The positioning accuracy cannot meet the needs of mobile mapping after the smoothing solution. In order to improve the positioning accuracy, this paper proposes a compensation method based on plane control. In this paper, the positioning error at the intermediate moment of GNSS loss-of-lock was first calculated in the area where the GNSS signal loss-of-lock using the control plane. Then, according to a relationship similar to a Gaussian function between the smoothed positioning error and the GNSS loss-of-lock time, the positioning error at other times is compensated for. As shown in
Figure 4, first of all, a total station is used to measure the coordinates of 15–30 measuring points which are evenly distributed on the space plane, and then the space plane equation of the plane is calculated. At the same time, the laser point cloud data of the same plane measured by the mobile measurement system and the smoothed position and attitude information are used to construct the laser point cloud positioning equation. Then, Based on the spatial plane equation and the laser point cloud positioning equation, a position error solving model based on plane control is established. Finally, by analyzing the characteristics of the smoothed position error and using the calculated position error to construct a Gaussian function error compensation model, the position error is compensated.
3.2. Positioning Error Solving Principle
The errors that affect the measurement accuracy of the mobile measurement system can be divided into three major components: POS error, laser scanner measurement error, and placement error after the integration of each sensor. With the development of laser scanner hardware, we must ensure that the measuring accuracy of the point position meets the specific measurement needs. The placement error between each sensor can reduce the measurement error by calibration. Therefore, the method in this paper ignores the influence of the laser scanner measurement error and the placement error after the integration of each sensor. Equation (1) can be expressed as follows:
The POS error includes three positioning errors
,
,
and three attitude errors
,
,
, which are added to Equation (2) to obtain the POS error:
in which
is the rotation matrix of the local level frame to the WGS-84 coordinate frame when there is a positioning error;
is the attitude error matrix; and
is the rotation matrix of the body frame to the local level frame when there is an attitude error.
Through the propagation of error, the error model of Equation (3) can be obtained:
In this equation, are the mean square errors of the laser point cloud in the X, Y, and Z directions, respectively, in the WGS-84 coordinate frame; are the mean square errors of the roll angle, pitch angle, and heading angle, respectively, and
are the mean square errors of the positioning in the X, Y, and Z directions, respectively, in the WGS-84 coordinate frame.
According to the official data provided by Novatel, after IE8.6 software processing, the mean square error of the pitch angle and roll angle is 0.005°, the mean square error of heading angle is 0.008°, the mean square error of the horizontal direction is 0.01 m, and the mean square error of the vertical direction is 0.015 m when the satellite signal is not loss-of-lock. The mean square error of pitch angle and roll angle is 0.006°, the mean square error of the heading angle is 0.006°, the mean square error of the heading angle is 0.010°, the mean square error of the horizontal direction is 0.011 m, and the mean square error of the vertical direction is 0.03 m when the satellite signal loss-of-lock lasts for 60 s.
It can be seen that as the satellite signal loses lock time increases, the positioning error increases significantly, while the attitude error varies little. In addition, when the error model is used to calculate the accuracy of the laser point cloud, the effect of the positioning error on the accuracy of the laser point cloud is much greater than the attitude error. The attitude error can be smoothed to meet the needs of movement measurement. Therefore, the influence of attitude error is not considered in the calculation, so Equation (3) can be simplified as follows:
The general form of the space plane is:
where
is the unit normal vector of the plane and
is the distance from the origin of the coordinate to the plane. Then the reference plane parameter
can be obtained using the eigenvalue method [
25].
The laser point cloud data on the control surface satisfy the plane equation of the control surface, so:
In Equation (7), , , , represent the plane parameter of the p-th control plane. is the point cloud data coordinates in the body frame. is the obtained positioning error.
3.3. Similar to the Gaussian Distribution Function Error Compensation Model
By analyzing the characteristics of the positioning error after smoothing, it can be observed that the relationship between the positioning error after smoothing and time is similar to the Gaussian function feature. Taking the GNSS signal loss-of-lock for 90 s as an example, the relationship between the smoothed position error curve and the Gaussian function curve is shown in
Figure 5. The red curve is the smoothed positioning error curve, and the green curve is the Gaussian function curve. It can be seen that the smoothed position error curve is similar to the Gaussian function curve.
It is assumed that the satellite signal is occluded at time
and the navigation information calculated after smoothing reaches the maximum value at the intermediate time
. The error and time in the satellite loss-of-lock period have the following Gaussian relationship:
where
is the positioning error value at time
, which can be obtained by the above method. The positioning error at other time points is calculated by a model similar to the Gaussian function, thereby compensating for the positioning error of the entire satellite signal during the loss-of-lock period.