Postural Balance (Equilibrium) Analysis Module

Overview

The Postural Balance module provides a robust elliptical equilibrium evaluation metric. It computes how well a subject's barycenter is centered between two base-of-support points (e.g., the left and right feet) to mathematically estimate the stability of a physical posture.

Theoretical Interpretation

  • Input Requirements: Expects a single frame of static position data spanning multiple joints. The algorithm exclusively relies on a 2D projection (specifically, the X and Y components) ignoring height (Z), as postural stability relative to the ground is a planar problem.
  • Value Interpretation:
    • An equilibrium value of 1.0 indicates perfect centering (the barycenter perfectly splits the base of support).
    • Values approaching 0.0 indicate extreme stretching, leaning, or falling, where the user's center of mass is breaching the edge of their support ellipse.
    • The algorithm also reports the angle of the base of support relative to the horizontal plane.

Margin Safeties

The mathematical ellipse is intentionally inflated by a margin_mm config to account for the physical footprint size extending beyond the single theoretical "foot joint" point tracked by typical pose systems.

Algorithm Details & Mathematics

The stable base of support is modeled as a 2D ellipse drawn around the left foot (\(P_{left}\)) and right foot (\(P_{right}\)).

  1. Center and Axes Formulation The center point \(C\) between the feet is derived as:
\[ C = \frac{P_{left} + P_{right}}{2} \]

The Euclidean distance between the feet is \(d = \|P_{right} - P_{left}\|\). The ellipse semi-major axis (\(a\)) incorporates the safety margin:

\[ a = \frac{d}{2} + margin \]

The ellipse semi-minor axis (\(b\)) determines the depth of the stability stance, weighted by \(y\_weight\):

\[ b = margin \times y\_weight \]

  1. Rotational Alignment The system computes the relative position of the barycenter to the geometric center \(C\), and rotates this relative vector to align linearly with the vector bridging the two feet. Let this rotated barycenter coordinate be \(B' = (x', y')\).

  2. Elliptical Evaluation The normalized distance equation inside the ellipse is computed as:

\[ norm = \left(\frac{x'}{a}\right)^2 + \left(\frac{y'}{b}\right)^2 \]

The final equilibrium score \(Value \in [0, 1]\) is extracted inversely from the norm:

\[ Value = \begin{cases} 1.0 - \sqrt{norm} & \text{if } norm \le 1.0 \\ 0.0 & \text{otherwise} \end{cases} \]

References

  1. Ghisio, S., Coletta, P., Piana, S., Alborno, P., Volpe, G., Camurri, A., ... & Ravaschio, A. (2015, June). An open platform for full body interactive sonification exergames. In 2015 7th International Conference on Intelligent Technologies for Interactive Entertainment (INTETAIN) (pp. 168-175). IEEE. ↩