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Abstract

The integrated navigation method based on star sensor celestial angles (altitude angle and azimuth angle) is proposed to serve the need for rapidly responsive, reliable, and precise of a hypersonic vehicle under a sophisticated environment. An integrated navigation algorithm suitable for large azimuth misalignment is established under launching point inertial coordinate and local geographical coordinate system based on altitude angle and azimuth angles. Meanwhile, a Bayesian method for data dropouts aided by the strapdown celestial angles is presented for the rapid variability in the celestial star angle with active galaxies. A nonlinear Bayesian filter is applied to implement the simulation on account of the nonlinear feature of the state and measurement equations. The simulation results showed that the Bayesian method for integrated navigation data dropouts could be accomplished by altitude angle and azimuth angle aiding in both launching point inertial coordinate and local geographical coordinate systems, which converge to 1′ in 10 s. The method indicated that the integrated navigation significant errors derived from initial localization and initial attitude alignment could be modified by the strapdown inertial navigation system (SINS) supported by the star sensor’s celestial angle in the local geographic coordinate system in the early launch stage. For the seconds of the flight phase, the integrated navigation aided by celestial angles in the launching point inertial coordinate system was guaranteed for the feasibility and validity. During the flight, the feasibility and validity of integrated navigation were guaranteed aided by celestial angles in launching point inertial coordinate system.

Keywords

Integrated navigation system strapdown celestial observation angles large misalignment inertial frame of launching point reference local geographical coordinates Bayesian method for data dropouts

Introduction

Due to the long-term flight of the vehicle in the adjacent space during a long-distance trip, it has unique characteristics in the aerospace field, and the adjoining area has become a hot spot in the world.¹ Navigation technology in the adjacent space has become a critical technology that is urgently needed to strengthen research. Particularly, hypersonic vehicles have the characteristics of fast flight speed, high attack power, and global reach and have high requirements for navigation system reliability, start-up time, and environmental conditions.^{2

–11} The strapdown celestial navigation measures natural celestial bodies to determine the vehicle location or heading, facilitating an independent navigation system. In the existing direction-finding technique, the star sensor is based on the star reference system and is the most accurate and minimally drifting attitude measuring component to date.^2

–5 The above advantages make the research and application of star sensors very active, which have been widely used in various fields such as Earth remote sensing, earth mapping, planetary mapping, interstellar communications, and intercontinental missiles.^6

–9 The star sensor and the inertial navigation system for integrated navigation have strong complementarity in performance. That integrated navigation can be used to calibrate the reference of inertial navigation devices and determine the attitude of the hypersonic vehicle.⁷

There is an essential feature of strapdown celestial navigation theory that the observation is the altitude or orientation of the celestial body. The star sensor is mounted on the carrier body, which can measure the strapdown celestial guidance of the altitude angle and azimuth angle about the body coordinate system directly.

Currently, the study addresses the geographic system as the navigation system to utilize the launcher facilities and other infrastructure systems in the ground station.¹⁰ Wang et al.^3,11 proposed the SINS/global navigation satellite system (GNSS)/celestial navigation system (CNS) integrated navigation scheme of the hypersonic vehicle with the geographic navigation system. It concluded that the attitude errors were improved after the navigation system assisted by the strapdown celestial angle. However, in previous work, an alignment error model on the assumption that the error angle of the Euler platform was small misalignment angle error. With the improvement of SINS technology, the navigation in the near-space hypersonic vehicle is limited by slight misalignment angle and linear error model, and the nonlinear error models of various large misalignment angles have received extensive attention. Previous studies of integrated navigation had emphasized the potential necessity for the navigation of hypersonic vehicles using the geographic scheme without small misalignment angle assumptions.

The responsive space lift requires the hypersonic vehicle carries a weapon capable of delivering an active payload to a user-specified orbit within hours. This contribution develops the conceptual mechanics of launching point missions in ballistic missile space orbit and presents results based on navigation strategy considering multiple celestial stars’ gravitational perturbations.^12

–18 The launching point inertial system is described as the long-range missile navigation reference to provide accurate spatial motion parameters and system control.^14,18 To obtain a consistent coordinate calculation of the ballistic missile, the coordinate system of a hypersonic vehicle platform is always parallel with the launching point inertial coordinate frame during the flight. Strapdown celestial guidance of single-star schemes of the strapdown celestial guidance in the launching point inertial system adopting “the altitude angle and the azimuth” matching is broadly applicable with the fast and accurate integrated navigation of the near-space hypersonic vehicle. The celestial altitude azimuth matching of the star sensor based on the body is proposed in the literature,^15

–19 and the rapid transit alignment method of the hypersonic vehicle with the launching point inertial system is proposed as the navigation system.

Meanwhile, concerning higher frequency vibrations and force gravitational fields influence of perturbations under complex environments of vehicle trajectories, rapid variability from active galaxies is susceptible to interference during stochastic measurement of celestial angles.²⁰ The setting of integrated navigation estimation is the Bayesian sequence since the measurements are randomly dropout, which involves a two-step prediction on the Chapman–Kolmogorov formula and two-step update on Bayes’ rules. In the case, a nonlinear Bayesian method for data dropouts aided by the strapdown celestial angles is presented.

The novel contributions are summarized as follows. Firstly, we provide optimizing models of the significant misalignment angle errors in two different navigational coordinate systems. Secondly, we fuse the star calendars’ knowledge with declination and the Greenwich hour angle and the meridian angle from star sensor in the dynamic observation equations. Finally, we develop celestial observation angle-assisted hypersonic vehicle integrated navigation methods in coordinate frames of launching point inertia system and geographical system. Our work proposes a Bayesian filtering method for data dropouts aided by the strapdown celestial angles in case of the rapid variability in the celestial star angle with active galaxies.

The article is organized in the following remainder. In the second section, we develop the large misalignment angle nonlinear error model of different navigation coordinates. In the third section, we describe the model of integrated navigation nonlinear equations. The fourth and fifth sections present efficient algorithms and simulation analysis in navigational coordinate systems in case of measurement data completeness and data dropouts. We conclude in the final section with a discussion.

The model of nonlinear error using large misalignment angle

Integrated navigation system model of state error propagation

The navigation system model under launching point inertial coordinate and local geographical coordinate can be obtained according to mathematical platform misalignment angle error equations, velocity, position error equations, and the process noise of propagation caused by those gyro and accelerator stochastic drift.

Launching point inertial system model of state error propagation

The symbols in the launch point inertial system model of state error propagation are as follows. The master’s inertial body coordinate system is defined as bm, for instance, the carrier coordinate frame or the star sensor carrier coordinate frame, and the slaver-inertial body coordinate system is bs. The $δ ω_{i b}^{b}$ and $δ f_{i b}^{b}$ are the measurement errors of gyroscope and accelerometer. The master-inertial conduction relative to slaver-inertial mounting error vector is described as μ= [μ_x μ_y μ_z ]^T, and the mathematical platform misalignment angle errors are $n = i$ and $\hat{n} = \hat{i}$ . The flexural deformation angle vector is φ = [φ_x φ_y φ_z ]^T. In the inertial system of the launching point, providing no limitations on a small-angle hypothesis, the error propagation equation of the SINS is as follows.

(1) Mathematical platform misalignment angle error equation ϕ

\dot{ϕ} = - C_{w}^{- 1} C_{b s}^{\hat{i}} δ ω_{i b s}^{b s}

and $C_{w}^{- 1}$ is that of misalignment angle error. The matrix $C_{b s}^{\hat{i}}$ is the transition matrix from the slaver-inertial body system to the launching point. In the case of a large misalignment angle, it is composed of the mathematical platform misalignment angle correlation matrix.

(2) Velocity error equation

δ {\dot{V}}^{i} = (I - C_{\hat{i}}^{i}) C_{b}^{\hat{i}} f^{b} + C_{\hat{i}}^{i} C_{b}^{\hat{i}} δ f_{i b}^{b} + δ g^{i}

where δVⁱ = [δv_x δv_y δv_z ]^T, f ^b is the measurement of the slaver navigation accelerometer, δf ^b is the accelerometer measurement error of the slaver INS, δgⁱ is the acceleration error of gravity, and $C_{\hat{i}}^{i}$ is the attitude transition matrix from the subordinate inertial navigation $\hat{i}$ to the master-inertial launching point i.

(3) Position error equation

The differential equation of position error in the inertial coordinate system is

δ {\dot{S}}^{i} = δ V^{i}

(4) Attitude transformation matrix

C_{b s}^{i} = C_{b m}^{i} C_{b h}^{b m} C_{b s}^{b h} = C_{b m}^{i} [I + (φ \times)] [I + (μ \times)]

where bm is the body of the master system, bh is a slaver-inertial horizontal coordinate system, $C_{b h}^{b m}$ is a transformation matrix from bh to bm, $C_{b s}^{b h}$ is a transformation matrix from bs to bh, and $C_{b m}^{i}$ is the attitude matrix of the master INS from bm to i.

The geographical system model of state error propagation

(1) Mathematical platform misalignment angle error equation

{\dot{ϕ}}^{n} = C_{w 2}^{- 1} [(I - C_{n}^{\hat{n}}) {\hat{ω}}_{c n}^{n} + C_{b s}^{\hat{n}} δ ω_{c n}^{n} - C_{b s}^{\hat{n}} δ ω_{c b s}^{b s}]

The symbol c is the geocentric inertial coordinate system. $C_{n}^{\hat{n}}$ is the transformation matrix from the master navigation n to the slaver navigation $\hat{n}$ , which is calculated by the mathematical navigation platform misalignment angle rotated around the axles. $C_{b s}^{\hat{n}}$ is the attitude transformation matrix of the slaver navigation coordinate bs to the geographic coordinate $\hat{n}$ . It is assumed that the system bs sequentially turns the roll angle γ, the heading angle ψ, and the pitch angle ϕ around the axis to obtain the $\hat{n}$ system.

(2) Velocity error equation

δ {\dot{V}}^{n} = [I - {(C_{n}^{\hat{n}})}^{T}] C_{b s}^{\hat{n}} {\hat{f}}^{b} + δ V^{n} \times (2 {\hat{ω}}_{c e}^{n} + {\hat{ω}}_{e n}^{n}) + {\hat{V}}^{n} \times (2 δ ω_{c e}^{n} + δ ω_{e n}^{n}) + {(C_{n}^{\hat{n}})}^{T} C_{b s}^{\hat{n}} δ f_{i b}^{b}

(3) Position error equation

δ \dot{L} = \frac{δ V_{n}}{R_{n} + h} - \frac{V_{n}}{{(R_{n} + h)}^{2}} δ h

δ \dot{λ} = \frac{sec L}{R_{e} + h} δ V_{e} + \frac{V_{e} sec L tan L}{R_{e} + h} δ L - \frac{V_{e} sec L}{{(R_{e} + h)}^{2}} δ h

δ \dot{h} = δ V_{u}

Due to the instability of the height channel of the inertial navigation system, external height damping information is introduced to resolve the divergence of the height channel. We do not describe the detail of the high damping effect. Attitude transformation matrix and wing flexural deformation error are similar to the previous work in launching point inertial system.

Integrated navigation system model of state equations

Navigation is used as the reference frame, and the error propagation equations of the launching point inertial system and those of the geographic system are concluded as follows: $x_{k} = {[ϕ, δ V, δ S, ε, \nabla, μ, φ, \dot{φ}]}^{T}$

Here, δS and δV represent the position and the velocity errors of the slaver navigation relative to the master navigation, respectively. The state equation of the system is $x_{k} = f (x_{k - 1}) + G_{k -} _{1} w_{k -} _{1}$ as a nonlinear system.

Nonlinear observation model matching of altitude angle + azimuth angle

The observation model of altitude angle and azimuth angles are established using launching point inertial coordinate and local geographical coordinate, respectively, which is suitable for significant misalignment integrated navigation.

The observation equations of “elevation angle + azimuth angle” in launching point inertial coordinate frame

The star sensor observation device is fixed on the flight carrier, and the starlight is tracked. Star patterns matching is observed by the star sensor and which fabricates the height angle (H_b ) and the azimuth angle (A_b ) of the star sensor system (i.e. the carrier coordinate bs). Physical measurements are altitude angles (H_ob ) and azimuth (A_ob ) corresponding to a navigation star.

H_{o b} = H_{b} + v_{h} A_{o b} = A_{b} + v_{a}

where v_h and v_a are the star sensor angle measurement noise.

Due to the position error, velocity error, installation error, and lever arm deflection effect of the slaver navigation system, there are calculation errors in the altitude angle (H_pb ) and the azimuth angle (A_pb ) solved by the slaver system

H_{p b} = H_{b} + δ h A_{o b} = A_{b} + δ a

The difference between H_pb and A_pb of the slaver solution and H_ob and A_ob actually measured by the star sensor is

δ h = H_{p b} - H_{o b} + v_{h} ​_{​} δ a = A_{p b} - A_{o b} + v_{a}

The altitude angles and the azimuth angles in different coordinate systems associated with the star sensor are shown in Figure 1.

Figure 1.

Star sensor’s altitude angles and azimuth angles.

The observation equations are calculated as follows:

Step (1): Altitude angle and azimuth of the inertial solution

The azimuth angle A_pt and the altitude angle H_pt in the local geographic system is obtained by the Longi_bs and the Latit_bs , which is solved by the navigational star calendar and the slaver navigation

H_{p t} = arcsin (sin D e c sin L a t i t_{b s} + cos D e c cos L a t i t_{b s} cos t_{E c})

A_{p t} = arccos [(sin D e c - sin H_{p t} sin L a t i t_{b s}) /(cos H_{p t} cos L a t i t_{b s})]

where Dec is the declination, Gha is the Greenwich hour angle, and t_Ec is the meridian angle

t_{E c} = \{\begin{cases} \begin{matrix} G h a + L o n g i_{b s} & if G h a + L o n g i_{b s} < π \end{matrix} \\ G h a + L o n g i_{b s} - 2 π if G h a + L o n g i_{b s} > π \end{cases}

A_{p t} = \{\begin{cases} A_{p t} if t_{E c} < 0 \\ 2 π - A_{p t} if t_{E c} > 0 \end{cases}

The solution of altitude angle H_pb and the azimuth angle A_pb are obtained by the attitude transformation matrix in the projective coordinate system.

The process is as follows:

(1) Calculate the starlight vector r_pt in the local geographic system from the altitude angle H_pt and the azimuth angle A_pt

r_{p t} = {[\begin{matrix} r_{p t x} & r_{p t y} & r_{p t z} \end{matrix}]}^{T} = {[\begin{matrix} cos H_{p t} cos A_{p t} & sin H_{p t} & cos H_{p t} sin A_{p t} \end{matrix}]}^{T}

(2) Using the attitude transformation matrix $C_{\hat{t}}^{b}$ to obtain the starlight vector of the projective coordinate system, $r_{p b} = C_{\hat{t}}^{b} r_{p t} = C_{\hat{i}}^{b s} C_{e}^{i} C_{\hat{t}}^{e} r_{p t}$ .

(3) Now we get the geometric relationship between the starlight vector, the altitude angle and azimuth angle and the attitude transformation matrix, the solution of altitude angle H_pb and the azimuth angle A_pb are obtained in the projective coordinate system

\begin{array}{l} H_{p b} = arcsin (r_{p b y} / \sqrt{r_{p b x}^{2} + r_{p b y}^{2} + r_{p b z}^{2}}) \\ A_{p b} = arctan (r_{p b z} / r_{p b x}) \end{array}

and

A_{p t} = \{\begin{cases} A_{p t} if t_{E c} < 0 \\ 2 π - A_{p t} if t_{E c} > 0 \end{cases}

Step (2): Establish a measurement model of the altitude angle and azimuth angle

The calculation errors of the altitude angle and azimuth of the star sensor coordinates are mainly caused by the initial orientation errors and position error. The orientation errors bring the misalignment angle of the mathematical platform. The position error of the inertial coordinate of the launching point is obtained by the compensation state variable, and the local latitude and longitude are obtained according to the conversion relationship between the inertial launching point and the local geographical coordinates.

We develop a new expression as the position error compensation of launching inertial point is corrected as $S_{i l} = S_{i} - δ S_{i}$ , the position of the inertial coordinate after compensation is $S_{c l} = C_{i}^{c} S_{i l}$ , and the position of the compensated Earth coordinate is $S_{e l} = C_{c}^{e} S_{c l}$ . Consequently, the local geographic longitude (Longi_otl ) and latitude (Latit_otl ) after compensation are determined, that is

L a t i t_{o t l} = arctan (S_{e l}^{z} / \sqrt{{(S_{e l}^{x})}^{2} + {(S_{e l}^{y})}^{2}}) L o n g i_{o t l} = arccos (S_{e l}^{x} / \sqrt{{(S_{e l}^{x})}^{2} + {(S_{e l}^{y})}^{2}})

then the altitude angle (H_otl ) and azimuth (A_otl ) in the compensated geographic coordinate system are obtained as

H_{o t l} = arcsin (sin D e c sin L a t i t_{o t l} + cos D e c cos L a t i t_{o t l} cos t_{E} ​_{o t l})

where

t_{E_{o t l}} = \{\begin{cases} \begin{matrix} G h a + L o n g i_{o t l} & G h a + L o n g i_{o t l} < π \end{matrix} \\ G h a + L o n g i_{o t l} - 2 π G h a + L o n g i_{o t l} > π \end{cases}

A_{o t l} = arccos [(sin D e c - sin H_{o t l} sin L a t i t_{o t l}) / (cos H_{o t l} cos L a t i t_{o t l})]

then, $A_{o t l} = A_{o t l}, if t_{E_{o t l}} < 0; A_{o t l} = 2 π - A_{o t l}, if t_{E_{o t l}} > 0$

The matrix $C_{i}^{c}$ is determined by the launching point’s Greenwich hour angle, the latitude and longitude, and the azimuth of the launching point. $C_{c}^{e}$ is determined by the angular correlation matrix of the Earth’s rotation and launching point.¹¹

The starlight vector $r_{o t l}$ in the local geographic coordinate can be obtained from the calculation of the above altitude and azimuth angles: $r_{o t l} = {[cos H_{o t l} cos A_{o t l} _{,} sin H_{o t l} _{,} cos H_{o t l} sin A_{o t l}]}^{T}$

The altitude angle and azimuth angle in the carrier coordinate can be obtained by the starlight vector $(r_{o b l})$ , the relationship of the altitude and the azimuth angle, and the attitude transformation matrix $C_{\hat{\hat{t}}}^{b m}$ again, that is $r_{o b l} = C_{\hat{\hat{t}}}^{b m} r_{c t}$ . The detailed process is as follows: the attitude matrix $C_{\hat{\hat{t}}}^{b m}$ is compensated, $C_{\hat{\hat{t}}}^{b m} = C_{b \hat{h}}^{b \hat{m}} C_{b \hat{s}}^{b \hat{h}} C_{\hat{i}}^{b s} C_{i}^{\hat{i}} C_{e}^{i} C_{\hat{\hat{t}}}^{e}$ where $C_{b \hat{h}}^{b \hat{m}}$ is represented by the estimated value of the filter deflection angle of the wing. $C_{b \hat{s}}^{b \hat{h}}$ is represented by the estimated value of the installation error angle, and $C_{\hat{i}}^{b s}$ is the attitude matrix of the slaver INS. $C_{\hat{\hat{t}}}^{e}$ is the compensated transition matrix from the geographic coordinates to the Earth coordinate system, which is determined by the Latit_otl and Longi_otl . Then the starlight vector (r_obl ) in the carrier coordinate is obtained, where is abbreviated as $r_{o b l} = {[r_{o b l x} r_{o b l y} r_{o b l z}]}^{T}$ .

Using the above attitude transformation relationship $C_{\hat{\hat{t}}}^{b m}$ , the starlight vector r _obl of the carrier coordinate frame, and the altitude and azimuth angle of the carrier coordinates can be obtained

H_{o b l} = arcsin (r_{o b l y} / \sqrt{r_{o b l x}^{2} + r_{o b l y}^{2} + r_{o b l z}^{2}})

A_{o b l} = arctan (r_{o b l z} / r_{o b l x})

According to the steps (1) and (2), the integrated navigation measurement model based on the star sensor is matched by altitude angle and the azimuth angle

δ h = H_{p b} - H_{o b l} + v_{h}, δ a = A_{p b} - A_{o b l} + v_{a}

The observation equations of geographic coordinate frame “elevation angle + azimuth angle”

We develop a new relationship between the starlight vector of the altitude angle/azimuth angle and SINS errors in the geographic coordinate frame.

The relationship between the starlight vector of the altitude angle/azimuth angle and SINS errors

The starlight vector $r_{i}^{m}$ in the inertial frame i of the launching point can be projected into the coordinate frame multiplied by the position matrix and the attitude matrix, the star direction vector

r_{b}^{s} = C_{n}^{b s} C_{e}^{n} C_{i}^{e} C_{b m}^{b s} r_{i}^{m}

$C_{b m}^{b s}$ is represented by the filter’s estimate of the wing deflection angle and the installation error angle.

Due to the error in the position and attitude provided by the inertial navigation, the starlight vector $r_{\hat{n} b}^{s}$ is resolved by slaver navigation

r_{\hat{n} b}^{s} = {\tilde{C}}_{n}^{b} {\tilde{C}}_{e}^{n} C_{i}^{e} C_{b m}^{b s} r_{i}^{m}

In equation (26), ${\tilde{C}}_{n}^{b}$ and ${\tilde{C}}_{e}^{n}$ , respectively, represent the matrix of the inertial navigation solution, which can be represented by the inertial model error term, where ${\tilde{C}}_{n}^{b}$ is

{\tilde{C}}_{n}^{b} = C_{n}^{b} C_{\hat{n}}^{n} = C_{n}^{b} {(C_{n}^{\hat{n}} (ϕ))}^{T}

Considering the error caused by the misalignment angle, we do not make the small-angle hypothesis $C_{n}^{\hat{n}} (ϕ)$ . The frame n successively undergoes three rotations to the frame $\hat{n}$

{\tilde{C}}_{e}^{n} = C_{n}^{\hat{n}} (s) C_{e}^{n}

where $C_{n}^{\hat{n}} (s)$ is caused by the position error $δ S$ to resolve by the slaver navigation, it is calculated as

δ S = [- δ L, δ λ cos L, δ L sin L] = [θ_{0}, θ_{1}, θ_{2}]

According to equations (25) and (26), $r_{\hat{n} b}^{s}$ is obtained as

r_{\hat{n} b}^{s} = {\tilde{C}}_{n}^{b} {\tilde{C}}_{e}^{n} C_{i}^{e} C_{b m}^{b s} {(C_{n}^{b} C_{e}^{n} C_{i}^{e} C_{b m}^{b s})}^{T} r_{b}^{s} = {\tilde{C}}_{n}^{b} {\tilde{C}}_{e}^{n} {(C_{n}^{b} C_{e}^{n})}^{T} r_{b}^{s}

Substituting equations (27)–(29) into (30) yields

r_{\hat{n} b}^{s} = {\tilde{C}}_{n}^{b} C_{n}^{\hat{n}} (s) C_{\hat{n}}^{n} (ϕ) {\tilde{C}}_{b}^{n} r_{b}^{s}

Using the position and attitude of the slaver navigation, and querying the right ascension and declination of the navigation star library, $H_{\hat{n} b}$ and $A_{\hat{n} b}$ can be obtained by expanding the starlight vector $r_{\hat{n} b}^{s}$ in equation (31), that is

r_{\hat{n} b}^{s} = {[r_{\hat{n} b x}^{s} r_{\hat{n} b y}^{s} r_{\hat{n} b z}^{s}]}^{T} = {[cos H_{\hat{n} b} sin A_{\hat{n} b},cos H_{\hat{n} b} cos A_{\hat{n} b},sin H_{\hat{n} b}]}^{T}

Starlight vector error model of the observation equation

Inertial navigation solves the existence of two types of error sources, position and attitude, which cause an error between the solution of the altitude angle $H_{\hat{n} b}$ and the azimuth $A_{\hat{n} b}$ of the above slaver navigation and the physical angles of the master navigation. Without a small-angle assumption, the relationship between the physical value $r_{b}^{s}$ and the inertial solution $r_{\hat{n} b}^{s}$ is expressed as

r_{b}^{s} = {(C_{\hat{n} b}^{b})}^{T} r_{\hat{n} b}^{s}

Comparing equations (31) and (33), we obtain $C_{\hat{n} b}^{b}$

C_{\hat{n} b}^{b} = {\tilde{C}}_{n}^{b} C_{n}^{\hat{n}} (s) C_{\hat{n}}^{n} (ϕ) {\tilde{C}}_{b}^{n}

Substituting equation (32) into equation (33) and expanding it, we get

[\begin{array}{l} r_{b x}^{s} \\ r_{b y}^{s} \\ r_{b z}^{s} \end{array}] = {[\begin{matrix} C_{\hat{n} b 00}^{b} & C_{\hat{n} b 01}^{b} & C_{\hat{n} b 02}^{b} \\ C_{\hat{n} b 10}^{b} & C_{\hat{n} b 11}^{b} & C_{\hat{n} b 12}^{b} \\ C_{\hat{n} b 20}^{b} & C_{\hat{n} b 21}^{b} & C_{\hat{n} b 22}^{b} \end{matrix}]}^{T} [\begin{array}{l} r_{\hat{n} b x}^{s} \\ r_{\hat{n} b y}^{s} \\ r_{\hat{n} b z}^{s} \end{array}]

expanding the third line of equation (35), it obtains

H_{b} = a r c sin (C_{\hat{n} b 02}^{b} cos H_{\hat{n} b} sin A_{\hat{n} b} + C_{\hat{n} b 12}^{b} cos H_{\hat{n} b} cos A_{\hat{n} b} + C_{\hat{n} b 22}^{b} sin H_{\hat{n} b})

From the first line and the second line of equation (35), it obtains

A_{b} = arcsin (\frac{C_{\hat{n} b 00}^{b} cos H_{\hat{n} b} sin A_{\hat{n} b} + C_{\hat{n} b 10}^{b} cos H_{\hat{n} b} cos A_{\hat{n} b} + C_{\hat{n} b 20}^{b} sin H_{\hat{n} b}}{1 - {(sin H_{\hat{n} b})}^{2}})

The observation is the difference between the slaver navigation solution of altitude/azimuth angles and those celestial angles of vehicle-mounted star sensor, which gives a detailed equation

δ h = H_{\hat{n} b} - H_{b}, δ a = A_{\hat{n} b} - A_{b}

According to the equations of the above nonlinear system, the two different navigation coordinates are used by the matching of star sensor celestial angle. The nonlinear Kalman filter algorithm, such as the sparse grid, is used to obtain the estimation.^21,22 The results of attitude misalignment angles and attitude estimation errors are presented to evaluate the alignment effect in the fourth section.

Bayesian method for data dropouts about the strapdown celestial angels

Dynamic variability from active galaxies is susceptible to atmospheric conditions during the stochastic measurement of celestial angles. Measuring and transmitting the celestial angles face dropout risk of all kinds unavoidably. The uncertainties in an astronomic measurement exist in model parameters and external disturbance seriously. In this case that the measurements are randomly drop out and uncertainty, a nonlinear Bayesian robust method for data dropouts aided by the strapdown celestial angles is presented. The estimation of measurement noise probability distribution function (PDF) is not available in the direct filter such as the extended kalman filter (EKF) or unscented kalman filter (UKF) Gaussian filter.

The system with measurement dropout and uncertainty is

x_{k} = f (x_{k - 1}) + w_{k - 1} z_{k} = h (x_{k}) + v_{k}

The dropout of measurements is a binary stochastic variable r_k ∈ R and r_k is set to 0 or 1 according to whether or not the measurement is arriving. The probability of p_k managing the dropout measurements from the star sensor is defined over the interval [0, 1]. $P (r_{k} = 1) = E [r_{k}] = 1 - p_{k}$ . The practical measure vector y_k is

y_{k} = (1 - r_{k}) z_{k} + r_{k} z_{k - 1}, k > 1; y_{1} = z_{1}

The measurement noise v_k is augmented in state vectors as follows. y_k is the measurement sequence at the noise and short for ${[z_{1}^{T} \dots z_{k}^{T}]}^{T}$ . The superscripts as the hat represent the estimate of random variables. The setting of the Bayesian sequence involves the predictions of the state and the noise on the Chapman–Kolmogorov formula²³

f (x_{k + 1}^{​}, y_{k +} ​_{1} | Y_{k}) = \int f (x_{k + 1}^{​} | x_{k}) f (x_{k}^{​}, y_{k +} ​_{1} | Y_{k}) d x_{k} = N (\begin{matrix} (\begin{array}{l} x_{k + 1}^{​} \\ y_{k + 1}^{​} \end{array}); & (\begin{array}{l} {\hat{x}}_{k + 1 | k}^{​} \\ {\hat{y}}_{k + 1 | k}^{​} \end{array}), & (\begin{matrix} {\hat{P}}_{k + 1 | k}^{​} & {\hat{P}}_{k + 1 | k}^{x y} \\ {({\hat{P}}_{k + 1 | k}^{x y})}^{T} & {\hat{P}}_{k + 1 | k}^{y y} \end{matrix}) \end{matrix})

and

f (v_{k + 1}^{​}, y_{k +} ​_{1} | Y_{k}) = \int f (v_{k + 1}^{​} | v_{k}) f (v_{k}^{​}, y_{k +} ​_{1} | Y_{k}) d v_{k} = N (\begin{matrix} (\begin{array}{l} v_{k + 1}^{​} \\ y_{k + 1}^{​} \end{array}); & (\begin{array}{l} {\hat{v}}_{k + 1 | k}^{​} \\ {\hat{y}}_{k + 1 | k}^{​} \end{array}), & (\begin{matrix} {\hat{P}}_{k + 1 | k}^{​} & {\hat{P}}_{k + 1 | k}^{v y} \\ {({\hat{P}}_{k + 1 | k}^{v y})}^{T} & {\hat{P}}_{k + 1 | k}^{y y} \end{matrix}) \end{matrix})

and a two-step update on Bayes’ rules is as follows: $f (x_{k + 1}^{} | Y_{k +} _{1}) \propto f (Y_{k + 1} | x_{k + 1}) f (x_{k + 1} | Y_{k}) = N (x_{k + 1}^{}; {\hat{x}}_{k + 1 | k + 1}^{}, {\hat{P}}_{k + 1 | k + 1}^{x x})$ and $f (v_{k + 1}^{} | Y_{k +} _{1}) \propto f (Y_{k + 1} | v_{k + 1}) f (v_{k + 1} | Y_{k}) = N (v_{k + 1}^{}; {\hat{v}}_{k + 1 | k + 1}^{}, {\hat{P}}_{k + 1 | k + 1}^{v v})$ .

The PDF of the state (2) is Gaussian as follows

p (x_{k + 1}^{a} | Y_{k +} ​_{1}) = N (x_{k + 1}^{a}; {\hat{x}}_{k + 1 | k + 1}^{a}, {\hat{P}}_{k + 1 | k + 1}^{a})

where

{\hat{x}}_{k + 1 | k + 1}^{a} = [\begin{array}{l} {\hat{x}}_{k + 1 | k + 1}^{​} \\ {\hat{v}}_{k + 1 | k + 1}^{​} \end{array}], {\hat{P}}_{k + 1 | k + 1}^{a} = [\begin{array}{l} \begin{matrix} {\hat{P}}_{k + 1 | k + 1}^{​} & {\hat{P}}_{k + 1 | k + 1}^{x v} \end{matrix} \\ \begin{matrix} {({\hat{P}}_{k + 1 | k + 1}^{x v})}^{T} & {\hat{P}}_{k + 1 | k + 1}^{v v} \end{matrix} \end{array}]

A nonlinear Bayesian robust for data dropouts is as follows.

The conditional mean ${\hat{x}}_{k | k}^{a}$ and its covariance $P_{k | k}^{a}$ and are initiated by

{\hat{x}}_{0}^{a} = [\begin{array}{l} {\hat{x}}_{0}^{​} \\ 0 \end{array}], {\hat{P}}_{0 | 0}^{a} = [\begin{array}{l} \begin{matrix} {\hat{P}}_{0 | 0}^{​} & 0 \end{matrix} \\ \begin{matrix} 0 & R_{0} \end{matrix} \end{array}], {\hat{P}}_{0 | 0}^{a} = S_{0 | 0}^{a} {(S_{0 | 0}^{a})}^{T}

1. Construct and propagate Sigma points, k = 1, 2,…

Factorize $P_{k | k}^{a} = S_{k | k}^{a} {(S_{k | k}^{a})}^{T}$ and construct the sigma points of ${\hat{x}}_{k | k}^{a}$ using the deterministic points $(ξ_{i}, w_{i}; i = 1 : N_{p}^{'})$ ²²

ξ_{i, k | k} = [\begin{array}{l} ξ_{i, k | k}^{x} \\ ξ_{i, k | k}^{v} \end{array}] = S_{k | k}^{a} ξ_{i} + {\hat{x}}_{k | k}^{a} i = 1 : N_{p}^{'}

χ_{i, k + 1 | k}^{x} = f_{k} (ξ_{i, k | k}^{x}), λ_{i, k + 1 | k}^{x} = h_{k} (ξ_{i, k | k}^{x}), i = 1 : N_{p}^{'}

{\hat{x}}_{k + 1 | k}^{a} = \sum_{i = 1}^{N p'} ω_{i} χ_{i, k + 1 | k}^{a} {\hat{x}}_{k + 1 | k}^{a} = \sum_{i = 1}^{N p'} ω_{i} χ_{i, k + 1 | k}^{a}

{\hat{P}}_{k + 1 | k}^{a} = \sum_{i = 1}^{N p'} ω_{i} χ_{i, k + 1 | k}^{a} {(χ_{i, k + 1 | k}^{a})}^{T} - {\hat{x}}_{k + 1 | k}^{a} {({\hat{x}}_{k + 1 | k}^{a})}^{T} + {\hat{Q}}_{k}

{\hat{P}}_{k + 1 | k}^{​} = S_{k + 1 | k}^{​} S_{k + 1 | k}^{T}, ξ_{i, k + 1 | k}^{x} = S_{k + 1 | k}^{​} ξ_{i} + {\hat{x}}_{k + 1 | k}^{​} i = 1 : N_{p}

θ_{i, k + 1 | k}^{​} = h_{k + 1} (ξ_{i, k + 1 | k}^{x}), i = 1 : N_{p}

2. State and measurement noise estimation update

{\hat{z}}_{k + 1 | k}^{​} = \sum_{i = 1}^{N_{p}} ω_{i} θ_{i, k + 1 | k}^{​}

{\hat{P}}_{k + 1 | k}^{z z} = \sum_{i = 1}^{N_{p}} ω_{i} θ_{i, k + 1 | k}^{​} θ_{i, k + 1 | k}^{T} - {\hat{z}}_{k + 1 | k}^{​} {\hat{z}}_{k + 1 | k}^{T} + {\hat{R}}_{k + 1}

{\hat{z}}_{k | k}^{​} = \sum_{i = 1}^{N_{p}} ω_{i} (λ_{i, k + 1 | k}^{x} + ξ_{i, k + 1 | k}^{v})

{\hat{P}}_{k | k}^{z z} = \sum_{i = 1}^{N_{p}} ω_{i} (λ_{i, k + 1 | k}^{x} + ξ_{i, k + 1 | k}^{v}) {(λ_{i, k + 1 | k}^{x} + ξ_{i, k + 1 | k}^{v})}^{T} - {\hat{z}}_{k | k}^{​} {\hat{z}}_{k | k}^{T}

{\hat{P}}_{k + 1 | k}^{x z} = \sum_{i = 1}^{N_{p}} ω_{i} ξ_{i, k + 1 | k}^{x} θ_{i, k + 1 | k}^{T} - {\hat{x}}_{k + 1 | k}^{​} {\hat{z}}_{k + 1 | k}^{T}

{\hat{P}}_{k + 1, k | k}^{x z} = \sum_{i = 1}^{N_{p}} ω_{i} χ_{i, k + 1 | k}^{v} {(λ_{i, k + 1 | k}^{x} + ξ_{i, k + 1 | k}^{v})}^{T} - {\hat{x}}_{k + 1 | k}^{​} {\hat{z}}_{k | k}^{T}

{\hat{P}}_{k + 1 | k}^{v y} = (1 - p_{k + 1}) {\hat{R}}_{k + 1}

{\hat{P}}_{k + 1 | k}^{y y} = (1 - p_{k + 1}) {\hat{P}}_{k + 1 | k}^{z z} + p_{k + 1} {\hat{P}}_{k + 1 | k}^{z z} + p_{k + 1} {\hat{P}}_{k | k}^{z z} + (1 - p_{k + 1}) p_{k + 1} ({\hat{z}}_{k + 1 | k} - {\hat{z}}_{k | k}) {({\hat{z}}_{k + 1 | k} - {\hat{z}}_{k | k})}^{T}

{\hat{K}}_{k}^{v} = {\hat{P}}_{k + 1 | k}^{v y} {({\hat{P}}_{k + 1 | k}^{y y})}^{- 1}

{\hat{y}}_{k + 1 | k} = (1 - p_{k + 1}) {\hat{z}}_{k + 1 | k} + p_{k + 1} {\hat{z}}_{k | k}

{\hat{P}}_{k + 1 | k + 1}^{v v} = {\hat{P}}_{k + 1} - {\hat{K}}_{k}^{v} {\hat{P}}_{k + 1 | k}^{y y} {({\hat{K}}_{k}^{v})}^{T}

{\hat{v}}_{k + 1 | k + 1}^{​} = {\hat{K}}_{k}^{v} (y_{k + 1} - {\hat{y}}_{k + 1 | k}^{​})

{\hat{P}}_{k + 1 | k}^{x y} = (1 - p_{k + 1}) {\hat{P}}_{k + 1 | k}^{x z} + p_{k + 1} {\hat{P}}_{k + 1, k | k}^{x z}

{\hat{K}}_{k}^{x} = {\hat{P}}_{k + 1 | k}^{x y} {({\hat{P}}_{k + 1 | k}^{y y})}^{- 1}

{\hat{x}}_{k + 1 | k + 1}^{​} = {\hat{x}}_{k + 1 | k}^{​} + {\hat{K}}_{k}^{x} (y_{k + 1} - {\hat{y}}_{k + 1 | k}^{​})

{\hat{P}}_{k + 1 | k + 1}^{​} = {\hat{P}}_{k + 1 | k}^{​} - [\begin{matrix} {\hat{P}}_{k + 1 | k}^{x y} & {\hat{P}}_{k + 1 | k} \end{matrix}] R_{e, k}^{- 1} [\begin{matrix} {({\hat{P}}_{k + 1 | k}^{x y})}^{T} \\ {\hat{P}}_{k + 1 | k}^{T} \end{matrix}]

R_{e, k} = [\begin{matrix} {\hat{R}}_{k} & 0 \\ 0 & - γ^{2} I \end{matrix}] + [\begin{matrix} {\hat{P}}_{k + 1 | k}^{y y} & {({\hat{P}}_{k + 1 | k}^{x y})}^{T} \\ {\hat{P}}_{k + 1 | k}^{x y} & {\hat{P}}_{k + 1 | k}^{​} \end{matrix}]

{\hat{P}}_{k + 1 | k + 1}^{x v} = - {\hat{K}}_{k}^{x} {\hat{P}}_{k + 1 | k}^{y y} {({\hat{K}}_{k}^{v})}^{T}

We can calculate the state vectors and the covariance matrices recursively for measurement dropouts. However, we need to be reminded that conditions for bounded exponents, including that, but not limited to, the probability p_k has an upper limit.

Analysis of the state celestial angle in two different navigation coordinates

Now we will present the performance of the models aided by strapdown celestial angles and the Bayesian method for data dropouts of celestial angles.

System simulation condition

The simulation was implemented using the missile described in Pehlivanoğlu and Ercan.²⁴ The initial site of the vehicle is at the longitude 118.8°, and the latitude 31.98° with 10 m error. The gyro and accelerator parameters are as follows: $\nabla = 0.05 g, ω_{\nabla} = N (0, {(0.001 g)}^{2}), ε = 0.1 g, ω_{ε} = N (0, {(0.01 g)}^{2})$ . We adopt two celestial navigation stars. The navigational ephemeris declinations are Dec1 = 30° and Dec2 = 55° and the Greenwich hour angles are Gha1 = 70° and Gha2 = 40°, respectively. The launch azimuth is 90°, and the large misalignment angles are up to 10°, which are nonlinear equations of state and equations of measurement. The vehicle is owing to the high-frequency vibration of the vehicle in the high-altitude environment.^24

–29 The vibration frequencies of the vehicle in the pitching and rolling directions are 100 Hz/80 Hz/60 Hz.

System simulation results

The models of the altitude and azimuth angles based on the star sensor

We aim at a reliable, accurate, and fast integrated navigation in complex environments. The error of the slave navigation of the missile carrier is corrected by the high-precision attitude information of the star sensor. The models of integrated navigation are proposed by the altitude and azimuth angles based on the star sensor.

The attitude misalignment error characteristics of star sensor/SINS using “altitude + azimuth angle” in two different navigation coordinates are shown in Table 1, and Figure 2. Those statistical error property in attitude angles is listed in Table 1. The means and the standard of error mean characteristics in launching point inertial coordinate are converged in 0.5 min of arc in the 30 s; those in local geographical coordinate are converged in 1 min of arc. The realizations of the convergence curve are shown in Figure 2.The integrated navigation of strapdown celestial guidance of estimation curves can converge within 10 s, and the estimated error means limited within 1 min of arc in the launching point inertia frame and the geographic coordinate frame.

Table 1.

Integrated navigation attitude estimate statistical properties in two different navigation coordinates with star sensor/SINS.

Attitude determination errors (′)	Launching point inertial coordinate		Local geographical coordinate
Attitude determination errors (′)	Mean	Standard of error mean	Mean	Standard of error mean
Pitch	0.14	0.12	0.56	0.37
Yaw	0.39	0.49	0.57	0.50
Roll	0.32	0.32	0.37	0.44

Figure 2.

(a) to (f) Attitude misalignment angles and attitude determination of hypersonic vehicle aided by star sensor celestial angles in different navigation coordinates.

Due to more consideration of the various perturbation factors in the launching point inertia frame, the model of the launching point is more accurate than the local geographic coordinate system model, and the estimation accuracy is better than the local geographic coordinate frame. The long-range missile and hypersonic vehicles often adopt the launching point inertia system, which helps to describe navigation benchmarks and provide accurate spatial motion parameters and system control. Therefore, the strapdown celestial guidance of the “altitude angle + azimuth angle” matching of the launching point inertial frame can achieve fast and accurate integrated navigation of the slaver navigation of the near-space hypersonic weapon platform.

The Bayesian method for data dropouts aided by the strapdown celestial angles star sensor

Measuring and transmitting the celestial angle will inevitably be challenged by the randomness and uncertainty of data loss. Consider the probability of dropout measurements p_k is 0.9. The attitude misalignment error characteristics are shown in Table 2 and Figure 3, which are general direct filter without data dropout,²¹ direct filter for data dropouts,³⁰ and the proposed Bayesian method for data dropouts. Direct filter without data dropouts is suitable for integrated navigation aided by the strapdown celestial angles. Next to dealing with data dropouts, the direct filter is weak convergence during the first 5 s, where the variance array of tremendous observation noise is added into the filter. It is defined as the measurement noise as infinity, which is equivalent to one-step state prediction, only propagating the previous state but no measurement estimation updating. It inevitably leads to the decline of filtering performance. Furthermore, the Bayesian filter is implemented correctly in the system state estimation and the measurement noise estimation, which has the performance close to direct filter without data dropouts.

Table 2.

Attitude estimate statistical properties for data dropouts.

Attitude determination errors (′)	Direct filter without data dropouts	Direct filter for data dropouts	Bayesian method for data dropouts
Pitch	0.30	0.83	0.32
Yaw	0.35	1.86	0.62
Roll	0.50	0.74	0.52

Figure 3.

(a) to (c) Attitude estimation errors for data dropouts.

However, the probability of data dropouts is too high, the proposed Bayesian filter will also diverge. The simulation experiment results are shown in Figure 4. When the probability of data dropouts is up to 95%, the proposed Bayesian filter is divergent during the 15 s and 20 s. We note that multi-constellation cooperative navigation aided by strapdown celestial angles can be adopted to improve the system’s dependence on a single constellation owing to dangerous loss probability.

Figure 4.

Performance comparison at different packet dropouts probability.

From the above-mentioned integrated navigation results, it is known that the vehicle trajectory is valid under the same parameter setting conditions. The strapdown celestial guidance of “altitude angle + azimuth angle” integrated navigation models of the local geographic system and the launching point inertial frame is practical. The initial localization and orientation errors occurred in the mobile launching of a ballistic missile, and the drifts of the reference coordinate of the inertia guidance system have existed in the launching point inertia frame. The two navigation mechanisms are arranged as follows. The strapdown celestial guidance of single-star schemes in local geographical coordinate is proposed to correct the initial localization and orientation errors that have occurred in the mobile launching of a ballistic missile in the first step. During the flying course, the strapdown celestial guidance in launching point coordinate is used to provide navigation reference to meet the requirement of quick response to spatial emission. Moreover, owing to the star celestial angle temporal malfunction with active galaxies, the proposed Bayesian method for integrated navigation data dropouts could be accomplished in 10 s of fast convergence within 1 min of arc.

Conclusion remarks

In this article, the simulation results show that the strapdown celestial observation angle-assisted hypersonic vehicle of two navigation coordinates can effectively achieve the fast and accurate integrated navigation in the near space. Before the launch, the celestial observation angles of the geographic navigation system are used to assist the vehicle to roughly correct the initial localization and orientation errors occurred in mobile launching of ballistic missile and the drift of the reference coordinate of the inertia guidance system. During the flying mission, the inertial navigation system of the launching point is used to provide the navigation reference for the vehicle, which responses quickly to spatial emissions. The proposed integration of the strapdown celestial angles and the Bayesian method for data dropout solves the problems of rapidly responsive, reliable, and precise of a hypersonic vehicle under a sophisticated environment. The proposed navigation approaches can be exploited to spatiotemporal cooperative navigation, including SINS, celestial angle measurements among multi-constellation, and the environmental knowledge for future work.

Footnotes

Acknowledgements

The authors thank the associate editor and the anonymous reviewers for their valuable suggestions on improving the text.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (no. U1804161, 51805148), the National Key Research and Development Program of China (no. 2017YFB0102603), Key Laboratory of Micro Inertial Instrumentation and Advanced Navigation Technology of Southeast University (Type B) Open Fund Project No. SEU-MIAN-201702, grants from NVIDIA and the NVIDIA DGX-Station/DRIVE PX 2 Auto Chauffeur, Henan Key Project of Science and Technology Research (nos 172102210214, 182102210095), the Science and Technology Innovation Planning Project of Ministry of Education of China, and the Fundamental Research Funds for the Central Universities.

ORCID iD

Hongmei Chen

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Integrated navigation approaches of vehicle aided by the strapdown celestial angles

Abstract

Keywords

Introduction

The model of nonlinear error using large misalignment angle

Integrated navigation system model of state error propagation

Launching point inertial system model of state error propagation

The geographical system model of state error propagation

Integrated navigation system model of state equations

Nonlinear observation model matching of altitude angle + azimuth angle

The observation equations of “elevation angle + azimuth angle” in launching point inertial coordinate frame

Step (1): Altitude angle and azimuth of the inertial solution

Step (2): Establish a measurement model of the altitude angle and azimuth angle

The observation equations of geographic coordinate frame “elevation angle + azimuth angle”

The relationship between the starlight vector of the altitude angle/azimuth angle and SINS errors

Starlight vector error model of the observation equation

Bayesian method for data dropouts about the strapdown celestial angels

Analysis of the state celestial angle in two different navigation coordinates

System simulation condition

System simulation results

The models of the altitude and azimuth angles based on the star sensor

The Bayesian method for data dropouts aided by the strapdown celestial angles star sensor

Conclusion remarks

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References