*2.4. Generalized Description of Kinematics and VGRF*

The collected gait signals in Section 2.2 are analyzed with the process shown in Figure 4.

**Figure 4.** The generalizing process of the lower limb displacement description.

Because the collected gait signals begin and end with standing, the initial and final effects should be eliminated by selecting data points from the median segment. Firstly, the starting point of stable walking and 2 entire gait cycles are selected for analysis. Then the fast Fourier transformation (FFT) is used to transform the signal into the frequency domain since gait is quasiperiodic. The frequency and amplitude of major harmonics are then recognized from the frequency domain, as displayed in Figure 5.

**Figure 5.** The vertical displacement and the spectrum of the joints for one subject. (**a**) The vertical oscillations of hip, knee, and ankle. (**b**) The spectrums of vertical oscillations of hip, knee, and ankle.

It can be observed in Figure 5b that the vertical oscillation of the hip is mainly accumulated at the first two harmonics, while the vertical oscillation of knee is mainly at the first three harmonics, and the vertical oscillation of ankle is composed mainly of the first four harmonics.. Therefore, the vertical displacement of the hip, knee, and ankle can be represented by the two, three, and four harmonics, respectively. The Fourier series is considered to fit the oscillation trajectory of the lower limb as follows:

$$S\_N \mathbf{x} = \frac{a\_0}{2} + \sum\_{n=1}^{N} \left( a\_n \cos 2\pi nx + b\_n \sin 2\pi nx \right). \tag{6}$$

The sine component and the cosine component of the same frequency can be synthesized into a sine component represented as:

1

$$S\_N x = \frac{a\_0}{2} + \sum\_{n=1}^{N} c\_n \sin(2\pi nx + \varphi\_n),\tag{7}$$

2 2 arctan where *c<sup>n</sup>* = p *an* <sup>2</sup> + *b<sup>n</sup>* <sup>2</sup> refers to the amplitude of each harmonic and *<sup>ϕ</sup><sup>n</sup>* <sup>=</sup> arctan *an bn* is the initial phase of the harmonic component in each order; *N* is the number of the harmonic order. The amplitude is assumed to be proportional to the leg length; therefore, the amplitude of each harmonic in the series is then divided by the leg length of the subject, and thus the ratio of amplitude to leg length is obtained. Then the mean of the ratio and the initial phase of all the subjects are calculated for a general description of lower limb displacements. Finally, the change in vertical displacement can then be derived as:

$$y = \sum\_{n=1}^{N} A\_n l \sin(2\pi nft + \varphi\_n),\tag{8}$$

where *A<sup>n</sup>* is the coefficient of each harmonic, *l* is the leg length of the subject, *f* refers to the real walking cadence, and it is the number of strides in one second; thus, it can also be calculated by the gait cycle time *T* since *f* = <sup>1</sup> *T* . 1

The theoretical displacement of one limb can also be derived from the contralateral limb since human walking has the characteristics of symmetry both in space and time. The locomotion of one limb lags a half-gait cycle compared to the contralateral limb. Thus, if a half-gait cycle is introduced to Equation (8), which means *<sup>t</sup>* in Equation (8) becomes (*<sup>t</sup>* <sup>−</sup> *<sup>T</sup>* 2 ), then the oscillation of the contralateral lower limb joints can be expressed as: 2

$$y\_r = \sum A\_i l \sin(2\pi i f t + \varphi\_i + (i - 1)\pi) + A\_j l \sin(2\pi j f t + \varphi\_j) \quad i = 1, 3, \dots; j = 2, 4, \dots, \tag{9}$$

where *i* represents the order of the odd harmonics, and *j* refers to the order of the even harmonics.

The measured VGRF is also a quasiperiodic signal, as displayed in Figure 4. Similar to the dealing process for kinematic signals, the VGRF can also be represented as:

$$F = \sum\_{n=1}^{N} A\_n M g \sin(2\pi nft + \varphi\_n),\tag{10}$$

$$F\_{\mathbf{r}} = \sum A\_i M g \sin(2\pi i f t + \varphi\_{\mathbf{i}} + (i - 1)\pi) + A\_j M g \sin(2\pi j f t + \varphi\_{\mathbf{j}}) \\ i = 1, 3, \dots; j = 2, 4, \dots, \tag{11}$$

where *F* refers to the VGRF of one foot and *F<sup>r</sup>* is the VGRF of the other foot, *M* is the mass of the body, and *M* = *m* + 2*m<sup>t</sup>* + 2*m<sup>s</sup>* . + 2 + 2

Walking is commonly studied as a repetitively periodic activity using the "gait cycle" [37]. The gait cycle is defined as the duration from the heel strike to the next heel strike of the same limb. It can also be subdivided into the stance phase (accounts for 60% of the gait cycle) and the swing phase (which accounts for 40% of the gait cycle). Moreover, the stance phase and the swing phase can be further subdivided, respectively. These phases can be determined based on the change in VGRF. The details of each gait phase and its corresponding VGRF are shown in Figure 6.

**Figure 6.** Gait cycles and corresponding ground reaction force. (**a**) Initial contact when heel strike, and it accounts for 2% gait cycle; (**b**) loading response that means foot flatting, and it accounts for 10% gait cycle; (**c**) midstance, and it accounts for 17% gait cycle; (**d**) terminal stance when heeling off, and it accounts for 19% gait cycle; (**e**) pre swing means toe-off, and it accounts for 12% gait cycle; (**f**) initial swing, and it accounts for 13% gait cycle; (**g**) mid swing, and it accounts for 12% gait cycle; and (**h**) terminal swing, and it accounts for 13% gait cycle; (**a'**,**b'**) are phases in the next gait cycle and their determination are the same as (**a**,**b**) respectively.
