# Incorporation of Torsion Springs in a Knee Exoskeleton for Stance Phase Correction of Crouch Gait
## 2 Design and Methods
### Mathematic Model of Human Gait  
The equations of motion for the gait cycle were determined using the Euler-Lagrange method. The lower extremity was modelled as three hinged rigid rods. In the stance phase of the gait cycle, the ground reaction forces (GRFs) are variable, based on the weight of the subject, speed, and type of gait, whereas the GRFs in the swing phase are zero for lack of contact with the ground. To account for this variability in the GRFs, [PERSON] analyzes the energy in the system. Since the human gait is cyclical, it is assumed that energy in the system is conserved; thus, Equation (1) was used to determine the angular accelerations for the hip, knee, and ankle joints. \[\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial\dot{q}_{i}}\right)-\frac {\partial\mathcal{L}}{\partial q_{i}}=0 \tag{1}\]  
Figure 2: Schematic for overall process of exoskeleton design. where \(\mathcal{L}\) represents the differences in the kinetic and potential energies in the system with respect to the generalized velocities \((\dot{q}_{i})\), and displacements \((q_{i})\) for \(i\) variables in the system. The thigh, shank, and foot were modeled as three hinged rods at each respective joint, as seen in Figure 3a. The three degrees of freedom (DOF) model can be seen in Figure 3b, with the origin at the top of the model, fixed to the upper body, wherein the motion for the upper body is considered negligible for the gait cycle. Translation and rotation are considered in the kinetic energies of the system with gravitational potential energy, and are converted to the generalized coordinates \(\gamma\), \(\beta\), \(\theta\), representing the angular displacements of the hip, knee, and ankle, respectively. The resulting equations of motion are given in Equations (2)-(4), where the angular accelerations of the hip, knee, and ankle are represented by \(\tilde{\gamma}\), \(\ddot{\beta}\), \(\text{ and }\ddot{\theta}\), respectively. \[\tilde{\gamma} =\underbrace{\left\{ \begin{matrix}0.443m_{s}l_{T}l_{S}\left[ \ddot{\beta}\alpha\cos(\gamma-\beta)-\dot{\beta}sin(\gamma-\beta)\left(\dot{ \gamma}-\dot{\beta}\right)-\dot{\gamma}\dot{\beta}sin(\gamma-\beta)\right]+m_ {f}l_{S}\left[\ddot{\beta}\cos(\gamma-\beta)-\dot{\beta}sin(\gamma-\beta) \left(\dot{\gamma}-\dot{\beta}\right)-\dot{\gamma}\dot{\beta}sin(\gamma-\beta )\right]\\ +0.436m_{f}l_{T}l_{f}\left[\ddot{\theta}sin(\gamma-\theta)\left(\dot{\gamma}- \dot{\theta}\right)-\ddot{\theta}\cos(\gamma-\theta)+\dot{\gamma}\dot{\theta} sin(\gamma-\theta)\right]+gl_{T}(m_{T}+m_{S}+m_{f})sin\gamma\\ l_{T}^{2}\left(0.9482m_{T}+m_{S}+m_{f}\right)\end{matrix} \right\}} \tag{2}\] \[\tilde{\beta} =\underbrace{\left\{ \begin{matrix}0.443m_{S}l_{T}l_{S}\left[ \gamma\alpha\cos(\gamma-\beta)-\dot{\gamma}sin(\gamma-\beta)\left(\dot{\gamma} -\dot{\beta}\right)+\dot{\gamma}\dot{\beta}sin(\gamma-\beta)\right]+m_{f}l_{T} l_{S}\left[\gamma\alpha\cos(\gamma-\beta)-\dot{\gamma}sin(\gamma-\beta) \left(\dot{\gamma}-\dot{\beta}\right)+\dot{\gamma}\dot{\beta}sin(\gamma-\beta )\right]\\ +0.436m_{f}l_{S}l_{f}\left[\ddot{\theta}\alpha\cos(\beta-\theta)-\dot{\theta} sin(\beta-\theta)\left(\dot{\beta}-\dot{\theta}\right)+\dot{\theta}\dot{ \theta}sin(\beta-\theta)\right]+gl_{S}(m_{S}+m_{f})sin\beta\end{matrix} \right\}} _{\text{$\tilde{\beta}$}(1.5295m_{S}+m_{f})}\] (3) \[\tilde{\theta} =\underbrace{\left\{ \begin{matrix}0.436m_{f}l_{S}l_{f}\left[ \ddot{\beta}\cos(\beta-\theta)-\dot{\beta}sin(\beta-\theta)\left(\dot{\beta}- \dot{\theta}\right)+\dot{\beta}\dot{\theta}sin(\beta-\theta)\right]+0.436m_{f }l_{T}l_{f}\left[\dot{\gamma}sin(\gamma-\theta)\left(\dot{\gamma}-\dot{\theta }\right)-\ddot{\gamma}\cos(\gamma-\theta)+\dot{\gamma}\dot{\theta}sin(\gamma- \theta)\right]\\ +m_{f}l_{S}l_{S}\dot{\theta}\dot{\theta}\\ 0.7133m_{f}l_{f}^{2}\end{matrix} \right\}} \tag{4}\]  
The masses of each segment are represented as \(m_{T},\ m_{S},\ \text{ and }m_{f}\), for the thigh, shank, and foot, respectively. Additionally, the lengths for the thigh, shank, and foot are represented by \(l_{T},\ l_{S},\ \text{ and }l_{f}\).