## 4 Motion Planning Application and Analysis
### Experimental Results and Analysis

Table 2: Road parameters of SCENARIO 2 and 3. Figure 9: _Cont._  
In SCENARIO 2, [PERSON] started from the initial position in the right lane. Since there was a low-speed obstacle in the same lane, [PERSON] changed to the adjacent lane and then performed lane keeping. The results in Figure 10a show that all planners could successfully complete the lane change. In Figure 10b,c, the control commands of the single-track model-based planner oscillates within the range of \(s\in[40,\ 80]\) m and has large longitudinal and lateral control errors as shown in Figure 10d,e. This is caused by the large coupling force, as illustrated in Figure 10g, which reaches 500 N. The single-track model didn't consider this coupling effect. Furthermore, it can be seen in Figure 10f that the tire's slip angle was small (the absolute value less than 1.2 degrees). Therefore, the performance of the kinematic model-based planner was similar to NMPC-SDM. Figure 9: Results of SCENARIO 1. (**a**) Longitudinal speed. (**b**) Lateral error. (**c**) Slip angle of front tire. (**d**) Longitudinal coupling component. Figure 10: Cont. In SCENARIO 3, since OV1 was in front of [PERSON] with a low speed, [PERSON] changed to the left lane. At the same time, OV2 cut into the left lane from the right, so [PERSON] changed to the right immediately. In Figure 11a, it can be seen that both NMPC-SDM and the single-track model-based planner could successfully overtake and finally return to the middle of the lane. It can be seen in Figure 11g that the dynamical coupling effect was weak (the peak of SDM is less than 85 N) due to the small steering angle required to overtake. Therefore, the planning results of the two planners were consistent, as compared in Figure 11b,c. However, as shown in Figure 11f, because of the large slip angle of the tire, which reached 5 degrees, the prediction error of the kinematic model was large. This resulted in a collision with the curb and large longitudinal/lateral control errors, as shown in Figure 11d,e. Figure 10: Results of SCENARIO 2. (**a**) Vehicle trajectories. (**b**) Longitudinal driving force. (**c**) Steering angle of front tire. (**d**) Longitudinal speed. (**e**) Lateral error. (**f**) Slip angle of front tire. (**g**) Longitudinal coupling components. Figure 11: _Cont._  
To verify the real-time performance of the proposed algorithm, we took SCENARIO 3 as an example to show the execution time of solving NMPC in Figure 12. Among the four planners, NMPC-kinematic had the shortest execution time, while NMPC-CDM had the longest execution time of \(100\) seconds. The NMPC-kinematic had the shortest execution time of \(100\) seconds. The NMPC-kinematic had the shortest execution time of \(100\) seconds. 2 The SDM proposed in this paper can characterize the vehicle longitudinal and lateral dynamical coupling effect and achieve a good balance between the accuracy and complexity;
3 Compared with the kinematic and single-track model-based planners, the proposed NMPC-SDM planner improves the performance of motion planning in normal driving. In the future work, the proposed framework will be validated under more challenging scenarios, such as obstacle avoidance with multiple obstacles, overtaking under uncertain disturbances, and so on. Moreover, field testing should be carried out to verify the effectiveness of the algorithm in real traffic environments. Conceptualization, D.D. and F.G.; methodology, D.D. ; software, D.D. ; validation, D.D., F.G. and Q.H. ; formal analysis, D.D. ; investigation, D.D. ; resources, F.G.; data curation, D.D. ; writing--original draft preparation, D.D. ; writing--review and editing, F.G.; visualization, D.D. ; supervision, F.G.; project administration, F.F. ; funding acquisition, F.G. All authors have read and agreed to the published version of the manuscript. This research was funded by the National key research and development program, grant number 2016 YFB0100900 and the Scientific Technological Plans of Chongqing under grant cstc2019 cjyj-zdxm0022. The authors declare no conflict of interest.