately stressing the solution until failure reveals deeper constraints. The current formulation, while adept at simulation and control, remains tethered to pre-defined loop closures. A compelling next step involves exploring methods for discovering optimal loop configurations autonomously – essentially allowing the robot to re-engineer its own mechanics during operation. This demands a move beyond kinematic constraints and towards a truly dynamic, learning-based approach to structural design.

Furthermore, the reliance on MuJoCo, a powerful but ultimately finite-element based simulator, introduces a subtle but critical limitation. While sufficient for many applications, the framework’s fidelity is capped by the simulator’s inherent approximations. The logical, if daunting, progression involves developing analytical methods – even if approximate – to directly validate and refine the energy-equivalent model, bypassing the need for extensive numerical computation. This pursuit will not only enhance simulation speed but also expose fundamental relationships currently obscured by computational complexity.

Ultimately, EquiMus represents a pivot – a move away from modeling reality and towards reproducing its essential energetic principles. The true test lies not in achieving increasingly realistic simulations, but in constructing robots that demonstrably outperform their conventionally-actuated counterparts – systems that exploit the very compliance once considered a hindrance, and that, ideally, surprise even their creators.

Original article: https://arxiv.org/pdf/2511.07887.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/