# 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

**Approximation 1.**

#### 2.2. Cascaded Composition

**Approximation 2.**

#### 2.3. Dynamical Simplification

## 3. Hybrid Periodic Orbit

#### 3.1. Choice of Poincaré Section

#### 3.2. Stride Map

#### 3.3. Stride Map Fixed Point

**Proposition 1.**

**Proof.**

#### 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

**Proposition 2.**

**Proof.**

## 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 |
---|---|

${\mathcal{H}}^{}:=({\mathcal{J}}^{},{\mathcal{T}}^{},{\mathcal{D}}^{},{\mathcal{F}}^{},{\mathcal{G}}^{},{\mathcal{R}}^{})$ | Hybrid system (3), (5), (6), (13), (17), (18) |

$\mathrm{F},\mathrm{D},\mathrm{R}$ | Hybrid modes (4) |

${D}_{i}^{},{G}_{i,j}^{},{R}_{i,j}^{},{F}_{i}^{}$ | Mode domains (7), guards (25), resets (30), vector fields (14) |

$t,y,\phi ,\tau $ | Time, mass-center height, body pitch, mode timer (10), Figure 2 |

$x,\phantom{\rule{4pt}{0ex}}{x}^{f},\phantom{\rule{4pt}{0ex}}{x}^{r}$ | Mass-center and front/rear toe horizontal positions (11), Figure 2 |

$\Delta {x}^{f}={x}^{f}-x,\phantom{\rule{4pt}{0ex}}\Delta {x}^{r}={x}^{r}-x$ | Front, rear horizontal leg splay distance with regard to the mass-center (12) |

${\mathit{x}}_{i}^{}:={({{\mathit{x}}_{i}^{\mathit{I}}}^{T},{{\mathit{x}}_{i}^{\mathit{H}}}^{T})}^{T}$ | Mode i state (9), with in-place (10) and horizontal (11) components |

${\mathit{x}}^{\mathit{I}}:={({\mathit{q}}^{\mathit{I}},{\dot{\mathit{q}}}^{\mathit{I}},\tau )}^{T},\phantom{\rule{4pt}{0ex}}{\mathit{q}}^{\mathit{I}}:={(y,\phi )}^{T}$ | In-place state, configuration (10) |

$m,\phantom{\rule{4pt}{0ex}}I,\phantom{\rule{4pt}{0ex}}g,\phantom{\rule{4pt}{0ex}}d$ | Physical model parameters (Figure 2) |

$\Delta {x}^{\mathrm{Avg}},\phantom{\rule{4pt}{0ex}}a,\phantom{\rule{4pt}{0ex}}{l}_{0}$ | Pseudo-physical simplifying parameters (22), (24), (26), Figure 2 |

${G}_{i,j}^{\mathit{I}}$ | In-place components of the guard set (25), (26) |

${y}^{{f}_{hip}}\left({\mathit{x}}^{\mathit{I}}\right),{y}^{{r}_{hip}}\left({\mathit{x}}^{\mathit{I}}\right)$ | Front/rear hip heights (29) |

${g}_{TD}\left({\mathit{x}}_{\mathrm{F}}^{\mathit{I}}\right),\phantom{\rule{4pt}{0ex}}{g}_{LO}\left({\mathit{x}}_{\mathrm{D}}^{\mathit{I}}\right)$ | Guard “control” functions for touchdown, liftoff events (26), (58) |

${\mathit{k}}^{\mathit{I}}={({{{\mathit{k}}^{\mathit{I}}}_{\mathrm{F}}}^{T},{{{\mathit{k}}^{\mathit{I}}}_{\mathrm{D}}}^{T})}^{T}$ | In-place guard control weights (26) |

${{y}^{{f}_{hip}}}_{{}_{i0}}\left({\mathit{x}}^{\mathit{I}}\right),{{y}^{{r}_{hip}}}_{{}_{i0}}\left({\mathit{x}}^{\mathit{I}}\right)$ | Front and rear initial hip height in mode i (59) |

${b}^{}={({{b}^{\mathit{I}}}^{T},\phantom{\rule{4pt}{0ex}}{{b}^{\mathit{H}}}^{T})}^{T}$ | “Bounding” symmetry map (41), (27), (33) |

${\mathcal{L}}_{f}V\left(x\right):=\frac{\partial}{\partial x}V\left(x\right)\xb7f\left(x\right)$ | Lie derivative (28) of scalar field V along vector field f at point x |

${R}_{i,j}^{\mathit{I}},{R}_{i,j}^{\mathit{H}}$ | In-place (31), horizontal (32) reset function components |

${r}_{\mathrm{F},\mathrm{D}}\left({\mathit{x}}_{\mathrm{F}}^{\mathit{H}}\right),\phantom{\rule{4pt}{0ex}}{r}_{\mathrm{D},\mathrm{R}}\left({\mathit{x}}_{\mathrm{D}}^{\mathit{H}}\right)$ | Reset “control” functions (32), (63) |

${\mathit{k}}^{\mathit{H}}:={({k}_{\mathrm{F}}^{\mathit{H}},{k}_{\mathrm{D},1}^{\mathit{H}},{k}_{\mathrm{D},2}^{\mathit{H}})}^{T}\in {\mathbb{R}}^{3}$ | Reset control weights (64) |

$\Delta {x}^{\mathrm{Nom}}$ | Nominal touchdown leg splay for front leg (32) |

$\overline{y}$ | Mass-center height Approximation 1 in pitching dynamics |

${u}_{y}\in (\frac{g}{2},g),\phantom{\rule{1.em}{0ex}}{{u}_{x}}_{i}\left({\mathit{x}}^{}\right)$ | Vertical (16), (20), (34), horizontal (16), (21) mass-specific |

ground reaction force applied from each hip | |

${\varphi}_{i}^{t}\left({\mathit{x}}^{\mathit{I}}\right),\phantom{\rule{1.em}{0ex}}{\widehat{\varphi}}_{i}^{t}\left({\mathit{x}}^{\mathit{H}}\right)$ | In-place (35), horizontal (36) mode-i flow |

${\mathit{c}}_{i}^{}$ | $(y,\phi )$ simplified acceleration vector for mode i (35) |

${C}_{\mathrm{F}},\phantom{\rule{4pt}{0ex}}{C}_{\mathrm{D}},\phantom{\rule{4pt}{0ex}}{C}_{\mathrm{R}}$ | Matrix components used in the description of ${\widehat{\varphi}}_{i}^{t}\left({\mathit{x}}^{\mathit{H}}\right)$ (36) |

${\mathsf{\Phi}}_{i,j},\phantom{\rule{4pt}{0ex}}{\mathsf{\Phi}}_{i,j}^{\mathit{I}},\phantom{\rule{4pt}{0ex}}{\mathsf{\Phi}}_{i,j}^{\mathit{H}}$ | Mode i-to-j map (38), with in-place, horizontal components (39) |

${T}_{i,j}^{\mathit{I}}\left({\mathit{x}}^{\mathit{I}}\right)$ | Mode i time-to-impact map (40) with guard ${G}_{i,j}^{\mathit{I}}$ |

${\tilde{D}}_{i}^{}:={\tilde{D}}_{i}^{\mathit{I}}\times {\tilde{D}}_{i}^{\mathit{H}}$ | Reduced ${D}_{i}^{}$ domain with horizontal, in-place components (42) |

${\tilde{\mathit{x}}}^{}:={(\tilde{\mathit{x}}{{}^{\mathit{I}}}^{T},\tilde{\mathit{x}}{{}^{\mathit{H}}}^{T})}^{T}$ | State on ${\tilde{D}}_{i}^{}$ with in-place and horizontal components (43) |

${\mathsf{\Pi}}^{}\left({\mathit{x}}^{}\right),{\mathsf{\Sigma}}^{}\left({\tilde{\mathit{x}}}^{}\right)$ | Projection and lift maps (44) |

${\mathsf{\Pi}}^{\mathit{I}}\left({\mathit{x}}^{\mathit{I}}\right),\phantom{\rule{4pt}{0ex}}{\mathsf{\Sigma}}^{\mathit{I}}\left({\tilde{\mathit{x}}}^{\mathit{I}}\right),\phantom{\rule{4pt}{0ex}}{\mathsf{\Pi}}^{\mathit{H}}\left({\mathit{x}}^{\mathit{H}}\right),\phantom{\rule{4pt}{0ex}}{\mathsf{\Sigma}}^{\mathit{H}}\left({\tilde{\mathit{x}}}^{\mathit{H}}\right)$ | In-place, horizontal projection, and lift maps (44) |

${S}^{},{H}^{}$ | Stride (45) and “flipped” half-stride (47) maps |

${\tilde{\overline{\mathit{x}}}}^{}={(\tilde{\overline{\mathit{x}}}{{}^{\mathit{I}}}^{T},\tilde{\overline{\mathit{x}}}{{}^{\mathit{H}}}^{T})}^{T}\in {\tilde{D}}_{\mathrm{F}}^{}$ | Fixed point of ${H}^{}$ (48) |

$\overline{\Delta {x}^{f}},\overline{\Delta {x}^{r}}$ | Leg splay components of ${\tilde{\overline{\mathit{x}}}}^{\mathit{H}}$ (50) |

${\overline{T}}_{\mathrm{Stance}},\phantom{\rule{4pt}{0ex}}\delta {\overline{x}}_{\mathrm{Stance}}$ | Total hip stance duration (54), leg-sweep distance (55) on the |

hybrid periodic orbit associated with ${\tilde{\overline{\mathit{x}}}}^{\mathit{H}}$ | |

${\overline{\mathit{x}}}^{}={\mathsf{\Sigma}}^{}\left({\tilde{\overline{\mathit{x}}}}^{}\right)\in {D}_{\mathrm{F}}^{}$ | Lift of ${\tilde{\overline{\mathit{x}}}}^{}$ (60) |

${\overline{T}}_{i,j},{\overline{\mathit{x}}}_{i0,j}^{\mathit{I}}$ | Mode i’s duration (52) and initial state (61) as it evolves into |

mode j under the hybrid execution from ${\overline{\mathit{x}}}^{\mathit{I}}$ | |

${\tilde{b}}^{\mathit{I}},D{\tilde{\mathsf{\Phi}}}_{i,j}^{\mathit{I}}$ | Simplified factors of ${H}^{}$’s in-place component (73) |

## Appendix B. Controller Stability Lemmas

**Lemma A1.**

**Proof.**

**Lemma A2.**

**Proof.**

**Lemma A3.**

**Proof.**

## Appendix C. Control Gain Selection Procedure

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**Figure 1.**The controller presented in this work is empirically demonstrated on the Inu robot [20]. Empirical bounding corresponding to the analytically predicted limit cycles derived in Proposition 1, using the simplified dynamics of Section 2.3, is documented in Section 5.

**Figure 2.**The simplified massless-leg representation of a quadrupedal robot bounding in the sagittal plane. The model’s configuration is shown in blue and is given by the body’s location in $SE\left(2\right)$ with mass-center position $(x,y)$ and body pitch $\phi $, as well as the horizontal location of the front and rear toes encoded either by their toe positions ${x}^{i}$ or splay distance $\Delta {x}^{i}$ from the mass center, $i\in \{f,r\}$. The physical parameters shown in green are the body’s mass m and moment of inertia I about its mass center, the body length d, and gravity’s acceleration g. Each leg in contact with the ground imparts a vertical (${u}_{y}$) and horizontal (${u}_{x}$) mass-specific ground reaction force law at each toe shown in red. Purple values relate to control parameters. The value ${l}_{0}$ is a nominal vertical leg length at the touchdown and liftoff events (used as a control parameter in (26)).

**Figure 4.**Cascaded hybrid dynamics achieved through the choice of force laws and hybrid guards and resets as well as Approximation 1. The choice of force laws (20) and (21) decouple the continuous dynamics of the hybrid system (3) into the cross product of “in-place” and “horizontal” vector fields representing the behavior of the “in-place” vertical and pitching states ${\mathit{x}}^{\mathit{I}}$ as well as the “horizontal” fore-aft mass-center and toe position states ${\mathit{x}}^{\mathit{H}}$. The isolated continuous dynamics—along with the hybrid guards being purely dependent on the in-place states (25) and the hybrid reset maps having a cascaded form (30)—endows a feedforward relationship between the in-place states and horizontal states in which a linearized stability analysis of a hybrid periodic orbit’s Poincaré map Jacobian has the separation-of-eigenvalues property indicated by (2), allowing for a more tractable analysis. A stable limit cycle is achieved by controlling the hybrid guards and the resets via (26), (31) and (32). In the vertical states, this is accomplished on the guards by vertically retracting the leg in stance to transition to flight and similarly by protracting the leg in flight to affect the onset of stance. In the horizontal states, this is accomplished on the resets by placing the toe position horizontally in flight in a similar fashion to Raibert’s neutral-point algorithm [13].

**Figure 5.**Traces of the predicted hybrid periodic orbit over a full stride using the parameters of Table 2 at a commanded speed of 1 m/s are provided so as to give the reader an early intuition of what the periodic orbits will look like in the later experimental section. These state variable traces characterize a useful steady-state bounding gait with realistically small oscillations in body height and forward speed. The readers will notice that the traces of the hybrid dynamical system are smooth everywhere except for points corresponding with the guards and resets in the next mode. The background color indicates the mode (4). Green is $\mathrm{F}$, blue is $\mathrm{D}$, and yellow is $\mathrm{R}$. In the $\Delta x$ graph, the blue trace gives $\Delta {x}^{r}$ while the orange trace gives $\Delta {x}^{f}$ (12). Notice that deviations in body height y and forward speed $\dot{x}$ are quite small, indicating a valid Approximation 1 as discussed in Section 3.4 and a small value of $\xi $ from Table 1.

**Figure 6.**Two slices of the numerically computed basin of attraction when the hybrid mode sequence is enforced, using parameters given in Table 2 (left—in the $(\phi ,y)$ plane; right—in the $(\dot{\phi},\dot{y})$ plane). The blue region indicates the basin, and the center orange dot corresponds with the fixed point ${\overline{\mathit{x}}}^{\mathit{I}}$ of the map ${H}^{\mathit{I}}$. The enforcement of the hybrid mode sequence is a very conservative assumption for real-world implementation, as the ability to move through transient hybrid mode sequences is an inherent affordance of legs that provides robustness and motivates their use on machines.

**Figure 7.**Robustness of deadbeat solution to perturbations in the parameters ${u}_{y}$ and the unitless a, as indicated by the value of the spectral radius of the Jacobian of ${H}^{\mathit{I}}$ when the true parameter values are varied from the parameter values used by the controller in Table 2, evaluated at the fixed point that results from this parameter perturbation. To give the reader an intuition on the range of a displayed, below the graph are cartoon representations of the robot for a generalized Murphy value a of $0.6$, $1.0$, and $1.4$, assuming all the robot mass is equally distributed at two point masses along the robot. The controller becomes unstable when the spectral radius exceeds unity, indicated by the red line. The parameters a and ${u}_{y}$ are the two parameters which are difficult to measure on the physical robot. The large distance from the unperturbed case (indicated by the orange dot) to the onset of destabilizing perturbations (indicated by the red line) suggests a large degree of robustness to uncertainty in these parameters.

**Figure 8.**Slices of the Jacobian spectral radius of ${H}^{\mathit{H}}$ evaluated at the appropriate fixed point with parametric perturbations in the parameters $\overline{y}$, ${\overline{T}}_{\mathrm{F},\mathrm{D}}$, and ${u}_{y}$—the only parameters entering into the Jacobian. This analysis uses numerical parameter values given in Table 2 as the unperturbed values. Here, the control is performed using the unperturbed parameters, showing the robustness of the control scheme to parametric uncertainty. The distance from the orange dot in the lower-left plot (representing the unperturbed parameter values) to the red line (indicating slices of the edge of stability) demonstrates that the controller can withstand sizable perturbations in parameter space before becoming unstable.

**Figure 9.**The in-place component of the controller implemented on the Inu robot shows good correspondence between the actual (blue) and analytically predicted (red) behavior of the robot over approximately 30 strides (10 s) of motion capture data. Here, the horizontal toe position is maintained through the use of a simple PD controller with relatively high-magnitude derivative term to dampen out fore-aft oscillations.

**Figure 10.**Depicted are the actual (blue) and desired (red) orbits and trajectories under motion capture using the full controller of Section 4 on the Inu robot over various running speeds up to Inu’s kinematic speed limit. As further discussed Section 5.2, we see a reasonable agreement with the desired limit cycle at lower speeds (top). At higher speeds (middle), we see the orbit of the pitch degree of freedom inconsistently sag during negative pitch values corresponding to when the front is in stance, as the front is slightly heavier than the rear. Approaching the speed limit imposed by Inu’s kinematics (bottom), Inu’s legs are commanded to lift off prematurely when they near their kinematic singularity as shown in Figure 11, which results in inconsistent trajectories. The lower time durations of the faster experiments are the result of the robot running faster through the motion capture area.

**Figure 11.**Toe kinematic trajectories for the front legs in the local hip frame show that at running speeds of $1.6$ m/s, the leg linkage is close to singularity. This represents a constraint on maximum running speed, as the leg runs out of workspace to sweep the leg backwards in stance. Faster running could be achieved by either using longer legs to increase the workspace or by achieving shorter stance durations through increasing the applied vertical stance force. In future work, we will investigate the addition of a spine morphology to provide this added workspace without detracting from the hip’s torque generation affordance.

**Table 1.**Minimum and maximum state values along the hybrid periodic orbit associated with the fixed point ${\tilde{\overline{\mathit{x}}}}^{}$ of Proposition 1.

State | Min Value on Orbit | Max Value on Orbit |
---|---|---|

y | ${l}_{0}+\frac{1}{8}{{\overline{T}}_{\mathrm{F},\mathrm{D}}}^{2}\frac{g-{u}_{y}}{2{u}_{y}-g}(\zeta -{u}_{y})$ | ${l}_{0}+\frac{1}{8}{{\overline{T}}_{\mathrm{F},\mathrm{D}}}^{2}\frac{g-{u}_{y}}{2{u}_{y}-g}\zeta $ |

$\zeta =2{u}_{y}(1-{a}^{-1})-g$ | ||

$\phi $ | $-\frac{g{u}_{y}{{\overline{T}}_{\mathrm{F},\mathrm{D}}}^{2}}{4ad(2{u}_{y}-g)},$ | $\frac{g{u}_{y}{{\overline{T}}_{\mathrm{F},\mathrm{D}}}^{2}}{4ad(2{u}_{y}-g)}$ |

$\dot{y}$ | $-\frac{g-{u}_{y}}{2}{\overline{T}}_{\mathrm{F},\mathrm{D}},$ | $\frac{g-{u}_{y}}{2}{\overline{T}}_{\mathrm{F},\mathrm{D}}$ |

$\dot{\phi}$ | $-\frac{{u}_{y}}{ad}{\overline{T}}_{\mathrm{F},\mathrm{D}},$ | $\frac{{u}_{y}}{ad}{\overline{T}}_{\mathrm{F},\mathrm{D}}$ |

$|\dot{x}|$ | $\sqrt{{\dot{\overline{x}}}^{2}-\xi},$ | $|\dot{\overline{x}}|$ |

$\xi =\frac{{u}_{y}}{\overline{y}}\xb7max\{{(\Delta {x}^{\mathrm{Avg}}-\overline{\Delta {x}^{f}})}^{2},$ | $\frac{1}{2}{(\Delta {x}^{\mathrm{Nom}}-\overline{\Delta {x}^{f}})}^{2}\}$ | |

$\Delta {x}^{r}$ | $-\Delta {x}^{\mathrm{Nom}},$ | $-(2\Delta {x}^{\mathrm{Avg}}-\Delta {x}^{\mathrm{Nom}})$ |

$\Delta {x}^{f}$ | $2\Delta {x}^{\mathrm{Avg}}-\Delta {x}^{\mathrm{Nom}},$ | $\Delta {x}^{\mathrm{Nom}}$ |

**Table 2.**Parameter values used in experiments. As explained near the end of Section 4.3, the nine control weights were used to place seven poles at the origin according to (A8), (A11) and (A4), fully determining both ${{\mathit{k}}^{\mathit{I}}}_{\mathrm{F}}$ and ${\mathit{k}}^{\mathit{H}}$ while leaving ${{\mathit{k}}^{\mathit{I}}}_{\mathrm{D}}$ constrained to a hypersurface. Having achieved infinitesimal deadbeat stability, we chose the remaining control parameters according to the constrained optimization procedure given in Appendix C to optimize various other performance metrics.

Numerical Parameters | Symbol | Value |
---|---|---|

Physical and pseudo- | d | $0.47m$ |

physical parameters | ${l}_{0}$ | $0.22m$ |

a | 1 | |

$\Delta {x}^{\mathrm{Avg}}$ | $\frac{d}{2}$ | |

$\overline{y}$ | $0.21m$ | |

g | $9.81\frac{m}{{s}^{2}}$ | |

Fixed-point parameters | ${u}_{y}$ | $8.5\frac{m}{{s}^{2}}$ |

${\overline{T}}_{\mathrm{F},\mathrm{D}}$ | $0.15s$ | |

$\dot{\overline{x}}$ | Varies by experiment | |

Control weights | ${{\mathit{k}}^{\mathit{I}}}_{\mathrm{F}}$ | ${(0.544,\phantom{\rule{4pt}{0ex}}-0.082,\phantom{\rule{4pt}{0ex}}0.299)}^{T}$ |

${{\mathit{k}}^{\mathit{I}}}_{\mathrm{D}}$ | ${(0.427,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}-0.314)}^{T}$ | |

${\mathit{k}}^{\mathit{H}}$ | ${(0.207,\phantom{\rule{4pt}{0ex}}-0.126,\phantom{\rule{4pt}{0ex}}0)}^{T}$ |

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## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Duperret, 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