Compass Star Tracker for GPS Applications

Transcription

1 AAS Compass Star Tracker for GPS Applications Malak A. Samaan Daniele Mortari John L. Junkins Texas A&M University 27th ANNUAL AAS GUIDANCE AND CONTROL CONFERENCE February 4-8, 2004 Breckenridge, Colorado Sponsored by Rocky Mountain Section AAS Publications Office, P.O. Box San Diego, California 92198

2 AAS Compass Star Tracker for GPS Applications 1 Malak A. Samaan 2, Daniele Mortari 3, and John L. Junkins 4 Spacecraft Technology Center TexasA&M University, College Station, Texas Abstract Star trackers are the most accurate attitude sensors to perform the complete 3-axis attitude estimation of spacecraft. However, even though they are specifically designed to estimate attitude, as it will be shown in this paper, they can also be used to estimate the local coordinates of the camera on the Earth as well as the direction of East. This requires us to align the camera with the local direction of the gravity and to have knowledge of the time, that can be provided by an accurate clock. The resulting system is the Compass Star Tracker, that is, a star tracker used as a global surface navigation system. This system would certainly not substitute the Global Positioning System which, in turn, does not suffer of the limitation night-only operation and in clear weather conditions. On the contrary, the Compass Star Tracker does not require any satellite information or other support from ground stations to be used in lieu of the traditional Global Positioning System. Introduction Star patterns, and even single identified stars, are very helpful for navigation. In ancient times, sailors navigated by the North Star, it was their reference point. By looking at it they could tell if they were approximately on right azimuth track or not. This is 1 Paper AAS presented at the 27th Annual AAS Guidance and Control Conference, February 4-8, 2004, Breckenridge, Colorado. 2 Post-Doc Research Associate, Spacecraft Technology Center, Room 127H, Texas A&M University, College Station, TX , Tel. (979) , samaan@tamu.edu 3 Associate Professor, Department of Aerospace Engineering, H.R. Bright Building, Room 741A. Tel. (979) , Fax (979) mortari@aero.tamu.edu 4 George J. Eppright Chair Professor, Department of Aerospace Engineering, Texas A&M University, College Station, TX , Tel: (979) , Fax (979) junkins@tamu.edu 1

3 similar to checking the position of our Sun, which rises in the East and sets in the West, and being able to tell in which direction we are heading. The Sun gives us a single point of reference for heading corrections. We mention that locating the elevation above the horizon of the high local noon position of the sun also measures latitude north of the equator. Sailors made frequent use of that fact to approximate north-south position. The next major developments in the quest for the perfect navigation were the magnetic compass and the sextant. The needle of a compass always points to local magnetic north, so it is always possible to estimate in what direction you are going. These historical navigation methods all relied upon locally observing mother nature to infer the navigation information. This paper introduces a novel way to use star tracker that we refer to as the Compass Star Tracker (CST). The primary objective of a CST is to determine longitude and latitude of the current location (ground or space) where the star image is taken. In order to give a correct output, the only condition required is that the optical axis of the star sensor is aligned, with respect to the local effective gravity field vector. When this occurs, then the picture contains all the needed information. The night sky image is then processed, and the star centroiding and star identification processes then allow the evaluation of the star tracker attitude with respect to the inertial frame. By knowing the current time of the captured star image, the longitude and latitude of the current position will be determined using some simple attitude transformations and the knowledge of time. Knowledge of the Earth s gravity model enables us to establish the latitude consistent with the star image. A primary limitation is the precision of the alignment of the tracker axis with the local gravity (plumb-bob) direction. This paper describes the technique developed to estimate latitude and longitude of the observing position and validates it by means of Monte-Carlo simulated images. Also, real night sky tests using a star camera and a gravity pendulum, a weight on the end of a rigid rod, have been accomplished to validate this idea. There are many applications for this instrument. One could be to use it for navigation if there are no satellite signals or as a back-up in the case of GPS failure. Of course the accuracy of this compass will depend on many factors which include: 1) the CCD resolution, 2) the centroiding accuracy, 3) the time precision, and 4) the deviation between the local vertical alignment of the tracker with respect to the direction of gravity. The latter is caused mostly by the deviation of the actual Earth with the adopted Earth model (sphere, ellipsoid, geoid). The idea of the Compass Star Tracker can easily be generalized for the Moon or the other planets. This requires, obviously, us to use a star catalog whose stars are provided in an inertial reference frame that has the planet spin axis as one of the reference frame. It would interesting to investigate the possibility to apply this technique when two bodies are visible, as for instance, the Earth and the Moon. From a strictly theoretical point of view, it should be possible to evaluate the absolute position in space with two observations, only. 2

4 Problem Formulation In order to define the problem of the compass star tracker, consider the celestial reference frame along with the camera body frame as shown in Fig. 1. In this figure, the camera optical axis is assumed to be aligned with the Earth local vertical. Let us assume the Earth shape to be spherical. In this research paper, the theory associated with Ellipsoidal or more accurate description of the Earth, will be studied later. Let us consider Fig. 1 where the parameters have the following meaning: λ and ϕ are the longitude and latitude of the camera body frame location, ψ is the angle between vernal equinox (ascending node of the geocentric ecliptic) and the Greenwich, which only depends on the current time, ε is the angle between the local East and the y-axis of the camera body frame, and x is the camera optical axis (the x-axis of the camera body frame). Figure 1: The Reference Frame for the Earth and the Optical Axis If A B/I is the attitude matrix for the camera body frame with respect to the inertial reference frame and A B/L is the attitude matrix for the camera body frame with respect 3

5 to the local reference frame, then A B/L R 3 (ε) = cos ε sin ε 0 sin ε cos ε (1) Also, if A L/G is the attitude matrix for the local reference frame with respect to the Greenwich reference frame, then cos ϕ 0 sin ϕ cos λ sin λ 0 A L/G R 2 ( ϕ) R 3 (λ) = sin λ cos λ 0 sin ϕ 0 cos ϕ = cos ϕ cos λ cos ϕ sin λ sin ϕ sin λ cos λ 0 sin ϕ cos λ sin ϕ sin λ cos ϕ Now, if A G/I represents the Earth attitude matrix with respect to J2000. This matrix is known by knowing the current time. It also will take into account the precession and the nutation of the Earth spin axis [1]. The matrix A G/I is calculated at the current time (t) by A G/I (t) = R 3 (ψ)np (3) where NP is the rotation matrix that accounts the precession and the nutation of the Earth, this will be studied in the next section. The three unknown parameters of the Compass Star Tracker are λ, ϕ and ε. These parameters could be solved as follows (2) A B/I = A B/L A L/G A G/I (4) By using Eqs. (1) and (2), we obtain (C x cos x and S x sin x) C ε S ε 0 C ϕ C λ C ϕ S λ S ϕ A B/L A L/G = S ε C ε 0 S λ C λ 0 = S ϕ C λ S ϕ S λ C ϕ C ε C ϕ C λ S ε S λ C ε C ϕ S λ + S ε C λ C ε S ϕ = S ε C ϕ C λ C ε S λ S ε C ϕ S λ + C ε C λ S ε S ϕ S ϕ C λ S ϕ S λ C ϕ But from Eq. (4) we can write that (5) A B/L A L/G = A B/I A I/G = A (6) where the attitude matrix A B/I is the one - known - provided by the star tracker and the matrix A I/G is also known from the current time. Finally the parameters ϕ, λ and ε can be computed directly from the expression of the attitude matrix A = A B/G. In fact, from this matrix we can compute the latitude ϕ as cos ϕ = A(3, 3) (7) 4

6 then the longitude λ from and the East direction ε from tan λ = tan ε = A(3, 2) A(3, 1) A(2, 3) A(1, 3) In these equations, it is important to use ATAN2 to resolve the quadrants of (λ, ε) correctly. While Eqs. (8) and (9) provide the result with no ambiguity, Eq. (7) provides the solution in term of ±ϕ. This ambiguity, is easily solved since the four equalities A(1, 1) = cos ε cos ϕ cos λ sin ε sin λ A(1, 2) = cos ε cos ϕ sin λ + sin ε cos λ (10) A(2, 1) = sin ε cos ϕ cos λ cos ε sin λ A(2, 2) = sin ε cos ϕ sin λ + cos ε cos λ must be all satisfied. (8) (9) Precession and Nutation of the Earth Neither the plane of the earth s orbit, the ecliptic, nor the plane of the earth s equator are fixed with respect to distant objects [1]. The dominant motion is the precession of the earth s polar axis around the ecliptic pole, mainly due torques on the earth cause by the moon and sun. The earth s axis sweeps out a cone of 23.5 degrees half angle in 26,000 years. The ecliptic pole moves more slowly. If we imagine the motion of the two poles with respect to very distant objects, the earth s pole is moving about 20 arcseconds per year, and the ecliptic pole is moving about 0.5 arcseconds per year. The combined motion and its effect on the position of the vernal equinox are called general precession. The predictable short term deviations of the earth s axis from its long term precession are called nutation. The nutation is the transformation for the periodic effects contributed by the sun and the moon. The precession-nutation matrix can be written as the product of four rotation matrices as NP = R 1 ( ɛ)r 3 ( ψ)r 1 (φ)r 3 (γ) (11) where φ and γ are the angles to specify the location of the ecliptic pole of date in the given inertial frame, ψ is the ecliptic angle of precession, and ɛ is the obliquity of the ecliptic at current time. The above four parameters in Eq. (11) are given in terms of the current time in degrees to be ɛ = ( t t t 3 )/3600 ψ = ( t t t 3 )/3600 φ = ( t t t 3 (12) )/3600 γ = ( t t t 3 )/3600 5

7 where t is the current time calculated by t = (JD T 0 )/T century (13) Here T century is the number of days in one century (=36525), T 0 is the Julian Date at J2000 (= ), and JD is the Julian Date at the current time. System Description The only condition required to validate the above equations for the current location longitude and latitude is that the star tracker should vertically aligned toward the sky with high precision. In order to satisfy this condition, the system introduced in Fig. 2 is designed to get a vertical alignment of the star camera focal axis with respect to the Earth surface. In Fig. 2, we use a simple gravity pendulum (a weight on the end of a rigid rod), aligned to the camera focal axis and has free rotation in all the three axes on the top of a Tripod. Figure 2: The Star Camera with Tripod and Pendulum Due to the Earth oblateness, the Earth is not a sphere. The equator bulges outward due to the planet s rotation. In geometric terms, the Earth is a geoid, not a sphere, and its polar and equatorial radii differ slightly [1]. According to the International Astronomical Union (IAU): polar radius r p = 6,378,140 meters equatorial radius r e = 6,356,755 meters The difference isn t much, but it does affect our view of the stars by changing our latitude slightly. Most maps show geodetic latitudes, based on lines perpendicular to a tangent plane on the Earth s surface. Another type of latitude, geographic, is calculated as the angle of a plumb line with the equator; both geodetic and geographic latitudes measure similar angles. If the Earth were a sphere, a plumb line (Pendulum direction) would point toward the center of the planet but the flattening of the planet makes this true only at the equator and poles. So neither geodetic nor geographic latitude is accurate from the perspective of astronomy, which calculates coordinates from the 6

8 center of the Earth. For tracking the stars, we need to calculate a geocentric latitude: a line drawn from the point of observation to the center of the Earth. Figure 3: geodetic (ϕ) and geocentric (ϕ ) latitude Figure 3 shows the relationship of geodetic (ϕ) and geocentric (ϕ ) latitude. difference is defined by the formula: The tan ϕ = r2 p tan ϕ = tan ϕ (14) re 2 In the Compass Star Tracker system the geocentric latitude (ϕ ) is used for our observing position. So, the correction of the current geocentric latitude using equation (14) should be taken into account when we used equation (7). We mention that the actual plumb-bob direction deviates locally from the normal to the reference ellipsoid (located by ϕ). These small deviations (less than 0.01 degrees) represents a limitation (our model will be revised to include these in the future design). However, we feel this small error is comparable to the alignment error we can routinely achieve, so we accept this approximation in the present paper. Compass Star Tracker Results Using Night Sky Tests A night sky test is done using the above approach to image stars using Star1000 camera with pixels, 109 mm focal length and 8 degrees field of view size. Fig. 4 7

9 shows one of the night sky images captured on Oct. 20, 2003 at College Station, TX. This image is captured such that, the focal axis of the camera is aligned vertically to the Earth surface. We mention that these results represent merely a demonstration rather than an indication that we established an instrument design for routine use. Figure 4: Night Sky Image Using Star1000 Camera The Centroiding algorithm [3, 4] is carried out for the above night sky image to get the resulting star centers as shown in figure 5. The star identification for the measured stars are done by using the Pyamid Star ID [5]. Once the measured and the cataloged vectors are determined the attitude estimation method ESOQ-2 [6] is used to estimate the camera attitude matrix A B/I with respect to the Earth inertial frame (J2000). A B/I = From Eq. (3), Eq. (11), and Eq. (12) with the known exact time of the measured image the matrix A G/I is calculated to be A G/I = Then, by using equation (6) the matrix A is calculated and then using Eq. (7), Eq. (8), and Eq. (9) the latitude ϕ, the longitude λ and the East direction ε are calculated 8

10 y (pixels) x (pixels) Figure 5: Star Centroids for Image 4 to be ϕ = o λ = o ε = o which have about o and 0.02 o errors from the actual values for the longitude and the latitude for College Station, TX. These errors indicate our alignment of the camera bore-sight with the local plumb direction was imperfect. Using Monte-Carlo Simulations A near real night sky image, obtained using the recently developed Virtual CCD software [2, 3], is created using the following parameters The optical axis of the camera body frame is Zenith up pointing, The camera has pixels, 109 mm focal length, and 8 FOV, The longitude and latitude of the camera body frame are and , respectively (Denver, CO), and The rotation of the Earth is assumed to be (2π/24 rad/hr). ε = 10. We also assume Now, using the above parameters and Eqs. (5, 6) the matrix A B/G is calculated and the resulting star image for this attitude matrix is shown in Fig. 6 The star centers are obtained using the image processing algorithm (Centroiding) [3, 4] and the star identification for the associated measurement stars are done by using 9

11 Figure 6: Star Image Using the Virtual CCD Software Y (mm) X (mm) Figure 7: The Centroiding for the Star Image in Fig. 6 10

12 the Pyamid Star ID [5]. The output for the centroiding algorithm is shown in Fig. 7 and star identification algorithm outputs are summarized in Table 1. Once the inertial vectors and the body vectors for the simulated star image are obtained, the optimal estimate of the attitude matrix A B/I e will be determined by using the ESOQ-2 optimal attitude estimator [6]. RA Dec Table 1: The Star Identification Output (deg) The estimated values for the longitude, latitude and the local east angle are calculated using Eqs. (7), (8), and (9) to be , , and , respectively. The same simulation is repeated 20 times with the same procedures as above. The output of these simulations with respect to the longitude, latitude and the local east of the camera body frame are shown in Fig λ (deg) φ (deg) ε (deg) Time (sec) Figure 8: The longitude, latitude and local east of the camera body frame These simulations provide basis for optimism for the approach, assuming the star sensor alignment with the local plumb-bob direction is satisfactorily achieved. 11

13 Conclusion In this paper, a novel method to determine Longitude, Latitude and local East direction of the current location using a star sensor, has been introduced. The proposed Compass Star Tracker system we presented in this paper can be used in place of the ordinary Global Positioning System. We note that such backup methods are useful under number of scenarios. The process involves the use of standard algorithms to process the star images (centroiding, star identification, and attitude estimation) and then the location of the star camera over the surface of the Earth can be found with a easy procedure. Alignment with the local effective gravity vector is revised, however. Precession and nutation of the Earth spin axis has been included in the procedure. It is obvious that the proposed system works can be used on the ground only during clear night operations, however. The proposed method has been finally validated by night sky tests and the Monte-Carlo simulations. Acknowledgements The authors thank Dr. David Boyle and Mr. Mike Jacox from the Spacecraft Technology Center (STC) for supporting this research. We sincerely appreciate the contributions of the Yellow Team at STC in validating these concepts using night sky tests. References [1] Vallado, D. A., Fundamentals of Astrodynamics and Applications, ISBN , McGraw-Hill Companies Inc. [2] Mortari, D., and Romoli, A. NavStar III: A Three Fields Of View Star Tracker, 2002 IEEE Aerospace Conference, Big Sky, MT, March 9-16, [3] Samaan, M. A., Pollock, T. C., and Junkins, J. L., Predictive Centroiding for Star Trackers with the Effect of Image Smear, Journal of the Astronautical Sciences, Vol. 50, 2002, pp [4] Berry, R., Burnell, J., The Handbook of Astronomical Image Processing, ISBN , Willman-Bell Inc., Richmond, VA. [5] Mortari, D., Junkins, J.L., and Samaan, M.A. Lost-In-Space Pyramid Algorithm for Robust Star Pattern Recognition, Paper AAS Guidance and Control Conference, Breckenridge, CO, 31. Jan. - 4 Feb [6] Mortari, D. Second Estimator of the Optimal Quaternion, Journal of Guidance, Control, and Dynamics, Vol. 23, No. 5, Sept.-Oct. 2000, pp