# Hardware-in-the-Loop Test of a Prosthetic Foot
## 2 State-of-the-Art
### Basic Human Gait Models  
Based on data from gait laboratories, various walking and running models have been proposed to describe the gait pattern. The HiL test requires a model of the amputee that implements basic gait characteristics. Furthermore, the model needs to be solved in real time, i.e., within the sample time \(\Delta\hat{T}\). For this first proof-of-concept study, a model with low complexity while still being able to represent basic gait characteristics was sought. Walking is a repetitive motion, and the movement and time interval between two successive events is called the Gait Cycle (GC). In this work, the GC starts when the Center of Mass (CoM) is above the left foot and terminates one stride later, i.e., at the next midstance of the left foot. During this GC, the left, as well as the right foot pass through the stance phase (60 % of the GC) and the swing phase (40 % of the GC). The CoM motion resembles a sinusoidal motion with a magnitude of \(0.032\pm 0.008\) m during a GC [14]. A further fundamental gait characteristic is the shape of the Ground Reaction Forces (GRFs) in the horizontal (\(x\)) and vertical (\(y\)) direction. The GRFs are the forces that the ground exerts on the body. In the horizontal direction, a point symmetric force profile is observed, where decelerating forces (negative \(x\) direction) are measured in the first part of the stance phase and accelerating forces are measured in the second half of the stance phase (positive \(x\) direction). Typically, the shape of the GRFs in the vertical direction exhibits a double-hump shape, with a minimum between the two humps (m-shape) [14; 15; 16; 17]. This work made use of the two-dimensional Virtual Pivot Point (VPP) model by [PERSON] et al. [18], which is based on the Spring-Loaded Inverted Pendulum (SLIP) template model [19]. The model is illustrated in Figure 3. The upper body is represented as a trunk, where the whole body mass is concentrated. The legs are modeled by massless springs. This model was based on the observations that the GRFs intersecting in one or more points above the CoM, resulting in a constellation similar to a physical pendulum [20]. This intersection point is observed in humans on treadmills (Note that the VP is only observed during the single support phase in humans [21]. However, the VPP model assumes that the GRFs are also redirected into the VPP during the double-support phase.) and animals such as dogs and chickens. The principle is named the Virtual Pendulum (VP), as it stabilizes the trunk orientation. The VP principle was implemented in the VPP model. In this model, the GRFs are actively redirected into the VPP by the implementation of an additional hip torque \(\tau_{\text{VPPj}}\) (with \(j=\{L,R\}\) for the left and right leg) [18; 22]. The resulting hip torque is calculated for each leg \(j=\{L,R\}\) independently during contact with the ground by:  
\[\tau_{\text{VPPj}}=F_{\tau\text{j}}\text{l}_{\text{j}}\qquad\text{with}\qquad F _{\tau\text{j}}=F_{\text{sj}}\tan\bigl{(}\gamma_{\text{VPPj}}\bigr{)}=k_{ \text{j}}(\text{l}_{\text{0j}}-\text{l}_{\text{j}})\tan\bigl{(}\gamma_{\text{ VPPj}}\bigr{)}. \tag{1}\]  
Here, \(\text{l}_{\text{0j}}\) and \(\text{l}_{\text{j}}\) describe the initial and current leg length, and the leg stiffness is denoted by \(k_{\text{j}}\). The required \(F_{\tau\text{j}}\) to redirect the leg forces \(F_{\text{sj}}\) into the VPP depend on the angle \(\gamma_{\text{VPPj}}\) between the leg axis and the connection of the foot point \(x_{\text{FPj}}\) to the VPP location (the equations describing the calculation of \(\text{l}_{\text{j}}\), \(\gamma_{\text{VPPj}}\), etc., are given in Appendix A).