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University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 40: Elements of Attitude Estimation.

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Presentation on theme: "University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 40: Elements of Attitude Estimation."— Presentation transcript:

1 University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 40: Elements of Attitude Estimation

2 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

3 University of Colorado Boulder 3 Project Q&A

4 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.

5 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

6 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

7 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

8 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

9 University of Colorado Boulder  What are some example torques acting on a spacecraft that we could model? 9

10 University of Colorado Boulder  A star’s 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

11 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 j’s location x j and y j in the camera frame 11

12 University of Colorado Boulder  It may then be shown that 12

13 University of Colorado Boulder  Least-squares cost function  Subject to the constraint  Hence, we instead have a constrained least squares problem 13

14 University of Colorado Boulder  We may instead write the cost function as  Due to constants, we can instead maximize 14

15 University of Colorado Boulder  Quaternions leverage Euler’s theorem to yield a non-singular attitude state representation  Constraint on magnitude:  Matrix/quaternion relationship: 15

16 University of Colorado Boulder 16

17 University of Colorado Boulder  Using the relationship between the attitude matrix and a quaternion  Combined with the previous identities 17

18 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

19 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

20 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

21 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

22 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

23 University of Colorado Boulder 23 FCQs


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