Modified Observation Model In Tightly鄄coupled Ins/ Gps Integration

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Journal of Beijing Institute of Technology, 2017, Vol. 26, No. 1 Modified Observation Model in Tightly鄄Coupled INS / GPS Integration Guochao Fan,摇 Dan Song and Chengdong Xu 苣 ( School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China) Abstract: The conventional Kalman filter ( CKF) is widely used in tightly鄄coupled INS / GPS integrat鄄 ed navigation systems. The linearization accuracy of the CKF observation model is one of the decis鄄 ive factors of the estimation accuracy and therefore navigation accuracy. Additionally, the conven鄄 tional observation model ( COM) used by the filter may be divergent, which would result into some terrible accuracies of INS / GPS integration navigation in some cases. To improve the navigation accu鄄 racy, the linearization accuracy of the COM still needs further improvement. To deal with this issue, the observation model is modified with the linearization of the range and range rate equations in this paper. Compared with COM, the modified observation model ( MOM) further considers the differ鄄 ence between the real user position and the position calculated by SINS. To verify the advantages of this model, INS / GPS integrated navigation simulation experiments are conducted with the usage of COM and MOM respectively. According to the simulation results, the positions ( velocities) calculat鄄 ed using COM are divergent over time while the others using MOM are convergent, which demon鄄 strates the higher linearization accuracy of MOM. Key words: INS / GPS integration; linearization; modified observation model ( MOM) ; conventional observation model ( COM) CLC number: V 249郾 32摇 摇 Document code: A摇 摇 Article ID: 1004鄄 0579(2017)01鄄 0016鄄 13 摇 摇 Strapdown inertial navigation system ( SINS) and global positioning system ( GPS) integration is widely used in aircraft navigation. There are three common modes for SINS / GPS integration, loosely coupled and tightly coupled in terms of software, and deeply coupled in terms of hard鄄 ware [1] . Compared to loosely coupled mode, the navigation accuracy of tightly coupled mode is higher. Compared to deeply coupled mode, the tightly coupled mode is more flexible because it is implemented in the software [1] . Thus there have Fig. 1摇 Structure of tightly鄄coupled INS / GPS integration As shown in Fig. 1, the output navigation pa鄄 rameters are position P In , velocity V In and attitude A In . P In = ( L I , 姿 I , h I ) , where the three elements of P In represent latitude, longitude, and height re鄄 been extensive theoretical studies dedicated to spectively. V In = ( v Ie , v In , v Iu ) , where the sub鄄 al structure is illustrated in Fig. 1. navigation frame respectively. A In = ( 兹 I , 酌 I , 渍 I ) , tightly鄄coupled INS / GPS integration [2] . Its gener鄄 scripts e, n, u represent the three axes of the where 兹 I is pitching angle, 酌 I is rolling angle and Received摇 2016鄄 06鄄 22 Supported by the National Natural Science Foundation of Chi鄄 na(61502257, 41304031) 苣 Author for correspondence, professor, Ph. D. E鄄mail: xucd@ bit. edu. cn DOI: 10. 15918 / j. jbit1004鄄0579. 201726. 0104 渍 I is yawing angle. The output parameters of GPS are the pseudoranges 籽 G and the pseudorange ·G rates 籽 between the user and the visible satel鄄 lites. By inputting above parameters to the filter, — 16 — Guochao Fan et al. / Modified Observation Model in Tightly鄄Coupled INS / GPS Integration the corrections of SINS navigation parameters can computational burden of nonlinear filter is much correction 啄P = ( 啄L,啄姿,啄h) , the velocity correc鄄 Other researchers applied adaptive filters to be obtained via filtering, including the position higher than that of CKF. tion 啄V = ( 啄V e , 啄V n , 啄V u ) and the misalignment eliminate SINS attitude correction 啄A = ( 啄兹,啄酌,啄准) . By u鄄 an adaptive Kalman filter to improve the filter per鄄 椎 = ( 准 E , 准 N , 准 U ) , which can be transformed to sing these corrections to revise the SINS naviga鄄 tion parameters, the SINS / GPS navigation param鄄 eters, the position P, the velocity V and the atti鄄 tude A, can be obtained [1] . The filter shown in Fig. 1 is usually called conventional Kalman filter ( CKF) [3] . Generally, the statistics of the system noise are assumed to be known in CKF. As long as the noise character鄄 istics are correctly known, the filter can produce optimal estimates for system states, which has high navigation accuracy. Moreover, the state model and the observation model of CKF should be linear. In tightly鄄coupled SINS / GPS integra鄄 tion, the observation model, including the pseud鄄 orange and pseudorange equations, should be lin鄄 tion [3 - 4,11 - 13] impacts caused by lineariza鄄 . For example, Ding and Wang used formance and restrain the filtering divergence in INS / GPS integrated navigation; Jwo et al. com鄄 bined an adaptive Kalman filter with the innova鄄 tion鄄based adaptive estimation approach to make the matrices in the KF loop be positive defi鄄 nite [11 - 13] ; Gao et al. applied a Helmert variance component estimation based adaptive Kalman fil鄄 ter in INS / GPS / BDS / GLONASS to weaken the impacts of the GNSS observations with low quality and the inaccurate state model [14] ; Jwo et al. used FASTUKF to evaluate the performance of GPS / INS integrate [15] . Thus good computational efficiency and numerical stability in integrated navigation can be achieved. However, these studies all focus on the mod鄄 earized before filtering. However, linearization ification of filtering algorithm. Little attention has the noise with unknown statistical properties. It model by far. If the linearization accuracy of the would bring model error, which is equivalent to may cause divergence of filter results or accuracy loss [4] . Many researchers are dedicated to solve this problem which exists in the conventional ob鄄 servation model ( COM) of tightly鄄coupled SINS / GPS integration navigation [5 - 6] . Some researchers are interested in nonlinear filtering [5 - 7] , in which the linearization of obser鄄 vation model is avoided. For example, Chang et al. applied a marginal unscented KF in INS / GPS integration; Zhou et al. improved an unscented particle filter in INS / GPS integration [6 - 7] ; Meng et al. provides an adaptive UKF with a noise sta鄄 tistic estimator to improve the nonlinear filter [8] ; Gong et al. improved the TFS in SINS / GPS inte鄄 gration [9] ; Tseng Chien鄄Hao et al. ACKF filter in INS / GPS integration applied a been paid to the modification of the observation observation model was high enough, the CKF would not be divergent and high accuracy naviga鄄 tion parameters would be obtained without the fil鄄 ter modification. In this paper, a modified obser鄄 vation model ( MOM) is derived with the linear鄄 ization of the range and range rate equations be鄄 tween the user and satellites. According to the simulation results, the filters using COM are di鄄 vergent over time while those using MOM are con鄄 vergent. The navigation accuracy of the SINS / GPS integrated navigation system which uses MOM is superior. 1摇 Derivation of MOM MOM consists of the pseudoranye equation . By using a which is used to observe the user position and the vented. Thus the filter convergence and the navi鄄 serve the user velocity. They are derived from the [10] nonlinear filter, the linear model error is circum鄄 gation accuracy are guaranteed. However, the pseudorange rate equation which is used to ob鄄 range and range rate equations between a satellite — 17 — Journal of Beijing Institute of Technology, 2017, Vol. 26, No. 1 position and the user. Symbols used in derivation are explained in Tab. 1. Tab. 1摇 Symbols used in derivation Symbol 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Meaning P Ie = ( x I ,y I ,z I ) User position calculated by SINS in ECEF ·I ·I V Ie = ( v Ix ,v Iy ,v Iz ) = ( ·I x ,y ,z ) User velocity calculated by SINS in ECEF P Ge = ( x G ,y G ,z G ) User position calculated by GNSS in ECEF P In = ( L I ,姿 I ,h I ) User position calculated by SINS in the navigation frame V In = ( v Ie ,v In ,v Iu ) User velocity calculated by SINS in the navigation frame P Gn = ( L G ,姿 G ,h G ) User position calculated by GNSS in the navigation frame ·G ·G ·G V Ge = ( v Gx ,v Gy ,v Gz ) = ( x ,y ,z ) User velocity calculated by GNSS in ECEF P e = ( x,y,z) User real position in ECEF V Gn = ( v Ge ,v Gn ,v Gu ) P n = ( L,姿,h) User velocity calculated by GNSS in the navigation frame User real position in the navigation frame · · · V e = ( v x ,v y ,v z ) = ( x ,y ,z ) User real velocity in ECEF P s = ( x s ,y s ,z s ) Position of a satellite in ECEF V n = ( v e ,v n ,v u ) User real velocity in the navigation frame ·s ·s V s = ( ·s x ,y ,z ) Velocity of a satellite in ECEF 摇 摇 In Tab. 1, the subscript “ e冶 presents the vec鄄 tor in the earth centered earth fixed coordinate ( z - zs ) ( xI - xs ) 2 + ( yI - ys ) 2 + ( z - zs ) 2 frame ( ECEF) ; the subscript “ n冶 means the vec鄄 tor in the navigation frame. 1郾 1摇 Linearization of the range equation 籽 Ix = Range between a satellite and the user爷 s position calculated by SINS is expansion, thus 籽 I can be expanded at P e = ( x,y, z) as 籽 I = r + a x ( x I - x) + a y ( y I - y) + a z ( z I - z) ax = 鄣籽 = 鄣x I | x I = x (x - x ) s (2) I (x - x ) + (y - y ) + (z - z ) s 2 I s 2 I 鄣籽 I ay = I = 鄣y | y I = y (y - y ) s 2 = (x - x ) 籽 Ix s (3) az = 鄣籽 I = 鄣z I | z I = z (5) 籽 Iy = 籽 Iz = ( x - x s ) 2 + ( y I - y s ) 2 + ( z I - z s ) 2 (6) ( x I - x s ) 2 + ( y - y s ) 2 + ( z I - z s ) 2 (7) ( x I - x s ) 2 + ( y I - y s ) 2 + ( z - z s ) 2 (8) The difference between P Ie and P e is recorded as 啄P e [ xI - x 啄P e = P Ie - P e = yI - y T z I - z] = [ 啄x 啄y s 啄z] T (9) According to Eq. (9) , Eq. (2) can be rewrit鄄 ten as 籽I = r + [ ax where r = ay a z ] [ 啄x 啄y 啄z] T (10) ( x - xs ) 2 + ( y - ys ) 2 + ( z - zs ) 2 . 1郾 2摇 Linearization of the range rate equation Range rate [16 - 17] between a satellite and the user position calculated by SINS is (y - y ) = I s 2 s 2 I s 2 籽 Iy (x - x ) + (y - y ) + (z - z ) s ( z - zs ) 籽 Iz In Eq. (3) - Eq. (5) , [16 - 17] 籽 I = ( x I - x s ) 2 + ( y I - y s ) 2 + ( z I - z s ) 2 (1) Expand 籽 I at P e = ( x, y, z) based on Taylor = ·I 籽 = (4) (xI -xs )( x - x ) +(yI -ys )( y - y ) +(zI -zs )( z - z ) ·I ·s ·I ·s (xI -xs )2 +(yI -ys )2 +(zI -zs )2 ·I ·s (11) Expand 籽 at P e = ( x,y,z) and V e = ( x,y,z ) ·I · · · ·I to Taylor expansion. Thus 籽 can be recorded as — 18 — Guochao Fan et al. / Modified Observation Model in Tightly鄄Coupled INS / GPS Integration 籽 = r + ( E x ( x I - x) + E y ( y I - y) + E z ( z I - z) ) + ·I · bx = ( b x ( x I - x) + b y ( y - y) + b z ( z - z ) ) (12) · · ·I Ex = · ·I · ( xI - xs ) 鄣籽 ·I ·s = (x - x ) / 鄣x I | x I = x ·I ( xI - xs ) 2 + ( yI - ys ) 2 + ( zI - zs ) 2 ( x - xs ) 2 + ( yI - ys ) 2 + ( zI - zs ) 2 ·I ·s ·I ·s [ ( x - xs ) ( x - x ) + ( yI - ys ) ( y - y ) + ·I ·s ( z I - z s ) ( z - z ) ]· ( x - xs ) / by = ( xI - xs ) 2 + ( yI - ys ) 2 + ( zI - zs ) 2 (13) According to Eq. (6) , Eq. (13) can be writ鄄 Ex = bz = (x - x ) ·I ·s - [ ( x - xs ) ( x - x ) + 籽 Ix ·I ·s ·I s ·s Ey = I s ·I ·I (15) According to Eq. (7) , Eq. (15) can be writ鄄 ten as ·I ·s (y - y ) ·I ·s ·I Ey = - [(xI - xs )(x - x ) + (y - ys )(y 籽Iy y ) + (zI - zs )(z - z )] ·I Ez = ·s y - ys (籽Iy ) 3 (16) 鄣籽 ·I ·s =(z - z ) / 鄣z I | z I = z ·I ·I ·I ·s ( 21 ) , = 籽I v Iz - v z ] = [ 啄v x T v Iy - v y 啄v y 啄v z ] T (22) According to Eq. (9) and Eq. (22) , Eq. (12) can be rewritten as 籽 I = r + [ Ex · · where Ey é 啄x ù ê ú E z ] ê 啄y ú + [ b x êë 啄z úû · by é啄 xù ê ú bz ] ê 啄 · yú êë · úû 啄z (23) (x -xs )(x - x s ) +(y -ys )(y - y s ) +(z -zs )( z - z s ) · · · · · · (x -xs ) 2 +(y -ys ) 2 +(z -zs ) 2 It is assumed that there are K visible satel鄄 lites in GPS constellation. The position and veloc鄄 ity of the k鄄th ( k = 1,…,K) satellite are P sk = ( x sk , y sk ,z sk ) and V sk = ( x k ,y k ,z k ) . Replacing P s = ( x s , ·s ·s ·s y s ,z s ) and V s = ( x ,y ,z ) with P sk = ( x sk ,y sk ,z sk ) ·s ·s ·s Eq. (21) servation equations (z - z ) ·I ·s Ez = - [ ( xI - xs ) ( x - x ) + 籽 Iz ( yI - ys ) ( y - y ) + ( z - zs ) ( z - z ) ] - ( zI - zs ) 籽I 1郾 3 摇 Pseudorange and pseudorange rate ob鄄 ( ( xI - xs ) 2 + ( yI - ys ) 2 + ( z - zs ) 2 ) 3 ·I = 啄V e = V Ie - V e = [ v Ix - v x r= (17) According to Eq. (8) , Eq. (17) can be writ鄄 ten as I ( xI - xs ) 2 + ( yI - ys ) 2 + ( zI - zs ) 2 . · ( xI - xs ) 2 + ( yI - ys ) 2 + ( z - zs ) 2 ·I ·s ·I ·s I [ ( x - xs ) ( x - x ) + ( yI - ys ) ( y - y ) + ·I ·s ( z - z s ) ( z - z ) ]· ( z - zs ) / (20) 鄣籽 = ·I 鄣 z | ·z I = ·z ( 19 ) as 啄V e ( ( xI - xs ) 2 + ( y - ys ) 2 + ( zI - zs ) 2 ) 3 ·s Eq. ( yI - ys ) 籽I The difference between V Ie and V e is recorded ( xI - xs ) 2 + ( y - ys ) 2 + ( zI - zs ) 2 ·I ·s ·I ·s [ ( xI - xs ) ( x - x ) + ( y - ys ) ( y - y ) + ·I ·s ( z I - z s ) ( z - z ) ]· ( y - ys ) / In (14) 鄣籽 ·I ·s = (y - y ) / 鄣y I | y I = y = ( xI - xs ) 2 + ( yI - ys ) 2 + ( zI - zs ) 2 ·s ( xI - xs ) 籽I (19) ( zI - zs ) x - xs (y - y )(y - y ) + (z - z )(z - z )] I 3 ( 籽x ) I 鄣籽 = ·I 鄣y | ·y I = ·y = I ( yI - ys ) ( ( x - xs ) 2 + ( yI - ys ) 2 + ( zI - zs ) 2 ) 3 ten as 鄣籽 I = ·I 鄣x | ·x I = ·x ·s ·s and V sk = ( x k ,y k , z k ) in Eq. (10) and Eq. (23 ) , ·s ·s ·s the linearized range and rang rate equations be鄄 z - zs 籽 Iz (18) tween the k鄄th satellite and the user can be ob鄄 tained, written as follows — 19 — Journal of Beijing Institute of Technology, 2017, Vol. 26, No. 1 籽 = r k + [ a xk a yk I k 籽 = r k + [ E xk ·I k E yk · éê 啄x ùú a zk ] ê 啄y ú êë 啄z úû (24) éê 啄 x ùú E zk ] ê 啄 y ú + êê úú ë 啄z û b yk The conversion equations between P Ie = ( x I , y I ,z I ) and P In = ( L I ,姿 I ,h I ) is [1] ìïx I = ( R NI + h I ) cos L I cos 姿 I ï íy I = ( R NI + h I ) cos L I sin 姿 I ï îz I = ( R NI (1 - f) 2 + h I ) sin L I · [ b xk (26) and Eq. (27) . éê 啄 x ùú ê· ú b zk ] ê 啄 y ú êê · úú ë 啄z û (25) And the conversion equations between P e = ( x,y, z) and P n = ( L,姿,h) is ìïx = ( R N + h) cos Lcos 姿 ï íy = ( R N + h) cos Lsin 姿 ï îz = ( R N (1 - f) 2 + h) sin L Thus the simultaneous range and rang rate e鄄 quations between all of K satellites and the user are [ r1 籽I … rk = [ 籽1 I … 籽 Ik … é a x1 ê ê左 T ê r K ] ,A = ê a xk ê左 êê ë a xK … (26) T 籽 IK ] , a y1 左 a yk 左 a yK = r a z1 ù ú 左ú ú a zk ú 左ú ú a zK úû é 啄x ù é啄 xù ê ú ê ú 籽 = r + E ê 啄y ú + B ê 啄 · yú ê 啄z ú ê ·ú ë û ë啄 z û (27) · ·I éê 籽 1 ùú ê左ú ê ·I ú · ·I where 籽 = ê 籽 k ú,r = ê左ú ê ú êë · 籽 IK úû éê b x1 b y1 b z1 ùú ê左 左 左ú ê ú B = ê b xk b yk b zk ú. ê左 左 左ú êê ú ë b xK b yK b zK úû Since navigation where R NI and R N are prime vertical radius calcu鄄 éê · r 1 ùú éê E x1 ê左ú ê左 ê· ú ê ê r k ú,E = ê E xk ê ú ê左 ê左ú êê êë · ë E xK r K úû E y1 左 E yk 左 E yK the ellipsoid rate. R NI and R N are calculated by the equation R N = R e ( 1 + f sin2 L ) , where R e is the earth equatorial radius. éê L I ê 姿I êê ë hI Putting · ·I E z1 ù ú 左ú ú E zk ú, 左ú ú E zK úû 啄y éê (RNI +h +啄h)cos (L +啄L)cos (姿 +啄姿) -(RN +h)cos Lcos 姿 ùú ê (RNI +h +啄h)cos (L +啄L)sin (姿 +啄姿) -(RN +h)cos Lsin 姿 ú êê ú 2 2 ë (R (1 -f) +h +啄h)sin (L +啄L) -(R (1 -f) +h)sin L úû NI N Simplifying Eq. ( 31 ) with cos 啄L 抑 1, cos 啄姿抑1, sin 啄L抑啄L, sin 啄姿抑啄姿, and ignoring higher order terms, the relationship between 啄P e parameters are in the navi鄄 啄z] T and 啄V e = [ 啄 · x 啄v n - L ù é 啄L ù é L I ù é L + 啄L ù ú ê ú ê ú ê ú - 姿 ú = ê 啄姿 ú圯 ê 姿 I ú = ê 姿 + 啄姿 ú (30) úú ê ú êê úú ê ú - h û ë 啄h û ë h I û ë h + 啄h û Eq. (28) , Eq. (29) , and Eq. (30) in鄄 (31) should be transformed to 啄P n = [ 啄L and 啄V n = [ 啄v e 啄P n = P In - P n = éê 啄x ùú éê x I - x ùú to ê 啄y ú = ê y I - y ú, it can be obtained that êë 啄z úû êê z - z úú ë I û éê 啄x ùú éê xI -x ùú ê 啄y ú = ê yI -y ú = êë 啄z úû êê z - z úú ë I û and 啄P n can be written as gation frame in SINS / GPS integration navigation, 啄P e = [ 啄x (29) lated with respect to L I and L respectively. f is éê 啄 x ùú 籽I = r + A ê 啄y ú êê úú ë 啄z û where (28) · 啄y 啄姿 啄 z] · T 啄h] T 啄v u ] T respectively in Eq. where — 20 — é 啄x ù é 啄L ù ê ú ê ú ê 啄y ú = Da ê 啄姿 ú + 啄Da êë 啄z úû êë 啄h úû Da = (32) Guochao Fan et al. / Modified Observation Model in Tightly鄄Coupled INS / GPS Integration éê -(RNI +h)sin Lcos 姿 -(RNI +h)cos Lsin 姿 cos Lcos 姿 ùú ê -(RNI +h)sin Lsin 姿 (RNI +h)cos Lcos 姿 cos Lsin 姿 ú êê ú 2 0 sin L úû ë (RNI (1 -f) +h)cos L éê 啄ve ùú 啄C + C (L ,姿 ) ê 啄vn ú êê ú ë 啄vu úû e n y ,z ) and V In = ( v Ie ,v In ,v Iu ) is ·I ·I I ·I éê v e ùú éê x ùú I e I I ·I ê y ú = C n ( L ,姿 ) ê v n ú êê I úú ê ·I ú ëz û ë vu û · · · they are é ve ù éê x ùú ê ú e · ê vn ú = C ( L,姿) y ê ú n êê úú ê·ú ëzû ë vu û · In Eq. (36) , éê - sin 姿 C (L,姿) = ê cos 姿 êë 0 e n - sin Lcos 姿 - sin Lsin 姿 cos L Replacing L, 姿 with L I , 姿 I in C en ( L, 姿 ) [1] , éê 啄v e ùú éê v ùú éê v e ùú éê v ùú éê v e + 啄v e ùú ê 啄v n ú = ê v In ú - ê v n ú圯 ê v In ú = ê v n + 啄v n ú (38) êê ú ê Iú ê ú ê Iú ê ú ë 啄v u úû êë v u úû êë v u úû êë v u úû êë v u + 啄v u úû I e Putting Eq. (35) , Eq. (36) , and Eq. (38) in鄄 · · · éê 啄 x ùú éê x I - x ùú · · to ê 啄 · y ú = ê y I - y ú, it can be obtained that ê · ú êê · · úú ë 啄 z û ë zI - z û ·I · I e éê ve + 啄ve ùú éê ve ùú e I I e Cn (L ,姿 ) ê vn + 啄vn ú - Cn (L,姿) ê vn ú = êê ú êê úú ë vu + 啄vu úû ë vu û e n (27) , then the range rate equation can be written as æ ö éê 啄L ùú ç ÷ 籽 = r + E ç Da ê 啄姿 ú + 啄Da ÷ + ç ÷ ê 啄h ú ë û è ø æ éê 啄v e ùú ö ç e ÷ e B ç 啄C n + C n ( L,姿) ê 啄v n ú ÷ êê úú ÷ ç è ë 啄v u û ø ·I · (42) Replacing r and r with the pseudorange ( 籽 G ) · and the pseudorange rate vectors ( 籽 ) as: ·G 籽 G = r + D t 啄t + V 籽 圯r = 籽 G - D t 啄t - V 籽 (43) 籽 = r + D f 啄f + V ·籽 圯 r = 籽 - D f 啄f - V ·籽 (44) where [ 籽1 ·G · 籽G … · = [ 籽1 G ·G k 籽 … … T 籽 Gk ·G … T 籽 GK ] , ·G 籽 = 籽 ] , 籽 Gk and ·G 籽 k are the pseu鄄 ·G K dorange and the pseudorange rate between the us鄄 er and the k鄄th satellite. 啄t and 啄f are the clock error and the clock drift of the GPS receiver. D t = D f = [1 … 1 … 1] T . V 籽 and V ·籽 are the observation noise of pseudorange and pseudor鄄 éê v ùú éê ve ùú éê 啄 x ùú éê x - x ùú I e I I e · ·I · ê 啄 y ú = ê y - y ú = Cn (L ,姿 ) ê vn ú - Cn (L,姿) ê vn ú = ê ú êê úú êë · úû êë ·I · úû êë vI úû 啄z z -z ë vu û u · I æ ö éê 啄L ùú ç ÷ 籽 = r + A ç Da ê 啄姿 ú + 啄Da ÷ (41) ç ÷ êë 啄h úû è ø Substituting Eq. (32) and Eq. (39 ) into Eq. ·G 啄V n = V - V n = I e I I C en ( L I ,姿 I ) of Eq. (35) can be obtained. I n e n range equation can be written as (36) cos Lcos 姿 ù ú cos Lsin 姿 ú sin L úû (37) (39) I éê v e ùú 啄C = [ C ( L ,姿 ) - C ( L,姿) ] ê v n ú (40) êê úú ë vu û Substituting Eq. (32) into Eq. (26) , then the e n (35) While between V e = ( x,y,z ) and V n = ( v e ,v n ,v u ) I In Eq. (39) , (33) éê cos Lcos 姿 ùú 啄Da = ( R NI - R N ) ê cos Lsin 姿 ú (34) ê (1 - f) 2 sin L ú ë û ·I The conversion equations between V Ie = ( x , e n ange rate respectively. D t , D f , V 籽 and V ·籽 are the matrices of K 伊 1. Then the pseudorange and the pseudorange rate equations used in SINS / GPS integrated navi鄄 gation system are constructed according to Eq. (41) and Eq. (42) . It should be mentioned that the modeling er鄄 rors in the pseudorange and the pseudorange rate, including the satellite clock error, the iono鄄 — 21 — Journal of Beijing Institute of Technology, 2017, Vol. 26, No. 1 spheric error and tropospheric error [16] , et al. , have been deduced. be rewritten as éê 啄L ùú 啄籽 = ADa ê 啄姿 ú + V 籽 êë 啄h úû Substituting Eq. (43) into Eq. (41) , then the pseudorange equation can be obtained é 啄L ù ê ú 籽 - 籽 - A啄Da = ADa ê 啄姿 ú - D t 啄t - V 籽 (45) ê 啄h ú ë û Substituting Eq. (44 ) to Eq. (42 ) , then the I é 啄v e ù éê 啄L ùú ê ú · 啄籽 = EDa ê 啄姿 ú + BC en ( L,姿) ê 啄v n ú + V ·籽 (48) êê úú ê 啄h ú ë û ë 啄v u û G pseudorange range equation can be obtained ·I ·G 籽 - 籽 - E啄Da - B啄C en = éê 啄v e ùú é 啄L ù ê ú e EDa ê 啄姿 ú + BC n ( L,姿) ê 啄v n ú êê ú êë 啄h úû ë 啄v u úû D f 啄f - V ·籽 MOM and COM (46) ·I G ·G E啄Da - B啄C as 啄籽 , Eq. (45) ,then Eq. (46) can · e n 2摇 Comparison Between MOM and COM 2郾 1摇 Comparison of pseudorange equations in Recording 籽 - 籽 - A啄Da as 啄籽 and 籽 - 籽 I (47) Both pseudorange equations in MOM and COM can be recorded in the form of Eq. ( 47 ) . However, the meanings of symbols for MOM and COM in Eq. (47) are different, as shown in Tab. 2. Tab. 2摇 Different meanings of each symbol in pseudorange equation in MOM and COM Symbol MOM 啄籽 a x1 éê 左 ê A = ê a xk ê左 êë a xK a y1 左 a yk 左 a yK a z1 ùú ú a zk ú 左 ú ú a zK û 左 COM 籽 - 籽 - A啄Da 籽 -籽 a xk = ( x - x sk ) / 籽 Ixk a xk = ( x - x sk ) / r k a zk = ( z - z sk ) / 籽 Izk a zk = ( z - z sk ) / r k I G I a yk = ( y - y sk ) / 籽 Iyk a yk = ( y - y sk ) / r k Replace R NI with R N in Eq. (33) , Da of COM is obtained Da refers to Eq. (33) Da éê 啄L ùú 啄籽 = ADa ê 啄姿 ú + V 籽 êë 啄h úû 摇 摇 In Tab. 2, x, y, z are the user real position coordinates on the three axes in ECEF. x , y , z s k s k s k are the k鄄th satellite position coordinates on the three axes in ECEF. 啄Da referring to Eq. (34) , rk = 籽 Ixk = 籽 Iyk = 籽 Izk = 2郾 2 摇 Comparison of pseudorange rate equa鄄 tions in MOM and COM Both pseudorange rate equations of MOM and COM can be recorded in the form of Eq. ( 48 ) . However, the meanings of symbols in pseudor鄄 ange rate equations for MOM and COM are differ鄄 ent, as seen in Tab. 3. éê 啄v e ùú é 啄L ù ê ú e 啄 籽 = EDa ê 啄姿 ú + BC n ( L,姿) ê 啄v n ú + V ·籽 êê ú êë 啄h úû ë 啄v u úû (x - x ) + (y - y ) + (z - z ) , s k 2 s k 2 s k G 2 · ( x - x sk ) 2 + ( y I - y sk ) 2 + ( z I - z sk ) 2 , ( x I - x sk ) 2 + ( y - y sk ) 2 + ( z I - z sk ) 2 , In Tab. 3, x I ,y I ,z I and x ,y ,z are the user ·I ( x I - x sk ) 2 + ( y I - y sk ) 2 + ( z - z sk ) 2 . As shown in Tab. 2, the denominators of a xk , a yk and a zk in MOM are 籽 Ixk , 籽 Iyk ,籽 Izk respective鄄 ·I ·I position and velocity calculated by SINS in ECEF. x,y,z is the user爷 s real position in ECEF. x sk ,y sk , ly while those are the same r k in COM. In addi鄄 z sk is the k鄄th satellite position in ECEF. r k , 籽 Ixk , MOM and COM. 籽 Ik = tion, R NI and R N are, respectively, used for Da in 籽 Iyk ,籽 Izk in Tab. 3 are the same as those in Tab. 2. — 22 — ( x I - x sk ) 2 + ( y I - y sk ) 2 + ( z I - z sk ) 2 . 啄C en re鄄 Guochao Fan et al. / Modified Observation Model in Tightly鄄Coupled INS / GPS Integration fers to Eq. (40) . The meanings of Da and 啄Da in In E xk of COM, parameters about the real us鄄 er position and velocity are used, including r k , x, the pseudorange rate equation are the same as those in the pseudorange equations of MOM. y,z and x,y,z . In E xk of MOM, parameters about · · · As shown in Tab. 3, the values of E xk , E yk the user position and velocity calculated by SINS and E zk in matrix E of MOM are different from are used, including 籽 Ixk ,x I ,y I ,z I and x ,y ,z , es鄄 ·I those in the matrix E of COM. Taken E xk as an The values of b xk , b yk and b zk in matrix B of E xk of MOM is MOM are different from those in matrix B of E xk = ( x - x k ) / 籽 Ixk - [ ( x - x sk ) ( x - x k ) + ·s ·I ·I ·s ·I ·s E xk of COM is ·s k · s k ·s · are 籽 Ik in MOM while those are r k in COM. · Moreover, compared with 啄籽 and 啄籽 of COM, ·s k ( y - y sk ) ( y - y k ) + ( z - z sk ) ( z - z k ) ] · used in COM. The denominators of b xk ,b yk and b zk x - x sk ( 籽 Ixk ) 3 E xk = ( x - x ) / r k - [ ( x - x ) ( x - x ) + · COM. x I ,y I ,z I are used in MOM while x,y,z are ·s ( y I - y sk ) ( y - y k ) + ( z I - z sk ) ( z - z k ) ] ·s ·I pecially those items marked in red. example, ·I ·I · 啄籽 and 啄籽 of MOM have additional terms of A啄Da and - E啄Da - B啄C en respectively. x -x ( rk ) s k 3 Tab. 3摇 Different meanings of each symbol in pseudorange range equations of MOM and COM Symbol MOM ·I 籽 - ·G 籽 - E啄Da - B啄C en 啄· 籽 ·I E xk = ( x éê E x1 左 ê E = ê E xk ê左 êë E xK éê b x1 左 ê B = ê b xk ê左 êë b xK E y1 左 E yk 左 E yK b y1 左 b yk 左 b yK COM E z1 ùú 左 ú E zk ú 左 ú ú E zK û b z1 ùú 左 ú b zk ú 左 ú ú b zK û - ·s xk) / 籽 Ixk - [(x ·I 籽 - ·G 籽 - x sk ) ( ·I x - ·s xk) E xk = ( · x - ·s x k ) / r k - [ ( x - x sk ) ( · x - ·s xk) + + ( y I - y sk ) ( ·I y - ·s x k ) + ( z I - z sk ) ( ·I z - ·s xk)] x - x sk ( 籽 Ixk ) 3 ( y - y sk ) ( · y - ·s x k ) + ( z - z sk ) ( · z - ·s xk)] x - x sk ( rk ) 3 E yk = ( ·I y - ·s x k ) / 籽 Iyk - [ ( x I - x sk ) ( ·I x - ·s xk) + E yk = ( · y - ·s x k ) / r k - [ ( x - x sk ) ( · x - ·s xk) + ( y - y sk ) ( ·I y - ·s x k ) + ( z I - z sk ) ( ·I z - ·s xk)] ( y - y sk ) ( · y - ·s x k ) + ( z - z sk ) ( · z - ·s xk)] y - y sk ( 籽 Iyk ) 3 y - y sk ( rk ) 3 E zk = ( ·I z - ·s x k ) / 籽 Izk - [ ( x I - x sk ) ( ·I x - ·s xk) + E zk = ( · z - ·s x k ) / r k - [ ( x - x sk ) ( · x - ·s xk) + ( y I - y sk ) ( ·I y - ·s x k ) + ( z - z sk ) ( ·I z - ·s xk)] ( y - y sk ) ( · y - ·s x k ) + ( z - z sk ) ( · z - ·s xk)] b xk = (x I I - x sk ) 籽 Ik z - z sk ( 籽 Izk ) 3 b xk = (x b yk = (y - y sk ) 籽 Ik b yk = ( y - y sk ) rk b zk = ( z I - z sk ) b zk = ( z - z sk ) rk 籽 Ik ( rk ) 3 rk 于The second part / module is the SINS simu鄄 3摇 Simulation To compare the performance of MOM and COM, some simulation experiments are conduc鄄 ted, which simulate the SINS / GPS integrated nav鄄 igation system. The system consists of four parts, as shown in Fig. 2. 淤The first part / module is the trajectory sim鄄 ulator, which generates the ideal trajectory ac鄄 cording to vehicle dynamic models. - x sk ) z - z sk lator. In this part, the ideal output of gyroscopes ( the attitude rate) , and accelerometers ( the spe鄄 cific force) are calculated according to the ideal trajectory. The noise of gyroscopes and acceler鄄 ometers is generated with noise models, such as constant, Marcov and GWN models. Adding these noises to the ideal output, the real output of gyro鄄 scopes and accelerometers are obtained. Input鄄 ting gyroscope and accelerometer data to the — 23 — Journal of Beijing Institute of Technology, 2017, Vol. 26, No. 1 Fig. 2摇 Structure of SINS / GPS integrated navigation system SINS navigation calculator, then the SINS naviga鄄 troposphere delay, the satellite clock error and 盂The third part / module is the GPS simula鄄 rors are disregarded as unknown parameters. In tion parameters are resolved. the relativistic effect error. The receiver clock er鄄 tor. The range between the vehicle and a satellite this mathematical simulation, the error models the satellite orbit is calculated according to the the models used for the error elimination before cluding ionosphere delay, troposphere delay, the dorange rate) in observation equations almost on鄄 is simulated with the vehicle ideal trajectory and used for the construction of the pseudorange and satellite ephemeris. The errors of the range, in鄄 filtering are the same. So the pseudorange ( pseu鄄 relativistic effect error, satellite clock error, and ly contains the range ( the range rate) , the receiv鄄 clock drift ) , are simulated by model errors measurement noise of the pseudorange ( pseudor鄄 receiver clock errors ( the clock error and the [16] . By adding these errors to the range, the pseudor鄄 ange is obtained. The pseudorange rate is the first er clock error ( clock drift ) and the receiver ange rate) . After the filtering process, the corrections of derivation of pseudorange. The receiver measure鄄 SINS navigation parameters are obtained. Consid鄄 ange rate are simulated by WGN. By adding the rate of GNSS are only associated with the position the pseudorange rate, the real observation data of tude error 啄A is inaccurate. Therefore, only posi鄄 榆The last part / module is the filter. The nav鄄 The state equation and observation equation ment noises of the pseudorange and the pseudor鄄 ering that the pseudorange and the pseudorange receiver measurement noise, the pseudorange and and the velocity, the filter estimation of SINS atti鄄 GPS, are finally generated. tion and velocity are concerned in this paper. igation parameters of SINS ( P ,V ,A ) , the nav鄄 of the filter are constructed as follows: servation data of GPS ( the pseudoranges 籽 G and vector X for SINS / GPS integrated navigation sys鄄 P Ge and V Ge are used to replace the user real posi鄄 bout SINS, X SINS and the state vector about GPS, It should be mentioned that the model errors state equation of SINS / GPS consists of the SINS I n I n I n igation parameters of GNSS ( P ,V ) and the ob鄄 G e G e Under the tightly coupled mode, the state the pseudorange rates 籽 ) are sent into the filter. tem is composed of two parts, the state vector a鄄 tion P e and V e in the construction of filter. X GPS . Corresponding to the two parts of X, the of pesudorange have been eliminated before filte鄄 and the GPS state equations. ·G ring, which include the ionosphere delay, the — 24 — · X = FX + G棕 (49) Guochao Fan et al. / Modified Observation Model in Tightly鄄Coupled INS / GPS Integration · éê X SINS ùú O ù éê X SINS ùú éê F SINS ú, where X = ê · ,X = ,F = êë X úû êë O F GPS úû êë X GPS úúû GPS O ù 棕SINS ù é G SINS úú, 棕 = éêê ú. X SINS consists of G = êê G û ë O ë 棕 úû · GPS GPS H v are constructed according to Eq. (47) and Eq. (48) . Z 籽 and Z ·籽 are 啄籽 of Eq. (47) and 啄籽 of Eq. · (48) respectively. V 籽 and V ·籽 are the same with those of Eq. (47) and Eq. (48) . In simulation, the parameters for MOM and the SINS navigation parameters corrections, in鄄 COM should be evaluated according to Tab. 2 and the velocity correction 啄V = ( 啄V e , 啄V n , 啄V u ) and velocity V e = ( x,y,z ) used in the construction of cluding the position correction 啄P = ( 啄L,啄姿,啄h) , the misalignment 椎 = ( 准 e ,准 n ,准 u ) , and the sen鄄 sor errors ( gyroscope and accelerometer errors) . X GPS includes two error states, the clock error 啄t n and the clock drift 啄f n . F SINS is constructed ac鄄 cording to the navigation parameter error equa鄄 tions of the attitude, the position and the veloci鄄 ty. F GPS is constructed according to differential e鄄 quations between 啄t and 啄f. G SINS and G GPS are the process noises driving matrices of SINS and GPS respectively. 棕SINS and 棕GPS are the process noises of SINS [1] . The observation equation for SINS / GPS inte鄄 grated navigation system can be recorded as Z = HX + V (50) é H籽 ù é Z籽 ù é V籽 ù úú, V = êê úú. H 籽 and where H = êê úú, Z = êê ë Hv û ë Z ·籽 û ë V ·籽 û Tab. 3. The real user position P e = ( x,y,z) and · · · the observation model are unknown. They should be replaced with the user position and velocity calculated by GPS, P Ge = ( x G , y G , z G ) and V Ge = ( x ,y ,z ) . ·G ·G ·G 3郾 1摇 Simulation scene To compare the performances of MOM and COM, a Monte Carlo simulation is designed in the same scene. The vehicle trajectory was described in Tab. 4 line by line, which imitated the trajecto鄄 ry of an aircraft. This trajectory was designed to cover different motion phases of an aircraft as much real as possible. The simulation began at 2015鄄3鄄31 0:00:00 of UTC. Simulation step was 1 s. The duration of the simulation was 1 200 s. 1 000 times simulation experiments are completed to ensure the validity of experimental results. Tab. 4摇 Simulation scene Motion Details Duration / s 1. Initial state North latitude 39毅,east longitude 116毅, height 6 000 m, velocity 200 m / s, heading north 2. Climbing Height change 1 800 m,climbing velocity 5 m / s 360 3. Straight Acceleration - 0郾 5 m / s2 60 4. Turning Steering angle 60毅 5. Declining Height change 300 m,declining velocity 5 m / s 60 6. Straight Acceleration 2郾 5 m / s2 60 7. Straight Acceleration 0 m / s2 About 60 More than 600 摇 摇 The sensor errors of SINS were added ac鄄 Tab. 5摇 Error of SINS cording to Tab. 5, which are typical values used Sensor craft. The noise added to the pseudoranges and Gyroscope in SINS / GNSS navigation simulation of an air鄄 the pseudorange rates were 10 times and 0郾 1 times of GWN for GPS. The WGN was randomly generated. Thus each start of the simulation environment could — 25 — Accelerometer Error type Value Constant 0郾 1( 毅) / h Marcov 0郾 05( 毅) / h GWN 0郾 01( 毅) / h Constant 1 伊 10 - 4 g Marcov 0郾 5 伊 10 - 4 g GWN 0郾 1 伊 10 - 4 g Journal of Beijing Institute of Technology, 2017, Vol. 26, No. 1 bring about different scene data, including the divergent obviously over time. There is no signifi鄄 parameters, the pseudoranges and the pseudor鄄 COM. SINS navigation parameters, the GNSS navigation ange rates. To exclude WGN interference, both cant divergence of the vertical velocity error of filters using MOM and COM calculated the user position and velocity with the same scene data. The simulation results are recorded in section 3郾 2. 3郾 2摇 Simulation result To display the experimental results clearly, one of the simulation results was arbitrarily se鄄 lected. The position and velocity errors of SINS / GPS integrated navigation using MOM and COM of this selected experiment can be seen in Fig. 3 and Fig. 4 respectively. Fig. 4摇 Velocity errors of MOM and COM In simulations, the same Kalman filter as well as the same state model are used, and the simulations for COM and MOM are completed simultaneously under the same scene, which means factors that affect simulation results are all excluded except the observation model differ鄄 ence. Thus it can be considered that the position ( velocity) difference is caused by differences be鄄 tween COM and MOM. Combined with theoretical analysis in section 2, the source of position and · velocity difference is mainly 啄籽 and 啄籽 in Tab. 2. Compared with COM, there is a additional item of Fig. 3摇 Position errors of MOM and COM The position ( velocity) error is the difference between the position ( velocity ) estimation of SINS / GPS integrated navigation system and the real position ( velocity) . Fig. 3 shows the position - A啄Da in 啄籽 and a additional item of - E啄Da · B啄C en in 啄籽 for MOM. 啄Da is the error compensa鄄 tion for the difference between R NI and R N . 啄C en is the error compensation for the difference between errors of three axes in the navigation frame. The C en ( L I , 姿 I ) and C en ( L, 姿 ) . ( R NI - R N ) and while that of COM in each axis becomes divergent difference between the real user position and the position error of MOM in each axis is convergent C en ( L I ,姿 I ) - C en ( L, 姿 ) are determined by the over time. The vertical position error of COM be鄄 user position calculated by SINS. Because the po鄄 the north position errors of COM begin to diverge ( R NI - R N ) and C en ( L I , 姿 I ) - C en ( L, 姿) will in鄄 earliest. Fig. 4 shows the velocity errors of three using MOM can keep convergence for the error MOM in each axis is convergent. However, the overtime. gins to diverge at about 400 s. And the east and sition error of SINS accumulates over time, both at about 600 s. The vertical position error diverges crease. Therefore, the position and velocity error axes in the navigation frame. The velocity error of compensations while those using COM divergent velocity errors of COM in east and north axes are Fig. 3 and Fig. 4 demonstrate that the linear鄄 — 26 — Guochao Fan et al. / Modified Observation Model in Tightly鄄Coupled INS / GPS Integration ization model error of COM affects the conver鄄 times of simulation. Both position ( velocity) er鄄 than COM on the linear approximation of pseuor鄄 ue, the standard deviation ( SD ) value and the gence of the filter. Thus MOM is more accurate ange and pseusorange rate equations. rors using MOM and COM contain the mean val鄄 root mean square ( RMS) value. As seen in Tab. It should be noted that the vertical velocity 6, the position errors of east and north axes are the vertical position error shown in Fig. 3 because COM. The position errors of vertical axis are less error shown in Fig. 4 seems to be conflicted with there is an integral relationship between position and speed, i. e. , the vertical position error of COM should be approximate to that of MOM be鄄 cause the vertical velocity errors of COM and MOM have small differences as shown in Fig. 4. However, in this paper, the observation equations of position and velocity are independent, which means the observed result for position is unrelat鄄 ed to that for velocity. Thus, the position ( veloc鄄 ity) error is only affected by the accuracy of the observation model. And Fig. 3 and Fig. 4 are irrel鄄 evant. Tab. 6 shows the statistical data for 1 000 less than 0郾 5 m for MOM and less than 1郾 5 m for than 2 m for MOM while they are less than 4 m for COM. Meanwhile, the velocity errors of each axis for MOM and COM are less than 0郾 05 m / s. How鄄 ever, the velocity errors of each axis for MOM is obviously less than that for COM except of verti鄄 cal axis. The vertical velocity error of MOM and COM is nearly the same. The statistical data in Tab. 6 is highly consistent with the data shown in Fig. 3 and Fig. 4. In addition, it should be noted that the position accuracy of vertical axis is lower than that of east and north axes because the posi鄄 tion accuracy of horizontal direction is better than that of vertical direction using GPS [16] . Tab. 6摇 Statistical data of position and velocity errors using MOM and COM Axis East North Vertical Method Position / m Mean SD RMS Mean MOM - 0郾 045 3 0郾 287 5 0郾 288 6 COM - 0郾 362 9 0郾 771 7 MOM 0郾 045 5 COM Velocity / ( m·s - 1 ) SD RMS - 0郾 011 9 0郾 006 8 0郾 012 4 0郾 852 4 0郾 013 5 0郾 011 2 0郾 017 5 0郾 173 2 0郾 180 5 0郾 002 3 0郾 005 5 0郾 005 9 0郾 651 4 0郾 931 0 1郾 136 0 0郾 009 0 0郾 011 6 0郾 014 7 MOM - 1郾 770 5 0郾 247 8 1郾 787 7 0郾 002 3 0郾 007 3 0郾 007 6 COM - 2郾 817 6 1郾 310 0 3郾 128 4 0郾 001 6 0郾 008 3 0郾 008 2 the position ( velocity) is convergent using MOM. 4摇 Conclusions In this paper, the MOM for SINS / GPS inte鄄 grated navigation is derived with the linearization Moreover, the position ( velocity ) accuracy of MOM is pretty good. It is verified that the improvement of filter al鄄 of the range and range rate equations between the gorithm is a good way to avoid filtering diver鄄 satellites of GPS. The linearization accuracy of GPS integration navigation. However, this im鄄 user position calculated by SINS and the visible MOM is much higher than that of COM which can be verified by the results of Monte Carlo simula鄄 tion experiments conducted in this paper. Ac鄄 cording to the simulation results, position ( veloc鄄 ity) is divergent using COM over time. However, gence and improve navigation accuracy in INS / provement is usually accompanied by the increase of computation cost. With more refined lineariza鄄 tion of the observation model, the filtering diver鄄 gence is avoided and the navigation accuracy is improved without higher computation and more — 27 — Journal of Beijing Institute of Technology, 2017, Vol. 26, No. 1 complex filter in this paper. This demonstrates precision airborne integrated GPS and inertial navi鄄 tering effect with a more accurate linearized ob鄄 Measurement, 2015, 64(12) :1 - 1. gation[ J] . 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