## 3 Modeling and Error Analysis of the Navigation Process in RINS
### Error Model of a Navigation System  
Compared with other navigation systems, the inertial navigation system has advantages of anti-interference, all-weather adaptability, good autonomy and complete information collection ability. Generally, inertial navigation works without external correction information; the resolved velocity, position and attitude diverge from actual motion states, and these navigation errors arise from several sources. For instance, errors would arise in the accelerometer and gyro measurements, physical parameter approximations and iteration rate limitations in navigation equations, as well as initialization errors. To make matter's worse, these errors would couple with each other, and the navigation accuracy could not be guaranteed and degrades the reliability of the navigation reliability. In order to eliminate the influence of the navigation errors, a brief introduction of the navigation error models is described in this section first. To explain the error model of the navigation algorithm, the measurement error of inertial sensors was analyzed first. For gyros and accelerometers, the sensor's measurement error of the IMU could be concluded into three parts: the scale factor error, zero drift and the installation error. The general error model of a gyro could be described as follows:  
\[ \begin{bmatrix}\omega_{ibx}^{b}\\ \omega_{ibx}^{b}\\ \omega_{ibx}^{b}\end{bmatrix} =C_{b_{g}}^{b} \begin{bmatrix}(1-\delta k_{gx}) \omega_{ibx}^{b_{g}}-\varepsilon_{x}^{b_{g}}\\ (1-\delta k_{gyy})\omega_{iby}^{b_{g}}-\frac{b_{y}}{y}\\ (1-\delta k_{gzz})\omega_{ibz}^{b_{g}}-\varepsilon_{z}^{b_{g}}\end{bmatrix}  \tag{13}\]  
where \(\omega_{ib}^{b}\) is ideal angular velocity in ideal b-frame. \(b_{g}\)-frame is the coordinate denoted by the actual direction of three gyros, which have small angular misalignments with b-frame. \(\omega_{ibx}^{b_{g}}\) is the angular measurement results measured from three gyros; \(\delta k_{gyy}\) is the scale factor errors caused by the temperature or system stability. \(\varepsilon_{x}^{b_{g}}\) is the gyro bias along the x, y and z directions in \(b_{g}\)-frame. We could denote the accelerometer measurement errors in a similar way. \[ \begin{bmatrix}f_{ibx}^{b}\\ f_{iby}^{b}\\ f_{ibx}^{b}\end{bmatrix} =C_{b_{g}}^{b} \begin{bmatrix}(1-\delta k_{gx})f_{ibx}^ {b_{g}}-\
abla_{x}^{b_{g}}\\ (1-\delta k_{gyy})f_{iby}^{b_{g}}-\
abla_{y}^{b_{g}}\\ (1-\delta k_{gzz})f_{ibz}^{b_{g}}-\
abla_{z}^{b_{g}}\end{bmatrix}  \tag{14}\]  
where \(f_{ibx}^{b}\), \(\delta k_{gxx}\), \(f_{ibx}^{b_{g}}\) and \(\
abla_{x}^{b_{g}}\) have similar definitions with the parameters in the gyro error models. The above analysis reveals that inertial sensors would generate inaccurate angular and acceleration measurement results. In the navigation algorithm, these errors would couple with the calculation process and are unsuitable for navigation resolving over an extended period of time. The following is a description of the position, attitude and velocity error models, and it shows how one error source can be coupled with other navigation results. Attitude error model: the attitude error equation could be expressed in the following equation:  
\[\dot{\phi}=\phi\times\omega_{in}^{n}+\delta\omega_{in}^{n}-\delta\omega_{ib}^ {n} \tag{15}\]  
where \(\dot{\phi}\) is the attitude error, \(\delta\omega_{in}^{n}\) is the calculated error in navigation frame and \(\delta\omega_{ib}^{n}\) is the measured gyro error. Velocity error model: the velocity error equation could be expressed in the following equation:  
\[\delta\dot{v}^{n}=f_{sf}^{n}\times\phi+v^{n}\times(2\delta\omega_{ie}^{n}+ \delta\omega_{en}^{n})-(2\omega_{ie}^{n}+\omega_{en}^{n})\times\delta v^{n}+ \delta f_{sf}^{n}+\delta g^{n} \tag{16}\]  
where \(\delta\ddot{v}^{n}\) is the velocity error in the navigation frame, \(\delta f_{sf}^{n}\) is the measured specific force error and \(\delta g^{n}\) is the gravity error.