## 3 Parameter Self-Calibration Method for Loops in Deeply Coupled Systems Based on Norm Analysis
### Norm Analysis to the IMU Error Propagation Properties  
Before the norm analysis referring to the INS error, the definition of the two kinds of norms should be made clear [13]. If \(A\) is a vector, the Euclid norm, or 2-norm, of \(A\) is \(\left\|A\right\|_{2}=(\sum\limits_{i=1}^{n}|A_{i}|^{2})^{1/2}\). If \(B\) is a matrix, the Frobenius norm, or \(F\)-norm, of \(B\) is \(\left\|B\right\|_{F}=(\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}\left|B_{ij} \right|^{2})^{1/2}\). Both of these two norms represent the meaning of spatial distance and satisfy the following axioms:  
1. \(\left\|X\right\|\geq 0\); further, \(\left\|X\right\|=0\) if, and only if, \(X=0\). (Positive-definiteness)
2. \(\left\|aX\right\|=\left|a\right|\left\|X\right\|\) for any scalar \(a\). (Homogeneity)
3. \(\left\|X+Y\right\|\leq\left\|X\right\|+\left\|Y\right\|\). (Triangle inequality)  
The constant coefficients theorem is described as follows: Let \(A\) be an matrix of \(m\times m\) scalars, \(X\) be an \(m\times n\) matrix of unknowns, and \(B\) be an \(m\times n\) matrix of given functions. Then the differential equation  
\[\dot{X}=A\cdot X+B, \tag{7}\]  
with initial value \(X(t_{0})=X_{0}\) has the solution  
\[X(t)=e^{(t-t_{0})A}X_{0}+\int_{t_{0}}^{t}e^{(t-t_{0})A}B(\tau)d\tau, \tag{8}\]  
where \(t_{0}\) denotes the initial moment. According to the theorem above, Equation (4) could be expressed briefly as the following differential equation:  
\[\dot{X}=F\cdot X+W, \tag{9}\]  
and the approximate analytical solution of Equation (7) is  
\[X(t_{0}+\Delta T)=e^{\Delta TF}X_{0}+\int_{t_{0}}^{t}e^{\Delta TF}Wd\tau, \tag{10}\]  
where \(\Delta T\) denotes the time interval between two consecutive estimations. From Equation (4), the velocity and position differential equation could be expressed as follows:  
\[\delta\dot{R}^{e}=\delta V^{e}, \tag{11}\]  
\[\delta\dot{V}^{e}=\Big{[}G-(\widehat{\Omega}^{e})^{2}\Big{]}\delta R^{e}-2 \widehat{\Omega}^{e}\delta V^{e}+\widehat{F}^{e}\delta y^{e}+C_{b}^{e}\delta A ^{b}, \tag{12}\]  
where \(\delta\dot{R}^{e}\) and \(\delta\dot{V}^{e}\) represent the derivatives of position and velocity error vectors, respectively. From Equation (11), the position error \(\delta R^{e}\) in one time interval should be \(\delta V^{e}\Delta T\). Thus, Equation (12) could be expressed as follows:  
\[\delta V^{e}=\Big{[}\Big{(}G-(\widetilde{\Omega}^{e})^{2}\big{)}\Delta T-2 \widetilde{\Omega}^{e}\big{]}\delta V^{e}+\widetilde{F}^{e}\delta\psi^{e}+C_{b }^{e}\delta A^{b}. \tag{13}\]  
Combining with Equation (10), the velocity error caused by the accelerometer error is  
\[\left(\delta V^{e}\right)_{acc-bias}=\frac{1}{F^{*}}e^{F+}C_{b}^{e}\delta A^{b}, \tag{14}\]  
\[F^{*}=(G-(\widetilde{\Omega}^{e})^{2})\Delta T-2\widetilde{\Omega}^{e}. \tag{15}\]  
Thus, from Equation (14), the velocity error caused by the accelerometer error is  
\[ \begin{split}(\delta V^{e})_{acc-bias}&\approx \left[\Delta TI-\Delta T^{2}\widetilde{\Omega}^{e}+\frac{\Delta T^{3}}{2}(G-( \widetilde{\Omega}^{e})^{2})+\frac{\Delta T^{3}((G-(\widetilde{\Omega}^{e}) ^{2})\Delta T-2\widetilde{\Omega}^{e})^{2}}{6}\right]C_{b}^{e}\delta A^{b}\\ &\approx\left[\Delta TI-\Delta T^{2}\widetilde{\Omega}^{e}+\frac {\Delta T^{3}}{2}G+\frac{\Delta T^{3}(\widetilde{\Omega}^{e})^{2}}{6}\right]C _{b}^{e}\delta A^{b}\end{split} \, \tag{16}\]  
and it should be noted that the small quantities of high order (\(\Delta T^{\geq 4}\)) in this step have been ignored.