Stability of a Groucho-Style Bounding Run in the Sagittal Plane
Abstract
:1. Introduction
1.1. Groucho Running
1.2. Cascade Compositions
1.3. Controlling on Hybrid Transitions
1.4. Outline
2. Model
2.1. Hybrid Dynamical System Description
2.2. Cascaded Composition
2.3. Dynamical Simplification
3. Hybrid Periodic Orbit
3.1. Choice of Poincaré Section
3.2. Stride Map
3.3. Stride Map Fixed Point
3.4. Constant Stance Height Approximation in Pitching Dynamics
3.5. Speed Limit
3.6. Cost of Enforcing a Cascade
4. Controller
4.1. Hybrid Guard Control
4.2. Hybrid Reset Control
4.3. Controller Stability Analysis
5. Empirical Demonstration of Controller
5.1. Setup
5.2. Results
6. Discussion
6.1. Infinitesimally Deadbeat Nature of Our Result
6.2. Controlling on the Hybrid Transitions
6.3. Cascade Compositions as Attracting Invariant Submanifolds
7. Conclusions
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Table of Symbols
Symbol | Description |
---|---|
Hybrid system (3), (5), (6), (13), (17), (18) | |
Hybrid modes (4) | |
Mode domains (7), guards (25), resets (30), vector fields (14) | |
Time, mass-center height, body pitch, mode timer (10), Figure 2 | |
Mass-center and front/rear toe horizontal positions (11), Figure 2 | |
Front, rear horizontal leg splay distance with regard to the mass-center (12) | |
Mode i state (9), with in-place (10) and horizontal (11) components | |
In-place state, configuration (10) | |
Physical model parameters (Figure 2) | |
Pseudo-physical simplifying parameters (22), (24), (26), Figure 2 | |
In-place components of the guard set (25), (26) | |
Front/rear hip heights (29) | |
Guard “control” functions for touchdown, liftoff events (26), (58) | |
In-place guard control weights (26) | |
Front and rear initial hip height in mode i (59) | |
“Bounding” symmetry map (41), (27), (33) | |
Lie derivative (28) of scalar field V along vector field f at point x | |
In-place (31), horizontal (32) reset function components | |
Reset “control” functions (32), (63) | |
Reset control weights (64) | |
Nominal touchdown leg splay for front leg (32) | |
Mass-center height Approximation 1 in pitching dynamics | |
Vertical (16), (20), (34), horizontal (16), (21) mass-specific | |
ground reaction force applied from each hip | |
In-place (35), horizontal (36) mode-i flow | |
simplified acceleration vector for mode i (35) | |
Matrix components used in the description of (36) | |
Mode i-to-j map (38), with in-place, horizontal components (39) | |
Mode i time-to-impact map (40) with guard | |
Reduced domain with horizontal, in-place components (42) | |
State on with in-place and horizontal components (43) | |
Projection and lift maps (44) | |
In-place, horizontal projection, and lift maps (44) | |
Stride (45) and “flipped” half-stride (47) maps | |
Fixed point of (48) | |
Leg splay components of (50) | |
Total hip stance duration (54), leg-sweep distance (55) on the | |
hybrid periodic orbit associated with | |
Lift of (60) | |
Mode i’s duration (52) and initial state (61) as it evolves into | |
mode j under the hybrid execution from | |
Simplified factors of ’s in-place component (73) |
Appendix B. Controller Stability Lemmas
Appendix C. Control Gain Selection Procedure
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State | Min Value on Orbit | Max Value on Orbit |
---|---|---|
y | ||
Numerical Parameters | Symbol | Value |
---|---|---|
Physical and pseudo- | d | |
physical parameters | ||
a | 1 | |
g | ||
Fixed-point parameters | ||
Varies by experiment | ||
Control weights | ||
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Duperret, J.; Koditschek, D.E. Stability of a Groucho-Style Bounding Run in the Sagittal Plane. Robotics 2023, 12, 109. https://doi.org/10.3390/robotics12040109
Duperret J, Koditschek DE. Stability of a Groucho-Style Bounding Run in the Sagittal Plane. Robotics. 2023; 12(4):109. https://doi.org/10.3390/robotics12040109
Chicago/Turabian StyleDuperret, Jeffrey, and Daniel E. Koditschek. 2023. "Stability of a Groucho-Style Bounding Run in the Sagittal Plane" Robotics 12, no. 4: 109. https://doi.org/10.3390/robotics12040109