# A Statistical Approach for Modeling Individual Vertical Walking Forces
## 3 Walking Force Models
### Proposed Force Model  
The proposed force model is based on the assumption that the single foot force presents the typical M shape. Hence, the dynamic force \(P_{m,k}(t)\) induced by the \(k\)th step is described by a Fourier series as follows:  
\[P_{m,k}(t)= \begin{cases}0&t<T_{i,k}\\ \sum_{n=1}^{4}A_{n,k}\cos(n2\pi f_{s,k}t+\Phi_{n,k})&T_{i,k}\leq t\leq T_{i,k}+ T_{c,k}\\ 0&t>T_{i,k}+T_{c,k}\end{cases}  \tag{1}\]  
where \(n\) is the harmonic number, \(A_{n,k}\) (N) and \(\Phi_{n,k}\) (-) are the amplitude and phase shift of the \(n\)th harmonic and \(f_{s,k}\) (Hz) is the walking frequency. The contribution of four harmonics is accounted for as we found that the first four harmonics allow for an accurate description of the measured foot forces. The variable \(t\) (s) denotes time, \(T_{i,k}\) (s) and \(T_{c,k}\) (s) are, respectively, the instant of application and the duration of the contact between the foot and the ground. Hereafter, the subscript \(k\) (if present) denotes the \(k\)th step; however, sometimes the subscript \(k\) is dropped to simplify notation and increase readability. The instant of application \(T_{i,k}\) of the \(k\)th step can be expressed as a function of the previous \(k-1\) steps:  
\[T_{i,k}=\sum_{i=0}^{k-1}T_{s,i} \tag{2}\]  
where \(T_{s,i}\) (\(s\)) is the time elapsed between the two successive heel-strikes of steps \(i\) and \(i-1\), with \(T_{s,0}=0\). It is worth pointing out that step duration \(T_{c}\) and walking period \(T_{s}\) are related by \(T_{c}=T_{s}+\Delta t\), where \(\Delta t\) (s) represents the period during which both feet are in contact with the ground. The continuous walking force \(F_{m}(t)\) in the time domain is modeled as a series of moving footsteps, each one described by Equation (1), i.e.,  
\[F_{m}(t)=\sum_{k=1}^{N_{p}}P_{m,k}(t) \tag{3}\]  
where \(N_{p}\) is the total number of considered steps. To describe the continuous walking force, spatial parameters also need to be introduced. These are the step length \(l\) and the step direction \(\theta\), which are the polar representation of the coordinates where the foot force is applied. In particular, the step length \(l\) represents the distance between two successive heel-strikes. According to the proposed load model, the pedestrian's walking is described by ten gait parameters, each being a random variable: the period \(\Delta t\), the pacing frequency \(f_{s}\), the amplitudes \(A_{1}\), \(A_{2}\), \(A_{3}\), \(A_{4}\), the phase shifts \(\Phi_{1}\), \(\Phi_{2}\), the step length \(l\), and the step direction \(\theta\). Note that the walking frequency \(f_{s}\) is directly related to the walking period \(T_{s}\) by \(f_{s}=1/T_{s}\) and the step duration \(T_{c}\) can be evaluated from \(\Delta t\) and \(T_{s}\). The phase shifts \(\Phi_{3}\), \(\Phi_{4}\) are selected so that the walking forces at \(P_{m}(0)\) and \(P_{m}(T_{c})\) are as close as possible to zero. The state of the Markov chain at the \(k\)th step is described by \(S_{k}=\left\{f_{s,k},A_{1,k},A_{2,k},A_{3,k},A_{4,k},\Delta t_{k},I_{k},\Phi_{1,k},\Phi_{2,k},\theta_{k}\right\}\). The geometrical interpretation of the proposed model is illustrated in Figure 3.  
Figure 4 shows an example of the numerical single foot forces compared to the experimental ones. It is observed that the shape of the simulated step forces is qualitatively the same as the experimental ones, despite the differences due to the variability of the walking parameters. Formally, this Markov chain is continuous in the state space \(S\) and discrete in \"time\", since the process is represented by a countable sequence of steps. If we assume that the steps are independent and identically distributed, then the Markov chain is fully described by the transition _probability density function_ (pdf) \(p(S)\). Note that this Markov chain is stationary, i.e., the transition pdf does not depend on step index \(k\).