## 4 Error Analysis
### Error From Compensation Angle  
Based on analysis in chapter 2, and according to (1) and (4)-(7), the compensation angle \(\varepsilon\) is determined by geomagnetic information, inclination \(I\) and declination \(D\), attitude of projectile,and pitch \(\theta\) and yaw \(\psi\) (given by GPS receiver or other sensors in practice). In this part, they will be discussed. Generally, the range of a howitzer is \(\sim\)30 km, in which the declination and inclination changing slightly has little effect on the compensation angle except for some special locations such as the north pole and south pole. Take some place \(\mathrm{O_{L}}\) in Beijing as a launching site, the declination and inclination of which are \(-6.8997^{\circ}\) and \(59.3551^{\circ}\), respectively. Based on the pitch angle of 6-DoF trajectory (shown in Figure 10b), azimuth at launch are set as 0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), and 270\({}^{\circ}\), respectively. Then, compare the compensation angle based on original geomagnetic information of launching site \(\mathrm{O_{L}}\) and compensation angle based on updated information. \(\mathrm{N_{6}}\), \(\mathrm{E_{6}}\), \(\mathrm{S_{6}}\), and \(\mathrm{W_{6}}\) are locations 30 km away from launching site in north, east, south, and west direction, respectively. In the north, geomagnetic information is updated at sites \(\mathrm{N_{j}}\) (\(j\) = 1, 2, 3, 4, 5), locations between \(\mathrm{O_{L}}\) and \(\mathrm{N_{6}}\), and they are 5 km from each other, and so are \(\mathrm{E_{j}}\) (\(j\) = 1, 2, 3, 4, 5), \(\mathrm{S_{j}}\) (\(j\) = 1, 2, 3, 4, 5), and \(\mathrm{W_{j}}\) (\(j\) = 1, 2, 3, 4, 5). All the data mentioned above come from [25; 27; 32; 33]. The results of comparison of compensation angle are shown as follows. Figures 14 and 15 show that in the range of 30 km; the change of declination and inclination has little effect on the compensation angle. In the direction of east, south, and west, the deviation is within 0.15\({}^{\circ}\). However, in the north direction, the compensation angle changes dramatically at the last 5 km which results in a comparatively large deviation 3.5\({}^{\circ}\). It also indicates that deviation depends on the attitude of projectile in flight. Thus, if the accuracy of compensation angle cannot meet the requirements such as situation in north direction mentioned above, the information of geomagnetic field should be stored in advance and be exported to the tracking system to figure out compensation angle timely according to the location provided by GPS receiver. Yaw and pitch are given by GPS receiver (or by other sensors), and analysis of error from pitch and yaw is as follows. Based on the attitude information from the 6-DoF trajectory mentioned above and the geomagnetic information of location \(\mathrm{O_{L}}\), compensation angle obtained from the attitude information with and without noise are to be compared in the circumstance of different yaw angles at the launch site. Shown in Figure 16, SD and mean of deviation rise with the increasing SD of noise in each launching direction. This indicates that when the SD of noise of pitch and yaw is less than \(5^{\circ}\), the mean and SD of deviation are approximately \(0.5^{\circ}\) and \(6^{\circ}\), respectively. However, there are comparatively undesirable results in the north (yaw is \(0^{\circ}\)) and south (yaw is \(180^{\circ}\)) direction. When the SD of noise is \(5^{\circ}\) the SD of deviation reaches \(16^{\circ}\) and \(14^{\circ}\) in the north and south, respectively. Figure 17 describes the change of compensation angle and deviation of compensation angle when the SD of noise of yaw and pitch is \(3^{\circ}\). When the yaw is \(0^{\circ}\) (shown in (a) and (b)), the deviation increases in the last 10 s; when the yaw is \(180^{\circ}\) (shown in (c) and (d)) the deviation is comparatively large at first and then converges till the end.