## 4 Results and Discussion  
In order to verify the lateral stability control of the MPC algorithm proposed in the previous chapter, simulation experiments are performed using ADAMS-MATLAB. We set up two different experiments, including tracking a desired line with MPC algorithm and following a sinusoid with MPC and SMC. The initial condition of the simulation is set as follows: the rollator starts from the coordinate origin and tracks the expected curve \(y=2\) with the desired longitudinal velocity \(v=1\) m/s. The sampling time is 50 ms and the simulation time is 20 s. The simulation process includes reference trajectory generation, variable initialization, system matrix definition and controller design. The simulation results are shown in Figures 9-12. Figure 9-12 present the trajectory control performance of the elderly rollator based on the model prediction controller. The reference trajectory is a straight line as shown in Figure 9. The four-wheeled rollator is moved at a uniform speed of 1 m/s. The tracking performance of longitudinal position, lateral position and yaw angle to follow the desired trajectories are shown in Figure 10 based on the MPC control method. Figure 11 shows the longitudinal speed and lateral speed. Figure 12 demonstrates the longitudinal error, lateral error and leading angle error. It can be seen from Figures 9-12 that the rollator can quickly track the reference trajectory at 0-4 s as well as at 4-20 s. The curve is smooth and the fluctuation is small. The position control is within the constraint range, and the yaw angle is within the constraint range in particular. Therefore, the MPC algorithm proposed in this rollator system can be used to improve the stability of elderly rollator. Figure 11: Longitudinal velocity and lateral velocity. Figure 10: Longitudinal position, lateral position and yaw angle. In order to demonstrate how the controller response and the accuracy of theelgorithm, the co-simulation of rollator for ADAMS-MATLAB are carried out. The reference path is a circle shape with a radius of 25 m, and Figures 13 and 14 show the control effect of the rollator based on the MPC algorithm to track the circular trajectory. Figure 13 shows the tracking performance, and Figure 14 exhibits the longitudinal position, lateral position, angle relationship, speed change and heading angle of the rollator. It can be seen from the co-simulation results that the proposed MPC strategy can track the reference path well, and the rollator responds smoothly with position error within 0.1 m. The turning angle and speed can achieve stable tracking control within 5 s. There are no fluctuations in lateral position and longitudinal position. In addition, the heading angle is accurate and the speed is stable. Thus, the MPC algorithm can be used to improve the efficiency of the rollator system. To further illustrate the effectiveness of the proposed algorithm, a contrastive experiment of trajectory tracking is designed. The reference trajectory is set to a 5 m straight line and then a sinusoidal curve. The model predictive control algorithm and the sliding mode control algorithm are used for  
Figure 12: Longitudinal error, lateral error and yaw angle error. Figure 13: The tracking results of circle path. simulation respectively. The comparison results are shown in Figure 15. It can be concluded that both MPC and SMC can track the reference path well in a straight direction, but the MPC algorithm has a better effect in tracking the reference curve than SMC control in sinusoidal curve section. The deviation error of MPC control is almost zero, while the SMC is 0.3 m. Obviously, the MPC controller has faster response, higher tracking accuracy and smoother trajectory, meeting the travel requirements of the elderly rollator. Since the four-wheeled rollator is an integrated functional application for empowering and assisting human mobility for the elderly or disabled users.