2.4.5. Gait Parameters Estimation

After calculating orientation, position and determining FO and FC events, temporal and spatial parameters were estimated for each gait cycle n. Temporal parameters—stride, swing and stance duration—were determined as defined in [14]. Cadence was obtained as the inverse of stride duration, expressed in steps per minute. Spatial parameters (illustrated in Figure 2) were calculated using information of moving intervals, defined by the temporal bounds of tn and tn+1. To estimate SL and SW, we used trajectories on the horizontal plane, sxy, as determined by Equations (12) and (13), where s→n(t) represents a displacement vector relative to the final stride position at tn+1, obtained as sxy(tn+1)−sxy(t)). In Equation (13), the symbol ∠ denotes the angle between two vectors.

Gait speed was obtained by dividing SL by its corresponding stride duration.

To calculate MTC, we used a method inspired on the work by Kanzler, C. [9]. To estimate toe trajectory, we have first estimated the distance between the sensor and the toe, r, using as a basis the angle produced by the foot at FO, α(FOn), in each gait cycle n, as illustrated in Figure 4.

Variables involved in the calculation of toe trajectory.

To obtain the angle α(t), we did a series of vector transformations. First, we have converted the vertical vector [0,0,1] to sensor coordinates, using the quaternion at the beginning of the moving interval, i.e., at the foot flat at tn. Then, we transformed the resultant vector back to global coordinates using the quaternion at FO. This vector, v→w, was used to estimate the medio-lateral vector, l→w, using the cross product between v→w and [0,0,1]. The forward vector, f→w, was calculated using the cross product between [0,0,1] and l→w, which was then converted back to sensor coordinates using the quaternion at tn. This vector, f→s, parallel to the ground at foot flat and pointing forward towards the toes, was used to estimate the angle α(t), as depicted in Equation (14).

The distance between the sensor and the toe (r) was obtained by the average of the values determined in each stride n, as shown in Equation (15).

where N is the total number of strides and sz(t) represents the z component of the trajectory of the sensor (i.e., its vertical displacement). MTC was considered the minimum peak vertical toe displacement measured during the swing phase of walking. This vertical displacement, m(t), was estimated as shown in Equation (16), where r×sin(α(t)) represents the vertical distance between the sensor and the toe (shown as j(t) in Figure 4).

To calculate turning angles, we converted an arbitrary horizontal vector (e.g., the vector [0,1,0]) to sensor coordinates, using as basis the quaternions estimated at tn and at tn+1. The resultant vectors represented the orientation of the sensor in the horizontal plane, so that the angle between these two vectors corresponded to the turning angle.