2.2.1. Real-time coarse alignment

Assuming that the orientation of the IMU b-frame relative to the n-frame is unknown at start-up, we arbitrarily form the b ₀- and n ₀-frames by inertially fixing the b- and n-frames at the beginning of the alignment. Since both the b ₀- and n ₀-frames are fixed with respect to the inertial frame, the direction cosine matrix ${\bi C}_{b_{0}}^{n_{0}}$ is constant (Silson, Reference Silson2011). According to the chain rule of the attitude matrix, the time-varying attitude matrix ${\bi C}_{b}^{n}(t)$ can be decomposed as (Gu et al., Reference Gu, El-Sheimy, Hassan and Syed2008; Wu et al., Reference Wu, Wu, Hu and Hu2011):

(1)$${\bi C}^n_b(t)={\bi C}^n_{n_0}(t){\bi C}^{n_0}_{b_0}{\bi C}^{b_0}_{b}(t)$$

where ${\bi C}_{n_{0}}^{n}(t)$ represents the time-varying matrix of the n-frame relative to the inertial space as a function of the Earth rotation rate and position change relative to the Earth; ${\bi C}_{b}^{b_{0}}(t)$ represents the time-varying matrix due to the rotation of the b-frame relative to the inertial space as a function of inertial angular rates measured by the gyroscope and ${\bi C}_{b_{0}}^{n_{0}}$ represents the constant matrix between the initial b ₀- and n ₀-frames at t=0. The purpose of the DADCA process is to determine the initial constant matrix ${\bi C}_{b_{0}}^{n_{0}}$ instead of the time-varying attitude matrix ${\bi C}_{b}^{n}(t)$.

The navigation specific force equation in the n-frame is well known as (Qin, Reference Qin2014):

(2)$$\dot{\bi v}^n = {\bi C}^n_b{{\bi f}}^b-(2{\bf \omega}^n_{ie}+{\bf \omega}^n_{en})\times {{\bi v}}^n+{{\bi g}}^n$$

where vⁿ denotes the velocity relative to the Earth in the n-frame, f^b is the specific force in the b-frame, ${\bf \omega}^{n}_{ie}$ is the Earth rotation rate vector in the n-frame, ${\bf \omega}^{n}_{en}$ is the angular rate of the n-frame with respect to the e-frame and gⁿ is the gravity vector.

Using Equation (1), Equation (2) can be reorganised as:

(3)$${\bi C}^{n_0}_n[{{\bi g}}^n - \dot{\bi v}^n - (2{\bf \omega}^n_{ie} + {\bf \omega}^n_{en}) \times {{\bi v}}^n] = -{\bi C}^{n_0}_{b_0}({\bi C}^{b_0}_b{{\bi f}}^{\!b})$$

Without the external velocity and position information required for traditional in-motion coarse alignment, the in-motion alignment is simplified as a stationary alignment. It is equivalent to ignoring the part $-\dot{\bi v}^{n}-(2{\bf \omega}^{n}_{ie}+{\bf \omega}^{n}_{en})\times {{\bi v}}^{n}$ on the left-hand side of Equation (3). Even though many error sources will be introduced, such a simplification is an expedient without auxiliary sensors. Denoting the current time as t, we have:

(4)$${{\bi g}}^{n_0}(t)=-{\bi C}^{n_0}_{b_0}\tilde{{{\bi f}}}^{b_0}(t)$$

in which:

(5)$${{\bi g}}^{n_0}(t)={\bi C}^{n_0}_n(t){{\bi g}}^n$$

(6)$$\tilde{{{\bi f}}}^{b_0}(t)={\bi C}^{b_0}_b(t)\tilde{{{\bi f}}}^b(t)$$

where $\tilde{{{\bi f}}}^{b}(t)$ is the specific force measured by accelerometers in the b-frame, ${\bi C}^{n_{0}}_{n}(t)$ denotes the changes of the n-frame during the alignment process and can be calculated as:

(7)$${\bi C}^n_{n_0}(t)={\bi C}^n_e(t){\bi C}^e_{e_0}(t){\bi C}^{e_0}_{n_0}$$

with

$${\bi C}^{e_0}_{n_0}=\left[\matrix{0 & -\sin L_0 & \cos L_0\cr 1 & 0 & 0 \cr 0 & \cos L_0 & \sin L_0} \right], {\bi C}^{e}_{e_0}(t)=\left[\matrix{\cos \omega_{ie}t & \sin \omega_{ie}t & 0\cr -\sin \omega_{ie}t & \cos \omega_{ie}t & 0\cr 0 & 0 & 1} \right]\ \hbox{and} \ {\bi C}^n_e(t)=({\bi C}^{e_0}_{n_0})^{\rm T}$$

where ω_ie denotes the magnitude of the Earth rotation rate and L ₀ the initial geographic latitude.

In Equation (6), ${\bi C}^{b_{0}}_{b}(t)$ represents the orientation of the b ₀-frame relative to the b-frame and can be calculated as:

(8)$$\dot{{\bi C}}^{b_0}_b(t) = {\bi C}^{b_0}_b(t)(\tilde{{\bf \omega}}^b_{ib}\times)$$

where $\tilde{{\bf \omega}}^{b}_{ib}$ denotes the body angular rate measured by gyroscopes in b-frame, and the skew symmetric matrix ( · ×) is defined so that the cross product satisfies a × b = (a × )b for two arbitrary vectors (Wu and Pan, Reference Wu and Pan2013b).

The alternative of Equation (8) is the quaternion differential equation:

(9)$$\dot{{{\bi Q}}}^{b_0}_b(t)=\displaystyle{1 \over 2}{{\bi Q}}^{b_0}_b(t)\otimes \tilde{{\bf \omega}}^b_{ib}$$

where ⊗ denotes the quaternion product and the initial condition is ${{\bi Q}}^{b_{0}}_{b}(0)=[1\ 0\ 0\ 0]^{\rm T}$.

Since the initial attitude matrix ${\bi C}^{n_{0}}_{b_{0}}$ is constant, it can be determined by a Three-axis Attitude Determination-based (TRIAD) algorithm with two non-collinear vectors (James, Reference James2012; Jiang, Reference Jiang1998). It is obvious from Equation (4) that the vectors of ${{\bi g}}^{n_{0}}(t)$ and $\tilde{{{\bi f}}}^{b_{0}}(t)$ at different moments are suitable for analytic alignment. Moreover, in order to suppress the influence of vehicle motion disturbance, a second-order integration method is introduced to accomplish coarse alignment as follows

(10)$${\bi C}^{n_0}_{b_0}=\left[\matrix{({{\bi p}}^{n_0}_{t_1})^{\rm T}\cr ({{\bi p}}^{n_0}_{t_1}\times {{\bi p}}^{n_0}_{t_2})^{\rm T}\cr ({{\bi p}}^{n_0}_{t_1}\times {{\bi p}}^{n_0}_{t_2}\times {{\bi p}}^{n_0}_{t_1})^{\rm T}} \right]^{-1} \left[\matrix{(\tilde{\bi p}^{b_0}_{t_1})^{\rm T}\cr (\tilde{\bi p}^{b_0}_{t_1}\times \tilde{\bi p}^{b_0}_{t_2})^{\rm T}\cr (\tilde{\bi p}^{b_0}_{t_1}\times \tilde{\bi p}^{b_0}_{t_2}\times \tilde{\bi p}^{b_0}_{t_1})^{\rm T}} \right]$$

with

$$\eqalign{\tilde{\bi p}^{b_0}_{t_1}&=\int^{t_1}_0\int^t_0 {\bi C}^{b_0}_b(t)\tilde{{{\bi f}}}^b(t)d\tau dt,\quad \tilde{\bi p}^{b_0}_{t_2}=\int^{t_2}_0\int^t_0 {\bi C}^{b_0}_b(t)\tilde{{{\bi f}}}^b(t)d\tau dt,\cr \tilde{\bi p}^{n_0}_{t_1}&=\int^{t_1}_0\int^t_0 {\bi C}^{n_0}_n(t){{\bi g}}^n d\tau dt,\quad \tilde{\bi p}^{n_0}_{t_2}=\int^{t_2}_0\int^t_0 {\bi C}^{n_0}_n(t){{\bi g}}^n d\tau dt,}$$

where t ₁ and t ₂ are selected as t ₂=t _c, t ₁=t ₂/2.

From Equation (10), the initial matrix can be approximately obtained, and one of the main error sources comes from the assumption of stationary alignment. The misalignment of the initial attitude caused by ignoring the motion acceleration in Equation (3) is approximately equal to the gravity deflection associated with the travelled distance. Since the normal speed of a land vehicle is generally less than 35 m/s and the alignment time lasts less than 10 minutes, it is possible to align the initial attitude to an accuracy of several degrees.

2.2.2. Data record for BFA

During the real time DADCA process, attitude quaternion and velocity increments must be recorded for the BFA process. To reduce storage space and computation load, the required data should be extracted and recorded with a lower frequency such as 1 Hz, which is the same as the Kalman filter measurement update frequency in the subsequent BFA process. In summary, seven floating-point data should be recorded in every storage cycle including four-dimensional attitude quaternion ${{\bi Q}}^{b_{0}}_{b_{k}}$ and three-dimensional velocity increments $\Delta\tilde{{{\bi v}}}^{b_{0}}_{k}$. Note that the quaternion instead of the attitude matrix should be stored to reduce the data volume. The quaternion ${{\bi Q}}^{b_{0}}_{b_{k}}$ can be obtained from Equation (9), and the velocity increments can be calculated as

(11)$$\Delta \tilde{{{\bi v}}}^{b_0}_k=\int^{t_k}_{t_{k-1}}{\bi C}^{b_0}_b(t)\tilde{{{\bi f}}}^b(t)dt$$

where the integral time interval [t _k−1, t _k] is the same as the recording cycle, that is, one second in this paper. Given that the alignment time is 600 seconds, only 4,200 floating-point data additional storage space is needed, which can be easily afforded by ordinary SINS navigation computers.

2.3.1. Backtracking navigation update

Besides the initial velocity and position, the approximate initial attitude matrix ${\bi C}^{n_{0}}_{b_{0}}$ is available after the DADCA process, so the navigation parameters through the whole alignment process can be calculated based on backtracking navigation with the stored data. As the normal inertial navigation solution, the backtracking navigation includes attitude, velocity and position updates.

2.3.1.1. Backtracking attitude update

The time-varying attitude matrix, ${\bi C}^{n_{k}}_{b_{k}}$ which represents the orientation of the n-frame relative to the b-frame at time t _k, can be calculated as follows:

(12)$${\bi C}^{n_k}_{b_k}={\bi C}^{n_k}_{n_0}{\bi C}^{n_0}_{b_0}{\bi C}^{b_0}_{b_k}$$

where ${\bi C}^{n_{0}}_{b_{0}}$ is the alignment result of the DADCA process, and ${\bi C}^{b_{0}}_{b_{k}}$ can easily be converted from the stored quaternion ${{\bi Q}}^{b_{0}}_{b_{k}}$ (Titterton and Weston, Reference Titterton and Weston2004). As the current position is available, the calculation of the matrix ${\bi C}^{n_{k}}_{n_{0}}$ in the BFA process is somewhat different to Equation (7):

(13)$${\bi C}^{n_k}_{n_0}={\bi C}^{n_k}_{e_k}{\bi C}^{e_k}_{e_0}{\bi C}^{e_0}_{n_0}$$

while ${\bi C}^{e_{0}}_{n_{0}}$ is calculated the same way as in Equation (7), ${\bi C}^{n_{k}}_{e_{k}}$ and ${\bi C}^{e_{k}}_{e_{0}}$ are obtained as:

$${\bi C}^{n_k}_{e_k}=\left[\matrix{0 & 1 & 0\cr -\sin L_k & 0 & \cos L_k\cr \cos L_k & 0 & \sin L_k} \right]\!, {\bi C}^{e_k}_{e_0}=\left[\matrix{\cos \Delta & \sin \Delta & 0\cr -\sin \Delta & \cos \Delta & 0\cr 0 & 0 & 1} \right]$$

where Δ = λ _k − λ ₀ + ω _iet and λ₀ denotes the initial geographic longitude at time t=0. λ_k and L _k denotes the current longitude and latitude at time t _k.

Although the backtracking attitude update is not explicitly used in subsequent velocity and position updates, it is required to solve the state Equation (24) and the measurement Equations (25)–(26) in the backtracking SINS/NHC Kalman filter. Meanwhile, the corrected attitude result should be obtained as:

(14)$$\hat{{\bi C}}^{n_k}_{b_k} = \lsqb {\bf I} + (\hat{{\bf \phi}}^n\times)\rsqb {\bi C}^{n_k}_{b_k}$$

where $\hat{{\bf \phi}}^{n}$ denotes the attitude misalignment angles estimated in fine alignment.

2.3.1.2 Backtracking velocity update

The velocity in the n-frame is updated as follows:

(15)$${{\bi v}}^n_k={{\bi v}}^n_{k-1}+{\bi C}^{n_{k-1/2}}_{n_{k-1}} {\bi C}^{n_{k-1}}_{n_{0}}{\bi C}^{n_{0}}_{b_{0}} \Delta\tilde{{{\bi v}}}^{b_0}_k-(2{\bf \omega}^n_{ie,k-1}+{\bf \omega}^n_{en,k-1}) \times {{\bi v}}^n_{k-1} T_s+{{\bi g}}^n_{k-1}T_s$$

where the subscript k denotes the parameter of the current update cycle and k−1 is the last update cycle, so ${{\bi v}}^{n}_{k}$ and ${{\bi v}}^{n}_{k-1}$ denote the velocity at the current and last cycle, respectively. ${\bi C}^{n_{k-1}}_{n_{0}}$ can be obtained by Equation (13), $\Delta\tilde{{{\bi v}}}^{b_{0}}_{k}$ is the stored velocity increments in the b ₀-frame, and T _s=1 s denotes the one second update cycle. Since ${\bf \omega}^{n}_{in}$ changes slowly, it is reasonable to approximate the attitude matrix by ${\bi C}^{n_{k-1/2}}_{n_{k-1}}\approx {\bf I}+({\bf \varPhi}^{n}_{k/2}\times)$, where ${\bf \varPhi}^{n}_{k/2}\approx -1/2({\bf \omega}^{n}_{ie,k-1}+{\bf \omega}^{n}_{en,k-1})T_{s}$ denotes the n-frame rotation vector from t _k−1 to t _k−1+T _s/2.

2.3.1.3 Backtracking position update

Given that the velocity is obtained, the position [λ, L, h]^T can be updated as

(16)$$\left[\matrix{\lambda_k\cr L_k\cr h_k} \right]=\left[\matrix{\lambda_{k-1}\cr L_{k-1}\cr h_{k-1}} \right]+\displaystyle{1 \over 2}{{\bi R}}^c({{\bi v}}^n_{k-1}+{{\bi v}}^n_k)T_s$$

(17)$${{\bi R}}^c=\left[\matrix{1/[(R_N+h)\cos L_{k-1}] & 0 & 0\cr 0 & 1/(R_M+h) & 0\cr 0 & 0 & 1} \right]$$

where R _N and R _M are the transverse and meridian radius of curvature, respectively.

2.3.2. Fine alignment algorithm

Since the backtracking navigation is conducted in the n-frame instead of the inertial frame, it is easy to adopt the well-known non-holonomic constraints to improve the fine alignment accuracy. Unlike an air vehicle, the motion of a wheeled vehicle on the land surface is governed by two non-holonomic constraints. When the vehicle does not jump off or slide on the ground, velocity of the vehicle in the plane perpendicular to the forward direction is zero, which is a virtual two-dimensional velocity measurement. Under ideal conditions, there is no side slip along the direction of the rear axle and no motion normal to the road surface, the constraints are (Dissanayake et al., Reference Dissanayake, Sukkarieh, Nebot and Whyte2001):

(18)$$\left\{\matrix{v^m_x = 0\cr v^m_z = 0}\right.$$

In practical situations, these constraints are somewhat violated due to the presence of side slip during cornering and vibrations caused by the engine and suspension system. Approximately, it is possible to model the extent of such violations as Gaussian white noise with zero mean and variance $\sigma^{2}_{x}$ and $\sigma^{2}_{z}$, respectively (Dissanayake et al., Reference Dissanayake, Sukkarieh, Nebot and Whyte2001).

The SINS/NHC integrated navigation based on Kalman filter is a good solution to realise the in-motion fine alignment without external sensors. In practice, the mounting angles between the IMU and the vehicle will noticeably affect the position accuracy, and they are commonly treated as main error sources to be estimated in SINS/odometer integrated vehicle navigation (Wu, Reference Wu2014). However, for pure SINS/NHC navigation, which is often used to aid a low-cost SINS during GNSS signal blockages, the mounting angles are usually ignored (Dissanayake et al., Reference Dissanayake, Sukkarieh, Nebot and Whyte2001; Peng et al., Reference Peng, Lin and Chiang2013; Rothman et al., Reference Rothman, Klein and Filin2015). Taking the mounting angles into the error state vector of the SINS/NHC Kalman filter to be estimated, we have realised a high positioning accuracy of horizontal error to 16·6 m Circular Error Probability (CEP) and vertical error 14·9 m Probable Error (PE) with a navigation-grade SINS in an 80 km vehicle test (Fu et al., Reference Fu, Qin, Li and Wang2012).

Since the mounting angles are inevitable and non-negligible for high-accuracy positioning, they should be handled properly in the SINS/NHC integration. The velocity of the vehicle calculated by SINS can be converted to the m-frame as (Wu, Reference Wu2014):

(19)$$\tilde{\bi v}^m_I = \tilde{{\bi C}}^m_b\tilde{{\bi C}}^b_n\tilde{{{\bi v}}}^n_I$$

where $\tilde{{\bi C}}^{m}_{b}$ denotes the converted matrix formed by the three mounting angles, and $\tilde{{\bi C}}^{b}_{n}$ and $\tilde{{{\bi v}}}^{n}_{I}$ are the attitude matrix and velocity calculated by SINS, respectively.

In general, the mounting angles can be easily controlled to small angles, and the converted matrix $\tilde{{\bi C}}^{m}_{b}$ approximately satisfies:

(20)$$\tilde{{\bi C}}^m_b\approx {\bf I}+{\bf \alpha}\times$$

where ${\bf \alpha}=[\matrix{\alpha_{x} & \alpha_{y} & \alpha_{z}}]^{\rm T}$ denotes the three-dimensional mounting angle vector.

Considering the navigation errors of SINS and the non-holonomic constraints, Equation (19) can be rewritten as (Fu et al., Reference Fu, Qin, Li and Wang2012):

(21)$$\eqalign{\tilde{{{\bi v}}}^m_I&=({\bf I}+{\bf \alpha}\times){\bi C}^b_n({\bf I}+{\bf \phi}^n\times )({{\bi v}}^n+\delta {{\bi v}}^n_I)\cr &\approx {{\bi v}}^m+{\bi C}^b_n\delta {{\bi v}}^n_I-{\bi C}^b_n({{\bi v}}^n\times) {\bf \phi}^n-({\bi C}^b_n {{\bi v}}^n)\times {\bf \alpha}\cr &\approx {{\bi v}}^m+{\bi C}^b_n\delta {{\bi v}}^n_I-{\bi C}^b_n({{\bi v}}^n\times) {\bf \phi}^n-\left[\matrix{0 & 0 & v^m_y\cr 0 & 0 & 0 \cr -v^m_y & 0 & 0} \right]\times \left[\matrix{\alpha_x\cr \alpha_y\cr \alpha_z} \right]}$$

where ϕⁿ = [ϕ _E, ϕ _N, ϕ _U]^T and $\delta {{\bi v}}^{n}_{I}=[\delta v_{E},\delta v_{N},\delta v_{U}]^{\rm T}$ represent the attitude misalignment angle and velocity error vector, respectively.

It is obvious that the roll mounting angle α_y is totally irrelevant to the vehicular velocity $\tilde{{{\bi v}}}^{m}_{I}$, so it is an unobservable error state and should be disregarded in the SINS/NHC Kalman filter. The other two mounting angles α_x and α_z can be treated as unknown random constants which satisfy:

(22)$$\left\{\matrix{\dot{\alpha}_x=0\cr \dot{\alpha}_z=0}\right.$$

Combining the error model of SINS and the mounting angles, the 17-dimensional error state vector of SINS/NHC integration can be defined as:

(23)$${{\bi x}}(t)=[\matrix{({\bf \phi}^n)^{\rm T} & (\delta {{\bi v}}^n_I)^{\rm T} & (\delta {{\bi P}})^{\rm T} & ({\bf \varepsilon}^b)^{\rm T} & ({\bi \nabla}^b)^{\rm T} & \alpha_x & \alpha_z} ]^{\rm T}$$

where δ P = [δλ, δL, δh]^T denotes the position errors, ε^b and tnabla^b denote the gyro and accelerometer biases, respectively.

The corresponding system equation of the error-state Kalman filter is:

(24)$$\dot{{{\bi x}}}(t)=\left[\matrix{{{\bi F}}_{\rm INS} & {\bf 0}_{15\times 2}\cr {\bf 0}_{2\times 15} & {\bf 0}_{2\times 2}} \right]{{\bi x}}(t)+\left[\matrix{-{\bi C}^n_b{\bf \varepsilon}^b_w\cr {\bi C}^n_b{{\bi \nabla}}^b_w\cr {\bf 0}_{11\times 1}} \right]$$

where F_INS is the 15×15 state transition matrix based on typical SINS error model (Qin, 2013), ${\bf \varepsilon}^{b}_{w}$ and ${\bi \nabla}^{b}_{w}$ denote the noise of gyroscopes and accelerometers, respectively.

As stated above, the non-holonomic constraints can be regarded as virtual velocity measurements. From Equations (18) and (19), the two-dimensional measurement is formed as:

(25)$${{\bi Z}}_{\rm NHC}=\left[\matrix{\tilde{v}^m_x\cr \tilde{v}^m_z} \right]={{\bi M}}^{\rm T}_e\tilde{{\bi C}}^b_n\tilde{{{\bi v}}}^n_I$$

where ${{\bi M}}_{e}=[{\bf e}_{1}\ {\bf e}_{3}]$, in which e_i is a unit three-dimensional column vector with the ith element being one.

The corresponding Kalman filter measurement equation is:

(26)$${{\bi Z}}_{\rm NHC}={{\bi H}}_{\rm NHC}{{\bi x}}+{{\bi V}}_{\rm NHC}$$

with V_NHC denoting the measurement noise of NHC, and the measurement matrix is:

(27)$${{\bi H}}_{\rm NHC}=[\matrix{{{\bi M}}_1 & {{\bi M}}_2 &{\bf 0}_{2\times 9} & {{\bi M}}_3}]$$

where ${{\bi M}}_{1}=-{{\bi M}}^{\rm T}_{e}{\bi C}^{b}_{n}({{\bi v}}^{n}\times)$, ${{\bi M}}_{2}={{\bi M}}_{e}^{\rm T}{\bi C}^{b}_{n}$, ${{\bi M}}_{3}=-{{\bi M}}^{\rm T}_{e}[({\bi C}^{b}_{n}{{\bi v}}^{n})\times]{{\bi M}}_{e}$.

2.3.3. Observability of the error states

Observability of the system is above all the first question to investigate, because if a state is unobservable, we have no way to achieve satisfactory estimation even with perfect measurements (Wu, Reference Wu2014). Although an improved Kalman filter algorithm has been developed to accomplish the backtracking fine alignment, not all error states are observable under all conditions. In this section, we analyse the observability of the SINS/NHC integration filter depending on the system and measurement equations.

From Equation (21), the error equation is rearranged as:

(28)$$\delta \tilde{{{\bi v}}}^m_I= \delta \tilde{{{\bi v}}}^b_I - {{\bi v}}^b\times {\bf \phi}^b-{{\bi v}}^b\times {\bf \alpha}$$

where $\delta {{\bi v}}^{b}_{I}={\bi C}^{b}_{n}\delta {{\bi v}}^{n}_{I}$, ${{\bi v}}^{b}={\bi C}^{b}_{n} {{\bi v}}^{n}$, ${\bf \phi}^{b}={\bi C}^{b}_{n}{\bf \phi}^{n}$.

The measurement Equation (25) can be rewritten as:

(29)$${{\bi Z}}_{\rm NHC}=\left[\matrix{\delta v^b_x-(\phi^b_z+\alpha_z)v^b_y\cr \delta v^b_z-(\phi^b_x+\alpha_x)v^b_y} \right]$$

As time advances, the measurement values Z_NHC and all-order derivatives can be treated as known quantities. According to the global observability theory (Wu et al., Reference Wu, Wu, Hu and Hu2009), a state is said to be observable if it can be determined uniquely from the measurement equation for a finite time under a possible condition. From Equation (29), it can be seen that the measurement equation is irrelevant to position errors, so the position error states δ P are obviously unobservable. Even so, the SINS/NHC integrated navigation can mitigate the drift of position errors by estimating the velocity errors.

Under static conditions, the longitudinal velocity $v^{b}_{y}$ is zero, so the velocity error states $\delta v^{b}_{x}$ and $\delta v^{b}_{z}$ can be estimated but the forward velocity error $\delta v^{b}_{y}$ is not directly observable in Equation (29). Assuming that the x _b axis of the IMU points to east, the observable velocity error $\delta v^{b}_{x}$ corresponds to eastern velocity error δ v _E, while when the x _b axis turns to north, $\delta v^{b}_{x}$ corresponds to northern velocity error δ v _N instead. So, the vertical velocity error is observable and the horizontal velocity errors can be alternatively estimated if the vehicle has a turning manoeuvre.

Due to the attitude errors and mounting angles, the observability of forward velocity $\delta v^{b}_{y}$ cannot be directly decided from Equation (29) whether the vehicle is moving or not. Taking the derivative of Equation (28) with respect to time and considering the simplified velocity error equation $\delta \dot{\bi v}^{n}_{\rm INS}\approx {{\bi f}}^{n}\times {\bf \phi}^{n}+{\bi C}^{n}_{b} {\bi \nabla}^{b}$, we get:

(30)$$\eqalign{\delta \dot{\bi v}^b_I &= \dot{{\bi C}}^b_n\delta {{\bi v}}^n_I+{\bi C}^b_n\delta \dot{\bi v}^n_I-\dot{\bi v}^b \times {\bf \phi}^b-{{\bi v}}^b\times \dot{{\bf \phi}}^b-\dot{\bi v}^b\times \delta{\bf \alpha}\cr &\approx -{\bf \omega}^b_{nb}\times \delta {{\bi v}}^b_I+{\bi C}^b_n({{\bi f}}^n\times {\bf \phi}^n+{\bi \nabla}^n)-{\bi C}^b_n({\bi a}^n\times {\bf \phi}^n)-{\bi a}^b\times \delta{\bf \alpha}\cr &=-{\bf \omega}^b_{nb}\times \delta {{\bi v}}^b_I+{\bi C}^b_n[({{\bi f}}^n-{\bi a}^n)\times {\bf \phi}^n + {\bi \nabla}^n] - {\bi a}^b\times \delta{\bf \alpha}\cr &=-{\bf \omega}^b_{nb}\times \delta {{\bi v}}^b_I-{\bi a}^b\times \delta {\bf \alpha} +{\bi C}^b_n(-{{\bi g}}^n\times {\bf \phi}^n+{\bi \nabla}^n)}$$

where ${\bi a}^{b}=[a^{b}_{x},a^{b}_{y},a^{b}_{z}]^{\rm T}$ denotes acceleration in the b-frame, ${\bi a}^{b}=\dot{\bi v}^{b}$, ${\bi a}^{n}={\bi C}^{b}_{n}{\bi a}^{b}$, ${\bf \phi}^{b}={\bi C}^{b}_{n}{\bf \phi}^{n}$, ${\bi \nabla}^{n}={\bi C}^{n}_{b}{\bi \nabla}^{b}$ and ${\bf \omega}^{b}_{nb}=[\omega_{x},\omega_{y},\omega_{z}]^{\rm T}$ is the angular rate of the vehicle.

From Equation (30), the derivatives of measurement value satisfy:

(31)$$\dot{{{\bi Z}}}_{\rm NHC}=\left[\matrix{\omega_z\delta v^b_y-\omega_y\delta v^b_z-a^b_y\alpha_z-g\phi^b_y+{\bf \nabla}^b_x\cr \omega_y\delta v^b_x-\omega_x\delta v^b_y-a^b_y\alpha_x+{\bf \nabla}^b_z} \right]$$

As the left-hand side of Equation (31) can be treated as known quantities as time advances, the vertical acceleration bias ${\bf \nabla}^{b}_{z}$ can be observed in a stationary state. Provided that there are sufficient turning manoeuvres, the azimuth rate ω_z will change while other coefficients remain constant, so the forward velocity error is solvable and thus observable. While the vehicle is accelerating or decelerating, the forward acceleration a _y^b is not zero and the mounting angles α_x, α_z are observable. Furthermore, the accelerometer bias ${\bf \nabla}^{b}_{x}$ and level misalignments ϕ_E, ϕ_N can be further separated in angular motions when there is change of the attitude matrix.

The azimuth error ϕ_U has no effect in Equation (31) so it has lower observability. After the level misalignments and respective derivatives are solved, the azimuth angle error is determined by the attitude error equation of SINS:

(32)$$\dot{{\bf \phi}}^n\approx -{\bf \omega}^n_{ie}\times {\bf \phi}^n-{\bf \varepsilon}^n$$

It is obvious that the azimuth error is also estimated from the compass effect as stationary alignment, and the estimation accuracy is still restricted by the equivalent eastern gyro drift ε_E. So, angular excitation of a vehicle is helpful to improve the estimation accuracy of azimuth error by distinguishing the level gyro drifts $\varepsilon^{b}_{x}$ and $\varepsilon^{b}_{y}$. In contrast to SINS/GPS integrated navigation, gyroscopes with higher accuracy are necessary for accurate positioning and orientation in the SINS/NHC integrated alignment filter.