## 3 Calibration Method
### Virtual Auxiliary Vector  
In order to determine the position relationship between an INS coordinate system and a magnetometer coordinate system, we assume a virtual vector \(\mathbf{V}\) to expand the INS's angle to magnetic value. \[\mathbf{V}=\left[ \begin{array}{c}v_{x}\\ v_{y}\\ v_{z}\end{array} \right] \tag{4}\]  
We set \(\mathbf{V}\) as a different background magnetic field in the INS coordinate system. The INS coordinate system is based on a geographic coordinate system, i.e., the \(x\)-, \(y\)- and \(z\)-axes point east, north, and up, respectively. The auxiliary correction data can be obtained by rotating \(\mathbf{V}\). [PERSON]'s rotation method is used to generate auxiliary calibration data. A vector can be rotated to any angle by three rotations around a single coordinate axis [18; 19]. These three rotations are as follows: rotating around the \(z\)-, \(x\)- and \(y\)-axes, respectively. They can be expressed as:  
\[\mathbf{E}_{x}=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\gamma&-\sin\gamma\\ 0&\sin\gamma&\cos\gamma\end{array} \right] \tag{5}\]  
\[\mathbf{E}_{y}=\left[ \begin{array}{ccc}\cos\beta&0&\sin\beta\\ 0&1&0\\ -\sin\beta&0&\cos\beta\end{array} \right] \tag{6}\]\[E_{z}=\left[ \begin{array}{ccc}\cos\alpha&-\sin\alpha&0\\ \sin\alpha&\cos\alpha&0\\ 0&0&1\end{array} \right] \tag{7}\]  
where \(\alpha,\beta\) and \(\gamma\) denote heading, roll, and pitch angles. \(E_{x},E_{y}\) and \(E_{z}\) denote the rotation matrix around the \(x\)-axis, \(y\)-axis, and \(z\)-axis, respectively. According to [PERSON] rotation method, the analytical order of INS attitude data is shown in Figure 2:  
\[0-xyz\to o-x_{1}y_{1}z_{1}\to o-x_{2}y_{2}z_{2}\to o-x_{3}y_{3}z_{3}\]  
We set the virtual vector as the magnetic field value of the geographic coordinate system. Therefore, we can generate the auxiliary correction dates as Equations (8) and (9). \[E=E_{y}\cdot E_{x}\cdot E_{z} \tag{8}\]  
\[V_{v}=E\cdot V \tag{9}\]  
where the \"-\" donates the matrix multiplication. We can consider that the virtual vector and the real geomagnetic field are two different stable background magnetic fields. The auxiliary correction data and the measured data are obtained from the same rotation in these two stable background magnetic fields. The trajectories of \(V_{v}\) and measured magnetic data \(H_{m}\) should be parallel. However, these two trajectories are not parallel under the influence of misalignment error. The trajectory of the measured data is shown in Figure 3. \(N_{h}\) is the normal vector of the plane \((o-x_{m}y_{m})\) and \(T_{h}(i)\) is the tangent vector of the circle at this phase which can be calculated using Equations (10) and (11), respectively. \[N_{h}=(H_{m}(i)-H_{m}(j))\times(H_{m}(i+n)-H_{m}(j+n))\quad(i\
eq j) \tag{10}\]  
\[T_{h}(i)=H_{m}(i)\times N_{h} \tag{11}\]  
where the \"\(\times\)\" donates the cross product. Figure 2: The diagram of [PERSON]’s rotation. The position relationships among the trajectories of actual measurement data and virtual data are shown in Figure 4. \(N_{v}\) is the normal vector of the plane \((o-x_{i}y_{i})\) and \(T_{v}(i)\) is the tangent vector of the circle at this phase which can be calculated using Equations (12) and (13), respectively. \[\mathbf{N}_{v}=(\mathbf{V}_{v}(i)-\mathbf{V}_{v}(j))\times(\mathbf{V}_{v}(i+n)-\mathbf{V}_{v}(j+n)) \quad(i\
eq j) \tag{12}\]  
\[0-xyz\to o-x_{1}y_{1}z_{1}\to o-x_{2}y_{2}z_{2}\to o-x_{3}y_{3}z_{3}\]0  
We can calculate the angle between \(\mathbf{T}_{h}(i)\) and the plane which is made up of vector \(\mathbf{V}_{v}(i)\) and \(z_{i}\)-axis by Equation (14). \[0-xyz\to o-x_{1}y_{1}z_{1}\to o-x_{2}y_{2}z_{2}\to o-x_{3}y_{3}z_{3}\]1  
We can rotate \(\mathbf{T}_{h}(i)\) to \(\mathbf{T}_{h^{\prime}}(i)\) about the \(z_{m}\)-axis by using Equations (15) and (16). \[\mathbf{M}(i)=\left[ \begin{array}{ccc}\cos(\Phi(i))&-\sin(\Phi(i))&0\\ \sin(\Phi(i))&\cos(\Phi(i))&0\\ 0&0&1\end{array} \right] \tag{15}\]  
Figure 4: The diagrams of measured data and auxiliary data. Figure 3: The diagram of measured data.