Human gait modelling and tracking based on motion functionalisation

Table 1 shows the experimental measurement results of the limb motion amplitude of the above nine subjects walking at different speeds [7]. We use height as the abscissa and motion amplitude as the ordinate and plot the measurement results at each speed on the same graph. As shown in Figure 3, the ‘*’ point corresponds to the data in Table 1, h is the height of the subject and a_y is the amplitude of limb movement. The minimum two multiplication method is used to fit the data to accurately express the relationship between height and amplitude at different speeds. The fitting process is realised by Matlab software. The straight line pointed by the arrow in Figure 3 is the primary fitting curve at the corresponding speed. The expression of the fitting function is (1) ay=ah+b {a_y} = ah + b

In the formula, a and b are the fitting coefficients.

Fig. 3

It can be seen from Figure 3 that there is a fitting error between the fitted curve and the measured value. The expression for calculating the fitting error Δe_y of the measuring point is (2) Δey=ay*−ay(h*) \Delta {e_y} = a_y^* - {a_y}({h^*})

The formula ay* a_y^* is the measured value and a_y(h^*) is the fitting function value obtained when the height is taken from the measured value. The expression for calculating the variance e_y of each set of fitting functions is (3) ey=∑Δey2m {e_y} = \sqrt {{{\sum \Delta e_y^2} \over m}}

In the above equation, m is the number of subjects, i.e. the number of measurement points at each speed. We take the average of the fitting variance of each speed, and the expression of the average variance Δe of all fitting functions can be obtained as (4) Δe=∑eyn \Delta e = {{\sum {e_y}} \over n} where n is the number of fitting functions, i.e. the number of speed values. The expression for calculating the average relative error ε_y is (5) εy=ΔeΔay* {\varepsilon _y} = {{\Delta e} \over {\Delta a_y^*}}

In the above equation, Δay* \Delta a_y^* is the average value of all amplitude measurements, and the expression is (6) Δay*=∑ay*nm \Delta a_y^* = {{\sum a_y^*} \over {nm}}

Table 2 shows the fitting coefficients a, b, variance e_y, average variance Δe and relative fitting error ε_y between the measured value and the fitted value at different speeds in Figure 3. With speed as the abscissa, and the fitting coefficients a and b as the respective ordinates, the position of the ‘*’ point is shown in Figure 4. The straight line is the fitting curve of the two, and the fitting function is as follows: (7) fa(v)=−73.07v+134.27 fa(v) = - 73.07v + 134.27 (8) fb(v)=106.51v−195.47 fb(v) = 106.51v - 195.47

Fig. 4

Similarly, the relative fitting error of coefficient a is 1.50%, and the relative fitting error of coefficient b is 1.86%. Substituting Eqs (7) and (8) into Eq. (1), it can be inferred that when healthy young people walk naturally on the treadmill, the mathematical model of the amplitude of the limbs in the left and right directions is as follows: (9) ay(h,v)=−73.07vh+134.27h+106.51v−195.47 {a_y}(h,v) = - 73.07vh + 134.27h + 106.51v - 195.47

From the experimental data, it can be concluded that during the walking process of the subject, the limb swings once in one gait cycle, which indicates that the limb movement cycle is consistent with the gait cycle [8]. Table 3 shows the same subject in the gait cycle, with the limb movement cycles at different walking speeds. Table 4 shows the limb movement cycles when subjects with different heights walk at a speed of 0.8 m/s.

We draw the experimental data with walking speed and height as the abscissa and the limb movement period as the ordinate. As shown in Figure 5, the ‘*’ points are the measurement data points. The least-squares method is also used for data fitting, and the period and walking are obtained. The fitting curve of the relationship between speed and height is shown in the Figure 5. From this, it can be seen that during the natural walking of healthy young people, the limb movement cycle is mainly affected by walking speed, and height has no effect on it. It is derived by the polynomial fitting method. The mathematical model of the relationship between the cycle and walking speed is as follows [Eq. (10)]. Similarly, the relative fitting error of the fitting function is 4.3%.

Fig. 5

The walking speed is taken as 0.8 m/s and substituted into the above formula, and the gait period is 1.25 s. The average value of the measurement data in Table 3 is 1.24 s, which shows that the measured value and the fitted value are the same.

It can be seen from Figures 1 and 2 that at the beginning of the gait cycle, the limb is located on the left side of the intermediate balance position (taking the left heel landing time as the gait cycle dividing point), and then the first wave peak is reached through the balance position; so is the initial phase of the sine curve [10]. Table 5 shows the initial phase φ_y of the limb movement trajectory of the same subject at different walking speeds. Table 6 shows that subjects with different heights all walk at a speed of 0.8 m/s. The initial phase φ_y of each time is calculated according to the method described above, and the initial phase expression of the limb swing at the beginning of the gait cycle is given by: (11) φy=0.48v−0.76 {\varphi _y} = 0.48v - 0.76

Similarly, the relative error of this fitting function is 1.8%.

The paper uses a sine function to establish a mathematical model of the limb movement trajectory, and the expression is (12) yp=aysin(2πTyt+φy) {y_p} = {a_y}\sin \left({{{2\pi} \over {{T_y}}}t + {\varphi _y}} \right)

The paper substitutes the models in Eqs (9), (10) and (11) of the amplitude, period and initial phase obtained in the above sections into Eq. (12), respectively, to obtain the trajectory of the limbs in the left and right directions during the natural walking of healthy young people. The mathematical model is as follows: (13) yp=(−73.07vh+134.27h+106.51v−195.47)⋅sin(2π1.52v2−3.24v+2.87t+0.48v−0.76) {y_p} = \left({- 73.07vh + 134.27h + 106.51v - 195.47} \right) \cdot \sin \left({{{2\pi} \over {1.52{v^2} - 3.24v + 2.87}}t + 0.48v - 0.76} \right)

When tracking the first frame, we don’t have any tracking history information to guide the estimation of the state. If the blind search is used, it will inevitably involve a heavy computational cost and affect the tracking efficiency. Therefore, we need a specific method to roughly estimate the state of the human body in the first frame [12]. The prior knowledge we can use is the periodic function model of joint angles in gait motion obtained through model learning. We select a particle set of 500 particles to be distributed according to Gaussian at the initialisation time.

The dynamic model uses continuity assumptions and prior knowledge (motion model and motion constraints) to predict the human pose represented by the particle state in the current frame according to the motion function of each joint [13]. A particle set of 500 particles is selected for error correction, and the corrected particle set has a μ(0, 0.5) Gaussian distribution.

We match the motion regions extracted from the video with the motion regions represented by the particles to detect the degree of matching for all particles. Regional matching of the head, upper body and arm parts is carried out. The movement of the head and the upper body torso is relatively simple, and the movement of the arm is more complicated, but the phenomenon of occlusion between the torso and the torso is serious when walking, and the functionalisation of human movement can be in the arm. We can predict the movement of the arm more accurately during occlusion [14]. Therefore, in the feature matching stage, we only use region matching for the head, upper body and arms. Subsequently, regional matching and edge matching of the legs is carried out. Among the changes of various parts of the human body during walking, the movement of the legs is the part that best reflects the characteristics of human walking. Therefore, in the field of human motion tracking, especially gait tracking, the accuracy of leg tracking is higher. In the research carried out in this paper, the feature matching of the legs uses both edge features and regional features, as well as the final region matching error Er, and edge matching error and feature error, which determine the proportion of regional matching error and edge matching error in the feature error.

The particle update process in this paper is different from the re-sampling update of the ordinary particle filter [15]. In the tracking study of this part, the particle state represents the error; so, the difference between the particle states at adjacent moments is not significant; thus, in the process of particle updating, the re-sampling operation is no longer needed. The generation of the new particle set is also sampled according to the Gaussian distribution. The mean of this Gaussian distribution is the output of the best particle state at the last moment, and the variance remains unchanged. The tracking effect is shown in Figure 6.

Fig. 6