Autonomous on-orbit Calibration Of Star Trackers

Transcription

1 Autonomous on-orbit Calibration Of Star Trackers Malak Samaan 1, Todd Griffith 2,Puneet Singla 3, and John L. Junkins 4 Department of Aerospace Engineering, Texas A&M University, College Station, TX Abstract This paper presents two calibration methods to obtain the estimates for the focal length and principal point offsets and the focal plane distortions for a star camera with a CCD array. These quantities can be obtained from star image frames where the stars have been identified. These estimation techniques can be used either on gourd or in flight. To model the optics of the camera, the pin-hole model, which is found to be very accurate for most star cameras, is used. The first method used is that of standard nonlinear least-squares optimal estimation. This method is used iteratively until it converges with a set tolerance. The second technique presented uses the on-orbit calibration corrections to calibrate the focal plane distortions. Introduction An important problem in spacecraft autonomy is the frequently occurring situation that instrument acting an other unpredictable effects result in significant changes in the star tracker. Recent missions, e.g. MSX have shown the consequence that attitude estimate precision is degraded. In the proposed paper, we show a novel method for solving this problem. The method makes use of residuals between measured inter star cosine angles and the interstar cosine angles between the corresponding cataloged stars (ˆv T i ˆv j = ŵ T i ŵj) to learn the calibration corrections on-orbit, starting from ground-based calibration results. 1 Graduate Student, Department of Aerospace Engineering, Texas A&M University, College Station, TX , mas1894@aero.tamu.edu, AIAA and AAS Member. 2 Graduate Student, Department of Aerospace Engineering, Texas A&M University, College Station, TX , dtg8332@aero.tamu.edu, AIAA and AAS Member. 3 Graduate Student, Department of Aerospace Engineering, Texas A&M University, College Station, TX , puneet@neo.tamu.edu, AIAA and AAS Member. 4 George Eppright Chair, Distinguished Professor, Department of Aerospace Engineering, Texas A&M University, College Station, TX , junkins@tamu.edu, AIAA and AAS Memeber. 1

2 Approach makes use of the truth that interstar angles are an invariant of rotational transformations, and therefore we do not require knowledge of the generally unknown spacecraft attitude. The calibration process is naturally divided into two major parts: Calibration of the principal point offset (x o, y o ) and focal length (f). Calibration of the focal plane distortions due to all other effects (lens distortion, misalignment, detector alignment, etc.) Both Batch (Ground-based) and sequential (On-orbit) algorithms have been developed. The paper will deal with algorithmic details for some simulation and early night sky experiments. The Ground (Batch) Calibration The main objective of the Ground calibration algorithms are to find accurate calibrated values for the focal length used for obtaining the CCD image and the offsets value for the x and y coordinates as shown in Fig. 1. Figure 1: The star coordinates and offsets Without the focal plane calibration the problem of the Star Identification (Ref [1]-[3]) may not give an accurate results due to the fact the the imaged stars have to be modified by the amount of the bore-sight offsets and the focal length used to get the star image. The method of least squares optimal estimation has proved to be an extremely powerful technique, and has been used extensively in the parameters identification algorithms (Calibration). To begin the ground calibration algorithm the stars of the CCD image has to be manually identified so that the inertial and the body direction vectors are given as an input to the algorithm. From the centroiding algorithm we have x i and y i for (i = 1,..., n) where n is the number of stars in the FOV. We can use initial guess of the focal length to get 2

3 ŵ i = 1 ((xi x o ) 2 p 2 sx + (y i y o ) 2 p 2 sy + f 2 ) (x i x o )p sx (y i y o )p sy f (1) where f is the camera focal length, (x o, y o ) are the bore-sight errors, and p sx and p sy are the pixel size in x - direction and y - direction respectively. image width p sx = No. of pixels of the CCD in x-dir. image height p sy = No. of pixels of the CCD in y-dir. So, f, x o and y o are the unknowns to be estimated using the Nonlinear Least Square Optimal Estimation. The Cataloged Vectors The inertial cataloged star direction cosines could be calculated using; ˆv i = cos α i cos δ i sin α i cos δ i sin δ i (2) (3) Where α i and δ i are the right ascension and declination for star i (i = 1,..., n) respectively. Now, to solve for the unknowns we can use the fact that the interstar angles for the imaged vectors and the cataloged vectors have to be the same, mathematically; Now, using equation (1) we can show that; Where ˆv T i ˆv j = ˆv T i ˆv j = ŵ T i ŵj (4) N D 1 D 2 = g ij (x o, y o, f) (5) N = (x i x o )(x j x o )p 2 sx + (y i y o )(y j y o )p 2 sy + f 2 D 1 = (x i x o ) 2 p 2 sx + (y i y o ) 2 p 2 sy + f 2 D 2 = (x j x o ) 2 p 2 sx + (y j y o ) 2 p 2 sy + f 2 (6) 3

4 By using the linearization about the nominal value (ˆx o,ŷ o, ˆf) we have x o = ˆx o + x, y o = ŷ o + y and f = ˆf + f (7) Substitute equation (7) in (5) to get ˆv T i ˆv j = g ij (ˆx o, ŷ o, ˆf) + [ x o y o f ] x o y o f (8) Let R ij = ˆv T i ˆv j g ij (ˆx o, ŷ o, ˆf) = [ x o y o f ] x o y o f (9) For (i = 1,..., n 1) and (j = i + 1,..., n) we can write equation (9) as {R} = [A]{ Z} (10) Where A = g 12 g 12 g 12 x o y o f g 13 g 13 g 13 x o y o f g n 1,n x o g n 1,n y o g n 1,n f R = R 12 R 13.. R n 1,n and Z = x o y o f (11) Least Squares Optimal Estimation The method of least squares is a powerful and widely applied tool from estimation theory. There are many excellent sources on the subject where derivations of the equations can be found in Ref. [4], [5]. By using the least squares we can show that the solution Z which minimize the residual of equation (10) in the least squares sense is given by; where k = 1, 2,..., No. of iterations { Z} k = [A T A] 1 A T {R} (12) 4

5 ˆx o ŷ oˆf k+1 = ˆx o ŷ oˆf k + x o y o f k The differentiation terms in equations (8), (9) are given by; (13) = D 1D 2 (2x o x i x j ) + N[(x i x o )D 2 /D 1 + (x j x o )D 1 /D 2 ] (14) x o (D 1 D 2 ) 2 = D 1D 2 (2y o y i y j ) + N[(y i y o )D 2 /D 1 + (y j y o )D 1 /D 2 ] (15) y o (D 1 D 2 ) 2 f = D 1D 2 (2f) Nf[D 2 /D 1 + D 1 /D 2 ] (D 1 D 2 ) 2 (16) Figure 2: Flow Chart for the Calibration Algorithm 5

6 Figure 3: Night Sky Image for TAURUS Figure 4: The Estimation of the Offsets and the Focal Length The Calibration Algorithm Results The flow chart of the MATLAB program used to solve the problem of the calibration of the focal length and the image coordinates offsets is given in figure 2. In order to run the calibration program, the image processing algorithm is applied for the TAURUS CCD image (3) to get the star coordinates (x i, y i ). Manual star identification is done to each one of the centroided stars to have the inertial position vector of all the imaged stars. Figure 4 shows the Calibration errors for the bore-sight position (x o, y o ) and the focal length ( f ) versus the number of iterations for the TAURUS image. It can be shown that the values of (x o, y o ) and f are converged after 2 or 3 iterations. The test results showed that the Calibrated Focal length = , the Calibrated Y-axis offset x o = and the Calibrated X-axis offset y o = The time consumed in Calibration sec. Figure 5 shows the effect of the calibration parameters on the identified stars of the TAURUS image. All the stars have been identified after including the results of the offsets and the focal length on the imaged vectors. 6

7 Figure 5: The Measured and the Identified Stars for TAURUS The Sequential (On-Orbit) Calibration Todd and Puneet Sections,... Conclusions This paper presents how to solve for the estimation of the three focal plane parameters, the focal length and the two principal point offsets using the standard nonlinear least-squares estimation. Also, the focal plane calibration (distortions) is also solved using Bla Bla Bla. References [1] Samaan, M.A., Mortari, D., and Junkins, J.L., Recursive Mode Star Identification Algorithms, Paper AAS AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, California, 11 Jan Feb [2] 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, Colorado, 31 Jan. - 4 Feb

8 [3] Ju, G., Kim, Y.H., Pollock, T.C., Junkins, J.L., Juang, J.N., and Mortari, D. Lost-In-Space: A Star Pattern Recognition and Attitude Estimation Approach for the Case of No A Priori Attitude Information, Paper of the 2000 AAS Guidance & Control Conference, Breckenridge, CO, Feb. 2-6, [4] Junkins, J. L. Optimal Estimation of Dynamical Systems, Sijthoff and Noordhoff International Publishers B. V., Alphan aan den Rijn, The Netherlands, [5] Kowalick, J. Methods for Unconstrained Optimization Problem, American Elsevier Publishing Company, New York, New York, [6] Mortari, D. ESOQ: A Closed-Form Solution to the Wahba Problem, Journal of the Astronautical Sciences, Vol. 45, No. 2, April-June 1997, pp [7] Mortari, D. Second Estimator of the Optimal Quaternion, Journal of Guidance, Control, and Dynamics, Vol. 23, No. 5, Sept.-Oct. 2000, pp [8] Junkins, J. L., and Kim, Y., Introduction to Dynamics and Control of Flexible Structures, AIAA Education Series, Reston, VA,