单频GPS卡尔曼滤波算法

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GPS卡尔曼滤波算法
Position Estimation in Single-Frequency GPS Receivers but its main problem is that it is an indirect measurement. It also gets limited with integer ambiguity. In this paper positioning in single-frequency gps receivers using kf with pseudo-range and carrier phase data and the corporation of pseudo-range capability and carrier phase in single-frequency GPS receivers will be discussed. This corporation makes a substitute for the pure pseudo-range observations and provides a high level of positioning accuracy Many researches have been done in this regard each of which has their own advantages and disadvantages. Our objective in this paper is to reach an acceptable positioning accuracy in high velocities(for example 3000 meter/second velocities) while keeping the simplicity of the filter and relations by using cheap single frequency receivers without using relative positioning(second receiver), at the same time by combining the data their benefits will be used simultaneously. As it is shown in the paper different methods have been compared with combined method and their efficiencies in high velocities have been analyzed Advantages of the proposed kF in our paper are that they are simple, low cost, and easy to design. They have the structure complexity less than for hardware implementation they also require less memory than for software implementation 2 Basic measurement 2.1 Calculation of the range between satellite and receiver To know the distance from the satellite is the main step in gPs. The main idea of it comes from the equation of light speed in delay time. GPS receiver measures the period of time in which it takes for the radio waves to reach the receiver from the satellite. as it is known radio waves move with the light speed. Therefore, the receiver can compute its distance to the satellite by multiplying the measured time to the light speed. Then to measure the signal's arrival time, we are dealing with very short periods of time, since the signals with light speed move very fast. There is a time gap between created GPS copy code in the receiver and the original received code from the satellite which by multiplying it to the light speed, pseudo range can be achieved. This method with both c/a and p methods is feasible. Produced codes in the receiver result from the receivers clock and the satellites sent codes are made by the satellite Time error in both receiver and satellite clocks causes that measured range vary from the geometric range between the satellite and receiver. These clocks are very precise and expensive Satellites to obtain more reliability level have got four atomic clocks. As they are very expensive. they cannot be used in receivers. As a result cheaper clocks are used which make slight differences in time measurement. However, because this difference symmetrically effects all the measurements, its effect in the computation can be assumed negligible. With some specific solutions, necessary corrections are carried though. The receiver computes its distance to the satellite with the existing error in its clock. Distances computed this way include many but coordinate errors; that is the error has the same effect on every four measurement. Therefore, these unreal but useful distances are called pseudo-range. Since positions of the satellites are precise, the error caused by pseudo-range can be computed with at least four equations. When the error is computed, position paramcters can be determined and receiver clock can be corrected [14] pringer M.R. Mosavi et al 2.2 Pseudo-Range measurement If data transfer time from j-th satellite is t/, data production time in receiver is ti, and satellite and receiver's clock crrors are respectively 8t/ and &ti, then pseudo-range p/ is obtained from Eg(1) t;+8t t1+8t E t;-t)c+(8t;-8f)c+E where ap is non-modeling errors. Ionosphere and troposphere delays(Ip, Tp) are raised, because their propagation speed is not always equal to C. Multi-path and satellite orbital errors for each satellite are analyzed in Eq(2): p+Stic-&tc+Tp+lp+me +e (2) Satellite orbital errors &/ are sent in navigation messages, but multi-path errors Mo depend on the receiver position. Pseudo-range equation without considering this error is computed in Eg. (3) P=+81c-87/c+Tn+l (3) where p is receivers precise range in ideal situation. If Xi=(xi, yi, zi) is receivers position and XJ=(r,y, z )is j-th satellite position for each j= 1, 2,.,J, then p in Eq (3)is computed like Eq (4) p=y(x-x)2+(y-y)2+(z1-z)2 4) 2. 3 Carrier Phase measurement The measurement which is more accurate than code is the received carrier phase from a satellite. Carrier phase measurement is the difference between produced carrier phase in receiver and received carrier from a satellite in a particular moment. Received signal phase in each moment is in accordance with the phase in the transfer time. Carrier phase measurement ill be fixed in a fraction of cycle and the range between satellite and receiver equals some numbers of whole cycles plus a fraction of the measured cycle The number of whole cycles, which is shown in carrier phase equation by Ni without considering errors, may change through time however this change is unknown for us. In case there are no clock and measurement errors, i carrier phase in cycle unit is like (t)=中()-9(t-)+N 5 where i(t) is carrier phase of produced code in receiver at the time t and o(t-t)is received carrier phase from a satellite at the time (t-t). so carrier phase based on meter dp! can be expressed as the following 入(1)=入中:(t) (t-τ)+入N→中=+81c-81′c+入 (6) Carrier phase equation based on meter which includes all the error sources is like Eq (7) +81;c-8t/c+Tp +M+入N;+ where p is the range between receiver and satellite, N; is integer ambiguity and n is carrier signal wavelength (LI wavelength equals 19 cm and L2 wavelength equals 24 cm) I Sprin Position Estimation in Single-Frequency GPS Receivers 3 Position Determination Using Kalman Filter 3. 1 Kalman Filter equations In the following, we will discuss position determination by KF using pseudo-range data carrier phase and the combination of pseudo-range and carrier phase. First, an introduction to KF is presented. The KF has been documented clearly [15-18]. KF is an optimal estimator which shows an optimal estimation of the system state by using the state space concept and system error model. One of the main features of Kf is its recursive calculations where b saving the latest calculations occupies a small volume of memory and updates the previous alculations with the new data. In order to explain KF's recursive relations, lets start with state and measurement equations. State equation in Kf is as the following [19, 20 Xk+1=¢kXk+Wk (8) where Xk is process state vector at the time tk, Xk+1 is process state vector at the time tk+, k defines state transition matrix from Xk to Xk+1 and Wk defines process error vector(a white sequence with a defined covariance function is presumed ). Measurement equation in Kf is as the following Zk= Hk xk t vu (9) where Zk defines measurement vector at time tk, Hk is a matrix which defines ideal relation (noiseless) between measurement vector and state vector at tk, Vk defines measurement error vector (it is presumed as a white sequence with a defined covariance and zero correlation with Wk sequence) Covariance matrix for Wk and Vk vectors are defined with the followin equations: ELWkW]=Ok; i=k ≠k (10) R k E[Ⅴkv] ≠k ElWKV=0; for all i and k (12 The summary of KF algorithm is as the following Step 1 Kalman gain computation: KF gain is computed as the following: kk=Pk Hk(hk pk Hk t rk (13) where Pk, whichis known for us, defines error covariance matrix corresponding with optimal estimation of system state vector Step 2 Updated estimation with measurement vector Zk: Updated estimation with measure ment vector Zk is done as the following: Xk=Xk+kk(Zk- HkX k) (14) Step 3 Error covariance computation for updating the estimation: Error covariance matrix corresponding with optimal estimation is computed as the following (I-Kk Hkp (15) pringer M.R. Mosavi et al Step 4 A propagation step forward: Finally optimal estimation of state and covariance matrix for the next time step is calculated as the following: Xk+1=中kXk (16) P+1=Pk中k+Qk (17) Matrix Eqs.(13)-(17)define Kalman recursive equations 3.2 Receivers Position Estimation Using KF with Pseudo-Range Data KF parameters' definition varies according to its function. To calculate KF, if receivers position(xk,δyk,8k), velocity(8k,δ,δ之k) and bias changes(8k,8k) are used, state vector X will be of eight elements where its relation with state transition matrix p in Eq (18) is shown with constant velocity for the receiver .k+1 100△T0000 6yk+1 0100△T 00 δ δxk 00100△T δx δk+1 00010000 δk 0000 (18 &yk δ之k 00000100 k δt 0000001△T|81k δtn 00000001 Sth where[8xk+1 Sy+1 82k+ dik+1 ik+1 Sik+1 1k+1 ik+1] is error state vector at epoch (k+l)and AT= l. State vector is according to the position and velocity errors of the receiver, clock bias and bias changes error state vector is estimated in measurement periods Then the estimated error state vector LSxk 8y+ SLk is added to the estimated quantities of the recciver's position, clock bias and bias changes ( l &kk ik lk ik according to the Eq(19)and for the next step, it is returned to zero xb+δx y+δ +8 +8 十δt In each step of measurement, state vector estimation is applied for linearization of shown observation matrix in Eq (20) ax I ax,I ay,I ay, I az, I ai.1 10 ax,2a,2ay,2a,2az,2a2,210 (20) L!x.tL元 aii az. i a; i 0 In Eq (20),ax, i, ay, i and az, i define the range between i-th satellite and receiver, and ax, i, av, i and ai. i define range changes velocity of i-th satellite to receiver. Covariance matrix P is initialized as an 8 x 8 diagonal matrix with large diagonal elements and zero non-diagonal 2 Springer Position Estimation in Single-Frequency GPS Receivers lements. Therefore, the position and the velocity error states can be assumed to have initial variances of(100m)and(100), respectively Measurement vector Z is the difference between the received observed pseudo-range vector from satellites and estimated pseudo-range according to the estimation of recciver's position (ik, yk, i k)and clock bias tk which is shown in Eq(21) )2+(y2-k)2+ )2+ Measurement covariance matrix R is a diagonal matrix with zero non-diagonal elements. The quantity of elements on original diameter depends separately on pseudo-range measurement variance of each satellite( 21, p2,.,00)which is shown in Eq(22) 320.0 尺 (22) 0 0 08 e is the process noise covariance matrix which describes uncertainty in dynamic model The goal of process noise covariance matrix is to define unknown parameters in time. White noise, random walk, and autocorrelation function are the three models which are usually used in data processing of GPS Elements of Q process noise covariance matrix are used to estimate un-modeled velocity variance and un-modeled clock variance besides preventing filter from falling asleep. These elements are not dependent on position. Q matrix equation is defined as Eq. (23) S△t0 △ At P P P Sn△tO Q (23) 0 0 △ p 3 p 2 Sn△tO 0 Sf△t+S △t 8 2 0 S。△t 8×8 3.3 Receivers Position Estimation USing KF with Single-Difference of Pseudo-Range Data In this method to compute receivers position, previous receivers position and velocity in KF are used. State vector X will be of nine elements which are defined in Eg (24) X=IPI, v, Pol (24) where P1=[x, y, z] and Po=Lx,y, z] are the current and previous position error vectors respectively, and the current velocity error vector is V=Vx, Vy, vzl. The KF must be modified to transfer the Pi clements to the Po in the state vector. The modified transition pringer M.R. Mosavi et al matrix p is defined as Eq (25) 100△t00100 0100△t001 00100△t001 0001 0000 0000 0000 1000 0100 010000 (25) 000 1000 0 0010 000 000 9×9 For the pseudo-range difference in i-th and j-th satellites, a linear equation based on the satellite's and receivers positions can be defined. As the single-difference is presumed, we can say. △H (26) △ △ △ △ △x △H= R R R R RI RJ 0,0,O, R R △ △ △z△ (27) R R/ R R. where Ax=x'-xr is the difference between x component of i-th satellite and receiver, Ax/=x-xr is the difference between x component of j-th satellite and receiver, with the same phrase the other components: R (△x)2+(△y2)2+(△z)2 is the best approxi- mation of geometric range from receiver to satellite (28 where Z defines measurement vector, Apm is measured pseudo-ranges difference and Ap defines difference of reconstructed pseudo-ranges. The measurement covariance matrix R is a diagonal matrix with zero non-diagonal elements. The covariance matrix P is a 99 diagonal matrix. The process noise covariance matrix Q is defined as Eq(29) qB△t3 △ q 0 △t0 qp△+qn30 0 4 0 △t0 4p△t+4v3-0 00qp△t 0 0 q△t0 0 0 Q 0g△t0000 q00g0 △t2 C 0 0 0 0 0 △t00 0 00 0 4p△t0 0 qn△t 0 0 004n△t 9×9 (29) where gp and y are the common spectral density for the position and velocity elements 2 Springer Position Estimation in Single-Frequency GPS Receivers 3. 4 Receivers Position estimation Using kf with Single-Difference of Carrier Phase data The single-difference phase across time can be modeled as △h1n2=H(x2-x1)+△ clock (30) where H vector is defined as H=Ar o Ay 0 As'010 and x12-xn is the vector of position differences between the current time(t2)and the previous time (t1) (31) where Z defines measurement vector, A pil, t defines i-th satellite measured phase data differ- ence in successive time intervals and 4 pil. t, defines i-th satellite reconstructed pseudo-ranges difference 3. s Receivers Position estimation using kf with double difference of carrier phase data For double-difference carrier phase with respect to time and satellite there is △Vq1,n2=△h12-△q12=△H"(xn2) (32 where Ai is j-th satellite measured phase data difference in successive time intervals and △ H 4J is defined as Eq.(33) AH(n)=H(n)-H(n); H A △ △ O.0,0. R Rl R (33) Z=(△12-△m,2)-(△qh,12-△1 (34) where Z defines measurement vector, Api. t, defines measured phase data difference in successive time intervals, Api, n defines i-th satellite reconstructed pseudo-range difference in successive time intervals and Api, t defines j-th satellite reconstructed pseudo-range difference in successive time intervals 3.6 Receivers Position Estimation Using kf with Double-Difference of Pseudo-Range and Carrier Phase Data In fact, this method is a combination of single-difference pseudo-range and double-difference carrier phase methods which is shown in Eq ( 35) (△qh1,-△12)-(mn)=△H(xn) (35) △ 0.0.0 R (36) RL RL R R R z=V△q12+P(tl)-(2) (37) where z defines measurement vector, Atl, l defines measured phase data difference in suc- cessive time intervals, pi defines measured pseudo-range by the system. Pj(t 1)is measured pseudo-range in the previous time and po(t2) defines estimated pseudo-range in the current time pringer M.R. Mosavi et al X change 3.0005 KF Code 30005 KF Code. Idiff KF Phase. diff E30005 KF Phase2d「 3.0005 KF Phase Code Ref 190 200 210 240 250 270 Time(second) Y change 5.5 KF Code 5 KF Code. diff KF Phase.1d「f 4.5 KF Phase- diff KF Phase-Code 0 200 D 400 500 700 00 900 Time( second Z change F Code b35 KF Code. diff KF Phase.ldr 三 3 KF PhaseCode 2.5 100 20 300 400 50 600 700 800 900 Time(second) Fig 1 Receiver position estimation using proposed KF methods(X component has been magnified) 4 Results and analysis Rohde schwarz GNSS simulator is used to generate raw gPs data such as pseudo range integrated carrier phase, Doppler shift and satellite ephemeris. The GNSS simulator in the r&s@ SMBV100A includes the ability to simulate realistic transmission conditions through the use of multipath signal generation and modeling of various atmospheric effects. The motion of an aircraft containing GNSS receiver, simulator is used to model effects that impact GNSS receiver performance, such as atmospheric conditions, multipath reflections, antenna characteristics and interference signals tt he results of receiver position estimation using KF with discussed methods are shown simultaneously in Fig. 1. In this figure, methods of smoothed positioning equations using KF with pseudo-range, carrier phase and the combination of these two are compared with a reference data. Also, Fig. 2 shows methods error. They show performance and accuracy of the proposed method As seen, the position estimation diagram by pseudo-range data has distortion. For the performance evaluating of the proposed method, root mean square error(rMse was used as[21-251: RMS ∑ Desired(i)-Output(i)] (38) i=1 2 Springer

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