On-board BDS dynamic filtering ballistic determination and precision evaluation

Shu-Qiang Zhao; Jia-Yu Chang

doi:10.1515/astro-2022-0193

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On-board BDS dynamic filtering ballistic determination and precision evaluation

Shu-Qiang Zhao and Jia-Yu Chang

Published/Copyright: October 6, 2022

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From the journal Open Astronomy Volume 31 Issue 1

Abstract

BeiDou navigation satellite system (BDS) receivers on-board cannot give real-time positioning precision in tracking missile and carrier rocket. In this article, the dynamic filtered optimal estimation theory is applied to the highly dynamic on-board BDS ballistic parameter solution, an optimally constrained geometric dilution precision (GDOP) constellation selection strategy is proposed, a mathematical model based on dynamic filtered estimation for the ballistic determination and precision estimation of the on-board BDS is established, and an analysis of the on-board BDS positioning precision is carried out using the observed data. The calculation results show that the dynamic filter positioning algorithm is simple, practical, and reliable, which can effectively suppress and reduce the random errors of ballistic parameters and significantly improve the positioning precision, fully satisfying the needs of high dynamic and high precision navigation and positioning users, and has good application prospects.

Keywords: on-board BDS; dynamic filtering; ballistic determination; precision evaluation

1 Introduction

In recent years, BeiDou navigation satellite system (BDS) has been widely used in the field of aerospace telemetry and control. Currently, the space range has successfully used the BDS compatible with telemetry ground station to carry out external ballistic tracking measurement of carrier rocket and missile flight (Zhao et al. 2015). BDS receivers on-board is a specially developed navigation receiver adapted to high dynamic characteristics, which can receive signals from more than four BDS satellites simultaneously and can calculate the position and speed of the launch vehicle in real time (Wan et al. 2000). In terms of precision estimation, the current BDS receiver on-board applied in the field of aerospace measurement and control can usually only give geometric dilution of precision (GDOP) or position dilution of precision, but not the positioning precision of each observation ephemeris. In terms of positioning algorithms, on-board BDS real-time positioning mostly adopts kinematic methods and uses the least squares method to solve the ballistic parameters of the rocket. This method has a simple model and does not need to understand the dynamic receiver dynamics model, which basically meets the data-processing precision requirements, but has the drawback of large random error in ballistic parameters. In the dynamic positioning on-board BDS, the dynamic receiver antenna is a dynamic system that follows a certain law of constant change (Zhang 2004). The state of the system is observed at different times, and there are certain interrelationships among the ephemeris of the observed quantities (Fu et al. 2003), and it is difficult for the least squares method to make full use of these laws and interrelationships among the ephemeris for data processing. In addition, to eliminate random errors in highly dynamic BDS positioning data, it is necessary to apply optimal estimation methods to estimate the true state in real time from various random disturbances. Kalman filtering is a recursive filtering method derived under the principle of linear unbiased minimum variance estimation (Yang et al. 2005). It introduces the concept of state space and recursively estimates the new state valuation based on the state estimate of the previous moment and the observation of the current moment with the help of the state transfer equation of the system (Wen 2009). Therefore, Kalman filter is more suitable for data processing of on-board BDS dynamic positioning. A dynamic model is developed to describe the motion of the BDS receiver on-board with the launch vehicle using a sequence of discrete observed quantities with noise errors obtained from the BDS satellite signal, where the estimated signal is a random response caused by a white noise excitation, the transfer structure between the excitation source and the response is known, and the functional relationship between the quantity measurement and the estimated quantity is also known (Chen et al. 2021). Therefore, dynamic filtering can be used for real-time positioning of BDS data in the dynamic data processing of on-board BDS, which can effectively reduce the random error of ballistic parameters and improve the positioning precision (Cui et al. 2019).

2 Mathematical modeling of dynamic filtering

2.1 Choice of reviewers

On-board BDS dynamic filter positioning generally treats the navigation satellite orbit as a known. The BDS constellation is a hybrid constellation consisting of MEO, IGSO, and GEO satellites. Among them, MEO and IGSO satellites can use broadcast ephemeris parameters to calculate the MEO and IGSO satellite orbits of BDS. Since the orbital inclination of GEO satellites is close to 0°, directly fitting GEO satellite orbits in the form of broadcast ephemeris parameters may not converge due to matrix singularities (Ruan et al. 2011), Liu (2012) proposed the method of coordinate rotation to solve the problem. To avoid the occurrence of singularities in the number of orbital roots after coordinate rotation and minimize the absorption of orbital inclination perturbation by other orbital roots perturbation, Wang et al. (2007) suggested that the coordinate rotation angle be set to a large value, generally 50°. The GEO broadcast ephemeris parameters can be obtained by fitting with the coordinate rotation method. When calculating the orbit of GEO satellite, users need to calculate the satellite position according to the calculation method of MEO satellite, and then carry out the corresponding coordinate inversion process, so that the position of GEO satellite in the earth-solid coordinate system can be obtained.

The dynamic model of the motion carrier (state equation) can be represented by the dynamic position, velocity, and acceleration. Since the dynamic model is difficult to represent by precise mathematical formulas (Xu 2001), in practical engineering applications, a simplified dynamic model is generally used under the premise of ensuring certain precision. In high dynamic on-board BDS filter positioning (Wang et al. 2017), the data sampling rate is generally 0.1 s or higher. In this case, the constant acceleration model can be used and the dynamic noise is assumed to be zero-mean Gaussian noise (Song 2006).

2.2 Construction of system state model

The launch vehicle state vector is chosen as follows:

(1) X k = [ x k , x ˙ k , x ¨ k , y k , y ˙ k , y ¨ k , z k , z ˙ k , z ¨ k , ξ k , ξ ˙ k ] T ,

where x k , y k , z k , x ˙ k , y ˙ k , z ˙ k , and x ¨ k , y ¨ k , z ¨ k are, respectively, the three-dimensional position, velocity, and acceleration components of the on-board BDS receiver antenna in the CGCS2000 coordinate system. ξ k is the pseudo-range deviation caused by the clock difference of the receiver, and ξ ˙ k is the rate of change of the pseudo-range deviation caused by the clock drift.

The dynamic equations is expressed as follows:

(2) X k = Φ k , k − 1 X k − 1 + W k − 1 .

In the aforementioned equation, Φ k , k − 1 is the nonsingular state one-step transfer matrix and W k − 1 is the state noise vector of the system at the time t k − 1 , which is a zero-mean Gaussian white noise random sequence, i.e.,

(3) E [ W k ] = 0 E [ W k W j T ] = Q k δ k j .

In the aforementioned equation, Q k is the variance matrix of the process noise vector sequence of the system, which is a symmetric nonnegative definite matrix, and δ k j is Kronecker- δ function, which is defined as follows:

(4) δ k j = 0 k ≠ j 1 k = j .

The state transition matrix is:

(5) Φ = Φ x 0 3 x 3 0 3 x 3 0 3 x 2 0 3 x 3 Φ y 0 3 x 3 0 3 x 2 0 3 x 3 0 3 x 3 Φ z 0 3 x 2 0 2 x 3 0 2 x 3 0 2 x 3 Φ ξ .

Among them,

(6) Φ x = Φ y = Φ z = 1 T T 2 / 2 0 1 T 0 0 1 ,

(7) Φ ξ = 1 0 0 1 .

Then, the system noise variance matrix is:

(8) Q k = 2 σ x ¨ 2 Q x ¨ τ x 0 3 x 3 0 3 x 3 0 3 x 2 0 3 x 3 2 σ y ¨ 2 Q y ¨ τ y 0 3 x 3 0 3 x 2 0 3 x 3 0 3 x 3 2 σ z ¨ 2 Q z ¨ τ z 0 3 x 2 0 2 x 3 0 2 x 3 0 2 x 3 2 σ ξ ˙ 2 Q ξ ˙ τ ξ ,

where τ x , τ y , τ z , τ ξ are the time constants associated with the target acceleration, σ x ¨ 2 , σ y ¨ 2 , and σ z ¨ 2 are the systematic interference variance of the acceleration relative to the moving target, and σ ξ ˙ 2 is the variance of the clock difference rate as the systematic noise.

(9) Q x ¨ = Q y ¨ = Q z ¨ = T 5 / 20 T 4 / 8 T 3 / 6 T 4 / 8 T 3 / 3 T 2 / 2 T 3 / 6 T 2 / 2 T ,

(10) Q ξ ¨ = T 3 / 3 T 2 / 2 T 2 / 2 T ,

where T is the observation ephemeris interval.

2.3 System measurement modeling

The pseudo-range and pseud-orange rate of n BDS satellites were observed simultaneously by the on-board BDS receivers at time k . The observation matrix can be expressed as follows:

(11) Z k = [ ρ 1 k , ρ 2 k , … , ρ n k , ρ ˙ 1 k , ρ ˙ 2 k , … , ρ ˙ n k ] T .

The linearized equation is expressed as follows:

(12) Z k = [ H k X k + V k ] ,

(13) H k = h x 1 0 0 h y 1 0 0 h z 1 0 0 1 0 ⋮ □ □ ⋮ □ □ ⋮ □ □ ⋮ □ h x n 0 0 h y n 0 0 h z n 0 0 1 0 0 h x 1 0 0 h y 1 0 0 h z 1 0 0 1 □ ⋮ □ □ ⋮ □ □ ⋮ □ □ ⋮ 0 h x n 0 0 h y n 0 0 h z n 0 0 1 .

Here,

(14) h x j = x k − x i k R i k , h y j = y k − y i k R i k , h z j = z k − z i k R i k .

R i k is the symmetric positive definite variance matrix of the system observation noise V k .

(15) R k = diag ( σ 1 2 , σ 2 2 , … , σ 2 2 n ) ,

where σ i 2 is the variance of the pseudo-range and pseudo-range rate measurement noise of n satellites. x k , y k , and z k are the coordinates of the launch vehicle in the CGCS2000 coordinate system at time k ; x i k , y i k , and z i k are the coordinates of the i th BDS satellite in the CGCS2000 coordinate system at time k . δ k is the clock deviation of the receiver; c is the speed of light; and v i is the sum of the measurement errors of the i th satellite, which is usually assumed to be white noise. R i k is the true distance between the launch vehicle and the i th BDS satellite at time k .

(16) R i k = ( x i − x i k ) 2 + ( y i − y i k ) 2 + ( z i − z i k ) 2 2 .

The nonlinear formula for the pseudo-range ρ i k is:

(17) ρ i k = R i k + c δ k + ε i k + v i k ,

where ε i k is the non white noise error of the receiver channel i and v i k is the measurement noise of channel i . The Doppler shift can be expressed as follows:

(18) ρ i k = ( x k − x i k ) ( x ˙ k − x ˙ i k ) R i k + ( y k − y i k ) ( y ˙ k − y ˙ i k ) R i k + ( z k − z i k ) ( z ˙ k − z ˙ i k ) R i k + c δ k ˙ ,

where x ˙ k , y ˙ k , and z ˙ k are the velocity coordinates of the receiver at time k ; x ˙ i k , y ˙ i k , and z ˙ i k are the velocity coordinates of the i th BDS satellite at time k .

3 Ballistic precision assessment

The precision of on-board BDS positioning depends on two aspects: (i) the precision of the observation quantity and (ii) the spatial geometric distribution of the observed satellites, which is usually called the geometric figure of satellite distribution (Liu et al. 2016). The on-board BDS positioning precision can be expressed by:

(19) σ = σ UERE × GDOP .

In the aforementioned equation, σ UERE is the equivalent ranging error; GDOP is the geometric dilution of precision, which reflects the scale factor between the pseudo-range measurement and the user position error due to the influence of satellite geometric relationship, and is the degree of amplification of the user ranging error. The analysis of positioning precision using GDOP is usually considered as independent equal precision among the observations, the weight matrix is P = I .

The premise of precision estimation based on the aforementioned method is based on equal precision observation, which is applicable to satellite navigation systems such as GPS and GLONASS, because the satellites in these systems have the same type and are all distributed at the same orbital altitude, and they have the same ranging error (Chen et al. 2017). The BDS is a hybrid constellation navigation system composed of heterogeneous satellites distributed at different orbital altitudes (Fang et al. 2019), and the satellites in different orbits have different orbital errors, so when analyzing the positioning precision of the system, the aforementioned method cannot truly reflect the actual situation (Huang et al. 2021).

For GEO satellites in the geosynchronous orbit, their geostationary characteristics make it difficult to separate the clock difference of the satellites during fixing, and the GEO satellites are greatly affected by the solar pressure, so the range error introduced by the ephemeris error of GEO satellites under the same conditions is about twice that of MEO satellites (Wang et al. 2016). The IGSO satellite has the best orbit determination precision in the local area compared with GEO and MEO (Yang 2017), but its orbit height is the same as GEO, so it is also affected by the solar pressure, and here it is assumed to have the same ranging precision as MEO (Yang 2006). In the case of independent observations, the covariance matrix of observations can be obtained by:

(20) Σ = σ MEO 2 I k 1 × k 1 □ □ □ σ GEO 2 I k 2 × k 2 □ □ □ σ IGSO 2 I k 3 × k 3 ,

where k 1 , k 2 , and k 3 represent the number of MEO, GEO, and IGSO satellites, respectively. Consider σ ˆ 0 2 = σ IGSO 2 = σ MEO 2 . According to the precision estimation formula of the least square method, the mean square error of the pseudo-distance observation value σ ˆ 0 2 is expressed by:

(21) σ ˆ 0 2 = ± V T V n − 4 2 = ± [ v v ] n − 4 2 , n > 4 ,

where [ v v ] is the sum of squares of the pseudo-range residuals.

(22) V T V = [ v v ] = ( v 1 ) 2 + ( v 2 ) 2 + … + ( v n ) 2 .

The covariance matrix D ˆ of the ballistic position parameters is:

(23) D ˆ = σ ˆ 0 2 Q = Q 11 Q 21 Q 31 Q 41 Q 21 Q 22 Q 23 Q 24 Q 31 Q 32 Q 33 Q 34 Q 41 Q 42 Q 43 Q 44 , = σ ˆ x 2 σ ˆ x y σ ˆ x z σ ˆ x t σ ˆ y x σ ˆ y 2 σ ˆ y z σ ˆ y t σ ˆ z x σ ˆ z y σ ˆ z 2 σ ˆ z t σ ˆ t x σ ˆ t y σ ˆ t z σ ˆ t 2 ,

where Q = ( H T P H ) − 1 is the weight inverse matrix of the ballistic position parameters and P is the weight matrix.

(24) P = σ ˆ 0 2 Σ − 1 .

Thus, the weighted GDOP value is obtained.

(25) GDOP = Q 11 + Q 22 + Q 33 + Q 44 .

In the case of unequal precision observation, the aforementioned formula assigns corresponding weights to the observed values according to different ranging errors of different types of satellites, and the geometric precision factors obtained through weighting calculation can be used to evaluate the positioning precision more objectively (Wang et al. 2015).

The precision of the ballistic position parameters is expressed as follows:

(26) σ ˆ x = σ ˆ x 2 ; σ ˆ y = σ ˆ y 2 ; σ ˆ z = σ ˆ z 2 ; σ ˆ t = σ ˆ t 2 .

The precision estimation method for the velocity parameter is the same as that for the position parameter.

4 Constellation selection strategy

From the positioning precision analysis, it can be seen that the spatial geometric distribution (directional cosine array) of the BDS satellite in the observed ephemeris is one of the main factors affecting the positioning precision, and the GDOP is the degree of the influence of the observed satellite geometry on the positioning precision (Chen et al. 2020), and the smaller its value, the higher the positioning precision (Yuan et al. 2019). According to the previous section, the positioning precision of the on-board BDS can be expressed as the product of the mean square error of the geometric precision factor and pseudo-range observation value, while the geometric precision factor has nothing to do with observation error and is only related to the configuration of satellite constellation (Zhang et al. 2021). The pseudo-range mean squared error is essentially the use equivalent range error, which includes satellite ephemeris error, satellite clock error, ionosphere, and troposphere residual error. After the BDS constellation is fully functional (Huang et al. 2020), the on-board BDS generally has 12–24 tracking channels and can use the “all-in-view” positioning mode, which makes use of the effective observation values of all satellites, and its advantage lies in the eliminating part of the systematic errors of satellite positioning. To minimize the positioning error, the constellation GDOP and its corresponding pseudo-range error should be considered comprehensively. Since the GDOP value is closely related to the elevation angle of the observation satellite, the elevation angle of the satellite not only affects the GDOP value but also affects the signal transmission error (Liu et al. 2022). Therefore, in the constellation selection, the satellite orbit should be taken into full consideration, based on the elevation angle of the satellite, the GDOP value as a measurement standard for satellite selection (Xu et al. 2021). For ground users, the elevation angle of BDS satellites is always positive, but for on-board BDS, negative elevation satellites may be observed, but the negative elevation satellites has much effect on the positioning precision. If there are redundant BDS satellites in the positioning, the negative elevation angle satellite should be removed and then positioning can take place.

An optimally constrained GDOP constellation selection strategy is as follows:

Select all available observation satellites in the field of view based on the tracking status code and signal-to-noise ratio, and eliminate those that do not meet the conditions.
Calculate the coordinates of the satellite at the moment of the epoch from the orbital parameters of the satellite in the broadcast message, and calculate the elevation angle of each satellite.
Select the satellite that meets the elevation angle threshold requirement and calculate the GDOP value for the constellation.
If the GDOP value of the constellation meets the positioning requirements, the constellation is used as the positioning constellation; otherwise, if there are redundant satellites, the satellite with the smallest elevation angle among the observed satellites is eliminated to form a new positioning constellation and its GDOP value is calculated.

Judge whether the GDOP value of the positioning constellation meets the need of positioning precision, otherwise repeat step ④ , until the GDOP value meets the requirement.

5 Example analysis

In a launch vehicle flight test mission, GPS and BDS dual-mode receivers were used to track and measure the external ballistic of the launch vehicle, and the BDS and GPS observation data were processed and analyzed by dynamic filtering, respectively. The observation data was in a special format with a sampling rate of 10 frames per second. The ballistic calculation process of on-board BDS is shown in Figure 1.

Figure 1

Ballistic solving process of on-board BDS.

The BDS, GPS ballistic, and their corresponding precision data are calculated through data processing, and their mutual difference comparison curves and precision curves are shown in Figures 2–9.

Figure 2

Difference between BDS and GPS position coordinates.

Figures 2 and 3 and Table 1, mean value statistics of coordinate residuals. The difference between the position coordinates of the on-board GPS and BDS in X and Y directions is about 3 m, while it is very close in the Z direction and the difference is basically around 0 m. The velocity difference between the two fluctuates around ± 0.02 m/s in the X , Y , and Z directions, and the mean of velocity residuals is less than 0.01 m/s in three directions, indicating that the velocity parameters of BDS and GPS are very close. The aforementioned results show that the ballistic parameters calculated by BDS are consistent with the trend of GPS, and the magnitude difference is very small (Ren 2019).

Figure 3

Difference between BDS and GPS velocity coordinate.

Table 1

Mean value statistics of coordinate residuals

Mutual difference mode	Mean value of coordinate residuals
	Mean of positions residuals (m)			Mean of velocity residuals (m/s)
	Δ X	Δ Y	Δ Z	Δ V X	Δ V Y	Δ V Z
GPS-BDS	2.578	3.209	− 0.163	0.001	0.007	0.007

As shown in Figures 4 to 6 and Table 2, the coordinates precision of rocket ballistic calculated by BDS and GPS dynamic filtering are all within 1 m in X and Z directions and within 2 m in Y direction. While the coordinate precision of GPS is slightly better than that of BDS in the X and Y directions, and about the same in Z direction.

Figure 4

Position precision of on-board BDS and GPS in X -direction.

Table 2

Mean value of positioning precision statistics

Positioning mode	Satellites	Mean value of positioning precision statistics
		Position precision (m)			Velocity precision (m/s)
		M X	M Y	M Z	M V X	M V Y	M V Z
GPS	11	0.417	1.050	0.582	0.012	0.027	0.015
BDS	10	0.743	1.299	0.683	0.010	0.017	0.009

Figure 5

Position precision of on-board BDS and GPS in Y -direction.

Figure 6

Position precision of on-board BDS and GPS in Z -direction.

As can be seen from Figures 7 to 9 and Table 2, the velocity precision of rocket ballistic calculated by BDS and GPS is all within 0.02 m/s in X and Z directions, and within 0.04 m/s in Y direction while the velocity precision of BDS is better than GPS in X , Y , and Z directions during the whole tracking.

Figure 7

Velocity precision of on-board BDS and GPS in X -direction.

Figure 8

Velocity precision of on-board BDS and GPS in Y -direction.

Figure 9

Velocity precision of on-board BDS and GPS in Z -direction.

6 Conclusion

The optimally constrained GDOP constellation selection strategy is adopted in the ballistic solution method of on-board BDS dynamic filtering, which makes the dynamic filtering positioning algorithm simple and practical. Compared with least squares, dynamic filtering does not perform iterative operations, and matrix calculation is relatively less, which can well meet the requirements of real-time positioning. Measured data calculation results show that this method can make full use of the available redundant satellite observation information, the obtained positioning results are reliable, and the ballistic precision is equivalent to that of GPS, which fully meets the requirements of the external ballistics measurement precision of the carrier rocket. Moreover, it has the characteristics of small computation, high precision, and low complexity and can be further popularized and applied to high dynamic BDS real-time positioning of missiles, spacecraft, and aircraft test flights.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2022-06-19

Revised: 2022-08-09

Accepted: 2022-08-11

Published Online: 2022-10-06

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/astro-2022-0193

Keywords for this article

on-board BDS; dynamic filtering; ballistic determination; precision evaluation

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