Variability Analysis of Therapeutic Movements using Wearable Inertial Sensors

Abstract

A variability analysis of upper limb therapeutic movements using wearable inertial sensors is presented. Five healthy young adults were asked to perform a set of movements using two sensors placed on the upper arm and forearm. Reference data were obtained from three therapists. The goal of the study is to determine an intra and inter-group difference between a number of given movements performed by young people with respect to the movements of therapists. This effort is directed toward studying other groups characterized by motion impairments, and it is relevant to obtain a quantified measure of the quality of movement of a patient to follow his/her recovery. The sensor signals were processed by applying two approaches, time-domain features and similarity distance between each pair of signals. The data analysis was divided into classification and variability using features and distances calculated previously. The classification analysis was made to determine if the movements performed by the test subjects of both groups are distinguishable among them. The variability analysis was conducted to measure the similarity of the movements. According to the results, the flexion/extension movement had a high intra-group variability. In addition, meaningful information were provided in terms of change of velocity and rotational motions for each individual.

Appendix: Time-domain features

The time-domain features are divided into two groups: central tendency (Eqs. 6 and 7) and dispersion (Eqs. 8–12), as described below.

In all the formulas detailed below: n is the size of the signal X, x is a value of X, th indicates the th-term of the ordered time serie X, and Q1 and Q3 are the lower and upper quartile of X.

• Arithmetic mean ($\bar {x}$):

  $$ \bar{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n} $$

  (6)

• Median (f(x)):

  $$ f(x) = \left\{\begin{array}{llllllll} \left( \frac{n+1}{2}\right)^{th} term & \text{when}\, n\, \text{is odd}\\ \frac{\left( \frac{n}{2}\right)^{th} term+\left( \frac{n+1}{2}\right)^{th} term}{2} & \text{when}\, n\, \text{is even} \end{array}\right. $$

  (7)

• Standard deviation (σ):

  $$ \sigma=\sqrt{\frac{1}{n}\left[\left( x_{1}-\bar{x}\right)^{2}+\left( x_{2}-\bar{x}\right)^{2}+...+\left( x_{n}-\bar{x}\right)^{2}\right]} $$

  (8)

• Variance (σ ²):

  $$ \sigma^{2}=\frac{1}{n}\left[\left( x_{1}-\bar{x}\right)^{2}+\left( x_{2}-\bar{x}\right)^{2}+...+\left( x_{n}-\bar{x}\right)^{2}\right] $$

  (9)

• Root mean square (rms):

  $$ rms=\sqrt{\frac{1}{n}\left( {x_{1}^{2}}+{x_{2}^{2}}+...+{x_{n}^{2}}\right)} $$

  (10)

• Interquartile range (iqr):

  $$ iqr=Q_{3}-Q_{1} $$

  (11)

• Mean absolute deviation (mad):

  $$ mad=\frac{1}{n}\left[\mid x_{1}-\bar{x} \mid +\mid x_{2}-\bar{x} \mid +...+\mid x_{n}-\bar{x} \mid \right] $$

  (12)

• Minimum is the smallest value of the sample

• Maximum is the largest value of the sample