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Exploring Nonlinear Body Oscillations for Natural Quadruped Gaits

Annika Schmidt, Davide Calzolari, Arne Sachtler, Florian Loeffl, Daniel Seidel, Milan Hermann, Robert Burger, Thomas Gumpert, Antonin Raffin, Tristan Ehlert, Maximilian Pries, David Wandinger, Florian Schmidt, Manuel Keppler, Jinoh Lee, Alin Albu-Schäffer

arXiv:2609.00539v2cs.RO

TL;DR

The paper addresses the difficulty of making compliant quadruped robots exploit multiple nonlinear resonances rather than relying on dedicated locomotion control. It develops and analyzes the elastic quadruped eBert, identifying six NNMs and using simple mode-specific excitation to generate gaits. Distinct gaits emerged in simulation and largely transferred to hardware, while hardware excitation showed hysteresis and some modes failed to sustain locomotion because of transmission slippage.

  • Problem

    Designing compliant legged robots whose mechanics support multiple useful gaits remains difficult because nonlinear dynamics are difficult to predict in advance.

    Source: §I · p. 2

  • Method

    The paper designs eBert around elastic mechanics, computes six NNMs using nonlinear mode analysis, and excites them with a simple state-switching controller extended with mode-matched foot lifting.

    Source: §I · p. 2, E. Robotic Design aided by Intrinsic… · p. 11

  • Results

    Distinct gaits emerged from the NNMs in simulation, and modes M1, M3, M4, and M6 produced predicted gaits in hardware.

    Source: C. From modes to locomotion · p. 9, §III · p. 9

  • Takeaways & Limitations

    The results support using purposefully designed nonlinear resonances and intrinsic mechanics as a foundation for multi-gait locomotion with minimal control.

    Source: §I · p. 2, §III · p. 9

    At larger deflections, linearized eigenvectors were inaccurate at the turning points, causing observable hysteresis during hardware excitation.

    Source: C. Limitations of Excitation Methods… · p. 10

Abstract

from arXiv · show

Animals' body morphology shapes the gait patterns they can perform, where mechanical resonance reduces the need for active control. By tuning posture and muscle stiffness, they leverage their embodied intelligence to achieve effective gaits for different speeds. In contrast, most quadruped robots are not specifically designed to exploit mechanical resonance due to the complexity of nonlinear dynamics and require dedicated locomotion controllers. To provide an alternative, we present a proof of concept framework making the nonlinear dynamics of a robot predictable in the design process and show how this knowledge can be leveraged such that multi-gait locomotion can emerge from nonlinear resonances, shaped by gravity, inertia, and elasticity. We present the highly compliant quadruped robot eBert, on which we identify six nonlinear normal modes (NNMs) using our new theoretical tools and validate their existence in simulation and hardware. With black-box optimization to determine step length, simulations show how each NNM naturally develops into a distinct gait, manifesting different speeds, which also largely transfers to the robotic hardware. Our experiments show that eBert can exploit its mechanics to generate task-specific movements which may serve as foundation for designing a new generation of agile and efficient robots leveraging embodied intelligence.

I. INTRODUCTION

The paper proposes designing compliant quadruped mechanics around nonlinear normal modes so multiple gaits can emerge with minimal control. In eBert, six computed modes were validated and linked to distinct locomotion patterns through mode-specific excitation and foot movements.

  • Most quadruped robots use stiff actuation and high-gain feedback, so controllers rather than mechanics largely dictate their gaits.Source: §I · p. 1
  • Animals tune posture, gait patterns, and muscle stiffness to exploit body resonance and maintain efficient movement across speeds.Source: §I · p. 1
  • Designing compliant robots with multiple gaits remains difficult because their nonlinear dynamics are hard to predict in advance.Source: §I · p. 2
  • Nonlinear mode theory provides a systematic way to compute periodic motions in complex, high-dimensional energy-conservative systems, including Nonlinear Normal Modes.Source: §I · p. 2
  • eBert uses highly elastic legs and series elastic actuators to encode multiple body oscillations, while a simple controller excites six NNMs and mode-matched foot lifting produces distinct gaits.Source: §I · p. 2, A. The highly elastic quadruped eBert · p. 2
  • The six NNMs span distinct body motions, including translation, rolling, pitching, yaw rotation, vertical movement, pure pitch, and pure roll, with shapes and frequencies varying with energy.Source: A. The highly elastic quadruped eBert · p. 5

B. Existence of modal oscillations on hardware

The predicted NNMs were sustained in simulation and hardware using a state-switching controller that compensates friction with brief energy injections. All six modes remained excitable, with resonant amplification and substantial passive mechanical work, although hardware altered frequencies and amplitudes.

Controller

  • A state-switching controller injected energy near oscillation turning points to compensate friction while preserving the robot’s natural dynamics.The controller used spring deflection and body velocity to determine when and how to apply brief motor commands.Source: B. Existence of modal oscillations on… · p. 5

Hardware validation

  • All six predicted NNMs were successfully excited in frictional simulation and hardware at distinct energy levels.The oscillations remained stable despite friction, hardware imperfections, and deviations from the conservative model.Source: B. Existence of modal oscillations on… · pp. 5–6
  • Frictional simulation frequencies were up to 6% higher than the ideal conservative model, while hardware frequencies were consistently lower.Despite these shifts, the predicted modal sequence from M1 to M6 was preserved.Source: B. Existence of modal oscillations on… · p. 6, Table I · p. 6

Mechanical response

  • Resonant amplification produced substantial passive mechanical work, with some modes reaching η-values up to 80%.Motor-position and link-deflection comparisons confirmed that springs amplified motion across the modes.Source: B. Existence of modal oscillations on… · p. 6

C. From modes to locomotion

Each NNM generated a characteristic foot coordination pattern, and minimal stepping control allowed these modal oscillations to develop into distinct gaits. The resulting gait speeds generally tracked modal frequency, but hardware reproduced some modes more successfully than others.

Foot patterns

  • Each mode’s eigenvector symmetry indicated a corresponding foot-pair coordination pattern, linking M1–M6 to pace, bound, trot, and hopping-like motions.M1 and M6 produced lateral rocking, M2 and M5 forward-backward motion, M3 diagonal coordination, and M4 vertical translation.Source: C. From modes to locomotion · p. 7

Stepping control

  • The extended controller sustained modal oscillations and commanded swing-leg motion at equilibrium crossings to generate forward steps.Each crossing shifted stepping to the opposite foot pair, matching the mode-specific coordination pattern.Source: C. From modes to locomotion · p. 7

Gait emergence

  • Each intrinsic mode evolved into a gait under minimal control, with forward velocity appearing to correlate with the underlying mode frequency.This relationship was observed across the simulated and hardware locomotion comparisons, with exceptions and transfer differences described below.Source: C. From modes to locomotion · p. 8

Gait outcomes

  • M2 produced a slow bounding gait and M5 a faster hopping-like bound, while M6 vibrated forward in simulation because foot contact was too brief.The M6 hardware motion stabilized into a gait, likely because friction increased ground contact.Source: C. From modes to locomotion · pp. 8–9
  • Hardware reproduced the predicted gaits for M1, M3, M4, and M6, whereas M2 and M5 failed to fully develop because hind-leg torques caused slippage.Hardware gait frequencies and step speeds were slightly lower than in simulation.Source: C. From modes to locomotion · p. 9

III. DISCUSSION AND OUTLOOK

The discussion presents eBert as a platform showing that purposefully designed nonlinear resonances can produce multiple gaits with simple control. It also identifies unresolved questions about efficiency, gait shaping, stability, and control limitations in highly nonlinear modes.

Discussion

  • eBert’s six NNMs could be excited in simulation and hardware, and matching foot-pair triggers allowed each mode to unfold into a gait without complex planning.The result supports using intrinsic dynamics as a basis for multi-gait locomotion.Source: §III · p. 9

Scope and limitations

  • The gaits reproduced biological temporal and spatial foot-contact symmetries, but their energy efficiency and comparisons with biological or robotic counterparts were not evaluated.The study therefore establishes gait emergence rather than comparative locomotion efficiency.Source: §III · p. 9

Outlook

  • Future work must address explicit shaping of step length and gait velocity and evaluate efficiency and stability across regimes and more complex environments.These directions define the practical boundary of the current proof of concept.Source: §III · p. 9

Design considerations

  • Lightweight compliant legs can vibrate during lift-off, limiting passive dynamics exploitation during flight phases.The discussion notes that quasi-direct drives with virtual elasticities could improve flight-phase foot coordination.Source: A. eBert as Test Platform for Embodied… · p. 9

Dynamics and control

  • The analysis identified six NNMs in the conservative grounded-leg model, with pronounced nonlinearity especially in M1 and M2 at large deflections.These large-deflection behaviors are not predictable through linearization alone and require active control in practice because friction dissipates energy.Source: B. Applicability of Conservative NNMs… · p. 10
  • The state-switching controller remained effective but produced hysteresis in highly nonlinear M1 and M2 because linearized eigenvectors do not remain valid at turning points.Alternative CPG- or learning-based excitation methods remain an open direction.Source: C. Limitations of Excitation Methods… · p. 10

D. Potential of Gaits emerging from NNMs

NNMs provide a mechanically grounded scaffold for diverse gait patterns, but their usefulness is bounded by hybrid contact dynamics, hardware transmission losses, and task-dependent trade-offs.

Potential of Gaits emerging from NNMs

  • NNMs can provide physics-grounded coordination priors linking structural compliance to feasible motion patterns, with gait shape largely shaped by the underlying mode.In eBert, foot-pair coordination emerged from linearized eigenvector symmetries, while only step width was learned through black-box optimization.Source: D. Potential of Gaits emerging from NNMs · p. 10
  • Conservative NNMs are invariant manifolds rather than standalone motion generators and are formally defined only for smooth, conservative systems with fixed boundary conditions.Quadrupedal locomotion involves intermittent contact, friction, and state-dependent energy exchange, so the conservative modes do not strictly equal locomotion cycles.Source: D. Potential of Gaits emerging from NNMs · p. 10
  • The framework does not cover all possible limit cycles or actuation and control strategies, but can reduce control burden and enhance robustness in compliant systems.Source: D. Potential of Gaits emerging from NNMs · p. 10
  • NNM analysis can reveal large-amplitude, multidirectional dynamics and help designers anticipate stable, exploitable modes before building the robot.This extends linear modal analysis, which describes behavior near equilibrium, to nonlinear behavior at higher energy levels.Source: E. Robotic Design aided by Intrinsic… · pp. 10–11
  • Analyzing only the conservative model omitted the full actuation chain, and belt slippage prevented hardware realization of modes M2 and M5 despite sufficient motor torque.The authors identify extending the analysis to full actuation chains as necessary for more robust hardware-oriented co-design.Source: E. Robotic Design aided by Intrinsic… · p. 11
  • High compliance enables diverse intrinsic dynamics and impact resilience but can reduce structural rigidity and complicate control, while NNM utility remains task-dependent.Source: E. Robotic Design aided by Intrinsic… · p. 11
  • The approach lacks rigorous evaluation across controllers, perturbations, and unstructured environments, leaving systematic comparison with stiff-control systems for future work.Source: E. Robotic Design aided by Intrinsic… · p. 11

B. Simulation model and pipeline

A multibody model derived from eBert’s CAD data was used for initial simulation and validation, with constrained feet for mode excitation and later free feet for locomotion.

Simulation model and pipeline

C. Nonlinear modal analysis

The nonlinear modal analysis identifies eBert’s natural oscillations by continuing linearized modes into the nonlinear regime, then uses mode-specific transformations and switching control to excite them.

Nonlinear modal analysis

  • The conservative model combines multibody dynamics with gravitational and elastic forces while fixing motor positions and feet to remove motor dynamics and ground motion.Source: C. Nonlinear modal analysis · p. 13
  • Starting from eigenvectors of the linearized system, numerical continuation computes nonlinear normal modes as energy increases.NNMs extend linear normal modes from small-amplitude behavior to nonlinear oscillations.Source: C. Nonlinear modal analysis · p. 13, D. Mode excitation controller · p. 13
  • A state-switching controller compensates for friction by orchestrating multiple joints through a one-dimensional control signal derived from multidimensional joint-space measurements.Source: D. Mode excitation controller · p. 13
  • Joint torques are calculated from measured motor and link positions using the spring stiffness matrix, then projected into a scalar control coordinate.Source: D. Mode excitation controller · p. 13
  • The controller detects return points from the combined Cartesian body velocity, injects energy there, and maps the resulting command back into joint-space position commands.Source: D. Mode excitation controller · pp. 13–14

E. Quantifying the exploitation of nonlinear resonance

The study quantifies how much motion is generated passively by defining an efficiency measure and deriving mode-specific foot-pair stepping patterns from modal symmetries.

Quantifying the exploitation of nonlinear resonance

  • Efficiency η is defined as the ratio of positive actuator work to positive joint work, quantifying the contribution of natural mechanical response to motion.Source: E. Quantifying the exploitation of… · p. 14
  • An η value of 1 denotes perfect exploitation of nonlinear passive dynamics, while 0 indicates that control does not use the mechanics.The metric represents the percentage of positive work provided passively in a cycle.Source: E. Quantifying the exploitation of… · p. 14
  • The stepping analysis used natural poses with the center of mass shifted forward and feet turned outward, searching over initial configurations with feet attached to the ground.Source: F. Initial joint configurations · p. 14
  • Foot pairings were extracted from symmetries of the linearized eigenvectors and encoded as binary stepping vectors.Source: G. Encoding of modal feet couplings · p. 14
  • Modes M2 and M5 suggested a bounding-like front-versus-back pattern, while M3 suggested diagonal pairing similar to a trot.Source: G. Encoding of modal feet couplings · p. 14
  • The controller triggered alternating forward steps when the robot shifted weight between foot pairs, using the combined torque metric τz as the trigger.Source: H. Stepping Controller · p. 14
  • A mode-specific step-length matrix changed the shoulder, hip, and knee positions for one foot pair while zero entries canceled commands for stance legs.Source: H. Stepping Controller · pp. 14–15

I. Optimizing step length

The study optimized modal step parameters in simulation and then transferred them to hardware to produce forward locomotion. The optimization adjusted hip and knee joint steps and motor-signal scaling while using forward velocity as its objective.

  • Hip and knee joint values were varied to create forward steps while shoulder motion remained constrained to its plane.Sagittal symmetry was used to assume identical left and right steps.Source: §I · p. 15
  • Motor-signal scaling was relearned because stepping forward required less energy injection at turning points than maintaining pure oscillations.Source: §I · p. 15
  • Each 8 s simulation trial optimized forward velocity after repeatedly initializing and exciting eBert in different mode oscillations.The optimization used empirically estimated starting parameters and the covariance matrix adaptation algorithm.Source: §I · p. 15
  • 100-200 simulation trials per mode found suitable step parameters for forward gait patterns, after which optimization was repeated directly on hardware.Hardware optimization started from the simulation-derived values.Source: §I · p. 15
  • The analysis and simulation code, raw simulation and hardware data, nonlinear modes, and figure-reproduction scripts were made available in a figshare repository.A separate repository contains the eBert-specific mode-calculation pipeline.Source: §VI · p. 15

APPENDIX A SUPPLEMENTARY METHODS

The supplementary methods reduce eBert’s constrained mechanics to a conservative six-degree-of-freedom system and compute nonlinear normal modes by numerical continuation. Each mode is represented as a family of periodic brake orbits parameterized by energy.

  • The conservative model combines multibody dynamics with gravitational and elastic potential energy to compute eBert’s nonlinear modes.Source: A. Nonlinear modal analysis · p. 17, Figure S1 · p. 18
  • Fixing the four feet to the ground and removing motor dynamics leaves six independent degrees of freedom represented by trunk coordinates.Inverse kinematics and constraint Jacobians express dependent leg variables through the independent coordinates.Source: A. Nonlinear modal analysis · pp. 17–18
  • The reduced system uses an inertia matrix, Coriolis and centrifugal forces, gravity, and total mechanical energy to describe its conservative dynamics.The energy combines kinetic energy with the potential V(x).Source: A. Nonlinear modal analysis · p. 18
  • Starting from small linear-mode perturbations, numerical continuation increases energy and adjusts turning points to preserve periodic orbits with two zero-velocity turning points.The resulting continuous family is called a nonlinear normal mode.Source: A. Nonlinear modal analysis · p. 18
  • The generators G_i±(E) specify the energy-dependent turning points, while collecting trajectories across energy levels forms a two-dimensional Eigenmanifold M_i.The generators act as nonlinear counterparts of eigenvectors.Source: A. Nonlinear modal analysis · p. 18

B. Quantifying Nonlinearity of the Modes

The study quantifies modal nonlinearity by testing whether each Eigenmanifold can be represented by a two-dimensional linear plane. PCA and singular values show that the first two modes require higher-dimensional representations.

C. Robotic Implementation

eBert is a compact, compliant quadruped built with serial elastic actuators and standalone electronics. Its locomotion design uses biologically inspired posture searches and tests candidate configurations across all nonlinear modes.

E. Control Parameters for the Mode Excitations

Mode excitation uses a state-switching controller to energize eBert’s natural oscillations, while later locomotion experiments optimize stepping parameters for each mode. A posture grid search identifies configurations that support forward motion across all six modes.

Mode excitation

Mode-based locomotion

Posture selection

  • 16 joint configurations varying hip and shoulder angles were evaluated with the bang-bang controller in realistic simulations with friction.After 40 s without further control, travelled distance was used to evaluate each posture.Source: Figure S4 · p. 21
  • Configurations 2, 10, and 14 achieved forward motion with all six modes.The result is reported as travelled distance across the tested initial joint configurations.Source: Figure S5 · p. 22

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