# Hardware-in-the-Loop Test of a Prosthetic Foot
## 3 Experimental Setup and Parameters
### Modeling an Amputee Using the VPP Model
#### 3.1.1 Model of the Amputated Leg  
To model the amputee, the right leg (index R) is cut at the ankle, which is the interface location in this HiL setup. The total leg length of the right leg is \(l_{\text{R}}\), which results from the height of the prosthetic foot \(h_{\text{f}}\) and the leg length of the remaining leg \(l_{\text{u}}\) (resting spring length \(l_{0,\text{a}}\); index \((\cdot)_{\text{u}}\) denotes the upper part of the amputated leg). In the HiL setup, the prosthetic foot needs to be moved according to the end point of the amputated leg. However, it is numerically not trivial to calculate this position. This is because the amputated leg and the prosthetic foot are as two springs in series, which have to experience the same force. Since the stiffness of the prosthetic foot is not known, the interface location between the amputated leg and the prosthetic foot cannot be calculated. Therefore, an additional DoF is inserted by adding a point mass \(m_{\text{IFP}}\) at the ankle [24]. Taking a very small value for that mass compared to the model mass (\(m_{\text{IFP}}\ll m_{\text{VPP}}\)), the system dynamics are barely modified. This point is called the Interface Point (IFP) at the location \((x_{\text{IFP}},y_{\text{IFP}})\). The mass IFP forms a vibratory system and introduces artificial high-frequency dynamics. To dampen these oscillations of this point mass, an additional damper with damping constant \(d_{\text{u}}\) is inserted in the remaining leg. The modified VPP model is shown in Figure 4.  
As described in Section 2.1, measured interface forces \(\mathbf{F}_{\text{int}}\) are the input to the simulation model (in this model, the external forces \(\mathbf{F}_{\text{ext}}^{\text{NUM}}\) only include the gravitational forces). As this model is a two-dimensional model, the vector \(\mathbf{F}_{\text{int}}\) comprises two forces and one torque. In this application, this is a force parallel to the leg axis \(F_{\text{int},\parallel}\), a force orthogonal to the leg axis \(F_{\text{int},\perp}\), and a torque about the \(z\) axis \(M_{\text{int}}\).  
Figure 4: Model of the amputee based on the VPP model. The right leg is cut, and a point mass \(m_{\text{IFP}}\) is added to serve as an interface point between the simulation and experimental system. A damper between the hip and point mass is added to reduce oscillations. The resting leg length of the remaining upper part of the right leg is \(l_{0,\text{u}}\). At the interface point (\(x_{\text{IFP}}\)-\(y_{\text{IFP}}\)), the measured forces/torque \(\mathbf{F}_{\text{int}}\) act on the model. Figure adapted from [23] (reprinted with permission from [23], 2020 [PERSON]).  
The motion of the interface mass point is restricted to the leg axis, and the gravitational forces acting on it are neglected. The internal dynamics of the leg with the added IFP and damper are written as:  
\[m_{\text{IFF}}\bar{l}_{\text{u}}=F_{\text{int},\parallel}-d_{\text{u}}\dot{l}_{ \text{u}}-k_{\text{u}}(l_{\text{u}}-l_{0,\text{u}})=F_{\text{int},\parallel}- F_{\text{u}}. \tag{8}\]  
In this equation, the forces of the spring-damper element are abbreviated as \(F_{\text{u}}\). Note that the dissipated energy of the damper is assumed to be negligible compared to the total system energy of the modeled amputee and therefore not compensated.