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Equilibrium Analysis Module

The Equilibrium module quantifies balance control by evaluating the barycenter position relative to an elliptical region defined by the two feet.

The metric produces a normalized equilibrium value in \([0, 1]\) indicating whether the barycenter lies within the ellipse, and the ellipse orientation angle in degrees.

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Algorithm Details

Ellipse Construction

  1. Inputs:

    • Left foot position \( p_s = (x_s, y_s) \)
    • Right foot position \( p_d = (x_d, y_d) \)
    • Barycenter \( b = (x_b, y_b) \)
    • Margin \( m \) (mm)
    • Y-axis weight \( w_y \)

  2. Bounding box with margin:

\[ \text{min} = \min(p_s, p_d) - m, \quad \text{max} = \max(p_s, p_d) + m \]
  1. Ellipse center:
\[ c = \frac{\text{min} + \text{max}}{2} \]
  1. Ellipse Semi-axes:
\[ a = \frac{\text{max}_x - \text{min}_x}{2}, \quad b = \frac{\text{max}_y - \text{min}_y}{2} \cdot w_y \]
  1. Ellipse orientation
\[ \theta = \arctan2(y_d - y_s, \; x_d - x_s) \]

Interpretation

The angle \( \theta \) aligns the ellipse relative to the X-axis (the line connecting the feet).

Normalized Value Computation

1.Relative position: Rotate barycenter into ellipse-aligned coordinates

\[ r = R(-\theta) \cdot (b - c) \]

with rotation matrix

\[ R(-\theta) = \begin{bmatrix} \cos(-\theta) & -\sin(-\theta) \\ \sin(-\theta) & \cos(-\theta) \end{bmatrix} \]
  1. Normalization
\[ \text{norm} = \left(\frac{r_x}{a}\right)^2 + \left(\frac{r_y}{b}\right)^2 \]
  1. Equilibrium value
\[ \text{value} = \begin{cases} 1 - \sqrt{\text{norm}}, & \text{if } \text{norm} \leq 1 \\ 0, & \text{otherwise} \end{cases} \]

Interpretation

  • Value = 1: barycenter at ellipse center (optimal balance).
  • Value = 0: barycenter outside ellipse (loss of balance).
  • Values in between indicate proximity to the center.

Usage Examples

Basic Equilibrium Evaluation

import numpy as np
from pyeyesweb.low_level import Equilibrium

# Initialize analyzer
eq = Equilibrium(margin_mm=120, y_weight=0.6)

# Define foot positions and barycenter
left = np.array([0, 0, 0])
right = np.array([400, 0, 0])
barycenter = np.array([200, 50, 0])

value, angle = eq(left, right, barycenter)

print(f"Equilibrium value: {value:.2f}")
print(f"Ellipse angle: {angle:.1f}°")