Algorithm for Temporal Gait Analysis Using Wireless Foot-Mounted Accelerometers

Abstract

We present a new signal processing algorithm that extracts five gait events: heel strike, toe strike, heel-off, toe-off, and heel clearance from only two accelerometers attached on the heels of the subjects usual shoes. This algorithm first uses a continuous wavelet-based segmentation that parses the signal of consecutive strides into motionless periods defining relevant local acceleration signals. Then, the algorithm uses versatile techniques to accurately extract the five gait events from these local acceleration signals. We validated, on a stride-by-stride basis, the extraction of these gait events by comparing the results with reference data provided by a kinematic 3D analysis system and a video camera. The accuracy and precision achieved by the extraction algorithm for healthy subjects, the reduced number of accelerometer units required, and the validation results obtained, encourage us to further study this system in pathological conditions.

Appendix

We present the piecewise-linear fitting method used to estimate the locations of the convex curvature in a signal (Sect. 2.4). For this, we consider a given signal $sig = sig(t_1 ),sig(t_2),\dots ,sig(t_N)$ defined in a time interval $I=t_1,t_2,\dots ,t_N$, where N is the total number of samples of sig. This method first computes the coefficients of piecewise-linear functions with two linear segments that best fit sig in the least-square sense, leading to the computation of least-square errors. The minimum of these least-square errors is then determined and the associated piecewise-linear function provides two linear segments that intersect at the breakpoint $(t_b,sig(t_b))$. The main steps to determine the breakpoint $(t_b,sig(t_b))$ are as follows:

• For each $k=1,\dots ,N$, one computes the coefficients $\alpha _1$, $\alpha _2$, $\beta _1$, and $\beta _2$ of a piecewise-linear function $p_k$ that best fits sig by minimizing

  $$\begin{aligned} E_k=\sum _{i=1}^{N} (sig(t_i)-p_k (t_i))^2, \end{aligned}$$

  (3)

  where

  $$\begin{aligned} p_k (t) = \left\{ \begin{array}{lll} \alpha _1*t+\beta _1, &{}\quad \textit{if}\; t\in [t_1,t_k],\\ \alpha _2*t+\beta _2, &{}\quad \textit{if}\; t\in [t_{k+1},t_N]. \end{array}\right. \end{aligned}$$

  (4)

  This error can be expressed as

  $$\begin{aligned} E_k=||A\,X_k-B||^2, \end{aligned}$$

  (5)

  where

  $$\begin{aligned} X_k=\left( \begin{array}{l} \alpha _1\\ \beta _1 \end{array}\right) \, A=\left( \begin{array}{ll} t_1\, &{} 1\\ \vdots \, &{} \vdots \\ t_k\, &{}1 \end{array}\right) \, B=\left( \begin{array}{l} sig(t_1)\\ \vdots \\ sig(t_k) \end{array}\right) \, if\, t\in [t_1,t_k], \end{aligned}$$

  and

  $$\begin{aligned} X_k=\left( \begin{array}{l} \alpha _2\\ \beta _2 \end{array}\right) \, A=\left( \begin{array}{ll} t_{k+1}\, &{} 1\\ \vdots \, &{} \vdots \\ t_N\, &{}1 \end{array}\right) \, B=\left( \begin{array}{l} sig(t_{k+1})\\ \vdots \\ sig(t_N) \end{array}\right) \, if\, t\in [t_{k+1},t_N], \end{aligned}$$

  The normal equations associated with (5) are

  $$\begin{aligned} A^{t}AX_k=A^{t}B. \end{aligned}$$

  (6)

  Solving (6) leads to the coefficients $\alpha _1$, $\alpha _2$, $\beta _1$, and $\beta _2$.

• Finally, one obtains the breakpoint $(t_b,sig(t_b))$ by determining the minimum of the least-square errors, i.e.,

  $$\begin{aligned} E_b=\underset{k=1,\dots ,N}{min} (E_k). \end{aligned}$$

  (7)