## 4 Application and Analysis
### Motion Planning Results and Analysis  
According to the characteristics of SDM and the natural driving conditions analyzed in Section 2.1, two typical driving scenarios are used to verify the improvement of NMPC-SDM in accuracy and real-time performance. * Scenario 1: The vehicle cruises on an S-shaped road and the relationship between the curvature and the desired speed is illustrated in Figure 12[37]. This scenario is designed to verify the improvement of the control accuracy. * Scenario 2: The vehicle moves at high speed and makes a lane change to avoid obstacles. The desired speed is 28 m/s and the obstacle vehicles move at 3 m/s and 10 m/s, respectively. As shown in Figure 13a, the maximum lateral error of SDM is only 0.08 m, which is reduced by 38% and 55% compared with the single-track model and the kinematic model, respectively. Moreover, the longitudinal speed error of SDM is only 0.11 m/s, which is also reduced by 71% and 74% (See Figure 13b). Summarizing the statistical results of tracking errors, it is concluded that NMPC-SDM has the best tracking control performance compared with the motion planners designed using the single-track model and kinematic model. There is no obstacle in scenario 1 and so the consumption of computation resources is much less. Besides the tracking control performance, the consumption of computation resources is further analyzed in Scenario 2, where the motion planner should find the optimal solution to avoid collisions with two obstacle vehicles, meanwhile tracking the desired speed as much as possible. The comparative simulations are shown in Figures 14 and 15. It is found from Figure 14a that the running time of NMPC-SDM is smaller than NMPC-CDM and larger than NMPC-KM overall. Compared with NMPC-CDM, the average running time of single cycle of NMPC-SDM is reduced from 0.91 s to 0.74 s (See Figure 14b). Though it is larger than NMPC-STM and NMPC-KM, both of the motion planners designed by the single-track and kinematic models sacrifice the tracking performance as shown in Figure 15. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Value** & **Symbol** & **Value** \\ \hline \(N\) & 60 & \(T\) & 3 s \\ \(Q\) & \(\text{diag}(0.844,\,1,40)\) & \(P\) & \(\text{diag}\left(10^{-5},\,62.5\right)\) \\ \(R\) & \(\text{diag}(10^{-4},\,90)\) & \(\overline{u}\) & \(\left[4000\text{ N }\,32\text{ deg}\right]^{\text{T}}\) \\ \(\Delta u\) & \(\left[4000\text{ N }/\text{s }63\text{ deg}/\text{s}\right]^{\text{T}}\) & \(x_{min}\) & \(\left[0\text{ m}/\text{s},-\infty\text{ m}/\text{s},0\text{ rad}\right]^{\text{T}}\) \\ \(x_{max}\) & \(\left[33\text{ m}/\text{s},\infty\text{ m}/\text{s},2\text{ pi rad}\right]^{\text{T}}\) & / & / \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of NMPC motion planner. Figure 12: Relationship between curvature and desired speed. As shown in Figure 15a-c, NMPC planners make lane change decision within \(s\in[100,\,200]\) m to avoid the obstacle vehicle (OV1) successfully. After overtaking the OV1, both NMPC-SDM and NMPC-STM return to the middle of the lane, but the NMPC-KM hits on the curb. Because the slip angle of tire reaches 5\({}^{\"}\) (See Figure 15d) and the longitudinal coupling force is nearly 250 N (See Figure 15e) that KM has a large predictive error under such a condition. The maximum lateral position error of NMPC-KM even reaches 2.8 m (See Figure 15g). This implies that the vehicle will invade other undesired lanes, which is very dangerous, especially under high speed conditions. The maximum lateral position error of NMPC-STM is 1.5 m (See Figure 15g), because a comparative large steering angle (See Figure 15c) is required to realize the collision avoidance while STM is only applicable to the conditions where the steering angle is small. Comparatively, NMPC-SDM has a good tracking performance of speed and reference lane, whose error are only 0.2 m/s and 0.7 m, respectively (See Figure 15f,g). Figure 14: Consumptions of computation resources: (**a**) running time of single cycle; (**b**) statistical results.