# Hardware-in-the-Loop Test of a Prosthetic Foot
## 3 Experimental Setup and Parameters
### Modeling an Amputee Using the VPP Model
#### 3.1.2 Center of Pressure Shift  
The idea of the VPP model is that the GRFs always point to the VPP. To calculate the required hip torque, the location where the GRFs are applied needs to be known. In the simple VPP model of Section 2.2, the point of application of the GRFs was easy to find since the leg was supposed to make contact with the ground at a single point \(x_{\text{FPJ}}\) corresponding to the intersection between the ground and the leg axis. When a foot is considered, the point of application of the forces is not necessarily anymore on the axis of the leg since the point of contact between the foot and the ground can travel during the stance phase. In human walking, a shift of this point from the heel to the toes during the stance phase can be observed, and this point is called the Center of Pressure (CoP) [25]. The CoP of the right foot is henceforth denoted by \(x_{\text{CoP,R}}\), and the foot point \(x_{\text{FPR}}\) marks the intersection of the leg axis with the ground. The location of the CoP is necessary for the calculations of the modeled amputee due to the required angle \(\gamma_{\text{VPP,R}}\). This angle is defined as the angle between the leg axis (here: connection \(x_{\text{CoP,R}}\) and \((x_{\text{h}},y_{\text{h}})\)) and the connection between the CoP and the VPP (here: connection \(x_{\text{CoP,R}}\) and \((x_{\text{VPP}},y_{\text{VPP}})\)). The geometry and the forces acting on the foot are illustrated in Figure 5.  
How the CoP shifts from the heel to the toes needs to be prescribed (The CoP location could also be determined from a torque equilibrium about the ankle joint using the measured forces and torques. This approach offers some difficulties: firstly, the inertial forces need to be determined, and secondly, our experiments revealed that the noise in the force measurement leads to a noisy calculation of the CoP location). This work assumed a CoP shift depending on the angle \(\alpha_{\text{R}}\) between the leg and the ground. The implemented relationship is written as:  
\[x_{\text{CoP,R}}=\begin{cases}\frac{l_{\text{h}}}{2-\alpha_{\text{R}}}\alpha_ {\text{R}}+c_{1},&\text{for}\ \alpha_{\text{R}}\leq\frac{\pi}{2}\\ \frac{l_{\text{H}}}{\alpha_{\text{TO}}-\frac{\pi}{2}}\alpha_{\text{R}}+c_{2},& \text{for}\ \alpha_{\text{R}}>\frac{\pi}{2},\end{cases} \tag{9}\]  
Figure 5: The implementation of the VPP model assumes that the GRFs act at the CoP. The figures were inspired by [23] (reprinted with permission from [23], 2020 [PERSON]). (**a**) The implementation assumes a linear shift of the CoP from the heel to the toes depending on the leg angle \(\alpha_{\text{R}}\). (**b**) The measured forces \(\mathbf{F}_{\text{int}}\) result from the GRFs. Inertial forces and gravitational forces are neglected.  
using the TD angle \(\alpha_{0}\), as well as the geometric dimensions of the foot \(l_{\text{h}}\) and \(l_{\text{ff}}\) (see Figure 5b). Furthermore, an assumption of the TO angle \(\alpha_{\text{TO}}\) is required. The constants \(c_{1}\) and \(c_{2}\) are found using \(x_{\text{CoP,R}}=x_{\text{F,R}}\) at \(\alpha_{\text{R}}=\frac{\pi}{2}\). This CoP shift is also implemented for the left, intact leg (\(x_{\text{F,L}}\) is replaced by \(x_{\text{CoP,L}}\) in the equations).  
Note that, in this model, the ankle joint is modeled as being stiff. This is because an amputee loses the bi-articular muscles at the ankle joint in an amputation and is not able to actively apply an ankle torque.