## 4 Simulation and Analysis
### Point Mass Model  
The simulation analysis is conducted using a point mass vehicle model in the MATLAB/Simulink R2022a platform, with a starting speed of 60 km/h on a high-friction road (\(\mu=0.9\)). Figure 3 compares the simulation results of fifth-order polynomial braking and seventh-order polynomial braking. The blue solid line represents the fifth-order polynomial results, while the red dashed line represents the seventh-order polynomial results. Panels (a) to (d) of Figure 3 show the responses in terms of the braking distance, vehicle speed, deceleration, and jerk, respectively. Given the same constraints, the minimum stopping distance for the fifth-order polynomial is 21.9 m, while, for the seventh-order polynomial, it is 27.4 m. The braking times are 3.29 s and 3.84 s, respectively. The times to reach peak deceleration are 1.11 s for the fifth-order polynomial and 1.54 s for the seventh-order polynomial. The maximum jerk values are 1.85 g/s at \(t=0\) for the fifth-order polynomial and 0.92 g/s at \(t=0.6\) s for the seventh-order polynomial. In conclusion, although the seventh-order polynomial requires a 25% longer braking distance compared to the fifth-order polynomial, it significantly reduces the maximum jerk value by approximately 50%, thereby improving the vehicle braking comfort. Figures 4 and 5 display the simulation results for fifth-order and seventh-order polynomial braking on a high-friction road (\(\mu=0.9\)), under different initial speeds. Panels (a) to (d) in both figures show the responses of the braking distance, vehicle speed, acceleration, and jerk. It is evident that, as the initial speed increases, the required braking distances for both polynomial types also increase. The braking time increases proportionally with the  
initial speed, while the maximum jerk value decreases as the initial speed rises. Detailed performance metrics can be found in Table 1. When traveling at 90 km/h, the fifth-degree polynomial yields a braking distance of 49.4 m, a braking time of 4.93 s, and a maximum jerk of 1.23 g/s. In contrast, the seventh-degree polynomial results in a braking distance of 61.7 m, a braking time of 5.76 s, and a maximum jerk of 0.61 g/s. Under the same initial conditions, the braking distance for the seventh-degree polynomial is 25% greater compared to the fifth-order polynomial, while its maximum jerk value is reduced by approximately 50%. Figure 6 compares bar charts of the fifth-order and seventh-order polynomials, using an initial speed of 90 km/h as an example. Figure 7 presents the simulation tests under the condition of a stationary leading vehicle, starting with a distance of 150 m between the trailing and leading vehicles, and the following vehicle travels at a starting speed of 80 km/h on a high-friction road (\(\mu=0.9\)). Figure 6a-e display the responses in terms of the distance between the two cars and the speed, acceleration, jerk, and TTC of the trailing vehicle. It is evident that the vehicle equipped with a seventh-order polynomial automatic braking control system can prevent rear-end collisions effectively, maintaining a distance of approximately 2 m between the two vehicles when the following car comes to a halt. The vehicle's braking deceleration initially increases and then decreases, reaching peak deceleration at \(t=6.5\) s. The following vehicle's speed drops to zero at around \(t=9.5\) s. Overall, the braking process is comfortable, with smooth jerk changes featuring both troughs and peaks, reaching \(-0.69\) g/s and 0.51 g/s at \(t=5.3\) s and \(t=7.8\) s, respectively. The TTC reaches its minimum value of 0.84 s at \(t=7.2\) s.