*3.3. The Vertical Stiffness of the Hip*

The empirical coefficients and initial phases of thighs and shanks shown in Table 1 are substituted into Equations (8)–(11) and Equation (4), and these equations are then substituted into Equation (5) to calculate the vertical stiffness of the hip. The obtained vertical stiffness of the left and right hips is shown in Figure 9a,b, respectively. The maximum value of hip stiffness reaches approximately 1 <sup>×</sup> <sup>10</sup><sup>6</sup> N/m, and its fluctuation accounts for half of the gait cycle. The stiffness in another half cycle is approaching zero, which seems unchanged. To study the unchanged section, the highly fluctuating section is hidden, as shown in Figure 9c. The theoretical GRF of both lower limbs is shown in Figure 9d to recognize the corresponding gait phase of the two sections.

The vertical stiffness of the hip is extremely high when its corresponding leg goes from the mid-stance phase to the mid-swing phase. From the mid-swing phase to the mid-stance phase, the vertical stiffness of the hip is rather small at about 5 N/m but with a regular shape like 'w'. There is a discontinuity of the first kind at the mid-stance phase and the mid-swing phase. Furthermore, the vertical stiffness between the right and left hips, like the knee, has a time delay for half of the gait cycle. During walking, the vertical stiffness of the knee and hip varies with the gait phase (time).

**Figure 9.** Vertical stiffness of the hip during walking. (**a**,**b**) denote the left and right hip, respectively, and it is evident that the fluctuation is rather high with a magnitude of 10<sup>6</sup> ; (**c**) shows the hip stiffness at the section of low value; and (**d**) is the vertical GRF of the double lower limbs. The green rectangles stand for half of the gait cycle.
