ASEN 5070 LECTURE 24 10/23/09

1. STATISTICALORBIT DETERMINATIONkalman filter example Orthogonal transformationsGivens Transformation ASEN 5070 LECTURE 24 10/23/09

2. Homework Problem 8.2

3. Homework Problem 8.2

4. Homework Problem 8.2

5. Kalman Filter Example A particle is moving up the ramp, with slope θ, at constant velocity v. Given range observations, Set up the Kalman Filter to estimate . Because the observation-state equation is nonlinear in , we must use state and observation deviation vectors. Hence,

6. Kalman Filter Example A particle is moving up the ramp, with slope θ, at constant velocity v. At the kth stage the position is given by: (These equations are used to propagate the reference solution) Assume that at t0=0, x=x0, y=0 Given range observations, Set up the Kalman Filter to estimate . Because the observation-state equation is nonlinear in , we must use state and observation deviation vectors. Hence,

7. Kalman Filter Example The state deviation vector at the kth stage is given by The range observation is: The matrix is given by:

8. Kalman Filter Example

9. Kalman Filter Example (Cont.) Hence, Set up the Kalman Filter to process range observations at the kth stage assuming results are available from the k-1st stage

10. Kalman Filter Example 1. Do the time update to , let

11. Kalman Filter Example 2. Do the measurement update at To implement the EKF at each stage set Set Go to step #1.

12. Batch Processor Example Assume we wish to process both range and range rate observations and estimate using a batch processor.

13. Batch Processor Example where

14. Batch Processor Example Cont. let

15. Batch Processor Example Assume we have m observations Finally,