Smoothness Analysis Module

Overview

The Smoothness module quantifies control and fluidity of movement using established motor control metrics. Smooth movements are characterized by continuous, coordinated trajectories with minimal abrupt changes.

The module implements two primary metrics validated in motor control research 12:

Signal Filtering

Raw derived speed profiles are notoriously noisy. By default, the Smoothness module applies a Savitzky-Golay filter to clean the kinematic sequence prior to extracting SPARC or Jerk, avoiding noise amplification during differentiation.

Algorithm Details & Mathematics

The module implements two primary metrics validated in motor control research.

1. Spectral Arc Length (SPARC)

The Spectral Arc Length (SPARC) quantifies movement smoothness by measuring the arc length of the normalized Fourier magnitude spectrum of the speed signal \(s(t)\).

  1. Compute the Fourier magnitude spectrum: Take the Fast Fourier Transform (FFT) of the input signal and keep only the positive frequencies:
\[ Y(f) = |\text{FFT}[s(t)]|_{f > 0} \]
  1. Normalize the spectrum: Normalize the magnitude by its maximum value:
\[ \hat{Y}(f) = \frac{Y(f)}{\max(Y(f))} \]
  1. Calculate geometric arc length: Compute the total Euclidean arc length of this curve across frequencies:
\[ L = \sum_{i=1}^{N-1} \sqrt{(f_{i+1} - f_i)^2 + (\hat{Y}_{i+1} - \hat{Y}_i)^2} \]
  1. Return SPARC:
\[ \text{SPARC} = -L \]

2. Jerk Root Mean Square (RMS)

The Jerk Root Mean Square (RMS) measures smoothness as the average magnitude of the finite-difference derivative of the input signal. Because the input \(s(t)\) is a speed profile, its first derivative represents acceleration, and the algorithmic logic correctly surfaces Jerk.

  1. Discrete Derivative: Approximate the temporal derivative using finite differences with sampling rate \(f_s\):
\[ j_i = \frac{s_{i+1} - s_i}{1 / f_s} \]
  1. Root Mean Square:
\[ \text{RMS}_j = \sqrt{\frac{1}{N} \sum_{i=1}^{N} j_i^2} \]

References

  1. Mazzarino, B., & Mancini, M. (2009). The need for impulsivity & smoothness: improving hci by qualitatively measuring new high-level human motion features. In Proceedings of the International Conference on Signal Processing and Multimedia Applications (IEEE sponsored). ↩

  2. Balasubramanian, S., Melendez-Calderon, A., Roby-Brami, A., & Burdet, E. (2015). On the analysis of movement smoothness. Journal of neuroengineering and rehabilitation, 12(1), 112. ↩

  3. Melendez-Calderon, A., Shirota, C., & Balasubramanian, S. (2021). Estimating movement smoothness from inertial measurement units. Frontiers in bioengineering and biotechnology, 8, 558771. ↩