# The Exoskeleton Balance Assistance Control Strategy Based on Single Step Balance Assessment
## 2 Single Step Balance Assessment Method
### Method for Single Step Balance Assessment  
To predict the position of the swing foot, an inverted pendulum model with a finite sized foot [22] and an ankle torque model are utilized in this paper. The inverted pendulum model with a finite sized foot is given by (2), and the ankle torque model is given by (3). \[\ddot{x}=\frac{g}{l}(x-x_{ankle})-\frac{\tau_{ankle}}{ml} \tag{2}\]  
where \(x\) is the position of CoM, \(\ddot{x}\) is the acceleration of CoM, \(x_{ankle}\) is the ankle position of the support foot, \(\tau_{ankle}\) is the ankle torque of the support foot, m is the total mass of the human body and \(l\) is the leg length. \[\tau_{ankle}=\left\{ \begin{array}{ll}-\tau_{1}\sin\frac{\pi x}{x_{ankle}}& \text{if }x-x_{ankle}\leq 0\\ \tau_{2}\sin\frac{\pi}{2\Gamma_{2}}(x-x_{ankle})&\text{if }x-x_{ankle}>0\end{array} \right. \tag{3}\]  
where \(\tau_{1}\) and \(\tau_{2}\) are the peak torque of flexor and extensor, respectively. \(L_{2}\) is the average stride length of quasi-period walking. Previous research [23] has proposed the relationship between ankle torque and gait cycle during quasi-period walking. As shown in Figure 2(a), the relationship between the position of CoM and ankle torque can be calculated according to the velocity of CoM. To establish the torque model, the ankle torque profile shown in Figure 2(a) can be described by the sine curve function given by (3), and the fitting error is below 10% which is precise enough to predict the ankle torque of the support foot during quasi-periodic walking. According to model (2) and (3), the state of CoM (\([x,\dot{x}]\)) can be predicted. In order to verify the feasibility of prediction, the predicted state of CoM are compared with the experimental data obtained under the same experiment setup. The prediction results and experiment data are shown in Figure 4.  
Figure 3: The relationship between the position of the center of mass (CoM) and ankle torque. In Figure 4, it is obvious that the predicted positions of CoM are close to the experimental value (Mean error: 0.03(0.02) m), and the maximum error between predicted velocity and real velocity of CoM is within 10%. Hence, the model (2) and (3) can be used to predict the state of CoM and the position of CP can be predicted simultaneously. The prediction of the swing foot position is based on the law of the swing foot. Previous research has shown that the swing foot accelerates at the beginning of the swing phase and decelerates at the end of the swing phase [24]. The measured acceleration of the swing foot is shown in Figure 5a, from which it can be seen that the fluctuation of the swing foot acceleration and deceleration are small. Hence within the allowable error, the motion of the swing foot can be considered as a uniformly accelerated motion and a uniformly decelerated motion. The equivalent acceleration of the swing foot is shown in Figure 5b, and based on this equivalent acceleration and the remaining time for the swing foot chasing CP, the swing foot position can be predicted. The equivalent acceleration \(a_{a}\) and \(a_{d}\) can be obtained from the statistic data of human walking. Due to biomechanical constraints and personal walking habits, the time for the swing foot chasing CP is limited. As shown in Figure 6, the angle between the support leg and the vertical line when the swing foot is touching down is noted as \(\theta_{f}\) or \(\theta_{b}\), here \(\theta_{f}\) is defined as the forward angle and \(\theta_{b}\) is defined as the backward angle. The time of reaching \(\theta_{f}\) or \(\theta_{b}\) are denoted as \(t_{f}\) and \(t_{b}\), respectively. According to (2), \(t_{f}\) and \(t_{b}\) can be calculated. The time the swing foot catches up with CP before touching down is denoted as \(t_{g}\), and \(x(t_{f})-x(t_{g})\) or \(x(t_{g})-x(t_{b})\) is denoted as the degree of single balance.