University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 40: Elements of Attitude Estimation University of Colorado Boulder Exam 3 In-class Students: Due December 11 by 5pm CAETE Students: Due 11:59pm (Mountain) on 12/13 Final Project Due December 14 by noon 2 University of Colorado Boulder 3 Project Q&A University of Colorado Boulder 4 Elements of Attitude Estimation Follows (with slight changes in notation to match class): Crassidis and Junkins, Optimal Estimation of Dynamic Systems, Chapman and Hall, Section 4.2, 2004. University of Colorado Boulder Like the orbit determination problem, we must estimate the attitude of a spacecraft to meet requirements Why would we require an accurate attitude estimate? 5 University of Colorado Boulder Attitude estimation can be difficult Highly non-linear dynamics State ambiguities (3, 4, or 9 estimated states?) More design considerations based on the dynamics of your problem (spacecraft/mission dependent!) Different sensor types providing different information Direct attitude information Star trackers Sun sensors Direct attitude rate information Rate gyros 6 Image Credit: Crassidis and Junkins, Fig. 4.1 University of Colorado Boulder For the current discussion, assume we are estimating the attitude matrix: This matrix is orthogonal (AA T =I) with differential equation 7 University of Colorado Boulder Dynamics of the angle rates is described by the Euler equations The Euler equations are the primary source of nonlinearity in the dynamics 8 University of Colorado Boulder What are some example torques acting on a spacecraft that we could model? 9 University of Colorado Boulder A stars location may be described by its right ascension and declination These angles are a projection of the position onto the unit sphere. 10 Image Credit: Wiki Commons University of Colorado Boulder Assume A is the transformation from the inertial frame to the camera frame (z along the boresight) If we estimate A, then we know our attitude Given star js location x j and y j in the camera frame 11 University of Colorado Boulder It may then be shown that 12 University of Colorado Boulder Least-squares cost function Subject to the constraint Hence, we instead have a constrained least squares problem 13 University of Colorado Boulder We may instead write the cost function as Due to constants, we can instead maximize 14 University of Colorado Boulder Quaternions leverage Eulers theorem to yield a non-singular attitude state representation Constraint on magnitude: Matrix/quaternion relationship: 15 University of Colorado Boulder 16 University of Colorado Boulder Using the relationship between the attitude matrix and a quaternion Combined with the previous identities 17 University of Colorado Boulder Okay, we have: Along with our constraint: When we have an equality constraint, what can we do to find the maximum of J()? 18 University of Colorado Boulder We incorporate our constraint in the cost function: Now, we solve for the multiplier Do we recognize the equation on the right? 19 University of Colorado Boulder We know that the quaternion needs to be an eigenvector of K and the Lagrange multiplier is an eigenvalue of K Substituting the previous equation into the original cost function: We maximize J() when we pick the eigenvector with the maximum eigenvalue! 20 University of Colorado Boulder It can be shown that, if there are at least two non-collinear observations then we have at least two unique eigenvalues In other words we need two stars with adequate separation in the field of view to fully estimate the attitude Each observation is 2-D, so we need two independent observations to estimate the 4-D quaternion Improved accuracy with larger separation of the stars (similar to measurement diversity) and more measurements (measurement volume) 21 University of Colorado Boulder Here, we discussed a relatively easy attitude problem No propagation, i.e., more observations than estimated states at a single time Relatively simple solution to constrained estimation problem Did not incorporate angle rates, modeling of torque, etc. More complex methods typically need an EKF or other nonlinear filter to be tractable 22 University of Colorado Boulder 23 FCQs