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Page 8 of 20 Huang et al. Complex Eng Syst 2023;3:2 I http://dx.doi.org/10.20517/ces.2022.43

Therefore, lane parameters and point sequences are stored in the map file in this study. Optimizing the curve
equation after parameter does not have a physical meaning. To make the spacing of sampling points on each
section of the lane consistent, the arc length of the curve needs to be calculated and used to re-extract the
equidistant sampling points.

The CHS arc lengths can be calculated from the following equation:

¹ ¹ q ¹
1 1 1 p
2 2 2 ′T T T ′ (9)
′
= ( ) dt = ( ) + ( ) + ( ) dt = dt
′
′
0 0 0
(9) is an elliptic integral, which is difficult to calculate by ordinary methods. In this study, the Gauss-Kronrod
quadrature method [33] is used to simplify the integration calculation process.
We use the G7-K15 method, a 7-point Gauss rule with a 15-point Kronrod rule, apply it to (9), and use the
rules of the upper and lower limits of the integral transformation to calculate the arc length from 0 to 1:

¹ ¹ 1 15
1 Õ
1 − 0 + 1 1 − 0 + 1
( ) dx = ( 1 − 0 ) + 0 dx ≈ ( 1 − 0 ) + 0 (10)
0 −1 2 2 2 2
=1
(10) can be used to calculate the arc length of the lane curve, which is not only used for equidistant sampling
but also in intersection steering scenarios. The arc length can also be used to calculate curvature, which is
convenient for planning.

5. LOCALIZATION BASED ON AN HD MAP
There are a variety of complex road environments in the cities, such as tunnels, overpasses, and urban canyons.
TheseenvironmentsmakeGNSS-basedlocalizationlessreliable. Someodometryfusingmethodshaveemerged
to solve the problem of GNSS failure. However, due to odometry drift, these methods cannot meet the local-
ization requirements in scenarios where there is a long-term lack of effective global position information [34] .
Although point cloud map relocalization based on the iterative closest point (ICP) [35] , normal distribution
transform (NDT) [36] and other methods is very effective, a very large point cloud map becomes a major chal-
lenge that affects practical use. An HD map contains various semantic features, while lane lines and traffic
signs have good recognition both day and night. To explore the global localization method combining an HD
map and IMU, two problems need to be solved. First, the elements in the HD map are associated with the
elements detected using other sensors. Second, the pose is estimated based on the feature association results.

5.1. Reprojection
Reprojection refers to projecting the coordinates of a corresponding point in 3D space back to the pixel plane
according to the currently estimated pose. The error between the reprojected and actual pixel coordinates is
called the reprojection error and is often used as an indicator to evaluate the pose. Based on the position of
the lane in the map, the known a priori knowledge of the HD map is projected onto the camera image by
combining the intrinsic and extrinsic parameters of the camera. The evaluation of a pose metric is obtained by
differencing the a priori map element positions and the coordinates of the matching perceptual results. Ideally,
the distance between the two should be zero. The optimal camera pose can be obtained by optimizing the
camera pose using a nonlinear optimization method to minimize this evaluation metric so that the optimal
vehicle pose can be calculated.

First, referring to the transcendental vehicle pose bw, combined with (3), the representation of a feature point