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Connecting Gaits in Energetically Conservative Legged Systems

Maximilian Raff, Nelson Rosa, C. David Remy

TL;DR

The paper develops a nonlinear dynamics framework for connecting gaits in energetically conservative legged systems by formulating Energetically Conservative Hybrid Dynamical Systems (ECM) and proving that gaits form 1D manifolds (generators) parameterized by energy $\bar{E}$. It introduces a phase-anchored gait space $\mathcal{G}$, uses energy conservation and the monodromy matrix $\boldsymbol{\Phi}_T$ to characterize periodic motions, and employs numerical continuation and a dissipative embedding to compute connected components of gaits, including bifurcations (BP) and turning points (TP). The approach is demonstrated on a 4-DOF one-legged hopper, where multiple vertical, forward, and backward hopping gaits are shown to lie on a single connected gait component $\mathcal{V}$ connected through simple BP. The framework provides a principled template for energy-efficient locomotion in robotics and offers insights into passive gait structures relevant to biology, with future work aimed at relaxing fixed phase sequences and mapping ECM templates to more realistic, energy-lossy systems.

Abstract

In this work, we present a nonlinear dynamics perspective on generating and connecting gaits for energetically conservative models of legged systems. In particular, we show that the set of conservative gaits constitutes a connected space of locally defined 1D submanifolds in the gait space. These manifolds are coordinate-free parameterized by energy level. We present algorithms for identifying such families of gaits through the use of numerical continuation methods, generating sets and bifurcation points. To this end, we also introduce several details for the numerical implementation. Most importantly, we establish the necessary condition for the Delassus' matrix to preserve energy across impacts. An important application of our work is with simple models of legged locomotion that are often able to capture the complexity of legged locomotion with just a few degrees of freedom and a small number of physical parameters. We demonstrate the efficacy of our framework on a one-legged hopper with four degrees of freedom.

Connecting Gaits in Energetically Conservative Legged Systems

TL;DR

The paper develops a nonlinear dynamics framework for connecting gaits in energetically conservative legged systems by formulating Energetically Conservative Hybrid Dynamical Systems (ECM) and proving that gaits form 1D manifolds (generators) parameterized by energy . It introduces a phase-anchored gait space , uses energy conservation and the monodromy matrix to characterize periodic motions, and employs numerical continuation and a dissipative embedding to compute connected components of gaits, including bifurcations (BP) and turning points (TP). The approach is demonstrated on a 4-DOF one-legged hopper, where multiple vertical, forward, and backward hopping gaits are shown to lie on a single connected gait component connected through simple BP. The framework provides a principled template for energy-efficient locomotion in robotics and offers insights into passive gait structures relevant to biology, with future work aimed at relaxing fixed phase sequences and mapping ECM templates to more realistic, energy-lossy systems.

Abstract

In this work, we present a nonlinear dynamics perspective on generating and connecting gaits for energetically conservative models of legged systems. In particular, we show that the set of conservative gaits constitutes a connected space of locally defined 1D submanifolds in the gait space. These manifolds are coordinate-free parameterized by energy level. We present algorithms for identifying such families of gaits through the use of numerical continuation methods, generating sets and bifurcation points. To this end, we also introduce several details for the numerical implementation. Most importantly, we establish the necessary condition for the Delassus' matrix to preserve energy across impacts. An important application of our work is with simple models of legged locomotion that are often able to capture the complexity of legged locomotion with just a few degrees of freedom and a small number of physical parameters. We demonstrate the efficacy of our framework on a one-legged hopper with four degrees of freedom.
Paper Structure (12 sections, 2 theorems, 27 equations, 4 figures, 2 algorithms)

This paper contains 12 sections, 2 theorems, 27 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Theory
  3. Dynamics of Legged Systems
  4. Periodic Solutions in Energetically Conservative Hybrid Dynamical Systems
  5. Connected Components of Energetically Conservative Gaits
  6. Implementation
  7. Constructing Conservative Models
  8. Numerical Exploration
  9. Example: One-Legged Hopper
  10. Model Description
  11. Results
  12. Discussion & Conclusion

Key Result

Lemma 2.1

Outside of an equilibrium, where $\nabla E(\boldsymbol{x}_0)$ and $\boldsymbol{f}_1(\boldsymbol{x}_0)$ are non-zero for a mechanical system, these vectors are also perpendicular.

Figures (4)

  • Figure 1: Different generators are connected by bifurcation (BP) and turning (TP) points and constitute the connected component $\mathcal{V}$ of the gait space $\mathcal{G}$. Isolated generators and generators (red) that only connect to inadmissible points (IP), including equilibira (EQ), are disjoint. Hence, they are part of different connected components.
  • Figure 2: An energetically conservative one-legged hopper with a torsional hip spring. It's planar configuration is described by $\boldsymbol{q}=[x~y~\alpha~l]^{\mathop{\mathrm{T}}}$.
  • Figure 3: Visualization of connected generators $\mathcal{M}_0$ - $\mathcal{M}_6$ of the one-legged hopper. Vertical hopping in-place motions are contained in $\mathcal{M}_0$, $\mathcal{M}_1$ and $\mathcal{M}_4$, while $\mathcal{M}_2$, $\mathcal{M}_5$ and $\mathcal{M}_3$, $\mathcal{M}_6$ are a collection of forward, backward gaits, respectively. All generators constitute the connected component $\mathcal{V}$ since they are connected by simple bifurcation points (BP). The equilibrium (EQ) and the contact sequence transition with vanishing flight duration are inadmissible points (IP). The locally defined 1D manifold of linear bouncing-in-place oscillations (red) is thus not in $\mathcal{V}$.
  • Figure 4: Key frames from periodic solutions of the SLIP model at energy level $\bar{E}=1.8~m_ogl_o$. Starting from apex transit ($\dot{y}_0=0$), three gaits from the generators $\mathcal{M}_4$, $\mathcal{M}_2$ and $\mathcal{M}_5$ are illustrated in the contact sequence $\{\mathrm{F},\mathrm{S},\mathrm{F}\}$. The stance duration differs between these gaits with: $t_\mathrm{S}^{\mathcal{M}_4}\approx 0.54~\sqrt{l_o/g}$, $t_\mathrm{S}^{\mathcal{M}_2}\approx 0.49~\sqrt{l_o/g}$, $t_\mathrm{S}^{\mathcal{M}_5}\approx 0.53~\sqrt{l_o/g}$.

Theorems & Definitions (16)

  • Remark 2.1
  • Definition : Energetically Conservative Model
  • Definition : Hybrid Periodic Flow
  • Definition : Monodromy Matrix
  • Lemma 2.1
  • proof
  • Definition : Gait
  • Theorem : Family of Gaits
  • proof
  • Remark 2.2
  • ...and 6 more