## 4 Motion Planning Application and Analysis
### Experimental Results and Analysis  
The parameters of the vehicle model and NMPC are presented in Table 1.  
To verify the improvement of the motion planning performances by considering the dynamical coupling, the driving scenarios with strong coupling were preferred, such as cruising on a curved road, lane changing, and so on:  
1. SCENARIO 1 was designed to verify the effectiveness of NMPC-SDM when the vehicle is cruising on an S-shaped road. It consists of continuous bends with decreasing curvature and increasing desired speed, as illustrated in Figures 8 and 9a. 2. SCENARIO 2 verifies the effectiveness of NMPC-SDM when the vehicle makes a lane change in a U-turn with low speed, as shown in Figure 10a. It consists of two 20 m straight roads at the start and the end, and an 80 m bend in the middle (Curvature of the bend is 0.04 m\({}^{-1}\)). An obstacle vehicle denoted by OV1 drives along the right lane. 3. SCENARIO 3 is an overtaking process with high speed on several successive bends as shown in Figure 11a. It consisted of a 30 m straight road, a 100 m counter-clockwise bend and a 100 m clockwise bend (Curvature of both bends is 0.002 m\({}^{-1}\)). Obstacle 1, denoted by OV1, ran along the right lane and obstacle 2, denoted by OV2, cut into the left lane from the right. The roads in SCENARIO 2 and 3 were composed of two lanes, whose parameters are shown in Table 2.  
SCENARIO 3 was the most complicated one among the three scenarios, which included two dynamical obstacles. When the scenario became more complex, such as overtaking with multi-obstacles, complex urban scenario and so on, the constraint number of obstacle in Equation (13) increased. The proposed planner can also be applied to more complicated driving situations. Here we focus on the validation of the improvement of planning performances by considering the coupling between lateral and longitudinal dynamics. The experimental results from SCENARIO 1-3 are given in Figures 9-11, respectively. In each figure, the solid lines are the results of NMPC-SDM (denoted by \"SDM\"), the dashed lines are the results of NMPC-single-track model (denoted by \"Single-track\"), and the dash-dotted lines are the results of NMPC-kinematic model (denoted by \"Kinematic\"). Since there was no obstacle vehicle in SCENARIO 1, the subject vehicle (SV) runs with the maximum speed in the center of lane. During the low-speed stage with a larger curvature, i.e., \(s\in[0,\,80]\) m, the longitudinal coupling component of lateral force was large (which the peak reaches 252 N as shown in Figure 9d) and the tracking performance of the single-track model based planner was worse (See Figure 9a,b). As shown in Figure 9c, with the increase of the speed and the decrease of the curvature, the lateral force in the longitudinal direction became smaller gradually, but the slip angle increased. This results in a gradual increase of the control error of the kinematic model-based planner, which the peak reached 0.39 m. However, the control error of the single-track model-based planner decreased gradually. Based on the above results, it can be seen that the kinematic model based planner performed poorly under high-speed conditions, where the slip angle of the tire was over 1.6 degree, and the single-track model-based planner performed poorly under low-speed and large steering angle conditions, in which the coupling force is over 130 N. Because SDM takes into account both dynamical coupling and tire cornering characteristics, NMPC-SDM exhibited better performance than other two planners. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Initial Speed** & **Desired Speed** & **Obstacle** \\ \hline
**2** & 0 & 5.8 m/s & OV1: \(s_{\text{o}\text{e}\text{l}}=55\) m, \(v_{\text{o}\text{e}\text{l}}=3\) m/s. \\
**3** & 10 m/s & 28 m/s & OV1: \(s_{\text{o}\text{e}\text{l}}=80\) m, \(v_{\text{o}\text{e}\text{l}}=3\) m/s; \\  & & & OV2: \(s_{\text{o}\text{e}\text{l}}=140\) m, \(v_{\text{o}\text{e}\text{l}}=10\) m/s. \\ \hline \hline \end{tabular}
\end{table}