Exploring Nonlinear Body Oscillations for Natural Quadruped Gaits

The gist: The research demonstrates how the analysis of nonlinear normal modes aids to develop natural, multi-gait locomotion in a compliant quadruped robot leveraging its inherent mechanics, offering a design tool to shape embodied intelligence in robotics.

Introduction and Motivation

Animals’ body morphology shapes the gait patterns they can perform, where mechanical resonance reduces the need for active control <ref:2609.00539#pg2>. Most quadruped robots are not specifically designed to exploit mechanical resonance due to the complexity of nonlinear dynamics and require dedicated locomotion controllers <ref:2609.00539#pg2>. The paper presents a proof-of-concept framework making the nonlinear dynamics of a robot predictable in the design process and shows how this knowledge can be leveraged such that multi-gait locomotion can emerge from nonlinear resonances—shaped by gravity, inertia, and elasticity <ref:2609.00539#pg2>. This research is motivated by the biomechanical insight that an animal’s body structure determines which gaits it can perform <ref:2609.00539#pg2>.

Robot Design and Nonlinear Mode Identification

The highly compliant quadruped robot eBert was designed with 12 degrees of freedom (DOF) to encode multiple functional body oscillations at frequencies and amplitudes suitable for gait generation <ref:2609.00539#pg2>. The hardware elasticity was implemented using Series Elastic Actuators (SEAs) with springs having the respective stiffness at each joint, which can mimic the energy storage of animals’ muscles and tendons <ref:2609.00539#pg2>. Using mathematical tools based on differential geometry and algebraic topology, the researchers identify six Nonlinear Normal Modes (NNMs) of the conservative dynamics <ref:2609.00539#pg2>. These NNMs are computed from the conservative model, which is based solely on the interaction of multi-body dynamics with gravitational and elastic forces <ref:2609.00539#pg2>.

Excitation and Gait Emergence

The six computed NNMs are excited by a deliberately simple feedback-based state-switching controller to compensate for friction <ref:2609.00539#pg2>. The controller functions analogously to pushing a playground-swing when it naturally comes to a halt on either side, avoiding motor commands during the system’s motion flow <ref:2609.00539#pg2>. Applying the state-switching controller in a eBert simulation with friction, and its hardware, successfully excited all six expected NNMs at distinct energy levels <ref:2609.00539#pg2>. When this controller was extended to trigger steps in distinct foot pairs matching respective modal oscillations, on flat ground each NNM naturally unfolded into a gait without complex planning or precise foot placement in simulations <ref:2609.00539#pg2>.

Mode-to-Locomotion Mapping

The paper shows that the identified NNMs serve as proxies for the robot’s natural movement tendencies, and that foot-pair coordination that enabled forward stepping emerged directly from linearized eigenvector symmetries in joint space <ref:2609.00539#pg2>. For example, the lateral rocking motions of M1 and M6 resembled the side-toside swing of pace-gait seen in camels <ref:2609.00539#pg2>. The forward-backward motions of M2 and M5 aligned with simultaneous front-hind leg movements, typical of bounding in squirrels <ref:2609.00539#pg2>. Diagonal coordination in M3 suggested trotting as in horses <ref:2609.00539#pg2>.

Conclusion and Future Directions

The combined findings support the hypothesis that intrinsic system dynamics, like NNMs, provide a promising foundation for generating distinct gaits in purposefully designed robots such as eBert <ref:2609.00539#pg2>. The resulting gaits follow biological principles where body resonance naturally organizes multi-legged coordination without explicit scripting <ref:2609.00539#pg2>. Future investigations on explicit control are needed to address how the different emerging gaits can be shaped to vary characteristics like step length and gait velocity along with performance evaluation of efficiency and stability across regimes and in more complex environments <ref:2609.00539#pg2>.

How it works

1. The robot eBert was designed with lightweight elastic legs and large enough compliance to encode multiple functional body oscillations at frequencies and amplitudes suitable for gait generation <ref:2609.00539#pg2>.

2. The researchers analyzed the inherently nonlinear system dynamics of eBert enabled by their recently developed numerical methods to identify the system’s Nonlinear Normal Modes (NNMs) <ref:2609.00539#pg2>.

3. A state-switching controller was implemented that injects energy near a system’s equilibrium where linearized mode vectors characterize motion direction <ref:2609.00539#pg2>.

4. The controller is extended to trigger steps in distinct foot pairs matching respective modal oscillations, which is done by crossing a threshold of the combined torque metric τz <ref:2609.00539#pg2>.

5. The step command was multiplied by a 4 × 3 matrix encoding the position change for the three joint motors of each leg to command a step <ref:2609.00539#pg2>.

6. Black-box optimization was used to determine the step length fitting each modal oscillation, which resulted in gait patterns realizing locomotion forward <ref:2609.00539#pg2>.

Quantifying Nonlinearity

The nonlinearity of the modes is quantified by measuring the error introduced by a linear approximation, where if a mode Mi is linear, all points will lie on a twodimensional plane in the state space; and if the mode is nonlinear, the Eigenmanifold will be curved and require a volume of higher dimension to contain all points <ref:2609.00539#pg2>. This was quantified by applying Principal Component Analysis (PCA) to the point cloud of the Eigenmanifold Mi <ref:2609.00539#pg2>.

Robotic Implementation Challenges

The analysis revealed limitations when applying the conservative NNM framework to real systems, as friction and motor dynamics distort system behavior, necessitating active control to drive oscillations <ref:2609.00539#pg2>. The paper noted that some modes, like M2 and M5, could not fully develop into gaits in hardware because high torques in the hind legs’ hip and knee joints caused unforeseen slippage in the belt transmissions <ref:2609.00539#pg2>.

Design Insights

Analyzing natural dynamics like NNMs during mechanical design offers new avenues for robot optimization by accounting for physical constraints and design limitations early on <ref:2609.00539#pg2>. The NNM framework allows designers to anticipate which modes will be stable and exploitable for intended motion patterns before building the robot <ref:2609.00539#pg2>.

Data Availability

All data needed to evaluate the conclusions of the paper are available in the paper or the Supplementary Materials <ref:2609.00539#pg2>. Additionally, raw data of robot trials in simulation and hardware are deposited on figshare accessible under https://doi.org/10.6084/m9.figshare.29514518 [72] <ref:2609.00539#pg2>.

Code Availability

The code to analyze the provided data can likewise be found in the same figshare repository accessible over https://doi.org/10.6084/m9.figshare.29514518 [72] <ref:2609.00539#pg2>. This repository contains the full model and analysis pipeline for eBert <ref:2609.00539#pg2>.

Acknowledgements

The authors thank Yannik P. Wotte (University of Twente, Bergische Universität Wuppertal), Filip Bjelonic (ETH Zurich), and Dr. Cosimo Della Santina (TU Delft) for their support in developing algorithms used for the nonlinear modal analysis <ref:2609.00539#pg2>.

Funding Statement

This research was supported by the European Research Council (ERC) through the European Union’s Horizon 2020 Research and Innovation Programme under Grant 835284 (MRunners) <ref:2609.00539#pg2>.

Competing Interests

The authors declare that they have no competing interests <ref:2609.00539#pg2>.

References

[1] P. Biswal and P. K. Mohanty, “Development of quadruped walking robots: A review,” Ain Shams Engineering Journal, vol. 12, no. 2, pp. 2017–2031, 2021 <ref:2609.00539#pg2>.

[7] R. Pfeifer, M. Lungarella, and F. Iida, “Self-organization, embodiment, and biologically inspired robotics,” science, vol. 318, no. 5853, pp.

1

No active control required for locomotion in compliant robots by exploiting intrinsic mechanics. The improved AI system can achieve natural multi-gait locomotion by leveraging nonlinear normal modes (NNMs) identified from hardware dynamics, allowing it to exploit its mechanics to generate taskspecific movements which may serve as foundation for designing a new generation of agile and efficient robots.

2

Design of quadruped robots where mechanical structure and control co-evolve. The AI system can be optimized by using NNMs as dynamic priors or stability constraints to parameterize the search space over leg compliance, mass distribution, and joint coupling.

3

Predictive design for large amplitude, multi-directional dynamics. The improved AI system can anticipate which modes will be stable and exploitable for intended motion patterns before building the robot by analyzing nonlinear normal modes during mechanical design.

4

Gait pattern generation from intrinsic modal oscillations without explicit scripting. By encoding foot-pair coordination that enabled forward stepping emerged directly from linearized eigenvector symmetries, the AI can generate gaits where foot-pair coordination that enabled forward stepping emerged directly from linearized eigenvector symmetries.

5

Task-aware co-evolution of robot body and brain. The system can be designed to achieve a state where motion arises from physical intelligence rather than from independently scripted commands by aligning with embodied AI paradigms seeking to close the loop between physical design, intrinsic dynamics, and adaptive control.