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An algorithm for astro-inertial navigation using CCD star sensors
Aerospace Science and Technology 10 (2006) 449–454 www.elsevier.com/locate/aescte
An algorithm for astro-inertial navigation using CCD star sensors Jamshaid Ali ∗,1 , Zhang Changyun 2 , Fang Jiancheng 3 Beijing University of Aeronautics and Astronautics, Beijing 100083, China Received 25 July 2005; received in revised form 11 January 2006; accepted 18 January 2006 Available online 17 February 2006
Abstract A self-contained suite of astro-inertial navigation system is capable of autonomous mission and is operationally reliable. The typical astronavigation system (ANS) makes use of star-trackers, which are expensive and complex. To make the system cost effective and less complex, the star-tracker is replaced by a charge coupled device (CCD)-based star sensor, rigidly mounted on a strapdown inertial measurement unit (SIMU) of the system. This electro-optical star sensor is compact and easy to use with an ANS that utilizes efficient star identification techniques. This paper designs an algorithm that estimates axes misalignment angles of strapdown inertial navigation system (SINS) that makes stars’ observations utilizing a CCD star sensor. Mathematical modeling of the suggested scheme was carried out and transformations between different frames were exercised. From the image projection geometry, stars’ right ascensions and declinations, relative to the body frame, were estimated. Lastly, from the known stars’ position vectors in mathematical platform and reference frames, axes misalignment matrix representing SINS attitude errors can be estimated employing the derived relationship. © 2006 Elsevier SAS. All rights reserved. Keywords: Strapdown inertial navigation system; Misalignment; Astronavigation; Star sensor
1. Introduction The purpose of the inertial navigation system is to measure the trajectory of the vehicle in space and thereby specify the changes required in the trajectory to bring back the vehicle into the path of the predetermined target. The trajectories are functions of the position and velocity vectors of the vehicle. These vectors are usually observed with reference to some reference frame. Powered flight phase of the vehicle is the most decisive phase during which, with the help of navigation information, the vehicle is placed on a trajectory with flight conditions that are appropriate for the desired target. Therefore, for navigation accuracy and reliability, an autonomous vehicle requires high * Corresponding author. Tel.: +86-10-82330114; fax: +86-10-82338058.
E-mail addresses: [email protected] (J. Ali), [email protected] (J. Fang). 1 Working towards Ph.D., School of Instrumentation Science and Optoelectronics Engineering. 2 Associate Professor, School of Automation Science and Electrical Engineering. Tel.: +86-10-82315809. 3 Professor, School of Instrumentation Science and Optoelectronics Engineering. Tel.: +86-10-82316548. 1270-9638/$ – see front matter © 2006 Elsevier SAS. All rights reserved. doi:10.1016/j.ast.2006.01.004
precision navigation solutions and unfailing data yielding system [6]. With rapid advances in inertial sensor technology and computing power, strapdown inertial navigation system (SINS) has supplanted the conventional gimbaled systems. However, small error sources in the inertial instruments, together with capricious variations in the gravitational field forces, may culminate in a miss distance at the target. The extended duration of the vehicle’s flight and absence of updates from the ground sources lead to a greater probability of errors in the navigation solution. Therefore, for precision guidance, the navigation system has to be augmented with external aid. Global Positioning System (GPS) is the prevailing choice for SINS augmentation but its vulnerability to jamming and degraded accuracy in hostile environments makes it a poor choice [8]. The astronavigation system (ANS), which has long been used in the past, still finds its application in the modern space missions. Therefore, SINS, augmented with ANS to estimate attitude errors or axis misalignment angles, is chosen as the subject of this paper. In short and medium range guided munitions, gyro drift contributes significantly to navigation error. ANS can effectively estimate in-flight gyros’ drift, also the velocity and position er-
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rors contributed by the drift [3]. Unlike the conventional GPS, ANS is not vulnerable to disturbances or jamming. As its intended use is with spacecrafts, there can be no hindrance to its stars’ observations. Thus, ANS is an all-weather and alltime navigation aid. With progress in technology and image processing techniques, ANS has become a camera based navigation system in which a charge coupled device (CCD) array, mounted in strapdown configuration, is used as a star sensor. Current CCD sensors provide an inexpensive means to image sky and extract the required information. These sensors, fixed to the body, can scan a section of the sky and determine the association between the scanned stars and the catalog of reference stars stored onboard using some star identification techniques. Stars’ identification leads to attitude error estimation [1].
(2)
(3)
2. System configuration Stars’ observations are made by video cameras using CCD electro-optical star sensor, mounted on SIMU. The cameras generate a two-dimensional image of a small section of the sky, equal to the field-of-view (FOV) of the star sensor. The image output from the camera goes to the onboard computer (Fig. 1). Computer programs, through some video thresholding schemes, limit the number of stars in a field that can be processed. Stored in the computer memory is a star catalog, with the celestial coordinates of each star in some reference system, which will be used for star field recognition. The recognition is accomplished through some stars identification techniques. After identification, a star’s right ascension and declination with respect to the body frame are computed. Then, from the known star position in the real and reference image, axes misalignment angles are estimated to construct the attitude error matrix [1].
(4)
(5)
(6)
3. Simulation algorithm for SINS misalignments (7) 3.1. Definitions of coordinate frames Inertial navigation system theory needs precise description of the coordinate frames. Definitions of the frames employed in this paper are as follows [1–3,7]:
The x-axis points to the point of intersection of mean Greenwich meridian and equatorial plane; z-axis is perpendicular to mean equatorial plane pointing towards the North Pole; and y-axis is defined by right-hand rule. The geocentric inertial frame (i-frame) has its origin at the center of the earth and is non-rotating with respect to the fixed stars. Its x-axis is in the equatorial plane and z-axis is normal to that plane; and y-axis complements the righthanded system. In this paper, the x-axes of e- and i-frames are so arranged that they are coincident at the navigation starting time, t = 0. After the launch of the vehicle, for time t > 0, the angle (α) between them varies in accordance with α = ωie t, where ωie is the earth’s rate. The launch-inertial frame (il-frame) has its origin at launch point. Its z-axis is vertical upward and normal to the reference ellipsoid; y-axis is horizontal and lies in nominal launch plane; and x-axis is horizontal and perpendicular to nominal launch plane, rightward. The body frame (b-frame) has its origin at the center of mass of the vehicle. Its x-axis points along longitudinal axis of the vehicle; z-axis is perpendicular to the longitudinal plane of symmetry; and y-axis complements the right-handed system. Here, the longitudinal plane of symmetry is closest to the nominal plane of launch. That is, in a normal situation, the xy planes of b-frame and il-frame are very near to each other. Observed star frame (j -frame) is defined for the j th observed star that has been identified as guide star in the reference star catalog. Fig. 2 depicts the relationship between i- and j -frames. The angles Θj and Γj represent j th star’s right ascension and declination, respectively. Local level or geographic frame (g-frame) has its origin at the location of navigation system of vehicle and its axes are aligned with the east, north and upward directions. Upward is defined to be normal to the reference ellipsoid. Star sensor frame (s-frame) has its origin at the center of focal plane of the star sensor with z-axis normal to the focal plane. Its x and y axes complement the right-handed system and lie in the focal plane.
3.2. Simulation data
(1) The earth-centered earth fixed frame (e-frame) is used for defining position. It has its origin at the center of the earth.
Astro-inertial navigation technique aims at estimating the attitude error matrix in the midcourse trajectory of a ballistic missile. Therefore, current navigation information from
Fig. 1. SINS/ANS configuration.
Fig. 2. Definition of frames.
J. Ali et al. / Aerospace Science and Technology 10 (2006) 449–454
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SINS simulation is used as an input to the proposed algorithm. This navigation information includes time elapsed after launch (tm ), roll (γm ), yaw (ψm ), pitch (θm ), longitude (λm ), il ), longitude of latitude (ϕm ), estimated attitude matrix (Cb,m launch point (λL ), star sensor to body frame orientation (Csb ) and inertial to launch inertial frame orientation (Ciil ). 3.3. Transformation between b- and i-frames Transformation between body and inertial frames is effected through navigation information provided by the SINS as g T il T il i Cb,m Cg C b , = Cei Ce (1) where Cei is computed from the earth’s rate and time inforg mation, Ce is obtained through three successive rotations of e-frame by (λm − λA + 90◦ ) and (90◦ − ϕm ) about z and y axes respectively, Cgil through a rotation of g-frame by launch plane azimuth about z axis, and Cbil through navigation algorithm from gyros angular rates. 3.4. Coordinates of a lead star The unit position vector along z-axis of s-frame in inertial frame is obtained as i r¯zs = Cbi Csb [ 0 0 1 ]T .
(2)
The stars’ coordinates generation for the proposed algorithm necessitates finding coordinates of the star named as lead star whose frame definition is the same as that of j -frame. To discriminate this from other j stars, ‘∗ ’ subscript is used in the transformation matrix. At this instant, it is supposed that this star lies nearest to the center of CCD array. As a result, the unit position vector for the lead star along z-axis of s-frame is expressed as i r¯zs
= C∗i [ 0
0
1] . T
(3)
If Θm and Γm represent orientation of the star sensor’s zaxis at time tm , transformation matrix Ci∗ is achieved by two successive rotations of i-frame by angles Θm and (90◦ − Γm ) about z and y axes respectively, i.e. ⎡ ⎤ sin Γm cos Θm sin Γm sin Θm − cos Γm ⎦. Ci∗ = ⎣ − sin Θm (4) cos Θm 0 cos Γm cos Θm
cos Γm sin Θm
sin Γm
Now, the unknown quantities Θm and Γm can be estimated by equating Eqs. (2) and (3). 3.5. Generation of stars’ coordinates In this step, a block of stars’ coordinates is generated in terms of right ascension and declination of stars using a Matlab function ‘rand’ with its center around Θ0 ≈ Θm [deg], and Γ0 ≈ Γm [deg]. The stars’ coordinates generated in this way are shown in Fig. 3.
Fig. 3. Stars positions in i-frame.
3.6. Stars’ position in reference image The ANS, based on stars’ identification concepts, uses a CCD star sensor to gather imagery of the stars as it flies en route to the target and processes these data with the reference star catalog stored in the onboard computer to determine the attitude. The star catalogs and astronomical almanacs express a star’s position in the inertial frame in terms of its right ascension Θj and declination Γj , whereas in real image from CCD star sensor, it is expressed in terms of pixels along x and y axes of the b-frame. To perform a star’s identification, its position in inertial frame should be transformed into reference image in terms of pixels along x and y axes. The relationship between inertial and reference image frames is derived through two successive rotations of i-frame by angles (Θ0 +180◦ ) and (90◦ −Γ0 ) about z and y axes, respectively, i.e. ⎡
− sin Γ0 cos Θ0 r Ci = ⎣ sin Γ0 cos Γ0 cos Θ0
− sin Γ0 sin Θ0 − cos Θ0 cos Γ0 sin Θ0
⎤ cos Γ0 0 ⎦,
(5)
sin Γ0
where (Θ0 , Γ0 ) is the chosen reference point corresponding to the center of the reference image. Positions of all other stars in reference image are determined by their right ascension and declination information with reference to the chosen point of reference. Fig. 4 illustrates this concept of reference image preparation. The location of any star in reference image is represented by angle Θr and distance O0 Oj , proportional to the angle between unit vectors in reference image and in i-frame. This angle is estimated as θ0j = cos−1 r¯0i · r¯ji /r¯0i r¯ji .
(6)
Now, Θr , the angle between x-axis of reference image and the position of a star in i-frame are determined. At the outset, j the matrix Cr is determined through two successive rotations
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Fig. 4. Reference image preparation.
Fig. 5. Orientation of s-frame and reference image.
of a reference frame by angles Θr and θ0j about z and y axes, respectively, i.e. ⎡ ⎤ sin θ0j cos Θr sin θ0j sin Θr − cos θ0j j ⎦. (7) Cr = ⎣ − sin Θr cos Θr 0 cos θ0j cos Θr cos θ0j sin Θr sin θ0j The unit position vector r¯j of j th star in the j -frame is expressed as j
r¯j = [ 0 0 1 ]T .
(8)
In addition, we know that j Cji r¯j
j
= Cri Cjr r¯j .
(9)
Pre-multiplication of Eq. (9) with j
Cir
yields
j
Cir Cji r¯j = Cjr r¯j .
(10) j
Substitution of values for Cir , Cji , Cjr and r¯j in Eq. (10) results in simplified form as
⎡ ⎢ ⎣
− sin Γ0 cos Θ0 cos Γj cos Θj − sin Γ0 sin Θ0 cos Γj sin Θj + cos Γ0 sin Γj sin Θ0 cos Γj cos Θj − cos Θ0 cos Γj sin Θj
⎤ ⎥ ⎦
cos Γ0 cos Θ0 cos Γj cos Θj + cos Γ0 sin Θ0 cos Γj sin Θj + sin Γ0 sin Γj ⎡
sin θ0j cos Θr
⎤
⎥ ⎢ = ⎣ sin θ0j sin Θr ⎦ , cos θ0j
Θr = tan−1 (sin Θ0 cos Γj cos Θj − cos Θ0 cos Γj sin Θj )/ (− sin Γ0 cos Θ0 cos Γj cos Θj − sin Γ0 sin Θ0 × cos Γj sin Θj + cos Γ0 sin Γj ) .
(11)
In this algorithm, it is presumed that the CCD array used in star sensor has F ◦ FOV and image plane with equal dimension of x × y pixels. From Fig. 5, it is possible to estimate a star’s coordinates in reference image plane as xr = f tan(θ0,j ) cos Θr,j . (12) yr = f tan(θ0,j ) sin Θr,j Also, the coordinates of guide star indicated by Θm and Γm are transformed into reference image as xrm and yrm (Fig. 6).
Fig. 6. Stars positions in reference image.
3.7. Orientation between xs and xr To carry out stars’ identification reliably, the axes of reference image and the s-frame must be perfectly aligned. This requires estimation of angular misalignment and displacement between the two. From the available information about matrices Csb , Cbi and Cir , matrix Csr is obtained as Csr = Cir Cbi Csb .
(13)
From the Csr matrix, horizontal misalignment between s-frame and reference image is obtained as (14) θsr = tan−1 Csr (1, 2)/Csr (1, 1) . Now, stars’ coordinates in the reference image are transformed into the estimated orientation of s-frame. This transformation is accomplished through displacement xrm , yrm and angular misalignment θsr as
sin θsr xs cos θsr xr − xrm = . (15) ys − sin θsr cos θsr yr − yrm
J. Ali et al. / Aerospace Science and Technology 10 (2006) 449–454
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Fig. 8. Error analysis schema.
For small angle approximations, the relationship between the mathematical platform frame and the reference navigation frame (il-frame) is given as ⎡ ⎤ 1 φz −φy p Cil = ⎣ −φz (22) 1 φx ⎦ = I − Φ, φy
3.8. Stars’ position in s-frame
p
From these angles, the transformation matrix Cjs , and then Cjb is obtained as (18)
3.9. Attitude error matrix j
Presuming that r¯j represents unit position vector of the j th star projected into j -frame, its transformation into i-frame using star’s right ascension and declination yields j
r¯ji = Cji r¯j .
(19)
Star’s right ascension and declination in the s-frame are obtained from the image projection geometry on two-dimensional plane as shown in Fig. 7. The orientation of j -frame with respect to b-frame, estimated from Eq. (18), is employed to compute the unit position vector of j th star in b-frame as j
r¯jb = Cjb r¯j .
(20)
Error analysis schema shown in Fig. 8 portrays the relationship between different frames. Here, c and p refer to computed and mathematical platform frames, respectively. It is obvious that ‘c’ follows ‘il’, ‘p’ follows ‘c’ and ‘p’ follows ‘il’ indirectly. There is misalignment between the ‘il’ and ‘p’ frames that is expressed by small angles φx , φy , φz . Therefore, estimated star’s position vector in the il-frame is expressed as p p r¯ˆilj = r¯j = Cb r¯jb = Cˆ bil r¯jb , p
p
r¯j = Cil r¯jil = (I − Φ)¯rjil .
As depicted in Fig. 7, a star’s position in star sensor frame is estimated by angles Θs and Γs as (16) Θs = tan−1 yjs /xjs , 2 2 xjs + yjs f . (17) Γs = 90◦ − tan−1
= Csb Cjs .
1
where φx , φy and φz represent SINS axis misalignment angles. The unit position vector of j th star in p-frame can also be expressed as
Fig. 7. Image projection geometry.
Cjb
−φx
(21)
where Cb , the estimated attitude matrix from SINS, can also be p p expressed as Cˆ bil = Cil Cbil = Cb .
(23)
If there are n number of stars in the FOV of star sensor identified as guide stars from the reference star catalog, Eqs. (21) and (23) can be equated to get the following relationship: p p p r¯1 r¯2 . . . r¯n (24) = (I − Φ) r¯1il r¯2il . . . r¯nil . In its simplified form, Eq. (24) can be written as Rp = (I − Φ)Ril , p where Rp = [¯r1 . . . r¯nil ].
(25) p r¯2
p r¯3
...
p r¯n ]
and Ril = [¯r1il
r¯2il
r¯3il
From Eq. (25), the desired SINS attitude error matrix is estimated as −1 p . (26) Cil = (I − Φ) = Rp RilT Ril RilT p
To ensure that rows and columns of Cil remain normal and orthogonal, an orthogonalization algorithm is included. If this attitude error matrix is to be orthogonal then the following simple relationship ought to hold [5]: 1 il −1 p p Cp,k Cil,k+1 = (27) + Cil,k , 2 p
p
where Cpil = (Cil )T , k = 0, 1, 2, . . . , and Cil,0 is the initial estimated matrix. This iterative method requires only three to four iterations for the matrix to become orthogonal. This orthogonalization method has the advantage of equal treatment of all the vectors without preference to any one of them. 4. Simulation and results The primary sensor subsystems used in the astro-inertial navigation system described in this paper are SINS and ANS. The SINS used in the simulation outputs information on position, velocity and attitude while ANS processes the sensed star images, in conjunction with SINS, to provide attitude error information. The objective of the proposed algorithm is to
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J. Ali et al. / Aerospace Science and Technology 10 (2006) 449–454
estimate the SINS attitude errors more accurately. The ANS augmentation comes into effect about 40 seconds after lift off of the missile when it attains more than 22-km altitude, because at that altitude, stars are observable and unobstructed [4]. The motorized segment of the missile’s flight is the most crucial stage to position the missile on a trajectory with flight states appropriate for the desired target. Hence, simulation of the presented algorithm has to be carried out during powered flight trajectory of the missile.
⎤ 0.084042 −0.015747 −0.042499 −0.011282 0.176607 0.174620 ⎦ , 0.996373
4.2. Astro-inertial simulation To validate the proposed algorithm, stars’ observation is made only once, that is 54 seconds after the vehicle’s lift off from launch point. Constant and random errors of inertial sensors have been added in the simulation. Only seven stars, shown in the solid block of Fig. 3, are supposed to be identified as guide stars from the reference star catalog. Using the derived relationship, the SINS attitude error is estimated by processing data for these identified stars. 4.3. Results The estimated attitude matrix from SINS at t = 54 seconds is p Cˆ bil = Cb ⎡ 0.9999572775 ⎣ = 0.0055900888
⎤ −0.0057193893 0.0070446581 0.9998276614 0.0177030632 ⎦ .
−0.0069916574 −0.0176617136 0.9998169182 The estimated unit position vectors of the identified stars in iland p-frames are ⎡ 0.036973 −0.054264 0.002306 0.048928 Ril = ⎣ −0.160347 0.006146 −0.018306 0.049280 0.986339 0.998487 0.999807 0.997564
0.983705
0.036707 −0.054436 0.002112 0.048778 Rp = ⎣ −0.159920 0.006104 −0.018331 0.049286 0.986450 0.998488 0.999827 0.997598 ⎤ 0.083870 −0.016012 −0.042831 −0.011318 0.177033 0.174999 ⎦ .
4.1. SINS mechanization In this paper, space-stabilized mechanization is employed for SINS realization. It is conceptually the simplest of all possible system implementations, because Newton’s laws are most austerely stated in an inertial frame of reference. The spacestabilized inertial navigation system outputs navigation parameters in an inertially non-rotating frame. Therefore, this mechanization is free from earth’s rotation and transport rate [2]. In the simulation of SINS, the gravitational acceleration is more accurately expressed in terms of ellipsoidal (or more precisely, spheroidal) earth model. Such a model accounts for the earth’s oblateness, as it includes the second order gravitational harmonic term of the earth’s gravitational field. Initial alignment and inertial sensors’ errors have also been incorporated in the simulation.
0.984142
⎡
0.996423
0.984070
0.983628
The attitude error matrix estimated through Eqs. (26) and (27) is p
Cil ⎡
⎤ 0.9999998984 0.0003307218 −0.0003063222 = ⎣ −0.0003306266 0.9999998970 0.0003106485 ⎦ . 0.0003064249 −0.0003105472 0.9999999048
5. Conclusion The algorithm for estimation of SINS attitude errors using astronavigation augmentation provides a method to estimate and compensate for the gyro drift. Gyro induced drift errors are the only error variables that contribute to attitude errors. Therefore, the gyro errors that the ANS updates are effective in estimating and compensating for the attitude errors, also the consequent position and velocity errors. References [1] J. Ali, Strapdown inertial navigation system aided by celestial image matching technique, MS thesis, Beijing University of Aeronautics and Astronautics, 2001. [2] K.R. Britting, Inertial Navigation Systems Analysis, Wiley-Interscience, New York, 1971. [3] M. Kayton, W.R. Fried, Avionics Navigation Systems, John Wiley and Sons, New York, 1997, pp. 551–596. [4] S. Levine, AGARDograph on advanced astroinertial navigation systems, AGARD-AG-331, in: Aerospace Navigation Systems, 1995, pp. 187–199. [5] R.W. Priester, E.D. Denman, Orthogonalization of a direction cosine matrix by iterative techniques, IEEE Trans. Aerospace Electron. Syst. 8 (1972) 692–694. [6] E.V.B. Stearns, Navigation and Guidance in Space, Prentice-Hall, Englewood Cliffs, NJ, 1963. [7] Y.L. Xiao, Foundation of Flight Dynamics-Modeling of Aerospace Vehicle Motion, Beijing University of Aeronautics and Astronautics Press, Beijing, 2003 (in Chinese). [8] A.V. Zbrutsky, O.I. Nesterenko, N.I. Lykholit, et al., Astroinertial navigation system for aircraft applications, in: Proceedings of the 9th Saint Petersburg International Conference on Integrated Navigation Systems, Russia, 2002, pp. 152–154.
An algorithm for astro-inertial navigation using CCD star sensors Jamshaid Ali ∗,1 , Zhang Changyun 2 , Fang Jiancheng 3 Beijing University of Aeronautics and Astronautics, Beijing 100083, China Received 25 July 2005; received in revised form 11 January 2006; accepted 18 January 2006 Available online 17 February 2006
Abstract A self-contained suite of astro-inertial navigation system is capable of autonomous mission and is operationally reliable. The typical astronavigation system (ANS) makes use of star-trackers, which are expensive and complex. To make the system cost effective and less complex, the star-tracker is replaced by a charge coupled device (CCD)-based star sensor, rigidly mounted on a strapdown inertial measurement unit (SIMU) of the system. This electro-optical star sensor is compact and easy to use with an ANS that utilizes efficient star identification techniques. This paper designs an algorithm that estimates axes misalignment angles of strapdown inertial navigation system (SINS) that makes stars’ observations utilizing a CCD star sensor. Mathematical modeling of the suggested scheme was carried out and transformations between different frames were exercised. From the image projection geometry, stars’ right ascensions and declinations, relative to the body frame, were estimated. Lastly, from the known stars’ position vectors in mathematical platform and reference frames, axes misalignment matrix representing SINS attitude errors can be estimated employing the derived relationship. © 2006 Elsevier SAS. All rights reserved. Keywords: Strapdown inertial navigation system; Misalignment; Astronavigation; Star sensor
1. Introduction The purpose of the inertial navigation system is to measure the trajectory of the vehicle in space and thereby specify the changes required in the trajectory to bring back the vehicle into the path of the predetermined target. The trajectories are functions of the position and velocity vectors of the vehicle. These vectors are usually observed with reference to some reference frame. Powered flight phase of the vehicle is the most decisive phase during which, with the help of navigation information, the vehicle is placed on a trajectory with flight conditions that are appropriate for the desired target. Therefore, for navigation accuracy and reliability, an autonomous vehicle requires high * Corresponding author. Tel.: +86-10-82330114; fax: +86-10-82338058.
E-mail addresses: [email protected] (J. Ali), [email protected] (J. Fang). 1 Working towards Ph.D., School of Instrumentation Science and Optoelectronics Engineering. 2 Associate Professor, School of Automation Science and Electrical Engineering. Tel.: +86-10-82315809. 3 Professor, School of Instrumentation Science and Optoelectronics Engineering. Tel.: +86-10-82316548. 1270-9638/$ – see front matter © 2006 Elsevier SAS. All rights reserved. doi:10.1016/j.ast.2006.01.004
precision navigation solutions and unfailing data yielding system [6]. With rapid advances in inertial sensor technology and computing power, strapdown inertial navigation system (SINS) has supplanted the conventional gimbaled systems. However, small error sources in the inertial instruments, together with capricious variations in the gravitational field forces, may culminate in a miss distance at the target. The extended duration of the vehicle’s flight and absence of updates from the ground sources lead to a greater probability of errors in the navigation solution. Therefore, for precision guidance, the navigation system has to be augmented with external aid. Global Positioning System (GPS) is the prevailing choice for SINS augmentation but its vulnerability to jamming and degraded accuracy in hostile environments makes it a poor choice [8]. The astronavigation system (ANS), which has long been used in the past, still finds its application in the modern space missions. Therefore, SINS, augmented with ANS to estimate attitude errors or axis misalignment angles, is chosen as the subject of this paper. In short and medium range guided munitions, gyro drift contributes significantly to navigation error. ANS can effectively estimate in-flight gyros’ drift, also the velocity and position er-
450
J. Ali et al. / Aerospace Science and Technology 10 (2006) 449–454
rors contributed by the drift [3]. Unlike the conventional GPS, ANS is not vulnerable to disturbances or jamming. As its intended use is with spacecrafts, there can be no hindrance to its stars’ observations. Thus, ANS is an all-weather and alltime navigation aid. With progress in technology and image processing techniques, ANS has become a camera based navigation system in which a charge coupled device (CCD) array, mounted in strapdown configuration, is used as a star sensor. Current CCD sensors provide an inexpensive means to image sky and extract the required information. These sensors, fixed to the body, can scan a section of the sky and determine the association between the scanned stars and the catalog of reference stars stored onboard using some star identification techniques. Stars’ identification leads to attitude error estimation [1].
(2)
(3)
2. System configuration Stars’ observations are made by video cameras using CCD electro-optical star sensor, mounted on SIMU. The cameras generate a two-dimensional image of a small section of the sky, equal to the field-of-view (FOV) of the star sensor. The image output from the camera goes to the onboard computer (Fig. 1). Computer programs, through some video thresholding schemes, limit the number of stars in a field that can be processed. Stored in the computer memory is a star catalog, with the celestial coordinates of each star in some reference system, which will be used for star field recognition. The recognition is accomplished through some stars identification techniques. After identification, a star’s right ascension and declination with respect to the body frame are computed. Then, from the known star position in the real and reference image, axes misalignment angles are estimated to construct the attitude error matrix [1].
(4)
(5)
(6)
3. Simulation algorithm for SINS misalignments (7) 3.1. Definitions of coordinate frames Inertial navigation system theory needs precise description of the coordinate frames. Definitions of the frames employed in this paper are as follows [1–3,7]:
The x-axis points to the point of intersection of mean Greenwich meridian and equatorial plane; z-axis is perpendicular to mean equatorial plane pointing towards the North Pole; and y-axis is defined by right-hand rule. The geocentric inertial frame (i-frame) has its origin at the center of the earth and is non-rotating with respect to the fixed stars. Its x-axis is in the equatorial plane and z-axis is normal to that plane; and y-axis complements the righthanded system. In this paper, the x-axes of e- and i-frames are so arranged that they are coincident at the navigation starting time, t = 0. After the launch of the vehicle, for time t > 0, the angle (α) between them varies in accordance with α = ωie t, where ωie is the earth’s rate. The launch-inertial frame (il-frame) has its origin at launch point. Its z-axis is vertical upward and normal to the reference ellipsoid; y-axis is horizontal and lies in nominal launch plane; and x-axis is horizontal and perpendicular to nominal launch plane, rightward. The body frame (b-frame) has its origin at the center of mass of the vehicle. Its x-axis points along longitudinal axis of the vehicle; z-axis is perpendicular to the longitudinal plane of symmetry; and y-axis complements the right-handed system. Here, the longitudinal plane of symmetry is closest to the nominal plane of launch. That is, in a normal situation, the xy planes of b-frame and il-frame are very near to each other. Observed star frame (j -frame) is defined for the j th observed star that has been identified as guide star in the reference star catalog. Fig. 2 depicts the relationship between i- and j -frames. The angles Θj and Γj represent j th star’s right ascension and declination, respectively. Local level or geographic frame (g-frame) has its origin at the location of navigation system of vehicle and its axes are aligned with the east, north and upward directions. Upward is defined to be normal to the reference ellipsoid. Star sensor frame (s-frame) has its origin at the center of focal plane of the star sensor with z-axis normal to the focal plane. Its x and y axes complement the right-handed system and lie in the focal plane.
3.2. Simulation data
(1) The earth-centered earth fixed frame (e-frame) is used for defining position. It has its origin at the center of the earth.
Astro-inertial navigation technique aims at estimating the attitude error matrix in the midcourse trajectory of a ballistic missile. Therefore, current navigation information from
Fig. 1. SINS/ANS configuration.
Fig. 2. Definition of frames.
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SINS simulation is used as an input to the proposed algorithm. This navigation information includes time elapsed after launch (tm ), roll (γm ), yaw (ψm ), pitch (θm ), longitude (λm ), il ), longitude of latitude (ϕm ), estimated attitude matrix (Cb,m launch point (λL ), star sensor to body frame orientation (Csb ) and inertial to launch inertial frame orientation (Ciil ). 3.3. Transformation between b- and i-frames Transformation between body and inertial frames is effected through navigation information provided by the SINS as g T il T il i Cb,m Cg C b , = Cei Ce (1) where Cei is computed from the earth’s rate and time inforg mation, Ce is obtained through three successive rotations of e-frame by (λm − λA + 90◦ ) and (90◦ − ϕm ) about z and y axes respectively, Cgil through a rotation of g-frame by launch plane azimuth about z axis, and Cbil through navigation algorithm from gyros angular rates. 3.4. Coordinates of a lead star The unit position vector along z-axis of s-frame in inertial frame is obtained as i r¯zs = Cbi Csb [ 0 0 1 ]T .
(2)
The stars’ coordinates generation for the proposed algorithm necessitates finding coordinates of the star named as lead star whose frame definition is the same as that of j -frame. To discriminate this from other j stars, ‘∗ ’ subscript is used in the transformation matrix. At this instant, it is supposed that this star lies nearest to the center of CCD array. As a result, the unit position vector for the lead star along z-axis of s-frame is expressed as i r¯zs
= C∗i [ 0
0
1] . T
(3)
If Θm and Γm represent orientation of the star sensor’s zaxis at time tm , transformation matrix Ci∗ is achieved by two successive rotations of i-frame by angles Θm and (90◦ − Γm ) about z and y axes respectively, i.e. ⎡ ⎤ sin Γm cos Θm sin Γm sin Θm − cos Γm ⎦. Ci∗ = ⎣ − sin Θm (4) cos Θm 0 cos Γm cos Θm
cos Γm sin Θm
sin Γm
Now, the unknown quantities Θm and Γm can be estimated by equating Eqs. (2) and (3). 3.5. Generation of stars’ coordinates In this step, a block of stars’ coordinates is generated in terms of right ascension and declination of stars using a Matlab function ‘rand’ with its center around Θ0 ≈ Θm [deg], and Γ0 ≈ Γm [deg]. The stars’ coordinates generated in this way are shown in Fig. 3.
Fig. 3. Stars positions in i-frame.
3.6. Stars’ position in reference image The ANS, based on stars’ identification concepts, uses a CCD star sensor to gather imagery of the stars as it flies en route to the target and processes these data with the reference star catalog stored in the onboard computer to determine the attitude. The star catalogs and astronomical almanacs express a star’s position in the inertial frame in terms of its right ascension Θj and declination Γj , whereas in real image from CCD star sensor, it is expressed in terms of pixels along x and y axes of the b-frame. To perform a star’s identification, its position in inertial frame should be transformed into reference image in terms of pixels along x and y axes. The relationship between inertial and reference image frames is derived through two successive rotations of i-frame by angles (Θ0 +180◦ ) and (90◦ −Γ0 ) about z and y axes, respectively, i.e. ⎡
− sin Γ0 cos Θ0 r Ci = ⎣ sin Γ0 cos Γ0 cos Θ0
− sin Γ0 sin Θ0 − cos Θ0 cos Γ0 sin Θ0
⎤ cos Γ0 0 ⎦,
(5)
sin Γ0
where (Θ0 , Γ0 ) is the chosen reference point corresponding to the center of the reference image. Positions of all other stars in reference image are determined by their right ascension and declination information with reference to the chosen point of reference. Fig. 4 illustrates this concept of reference image preparation. The location of any star in reference image is represented by angle Θr and distance O0 Oj , proportional to the angle between unit vectors in reference image and in i-frame. This angle is estimated as θ0j = cos−1 r¯0i · r¯ji /r¯0i r¯ji .
(6)
Now, Θr , the angle between x-axis of reference image and the position of a star in i-frame are determined. At the outset, j the matrix Cr is determined through two successive rotations
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Fig. 4. Reference image preparation.
Fig. 5. Orientation of s-frame and reference image.
of a reference frame by angles Θr and θ0j about z and y axes, respectively, i.e. ⎡ ⎤ sin θ0j cos Θr sin θ0j sin Θr − cos θ0j j ⎦. (7) Cr = ⎣ − sin Θr cos Θr 0 cos θ0j cos Θr cos θ0j sin Θr sin θ0j The unit position vector r¯j of j th star in the j -frame is expressed as j
r¯j = [ 0 0 1 ]T .
(8)
In addition, we know that j Cji r¯j
j
= Cri Cjr r¯j .
(9)
Pre-multiplication of Eq. (9) with j
Cir
yields
j
Cir Cji r¯j = Cjr r¯j .
(10) j
Substitution of values for Cir , Cji , Cjr and r¯j in Eq. (10) results in simplified form as
⎡ ⎢ ⎣
− sin Γ0 cos Θ0 cos Γj cos Θj − sin Γ0 sin Θ0 cos Γj sin Θj + cos Γ0 sin Γj sin Θ0 cos Γj cos Θj − cos Θ0 cos Γj sin Θj
⎤ ⎥ ⎦
cos Γ0 cos Θ0 cos Γj cos Θj + cos Γ0 sin Θ0 cos Γj sin Θj + sin Γ0 sin Γj ⎡
sin θ0j cos Θr
⎤
⎥ ⎢ = ⎣ sin θ0j sin Θr ⎦ , cos θ0j
Θr = tan−1 (sin Θ0 cos Γj cos Θj − cos Θ0 cos Γj sin Θj )/ (− sin Γ0 cos Θ0 cos Γj cos Θj − sin Γ0 sin Θ0 × cos Γj sin Θj + cos Γ0 sin Γj ) .
(11)
In this algorithm, it is presumed that the CCD array used in star sensor has F ◦ FOV and image plane with equal dimension of x × y pixels. From Fig. 5, it is possible to estimate a star’s coordinates in reference image plane as xr = f tan(θ0,j ) cos Θr,j . (12) yr = f tan(θ0,j ) sin Θr,j Also, the coordinates of guide star indicated by Θm and Γm are transformed into reference image as xrm and yrm (Fig. 6).
Fig. 6. Stars positions in reference image.
3.7. Orientation between xs and xr To carry out stars’ identification reliably, the axes of reference image and the s-frame must be perfectly aligned. This requires estimation of angular misalignment and displacement between the two. From the available information about matrices Csb , Cbi and Cir , matrix Csr is obtained as Csr = Cir Cbi Csb .
(13)
From the Csr matrix, horizontal misalignment between s-frame and reference image is obtained as (14) θsr = tan−1 Csr (1, 2)/Csr (1, 1) . Now, stars’ coordinates in the reference image are transformed into the estimated orientation of s-frame. This transformation is accomplished through displacement xrm , yrm and angular misalignment θsr as
sin θsr xs cos θsr xr − xrm = . (15) ys − sin θsr cos θsr yr − yrm
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Fig. 8. Error analysis schema.
For small angle approximations, the relationship between the mathematical platform frame and the reference navigation frame (il-frame) is given as ⎡ ⎤ 1 φz −φy p Cil = ⎣ −φz (22) 1 φx ⎦ = I − Φ, φy
3.8. Stars’ position in s-frame
p
From these angles, the transformation matrix Cjs , and then Cjb is obtained as (18)
3.9. Attitude error matrix j
Presuming that r¯j represents unit position vector of the j th star projected into j -frame, its transformation into i-frame using star’s right ascension and declination yields j
r¯ji = Cji r¯j .
(19)
Star’s right ascension and declination in the s-frame are obtained from the image projection geometry on two-dimensional plane as shown in Fig. 7. The orientation of j -frame with respect to b-frame, estimated from Eq. (18), is employed to compute the unit position vector of j th star in b-frame as j
r¯jb = Cjb r¯j .
(20)
Error analysis schema shown in Fig. 8 portrays the relationship between different frames. Here, c and p refer to computed and mathematical platform frames, respectively. It is obvious that ‘c’ follows ‘il’, ‘p’ follows ‘c’ and ‘p’ follows ‘il’ indirectly. There is misalignment between the ‘il’ and ‘p’ frames that is expressed by small angles φx , φy , φz . Therefore, estimated star’s position vector in the il-frame is expressed as p p r¯ˆilj = r¯j = Cb r¯jb = Cˆ bil r¯jb , p
p
r¯j = Cil r¯jil = (I − Φ)¯rjil .
As depicted in Fig. 7, a star’s position in star sensor frame is estimated by angles Θs and Γs as (16) Θs = tan−1 yjs /xjs , 2 2 xjs + yjs f . (17) Γs = 90◦ − tan−1
= Csb Cjs .
1
where φx , φy and φz represent SINS axis misalignment angles. The unit position vector of j th star in p-frame can also be expressed as
Fig. 7. Image projection geometry.
Cjb
−φx
(21)
where Cb , the estimated attitude matrix from SINS, can also be p p expressed as Cˆ bil = Cil Cbil = Cb .
(23)
If there are n number of stars in the FOV of star sensor identified as guide stars from the reference star catalog, Eqs. (21) and (23) can be equated to get the following relationship: p p p r¯1 r¯2 . . . r¯n (24) = (I − Φ) r¯1il r¯2il . . . r¯nil . In its simplified form, Eq. (24) can be written as Rp = (I − Φ)Ril , p where Rp = [¯r1 . . . r¯nil ].
(25) p r¯2
p r¯3
...
p r¯n ]
and Ril = [¯r1il
r¯2il
r¯3il
From Eq. (25), the desired SINS attitude error matrix is estimated as −1 p . (26) Cil = (I − Φ) = Rp RilT Ril RilT p
To ensure that rows and columns of Cil remain normal and orthogonal, an orthogonalization algorithm is included. If this attitude error matrix is to be orthogonal then the following simple relationship ought to hold [5]: 1 il −1 p p Cp,k Cil,k+1 = (27) + Cil,k , 2 p
p
where Cpil = (Cil )T , k = 0, 1, 2, . . . , and Cil,0 is the initial estimated matrix. This iterative method requires only three to four iterations for the matrix to become orthogonal. This orthogonalization method has the advantage of equal treatment of all the vectors without preference to any one of them. 4. Simulation and results The primary sensor subsystems used in the astro-inertial navigation system described in this paper are SINS and ANS. The SINS used in the simulation outputs information on position, velocity and attitude while ANS processes the sensed star images, in conjunction with SINS, to provide attitude error information. The objective of the proposed algorithm is to
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estimate the SINS attitude errors more accurately. The ANS augmentation comes into effect about 40 seconds after lift off of the missile when it attains more than 22-km altitude, because at that altitude, stars are observable and unobstructed [4]. The motorized segment of the missile’s flight is the most crucial stage to position the missile on a trajectory with flight states appropriate for the desired target. Hence, simulation of the presented algorithm has to be carried out during powered flight trajectory of the missile.
⎤ 0.084042 −0.015747 −0.042499 −0.011282 0.176607 0.174620 ⎦ , 0.996373
4.2. Astro-inertial simulation To validate the proposed algorithm, stars’ observation is made only once, that is 54 seconds after the vehicle’s lift off from launch point. Constant and random errors of inertial sensors have been added in the simulation. Only seven stars, shown in the solid block of Fig. 3, are supposed to be identified as guide stars from the reference star catalog. Using the derived relationship, the SINS attitude error is estimated by processing data for these identified stars. 4.3. Results The estimated attitude matrix from SINS at t = 54 seconds is p Cˆ bil = Cb ⎡ 0.9999572775 ⎣ = 0.0055900888
⎤ −0.0057193893 0.0070446581 0.9998276614 0.0177030632 ⎦ .
−0.0069916574 −0.0176617136 0.9998169182 The estimated unit position vectors of the identified stars in iland p-frames are ⎡ 0.036973 −0.054264 0.002306 0.048928 Ril = ⎣ −0.160347 0.006146 −0.018306 0.049280 0.986339 0.998487 0.999807 0.997564
0.983705
0.036707 −0.054436 0.002112 0.048778 Rp = ⎣ −0.159920 0.006104 −0.018331 0.049286 0.986450 0.998488 0.999827 0.997598 ⎤ 0.083870 −0.016012 −0.042831 −0.011318 0.177033 0.174999 ⎦ .
4.1. SINS mechanization In this paper, space-stabilized mechanization is employed for SINS realization. It is conceptually the simplest of all possible system implementations, because Newton’s laws are most austerely stated in an inertial frame of reference. The spacestabilized inertial navigation system outputs navigation parameters in an inertially non-rotating frame. Therefore, this mechanization is free from earth’s rotation and transport rate [2]. In the simulation of SINS, the gravitational acceleration is more accurately expressed in terms of ellipsoidal (or more precisely, spheroidal) earth model. Such a model accounts for the earth’s oblateness, as it includes the second order gravitational harmonic term of the earth’s gravitational field. Initial alignment and inertial sensors’ errors have also been incorporated in the simulation.
0.984142
⎡
0.996423
0.984070
0.983628
The attitude error matrix estimated through Eqs. (26) and (27) is p
Cil ⎡
⎤ 0.9999998984 0.0003307218 −0.0003063222 = ⎣ −0.0003306266 0.9999998970 0.0003106485 ⎦ . 0.0003064249 −0.0003105472 0.9999999048
5. Conclusion The algorithm for estimation of SINS attitude errors using astronavigation augmentation provides a method to estimate and compensate for the gyro drift. Gyro induced drift errors are the only error variables that contribute to attitude errors. Therefore, the gyro errors that the ANS updates are effective in estimating and compensating for the attitude errors, also the consequent position and velocity errors. References [1] J. Ali, Strapdown inertial navigation system aided by celestial image matching technique, MS thesis, Beijing University of Aeronautics and Astronautics, 2001. [2] K.R. Britting, Inertial Navigation Systems Analysis, Wiley-Interscience, New York, 1971. [3] M. Kayton, W.R. Fried, Avionics Navigation Systems, John Wiley and Sons, New York, 1997, pp. 551–596. [4] S. Levine, AGARDograph on advanced astroinertial navigation systems, AGARD-AG-331, in: Aerospace Navigation Systems, 1995, pp. 187–199. [5] R.W. Priester, E.D. Denman, Orthogonalization of a direction cosine matrix by iterative techniques, IEEE Trans. Aerospace Electron. Syst. 8 (1972) 692–694. [6] E.V.B. Stearns, Navigation and Guidance in Space, Prentice-Hall, Englewood Cliffs, NJ, 1963. [7] Y.L. Xiao, Foundation of Flight Dynamics-Modeling of Aerospace Vehicle Motion, Beijing University of Aeronautics and Astronautics Press, Beijing, 2003 (in Chinese). [8] A.V. Zbrutsky, O.I. Nesterenko, N.I. Lykholit, et al., Astroinertial navigation system for aircraft applications, in: Proceedings of the 9th Saint Petersburg International Conference on Integrated Navigation Systems, Russia, 2002, pp. 152–154.