Report

ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 20: Project Discussion and the Kalman Filter University of Colorado Boulder Homework 6 Due Friday University of Colorado Boulder 2 Project/Homework Discussion University of Colorado Boulder 3 Satellite state estimated and propagated in the inertial frame: Dynamics solve-for parameters are (fundamentally) not tied to a coordinate system: Ground-station locations are in the Earth-fixed frame: University of Colorado Boulder 4 Since the ground stations are in the Earth-fixed frame, we assume: Hence, we have: University of Colorado Boulder 5 The portions of the reference state requiring integration only includes the spacecraft position and velocity Strictly speaking, we only need to propagate a 6 × 9 matrix! University of Colorado Boulder 6 All of these need to be in the same reference frame! We recommend including this transformation in the measurement model: University of Colorado Boulder 7 How can we estimate the filter solve-for parameters since the observations do not seem to depend on them? How/why can we estimate these values? (conceptual and mathematical answers) University of Colorado Boulder The STM is a function of these values 8 Compare to solution online Results available as .txt and .mat ◦ Results generated for the .txt files did not use ode45()! ◦ Results in .mat file appear to have used Rel/Abs tolerances of 1e-11 Note: some elements of the project website need to be updated (suggestions and rubric) University of Colorado Boulder 9 We ask for relative differences to quickly identify differences between your result and the one online: Example: University of Colorado Boulder 10 Conventional Kalman Filter (CKF) University of Colorado Boulder 11 Given from a previous filter: We have new a observation and mapping matrix: We can update the solution via: University of Colorado Boulder 12 Is there a better sequential processing algorithm? ◦ YES! – The equations above may be manipulated to yield the Kalman filter University of Colorado Boulder 13 Today – Outline derivation from minimum variance estimator ◦ Demonstrates mathematical equivalence of CKF and Batch Wednesday – Derivation as a solution to Bayes theorem ◦ Demonstrates strengths of Kalman filter in context of probability/statistics ◦ Also helps to understand impacts of assumptions University of Colorado Boulder 14 Schur Identity (Appendix B, Theorem 4): (Yes, it will simplify things…) University of Colorado Boulder 15 Kalman Gain University of Colorado Boulder 16 University of Colorado Boulder 17 Instead inverting a p×p matrix Mathematically equivalent to the batch least squares Also provides a solution to the least squares minimization problem Yields a new set of problems in filtering (to be covered later) University of Colorado Boulder 18 Does not map to epoch time! Note the use of Htilde University of Colorado Boulder 19 Reinitialize integrator after each observation: Alternatively, we can use already generated output: University of Colorado Boulder 20 We have to invert a p×p matrix, which is likely more efficient and stable than a n×n matrix inversion Can we further reduce the computation overhead? Yes – under certain conditions… University of Colorado Boulder 21 University of Colorado Boulder 22 University of Colorado Boulder 23 Whitening Transformation Use new values in Kalman filter University of Colorado Boulder 24 Whitening Transformation University of Colorado Boulder 25 The Kalman Filter – Prediction Residuals University of Colorado Boulder 26 Previously, we have discussed the pre-fit and post-fit residuals: How can this change in the context of the CKF? University of Colorado Boulder 27 At each measurement time in the CKF, we can take a look at the prediction residual: Covariance of the prediction residual: University of Colorado Boulder 28 How might we use the prediction residual PDF? University of Colorado Boulder 29