## 3 Optimization of Lateral Vibration Parameters Based on the SSA
The initial position of the watcher is randomly generated in the population, and the position expression is shown in Formula (17):  
\[X_{i,j}^{t+1}=\left\{ \begin{aligned} X_{\text{best}}^{t}&+ \beta\cdot\left|X_{i,j}^{t}-X_{\text{best}}^{t}\right|,& if& f_{i}>f_{g}\\ X_{i,j}^{t}+K\cdot\left(\frac{\left|X_{i,j}^{t}-X_{\text{ worst}}^{t}\right|}{(f_{i}-f_{w})+\varepsilon}\right),& if& f_{i}=f_{g}\end{aligned} \right. \tag{17}\]  
where \(X_{best}\) denotes the current global optimal location, indicating that the food source here is the most abundant and the population is also the safest. \(\beta\) is a random number subject to a normal distribution with a mean value of 0 and a variance of 1, which is used here as the step control parameter and the direction of sparrow movement. \(K\in[-1,1]\) is a random number. \(\varepsilon\) is a non-zero constant, which is used to avoid the case where the denominator is zero. \(f_{i}\) is the fitness value of the current individual sparrow. \(f_{g}\) and \(f_{w}\) are the current global best and worst fitness values, respectively. \(f_{i}>f_{g}\) means that the sparrow is at the edge of the population and is extremely vulnerable to predators. \(f_{i}=f_{g}\) means that the sparrows in the middle of the population are aware of the danger and need to be close to other sparrows to minimize the risk of predation. During the foraging process of the sparrow population, the follower constantly monitors the actions and movement patterns of the discoverer. For followers they will replace discoverers if they find a better food source, and the identities will exchange. The updated position formula of followers is shown in Formula (18):  
\[X_{i,j}^{t+1}=\left\{ \begin{array}{l}Q\cdot\exp\left(\frac{X_{\text{wont}}-X_ {ij}^{t}}{\beta^{2}}\right),\ \ \ \ \ if\ \ \ i>\frac{u}{2}\\ X_{P}^{t+1}+\left|X_{i,j}-X_{P}^{t+1}\right|\cdot A^{+}\cdot\text{ L},\ \ \ \text{others}\end{array} \right. \tag{18}\]  
where \(X_{p}\) stands for the best position discovered by the discoverer. \(A\) is a row matrix of size \(1\times d\) where each element in the matrix is either \(1\) or \(-1\), and \(A^{+}=A^{\text{T}}\left(AA^{\text{T}}\right)^{-1}\). Ultimately, Figure 3 depicts the flow chart of the lateral vibration control strategy of a high-speed elevator, with the main steps as follows:  
1. Replace the traditional guide shoes in the cabin guiding device with electromagnetic active roller guide shoes to reduce the impact on the cabin caused by uneven guide rails. 2. To overcome the drawbacks of the electromagnetic active roller guide shoes, introduce a sky-hook damping control strategy to avoid the increase in cabin vibration due to the damping force generated by the electromagnetic active roller guide shoes when the cabin's motion speed and relative speed (cabin frame and cabin) are opposite. 3. Establish three different types of mathematical simulation models for cabin motion in high-speed elevators, and then construct Simulink simulation models based on the sensitivity range of the human body to elevator vibrations. 4. Introduce the SSA optimization algorithm and take the mean value of vibration amplitude within the sensitive range of the human body as the optimization objective. Optimize the damping coefficients \(c\) of the electromagnetic active roller guide shoes and the \(C_{\text{min}}\) and \(C_{\text{max}}\) in the sky-hook damping control strategy in all three motion modes of the simulation model to obtain the optimal solution. 5. Apply the optimized parameters obtained from SSA to actual high-speed elevators to verify the effectiveness and advancement of the proposed cabin vibration control strategy. Figure 3: The flow chart of lateral vibration control model of high-speed elevator.