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An Approach for Generating Families of Energetically Optimal Gaits from Passive Dynamic Walking Gaits

Nelson Rosa, Bassel Katamish, Maximilian Raff, C. David Remy

TL;DR

The paper addresses how to connect passive dynamic walking gaits to continuously parameterized families of energetically optimal gaits in impulsive biped models. It reformulates the parametric gait optimization as an implicit map whose roots are stationary solutions and uses numerical continuation to trace these solutions across operating points, seeded by passive gaits. A global-seed strategy yields continuous curves of gaits, demonstrated on a two-link compass-gait walker with hip actuation, thereby building libraries of gaits across slope and speed and revealing intersections with globally optimal passive solutions. This framework provides a principled way to explore the gait space, offering insights into the relationship between PW motions and actuated, energy-efficient gaits and enabling practical gait-library construction for robotic locomotion, with potential extensions to alternative actuation bases and inequality constraints.

Abstract

For a class of biped robots with impulsive dynamics and a non-empty set of passive gaits (unactuated, periodic motions of the biped model), we present a method for computing continuous families of locally optimal gaits with respect to a class of commonly used energetic cost functions (e.g., the integral of torque-squared). We compute these families using only the passive gaits of the biped, which are globally optimal gaits with respect to these cost functions. Our approach fills in an important gap in the literature when computing a library of locally optimal gaits, which often do not make use of these globally optimal solutions as seed values. We demonstrate our approach on a well-studied two-link biped model.

An Approach for Generating Families of Energetically Optimal Gaits from Passive Dynamic Walking Gaits

TL;DR

The paper addresses how to connect passive dynamic walking gaits to continuously parameterized families of energetically optimal gaits in impulsive biped models. It reformulates the parametric gait optimization as an implicit map whose roots are stationary solutions and uses numerical continuation to trace these solutions across operating points, seeded by passive gaits. A global-seed strategy yields continuous curves of gaits, demonstrated on a two-link compass-gait walker with hip actuation, thereby building libraries of gaits across slope and speed and revealing intersections with globally optimal passive solutions. This framework provides a principled way to explore the gait space, offering insights into the relationship between PW motions and actuated, energy-efficient gaits and enabling practical gait-library construction for robotic locomotion, with potential extensions to alternative actuation bases and inequality constraints.

Abstract

For a class of biped robots with impulsive dynamics and a non-empty set of passive gaits (unactuated, periodic motions of the biped model), we present a method for computing continuous families of locally optimal gaits with respect to a class of commonly used energetic cost functions (e.g., the integral of torque-squared). We compute these families using only the passive gaits of the biped, which are globally optimal gaits with respect to these cost functions. Our approach fills in an important gap in the literature when computing a library of locally optimal gaits, which often do not make use of these globally optimal solutions as seed values. We demonstrate our approach on a well-studied two-link biped model.
Paper Structure (14 sections, 1 theorem, 9 equations, 3 figures, 1 table)

This paper contains 14 sections, 1 theorem, 9 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Biped Model
  3. The Impulsive Dynamics
  4. Modeling Assumptions
  5. Optimality in the Trajectory Space
  6. Periodic Trajectories and Operating Points of Interest
  7. Cost Functions of Interest
  8. Numerical Continuation of Optimal Gaits
  9. Optimal Trajectories as Roots of a Map
  10. Passive Gaits as Seed Values
  11. Tracing Continuous Families of Optimal Gaits of the Compass Gait Walker
  12. An Example Constant-Velocity Slice of Optimal Gaits
  13. Growing the Library of Gaits Along Different Slices
  14. Discussion and Conclusion

Key Result

Proposition 1

If the pair $(c^\ast, \lambda^\ast)$ is a regular point of $\Popt$ and a solution of $\OP$ for fixed $p$, then it is an isolated point in $\Popt^{-1}(0)$.

Figures (3)

  • Figure 1: A demonstration of our approach on (a) a two-link biped robot with an actuator at the hip using (b) a passive (i.e., unactuated) dynamic walking motion of the biped model to compute continuous sets of energetically optimal actuated gaits, including (c) gaits that walk on level ground. (d) Using a single passive gait as a seed value from a family of unactuated gaits (green curve), we are able to generate a curve of gaits with the same walking speed across a range of slopes (black curve in gray plane) and then switch to generating a curve of gaits that walk on level-ground across a range of walking speeds (black curve in yellow plane). (e) Examples of locally optimal actuation profiles generated for a fixed speed and range of slopes in between gaits depicted in (b) and (c). The example gaits are highlighted in (d) using a green to yellow color gradient between the seed value and level-ground walking gait. The model is scaled with time measured in units of $t_0$, torque in units of $u_0$, and speed in units of $v_0$ (see Table \ref{['tab:parameters']}).
  • Figure 2: (a) A constant-velocity slice of optimal gaits projected onto a slope-step-duration subspace; the seed value is labeled with a (1). (b) The optimal cost as a function of slope along the curve in the constant-velocity slice; the first through fourth quartiles define the color coding of this plot (see inset legend). (c)--(f) Example gait motions of labeled points in the plots.
  • Figure 3: (a) A constant-slope slice of optimal gaits projected onto a slope-step-duration subspace; the level-ground gait from the constant-velocity slice of gaits of Figure \ref{['fig:velocity']} is labeled with a (0). (b) The optimal cost as a function of slope along the curve in the constant-velocity slice; the first through fourth quartiles define the color coding of this plot (see inset legend).

Theorems & Definitions (10)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 2
  • Definition 6
  • Proposition 1
  • proof