*3.1. The Empirical Parameters of Unified Representation*

As obtained from Section 2.4, all the vertical oscillations of lower limb joints and segments can be obtained with amplitude coefficients and initial phases as represented in Equations (8) and (9). Furthermore, Equations (10) and (11) represent the VGRF with amplitude coefficients, initial phases, walking cadence, and body weight. Their average value across all the subjects is obtained as illustrated in Section 2.5, and they are displayed in Table 1. The vertical displacement of the lower limb can be expressed directly with leg length and walking cadence using these parameters. Moreover, the estimated vertical oscillations were compared to the measured data, as shown in Figure 7. It can be seen that the unified equation with the empirical parameters obtained in Table 1 fits the measured oscillation of the lower limbs well.


**Table 1.** The parameters of the vertical displacement of the lower limbs.

**Figure 7.** Comparison of vertical displacement between the measurement and the estimation by the individual parameters.

This unification of the quantitative description of human lower limb oscillation during overground walking helps to establish a general representation of the dynamic characteristics such as stiffness.

#### *3.2. The Vertical Stiffness of the Knee*

The vertical displacements of the thigh and shank can be represented by the leg length and walking cadence, as illustrated in Equation (12). Figure 8a shows the obtained vertical stiffness of the left knee across several gait cycles after substituting the empirical coefficients and initial phase shown in Table 1 into Equation (12) and then calculating it with MATLAB 2019 a. With the same process, the vertical stiffness of the right knee is calculated and displayed in Figure 8b. In addition, the corresponding ground reaction force is shown in Figure 8c. It can be observed that the vertical stiffness in the knee experienced three changing stages in one stride cycle.

As shown in Figure 8, the vertical stiffness of the knee fluctuated around zero during the first 40% of the gait cycle, from the loading response phase to the terminal stance phase. Moreover, this duration equals the swing duration of the contralateral leg. At the terminal stance phase, the vertical stiffness of the knee appears as the discontinuity point of the first kind, and then it maintains a wide 'U' shape until the mid-swing phase with the duration of 30% of the gait cycle. There is also a discontinuity point of the first kind at the mid-swing phase, and a curve similar to a sinusoid is produced from the mid-swing phase to the loading phase with a duration of 30% of the gait cycle. The duration of the 'U' shape and the sinusoid stiffness curve is the exact stance duration of the contralateral leg. Furthermore, the changing tendencies of the two double support phases differ. When the limb is preparing to swing, there is a discontinuity, and when the limb is preparing for stance, the stiffness variation is continuous. Additionally, the vertical stiffness of the right knee is delaying or ahead of the left knee by half of the gait cycle.

**Figure 8.** Vertical stiffness of the knee during walking. (**a**,**b**) denote the left and right knees, respectively, and (**c**) is the corresponding vertical GRF. The yellow rectangle shows the loading response phase to the terminal stance phase of the right leg and the mid-swing phase of the left leg. The purple rectangle represents the terminal stance phase to the loading response phase and is equally separated by the mid-swing phase of the right leg as well as the stance phase of the left leg. The blue rectangles are the double support phase.
