Least Square Estimation of Spacecraft Attitude along with Star Camera Parameters Madhumita Pal ∗ M. Seetharama Bhat ∗∗ ∗ Research associate at Aerospace Engineering Department, Indian Institute of Science, India (e-mail: meetmpal@gmail.com). ∗∗ Professor at Aerospace Engineering Department, Indian Institute of Science, India (e-mail: msbdcl@aero.iisc.ernet.in). Abstract: A methodology for determining spacecraft attitude and autonomous calibration of star camera, both independent of each other, is presented. In this paper, both attitude estimation and star camera calibration is done together, independent of each other, by directly utilizing the star coordinate in image plane and corresponding star vector in inertial coordinate frame. Both radial and decentering distortion of lens accounted in the analysis. Satellite attitude, camera principal point, focal length (in pixel), lens distortion coefficients are found by a simple three step method. In the first step, camera intrinsic parameters are estimated using a closed-form solution assuming lens is distortion free. In the second step lens radial distortion coefficient is estimated by linear least squares method using the solution of the first step to be used in the camera model that incorporates only radial distortion. These steps are applied in an iterative manner until the radial distortion coefficient converges. In third step, lens decentering distortion coefficients are calculated using the estimated camera parameters and lens radial coefficient estimated in the previous steps. The whole procedure is fast enough for onboard implementation. Keywords: Star camera calibration, lens distortion, attitude estimation, closed-form solution. 1. INTRODUCTION With the increasing demand for small inexpensive satellite for short period missions, the use of high band width and accurate star tracker are becoming relevant than the use of expensive gyroscope. But the accuracy of spacecraft attitude determination depends upon the accuracy of Star camera calibration. In literature all attitude estimation algorithms make use of the star vector pairs in camera coordinate system and inertial coordinate system. The difficulty of these methods is that attitude estimation de- pends upon the accuracy of star camera calibration as star vector in camera coordinate frame depends on camera pa- rameters. Though star camera is calibrated on the ground with high accuracy before the launch but due to many reasons like temperature, vibration, aging electronics etc, camera parameters get changed in orbit. This necessitates the fact of on-orbit camera calibration independent of attitude determination. The method proposed in Samaan [2003] for star camera calibration utilizes the fact that interstar angles are an invariant of spacecraft rotation thus there is no need of unknown spacecraft attitude to estimate calibration parameters that offset interstar angles. Singla and Junkins [2002] present both attitude dependent and attitude independent methods for star camera calibration. They have also studied relative merits of two algorithm for attitude determination and camera calibration. In re- alisability and robustness issues, they have shown that attitude independent algorithm performs better than at- titude dependent algorithm. But the method proposed for attitude determination in Singla and Junkins [2002] utilizes the measured star vector in camera coordinate frame which in turn depends on the camera calibration parameters. In Oshman and Markley [1999], Gai and Lemos [1995], the measured and corresponding inertial vectors are used for computing attitude and attitude rate using EKF. The difficulty in this method is that the estimated attitude is dependent on camera calibration parameters apart from the EKF convergence issue where the solution is far apart from the initial guess. In Pal and Bhat [2009], Pal [2013], calibration of principal point offset and focal length in pixel is done so as to estimate the internal parameters of the star camera. These parameters relate the star vectors in camera coordinate frame to the corresponding star image coordinates in the image plane. Attitude of the satellite is obtained as an estimation of the camera external parameters, which relate star vectors in the inertial coordinate frame to the corre- sponding star vectors in the camera coordinate frame. A simple closed form solution is used to get all the calibra- tion parameters of the star camera (internal parameters) and attitude of the satellite (external parameters) using Third International Conference on Advances in Control and Optimization of Dynamical Systems March 13-15, 2014. Kanpur, India 978-3-902823-60-1 © 2014 IFAC 20 10.3182/20140313-3-IN-3024.00233