Tracking Algorithm for Estimating the Orientation Angles of the Object Based on the Signals of Satellite Radio Navigation System

Email: russian-science@mail.ru Abstract: In the article, the theory of optimal filtering of information processes is used to synthesize optimal one-stage estimation algorithm for the object orientation angles using the signals of the satellite navigation system. The article also provides equations of optimal filtering, as well as their representation in the form of the tracking system comprising of an orientation angles discriminator and a smoothing filter. The synthesized algorithm has no tracking systems for the phases of the received signals (or phase differences) and the procedures for resolving the ambiguity of phase measurements that enhance its noise immunity while maintaining high precision in estimating the orientation angles. The article provides expressions for the calculation of the Cramer-Rao lower limit for the dispersion errors matrix for estimating object orientation angles when receiving signals of an arbitrary number of navigation satellites. The results of the simulation of the synthesized system are provided in the study.


Introduction
Satellite navigation systems are now widely used for a variety of technical problems, including extracting information from the phase of the received signals (Misra and Enge, 2012;Kaplan and Hegarty, 2006). One of such application is the determination of the orientation angles of the object on the signals of satellite radio navigation system, taken spaced points (Cohen, 1992;Parkinson and Spilker, 1996). The approach is because the signals coming from the navigation satellite in two spaced points have a phase shift depending on the angle between the reference line connecting the two points of reception and the direction of arrival of the satellite navigation signal. Measuring this angle and knowing the direction of satellite navigation, we can determine the orientation angle of the reference line. To implement this approach, navigation receivers are placed in the receiving points, which measure the phases of the received navigation signals. Further, the phase difference is formed, which carries information about the angle between the reference line and the direction of arrival of the signal from the navigation satellite. The main problem consists in that the measured phase difference is different from the true phase difference proportional to the phase difference between the arrival times of the signal phase edge in the receiving point on an integer number of periods of the high-frequency filling. That is, there is ambiguity of phase measurements. Various algorithms for ambiguity resolution of phase measurements are presented extensively in literature (Povalyaev, 2008;Lipkin, 2008;Boriskin et al., 2010;Perov and Kharisov, 2010;Knight, 1994;Hodgart and Purivigraipong, 2000;Lin et al., 2004;Verhagen and Teunissen, 2006).
The described ideology for determining the orientation angles using signals of satellite navigation systems is based on the principle of two-stage processing of signals from navigation satellites (Perov and Kharisov, 2010), in which the first stage is estimation of parameters of the received radio signals (delay, phase and the Doppler frequency shift) and the second stage solves the navigation task-formation of estimations of coordinates, velocity and orientation angles of the consumer. At the same time in relation to a standard navigation equipment, recently there are proposed and studied one-stage processing algorithms, where, in a single complex filter processing the signals of all visible navigation satellites, the desired estimations of position and velocity of the consumer (Perov, 2004) are formed. One-stage algorithm has no intermediate stage of where, A is the amplitude; ω i is the signal carrying frequency; д,i ω is the Doppler frequency shift due to the movement of the point A (O c ) of the triangle ABC; τ 0,i is the delay of the envelope curve of the signal at its distribution from the navigation satellite to receiver, ϕ 0,i is the initial phase of the signal received from the i-th navigation satellite, д,i ω is the Doppler signal frequency shift; ( ) дк,i h t is the function of modulation by the ranging code, ϑ HC, i are the navigation data taking the value of 0 or 1.
Representation (2) is valid for a certain time interval T in which the parameters д,i ω , τ 0,i can be considered as constant.
At points A, B and C, the received signals are shifted in phase relatively to similar signals received at the point O c into the phase angles: mean the distance between the exact O c and points А, В and С respectively. Therefore, it is possible to write (for the same time slot T) the following: where, ω A,ψ,i , ω B,ψ,i , ω C,ψ,i are the Doppler signals frequency shifts in points А, В and С respectively due to the rotation of the triangle ABC relatively the point O c , τ A,i , τ B,i , τ C,i is the delays of the envelope curve of the signal from the i-th NS in the receiving points А, В and С. Strictly speaking, the delays τ A,i , τ B,i , τ C,i depend on the vector of angles α. However, the consideration of this dependence has little effect on the accuracy of estimating the orientation angles of the phase of the carrier frequency (ψ A,i (α),(ψ B,i (α),(ψ C,i (α)). Thus, this dependence is not taken into account further.
When receiving n satellite radio signals, the total signal over small time intervals (for example, equal to the period ranging code) are described by the following expressions: Let us write the equations of observation in the points А, В and C: y t s t n t y t s t n t y t s t n t where, n A (t), n B (t) and n C (t) are independent white Gaussian noises with equal bilateral spectral densities N 0 /2. Let us introduce the vector surveillance y(t) = |y A (t) y B (t) y C (t)| T , for which let us write: white Gaussian noise vector with a matrix of bilateral spectral densities N 0 = I N 0 /2. Let us assume that receivers carry synchronous sampling of the input process in time, so that the processing system receives an implementation in discrete time t k,i (Fig. 2): where, t k,i = kT+iT d ; T = NT d is a step of discrete processing in the tracking system loop; T d is a discretisation period in an analogue-digital converter; n k,i is a vector of independent discrete white Gaussian noises where N 0 is one-sided power spectral density of the internal noise of the receiver.
The dynamics of change of the orientation angles is defined by equations in sampling times t k : where, ξα, k-1 is the vector of discrete white Gaussian noise with dispersion matrix D ξ .  Fig. 1) give information about the state vector X k-1 , relevant to the time t k-1 . Therefore, if we are interested in assessing the current state vector x k-1 according to the observations

Synthesis of Optimal Filtering Algorithm for Orientation Angles
it is necessary to consider the a posteriori probability density ( ) for which it is possible to write the equations (Tikhonov and Kharisov, 2004;Perov, 2012): Let us write the equations of optimal filtering of the vector X k in the Gaussian approximation (Tikhonov and Kharisov, 2004), assuming the delays τ A,i , τ B,i , τ C,i and Doppler frequency shifts д,i ω , Where: In (5), (6) the derivative of the scalar according to vector is understood as a row vector (Perov and Kharisov, 2010).
Let us write signal functions (4) in the discrete time for the moments t k,j ∈[t k,1 , t k,M ], where t k,j = t k +jT d , T d is the discretisation step for analogue-to-digital converter, MT d = T is the time interval, multiple to periods of the ranging code: In these equations, parameters ϕ 0,i,k and ϑ HC,i,k are not informative. Therefore, let us consider the likelihood function averaged by these parameters: where, I 0 (x) is the zero-order Bessel function of imaginary argument: , , дк, , ; , sin 1 sin 1 Let us represent (10) in the form of: Describe the multi-channel correlator, during the implementation of which, the Doppler frequencies д , i k ω , ω A,ψ,i,k , ω B,ψ,i,k , ω C,ψ,i,k shall be replaced by the corresponding estimations д , i k ω ɶ , , , , which are formed in independent tracking rings such as those described in (Perov and Kharisov, 2010).
Let us note that the correlation integrals (11) shall be calculated in a single timeline.
Substituting (9) into (7) and performing the necessary transformations, we obtain the following: Let us introduce the vector: And a matrix c such that α = cX. Let us transform the derivative: Let us consider the following: is obtained by differentiating (12) with respect to the corresponding parameter ψ A(B,C)i and, for example, for ψ Ai it has the following form: Let us write the phases ψ A,i (α k ), ψ B,i (α k ), ψ C,i (α k ) with regard to (1), (4) in the following form: Let us consider the derivative: Let us introduce the matrix:  (14), (15), the following expression can be written: Similar expressions can be written for derivatives: Let us note that in (16), (17) differentiation with respect to the orientation angles α shall be applied only to elements of the transformation matrix ( ) З c U α , e.g.: Then, let us represent (17) in the form of: where, the matrix c is determined by the ratio α = cX, i.e., c = |I 3×3 0 3×3 |, where I 3×3 is the identity matrix of 3×3 size.
Substituting (18) into (5), let us write an equation for estimating the state vector: Is the vector of discriminators of phase difference according to the signals of visible NS: Is the vector of discriminators for the orientation angles. Generalized block diagram of a single-stage filtration system for orientation angles of the object is shown in Fig. 3.
Let us note that in the diagram of Fig. 3 has a multichannel correlator.

The Potential Accuracy of the Estimation of Orientation Angles
It is known (Perov and Kharisov, 2010) that the discriminator is a device for which the parameters of the received signal and their evaluation are constant at the duration of the interval T. To characterize the quality of the estimates generated in the optimal discriminator, one can use the lower limit of the Cramer-Rao for the dispersion of the parameter estimates. Let us calculate this boundary for the problem. To simplify notations, let us turn to continuous time and consider the time interval [0, T]. Lower boundary for the matrix D pot,α of the dispersion of estimation errors of the orientation angles α is determined by the expression (Tikhonov and Kharisov, 2004;Perov, 2012): where, J is the Fisher's information matrix: where, ( ) 0 T p Y α is described by (8), E[*] is a sign of the mathematical expectation of a random variable in brackets.
Let us consider the derivative of the natural logarithm (7) by α: where, I 1 (x) is the first order Bessel function of imaginary argument.
In the standard signal/noise ratio corresponding to the Interface Control Document (n.a., 2011) Therefore, (21) takes the form of: Taking into account (22), let us write (20) in the form of: As X i is determined by synphase and quadrature components I j,i , Q j,i , then in order to perform the averaging in (23), it is necessary to have the mathematical expectations I j,i and Q j,i . Therefore, let us perform averaging in (11): where, q c /n 0 = P s /N 0 is the ratio of signal power to the power of spectral density of the internal noise. Taking into account (24) and the relations (9) and (3), let us calculate M[X i ] = 6q c /n 0 T.
Performing in (23) differentiation with respect to α and averaging the result obtained by realizations of noise, taking into account (24), let us write the expression for the matrix elements J: where, Let us consider the special case where the distance where: From (26) it follows that the elements J v,γ of the Fisher's information matrix contain the same factor , i.e., it is possible to write the following: Where: Substituting (27) into (19), let us write: About (28) is the source of well-known provisions that the potential accuracy of the estimation of the orientation angle is as higher as better is the signal/noise ratio q c /n 0 T and as higher is the ratio l/λ, which characterize the ratio of some characteristic linear dimension l of the receiving system to the wavelength of the received navigation signal. About (28) also means that the potential accuracy of the estimation of orientation angles of the object by the NS signals is dependent on the spatial angle between the lines connecting the receiving points and the direction of arrival of navigation signals from the number n of the processed NS signals and the number m of reception points.
To analyse the expressions (19) and (25), let us use the following procedure. Let the values of the vector of orientation angles α be independent random variables that take values from a given range, for example α∈[-30°, 30°], β∈ [-20°, 20°], γ∈[0°, 360°]. We also believe the location of navigation satellites relatively to the object to be random with uniform distribution law in the upper hemisphere. We are interested in the histogram of distribution of the values of standard errors estimation of the angular orientation vector D pot,α1 , D pot,α2 , D pot,α3 (19).
Let us assume q c /n 0 , expressed in decibels 0 c n q ɶ = 45 dBHz, T = 5 ms, λ = c/f 0 , f 0 = 1575.42 Hz. Let us consider the 3 reception point located in the plane O c X c Y c and lying at the vertices of an equilateral triangle inscribed in a circle of radius R = 1 m centred at the point O c . Figures 4-6 show histograms of the distribution of values of standard errors estimation of different angles at the reception n = 4 signals from navigation satellites. Figures 5 and 6 show that the histogram distributions for the various components of the angular orientation are close enough to each other. Therefore, we will only talk about one component, for example α. The most probable value of the mean square error of the angle estimation is σ α = 3.15 arc. min., mean square scatter of values is ~ 1 arc. min.
Let us consider the effect of the number of navigation satellites for distribution histogram of mean square errors for angle estimation. Figures 7 and 8 show distribution histograms of values of mean square error of α angle estimations calculated at n = 10 and n = 24 and when receiving signals in the same three points.
From the histogram it is clear that with the increase of the number of NS processed signals, there is a decrease of the most probable value of the mean squared error of angle estimation (σ α = 3.15 arc. min. for n = 4; σ α = 2 arc. min. for n = 10; σ α = 1.3 arc. min. for n = 24); the range of possible values of the mean square errors of the angle estimation is narrowed.

Simulation of the Synthesized Tracking System
To carry out a simulation of the synthesized tracking system for orientation angles, let us simplify the dispersion Equations 6.
The essence of this simplification is to replace the Equation 30 is a matrix equation with constant coefficients, which solution in the steady state shall be denoted as D X,steady .
Synthesized algorithm of optimal filtering for orientation angles of the object was simulated on a computer at a steady value of D X,steady .
For clarity, we present the results of a simplified version. Assume that the triangle ABC is isosceles with a , , 0,53 , 2 where, ∆f cc is the bandwidth of the Tracking System (TS). We assume that the errors on Doppler frequencies are not presented. Let us consider first the case of zero initial error in the angle and angular velocity. Assume that ∆f cc = 2 Hz and the number of NS is 8 and is uniformly distributed angularly in sight. Figure 9 shows the implementation of the tracking error in the angle. Mean square error of the estimation is σβ = 0.75 arc. min.
Let us narrow TS bandwidth ∆f cc = 0.5 Hz. Figure 10 shows the implementation of the tracking error in the angle. Mean square error of the estimation is σβ = 0.34 arc. min.
Assume ∆f cc = 2 Hz and the number of navigation satellites is 2 (the first two of the preceding 8). Figure 11 shows the implementation of the tracking error in the angle. Mean square error of the estimation is σβ = 5.48 arc. min.
Let us consider the effect of the initial error in the angular coordinate (with zero error in the angular velocity). Let us set the initial error in the angular coordinate нач 5 β ∆ =°, ∆f cc = 2 Hz and the number of navigation satellites is 8. Figure 12 shows the implementation of the tracking error in the angle. Mean square error estimation of the angle in that mode is σβ = 0.77 arc. min.

Discussion
From the simulation results, it is clear that the synthesized filter algorithm for orientation angles of the object is successfully working in various tactical situations when the initial error on the orientation angles not exceeding 5 degrees. We also note the following: • The synthesized estimation algorithm for orientation angles of the object has no tracking systems for the phase differences of the received signals, as was the case in the algorithm described in (Perov and Kharisov, 2010) • The synthesized estimation algorithm of orientation angles of the object has no procedures for resolving ambiguity of the phase measurements on signals of navigation satellites

Conclusion
In this study, the theory of optimal filtering of information processes was used to synthesize the optimal one-stage algorithm for estimation of orientation angles of the object by the signals of satellite radio navigation system.
Equations of optimal filtering are obtained, as well as their representation in the form of a tracking system including a discriminator for orientation angles and a smoothing filter.
The synthesized algorithms have no tracking systems for the phases of the received signals (or phase differences) and procedures for resolving ambiguity of phase measurements, which enhances its noise immunity, while maintaining high precision estimation of orientation angles.
Expressions are obtained for the calculation of the lower limit of matrix of dispersion estimation errors for orientation angles of the object when receiving signals from an arbitrary number of navigation satellites in the interval with duration of T, in which the orientation angles do not change over the signals of satellite navigation systems with an arbitrary number of navigation signals.
Examples of receiving from 4 to 24 signals from navigation satellites at three points show that an increase of the received signals decreases the most probable value of the mean squared error for estimation of angles (3 to 1.2 arc minutes respectively at the signal/noise ratio of 45 dBHz and T = 5 ms). The range of possible values of the standard errors for angles estimation is narrowed. The results of simulation are provided, from which it follows that at the length of the spaced receiving points of l = 1.5 m, there can be achieved a mean square error estimation of the orientation of angles in the steady state of 0.5...0.7 arc. min.