## 3 Modeling and Error Analysis of the Navigation Process in RINS
### The Navigation Algorithm Calculation Process  
In the inertial navigation process, the IMU is usually set on a stationary state during its initial alignment process. In a static state, the initial velocity is zero, and the initial position could be directly obtained from a GNSS. Therefore, the primary purpose of initial alignment is to obtain the initial azimuth and the pitch and roll angles of a carrier, which are expressed in a transformation matrix relating to the b-frame and n-frame, also known as \(C_{n}^{b}\). The process of the initial alignment process consists of a coarse alignment and a fine alignment. In a static condition, inertial sensors measure movement information resulting from earth's gravity and rotation. Since earth's gravity and rotating speed are known parameters in geophysical fields, the angular and acceleration signals obtained from sensors are mainly influenced and determined by the initial attitude of the carrier. Usually, initial alignment calculates the mean value of gyro and accelerometer measurement resultsfirst. The transformation matrix could be calculated by relating the movement information in b-frame and n-frame with the following equation. \[\left[ \begin{array}{c}f^{b}\\ \omega^{b}_{ib}\\ v^{b}\end{array} \right]=C^{b}_{n}\times\left[ \begin{array}{c}f^{n}\\ \omega^{n}_{ib}\\ v^{n}\end{array} \right] \tag{1}\]  
where \(f^{b}\) is the specific force in the body frame, \(\omega^{b}_{ib}\) is the angular rate in the body frame and \(v^{b}\) is the cross product of a specific force and angular rate in the body frame. \(f^{n}\) is the specific force in a navigation frame, \(\omega^{n}_{ib}\) is the angular rate in a navigation frame and \(v^{n}\) is the cross product of a specific force and angular rate in a navigation frame. \(C^{b}_{n}\) is the transformation matrix from n-frame to b-frame. After coarse alignment, an approximate carrier attitude can be obtained. However, coarse alignment can only obtain the rough attitude of a carrier, and it has a deviation with real attitude. If it is used directly as the initial motional state in the navigation process, the initial deviation of attitude would also lead to the dispersion of position-resolving accuracy. To solve this problem, the fine alignment process is conducted to improve the initial alignment accuracy. Fine alignment includes a filtering and optimization process. It usually uses [PERSON]'s filtering algorithm to integrate external navigation references with the inertial state, which could yield a more accurate navigation initial state. After obtaining the initial attitude, velocity and position of the carrier, the subsequent motion state of the carrier can be obtained by measuring angular increment and acceleration. Specifically, the attitude of the carrier in the whole navigation process could be iteratively calculated through the carrier's previous attitude state and the angular increment in inertial coordination. The velocity of the carrier could be calculated with known acceleration, measured from an accelerometer. Thus, the precise location of the carrier could be deduced with known attitude and velocity of the carrier. A brief attitude-, velocity- and position-updating process is introduced as follows:  
Attitude-updating algorithm:  
The classic dynamic-updating equation of the attitude of an IMU could be expressed as follows:  
\[C^{n(m)}_{b(m)}=C^{n(m)}_{n(m-1)}C^{n(m-1)}_{i}C^{i}_{b(m-1)}C^{b(m-1)}_{b(m)}= C^{n(m)}_{n(m-1)}C^{n(m-1)}_{b(m-1)}C^{b(m-1)}_{b(m)} \tag{2}\]  
where \(C^{n(m-1)}_{b(m-1)}\) and \(C^{n(m)}_{b(m)}\) denote the attitude transformation matrix between the b-frame and n-frame at successive time moments of t\({}_{m}\) and t\({}_{m-1}\). \(C^{b(m-1)}_{b(m)}\) denotes the attitude-updating matrix of the carrier in b-frame, while \(C^{n(m)}_{n(m-1)}\) denotes the attitude-updating in n-frame, assuming that the gyro has an increment of \(\phi^{b}_{it(m)}\) during one calculation period and supposing the earth has a rotation speed of \(\omega^{n}_{in}\) in n-frame.