Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlinear stiffness allows passive dynamic hopping for one-legged robots with an upright trunk

Dennis Ossadnik, Elisabeth Jensen, Sami Haddadin

TL;DR

Problem: Stabilizing an upright trunk during one-legged hopping is challenging, and existing template models with a linear hip spring do not yield passive limit cycles. Approach: the authors develop a hybrid floating-base model with non-negligible leg inertia and examine centered-hip and upright-trunk configurations, testing linear and nonlinear hip springs, and they search for passive gaits using a Poincaré-map Newton method. Key findings: a passive limit cycle is found for the upright-trunk model when using a nonlinear hip spring (cubic or exponential), though the cycles remain unstable; GRFs intersect above the CoM, consistent with a virtual pivot point mechanism. Significance: this is the first demonstration of a passive limit cycle for an upright-trunk template with leg mass, offering a pathway toward energy-efficient, passively stabilized hopping controllers and informing future stabilization and template-extension work.

Abstract

Template models are frequently used to simplify the control dynamics for robot hopping or running. Passive limit cycles can emerge for such systems and be exploited for energy-efficient control. A grand challenge in locomotion is trunk stabilization when the hip is offset from the center of mass (CoM). The swing phase plays a major role in this process due to the moment of inertia of the leg; however, many template models ignore the leg mass. In this work, the authors consider a robot hopper model (RHM) with a rigid trunk and leg plus a hip that is displaced from the CoM. It has been previously shown that no passive limit cycle exists for such a model given a linear hip spring. In this work, we show that passive limit cycles can be found when a nonlinear hip spring is used instead. To the authors' knowledge, this is the first time that a passive limit cycle has been found for this type of system.

Nonlinear stiffness allows passive dynamic hopping for one-legged robots with an upright trunk

TL;DR

Problem: Stabilizing an upright trunk during one-legged hopping is challenging, and existing template models with a linear hip spring do not yield passive limit cycles. Approach: the authors develop a hybrid floating-base model with non-negligible leg inertia and examine centered-hip and upright-trunk configurations, testing linear and nonlinear hip springs, and they search for passive gaits using a Poincaré-map Newton method. Key findings: a passive limit cycle is found for the upright-trunk model when using a nonlinear hip spring (cubic or exponential), though the cycles remain unstable; GRFs intersect above the CoM, consistent with a virtual pivot point mechanism. Significance: this is the first demonstration of a passive limit cycle for an upright-trunk template with leg mass, offering a pathway toward energy-efficient, passively stabilized hopping controllers and informing future stabilization and template-extension work.

Abstract

Template models are frequently used to simplify the control dynamics for robot hopping or running. Passive limit cycles can emerge for such systems and be exploited for energy-efficient control. A grand challenge in locomotion is trunk stabilization when the hip is offset from the center of mass (CoM). The swing phase plays a major role in this process due to the moment of inertia of the leg; however, many template models ignore the leg mass. In this work, the authors consider a robot hopper model (RHM) with a rigid trunk and leg plus a hip that is displaced from the CoM. It has been previously shown that no passive limit cycle exists for such a model given a linear hip spring. In this work, we show that passive limit cycles can be found when a nonlinear hip spring is used instead. To the authors' knowledge, this is the first time that a passive limit cycle has been found for this type of system.
Paper Structure (9 sections, 24 equations, 5 figures, 3 tables)

This paper contains 9 sections, 24 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Modeling
  4. Application to one-legged robots
  5. Gait search algorithm
  6. Software
  7. Results
  8. Discussion
  9. Conclusion

Figures (5)

  • Figure 1: Sketch of the considered models with parameter definition. The left model corresponds to the one-legged hopping robot of Hyon04, where the hip is coincident with the CoM ("centered-hip model"). The right model has the hip offset by the factor $d$ ("upright-trunk model").
  • Figure 2: Illustration of a hybrid periodic orbit $\boldsymbol{\Gamma}$. The solution is comprised of the individual contributions of each phase $\boldsymbol{x}_i$. At each cross-section $\Sigma_i$, the jump maps $\boldsymbol{g}_i(\boldsymbol{x}_i)$ are evaluated.
  • Figure 3: Normalized ground reaction force vectors in the CoM frame of reference for the CHLS (top), CHNS (middle) and UTNS (bottom) models. In all cases the vectors are directed well above the CoM.
  • Figure 4: Visualization of limit cycle jumping for the centered-hip nonlinear-spring model. The images were rendered with the open-source software MeshUp (https://github.com/ORB-HD/MeshUp).
  • Figure 5: Visualization of limit cycle jumping for the upright-trunk nonlinear-spring model. The images were rendered with the open-source software MeshUp (https://github.com/ORB-HD/MeshUp).