## 3 Novel Approach EKF Tuning
### Mathematical Modeling
#### 3.1.1 Inertial Sensor Error Characteristics  
In this subsection, the following equations are based on the work presented by [PERSON] [1].  
The accelerometer error model is given by Equation (32), which accounts for bias (\(\mathbf{b}_{a}\)), errors due to the scale factor and cross-coupling (\(\mathbf{M}_{a}\), as shown in Equation (35)), and random noise (\(\mathbf{w}_{a}\)):  
\[\mathbf{\tilde{r}}_{ib}^{b}=\mathbf{b}_{a}+(\mathbf{I}_{3\times 3}+\mathbf{M}_{ a})\mathbf{f}_{ib}^{b}+\mathbf{w}_{a}, \tag{32}\]  
where \(\mathbf{f}_{ib}^{b}\) is the ideal specific force expressed in the body frame, obtained from the trajectory, and \(\mathbf{\tilde{r}}_{ib}^{b}\) is the computed specific force from the IMU.  
The gyroscope error model is given by Equation (33), where the considered stochastic variables are bias (\(\mathbf{b}_{g}\)), errors due to the scale factor and cross-coupling (\(\mathbf{M}_{g}\), as shown in Equation (35)), g-dependent bias (\(\mathbf{G}_{g}\)), and random noise (\(\mathbf{w}_{g}\)):  
\[\tilde{\mathbf{\omega}}_{ib}^{b}=\mathbf{b}_{g}+(\mathbf{I}_{3\times 3}+\mathbf{M} _{g})\mathbf{\omega}_{ib}^{b}+\mathbf{G}_{g}\mathbf{f}_{ib}^{b}+\mathbf{w}_{g}, \tag{33}\]  
where \(\mathbf{\omega}_{ib}^{b}\) is the ideal angular rate expressed in the body frame, also obtained from the trajectory, and \(\tilde{\mathbf{\omega}}_{ib}^{b}\) is the computed angular rate vector from the IMU.  
When considering biases, it can be divided into static components, \(\mathbf{b}_{as}\) and \(\mathbf{b}_{gs}\), and dynamic components, \(\mathbf{b}_{ad}\) and \(\mathbf{b}_{gd}\), where  
\[\mathbf{b}_{a}=\mathbf{b}_{as}+\mathbf{b}_{ad}\quad\text{and}\quad\mathbf{b}_ {g}=\mathbf{b}_{gs}+\mathbf{b}_{gd}. \tag{34}\]  
The static component, also referred to as the fixed bias, turn-on bias, or bias repeatability, encompasses the run-to-run variation in each instrument bias along with the residual fixed bias that remains after sensor calibration. This component remains constant during the IMU's operation but can differ between runs. The dynamic component, also termed in-run bias variation or bias instability, changes over periods in the order of a minute and includes the residual temperature-dependent bias left after sensor calibration. Typically, the dynamic bias constitutes about 10% of the static bias [1].  
The scale factor and cross-coupling errors for a nominally orthogonal set of accelerometers and gyroscopes can be represented by the following matrices:  
\[\mathbf{M}_{a}=\begin{pmatrix}s_{a,x}&m_{a,xy}&m_{a,xz}\\ m_{a,yx}&s_{a,y}&m_{a,yz}\\ m_{a,zx}&m_{a,zy}&s_{a,z}\end{pmatrix},\quad\mathbf{M}_{g}=\begin{pmatrix}s_{ g,x}&m_{g,xy}&m_{g,xz}\\ m_{g,yx}&s_{g,y}&m_{g,yz}\\ m_{g,zx}&m_{g,zy}&s_{g,z}\end{pmatrix}, \tag{35}\]  
where the accelerometer and gyroscope scale factor errors within the IMU are represented by the vectors \(\mathbf{s}_{a}=(s_{a,x},s_{a,y},s_{a,z})\) and \(\mathbf{s}_{g}=(s_{g,x},s_{g,y},s_{g,z})\), respectively. Additionally, \(m_{a,\alpha\beta}\) represents the cross-coupling coefficient for the specific force along the \(\beta\)-axis sensed by the \(\alpha\)-axis accelerometer, and \(m_{g,\alpha\beta}\) represents the cross-coupling coefficient for the angular rate along the \(\beta\)-axis sensed by the \(\alpha\)-axis gyroscope. Scale factor and cross-coupling errors are dimensionless and are typically expressed in parts per million (ppm) or as a percentage.  
Once the inertial sensor data are obtained, their variances are calculated incrementally and the data are used in the INS mechanization equations.