## 3 Results  
### Simulation Analysis of Lateral Positioning Algorithm  
A trajectory of a car running from south to north is obtained by trajectory simulation using MATLAB, and the inertial device output information is obtained by using the back-propagation method. Assuming a lane width of 4.5 m and a total of three lanes, the vehicle enters the tunnel at the center of the middle lane. As shown in Figure 9a, the simulation diagram of the vehicle's trajectory, the distance to the change lane to the right is positive, and the distance to the change lane to the left is negative. The working conditions include the following four states: changing lanes to the left, changing lanes to the right, changing two lanes to the left, and driving in a straight line. The car travels at a speed of 18 m/s (about 65 km/h), and the car turns at an angle of 8\({}^{\circ}\) each time it changes lane, and the angular speed when turning is 40\({}^{\circ}\)/s. From this, the real-time lateral distance of the car while driving can be obtained, as shown in Figure 9b.  
MATLAB is used to simulate the pure inertial guidance solution, setting the bias of the gyroscope and accelerometer in strapdown inertial guidance to 0.1 deg/h and 100 ug respectively, and the noise of the gyroscope and accelerometer to 0.01 deg/sqrt (h) and 10 ug/sqrt (Hz) respectively, and the sampling frequency of inertial components is 50 Hz. The initial position error of latitude and longitude height is 1 m. The initial azimuth error is set to 0.5 arc second for the east and north directions, the speed error is 0 m/s, the time delay between the odometer data and IMU data is 0 s, and the odometer scale factor error is set to 1. Figure 10 shows the vehicle heading angle variation, Figure 11a shows the lateral distance variation under the inertial guidance solution, and Figure 11b shows the error of the lateral distance from the inertial guidance solution. It can be seen from Figure 11b that the error of vehicle lateral displacement distance solved by inertial navigation increases with the increase in time, which is in line with the characteristics of inertial navigation's positioning error accumulated with time, and the error is close to 9 m in about 120 s, which obviously does not meet the requirements of vehicle lateral positioning.  
Figure 10: Vehicle heading angle information.  
Figure 9: Vehicle track information image: (**a**) Vehicle track information; (**b**) Lateral distance variation of the vehicle.  
As shown in Figure 12, it is a graph of the change in vehicle lateral distance based on the algorithm proposed in this paper. From Figure 12, it can be seen that the vehicle lateral distance solved based on the algorithm in this paper is basically consistent with the real distance, and the two data lines basically overlap. Figure 13 shows the error of the lateral distance of the vehicle based on the algorithm of this paper, with a maximum error of about 0.13 m. Since the four error peaks have typical coarse error characteristics and are generated during the lane change process, they can be eliminated in the actual system. For the whole process of lane change, the difference between the calculated distance and the real distance is less than 0.02 m at the end of the lane change (Figure 12b). It can be seen that when the accuracy of inertial components is not high, the algorithm proposed in this paper can still provide high positioning accuracy, so the algorithm has good robustness in practical applications and is less affected by the accuracy error of inertial components, which has high practicality.  
Figure 11: Inertial guidance solution information: (**a**) Lateral distance variation of the inertial guidance solution; (**b**) Lateral distance error of inertial guidance solution.  
Figure 12: Lateral distance variation based on the algorithm of this paper: (**a**) Lateral distance variation of vehicles based on lateral positioning algorithm; (**b**) Local enlargement.