colorado center for astrodynamics research the university of colorado statistical orbit...

Colorado Center for Astrodynamics Research The University of Colorado

STATISTICALORBIT DETERMINATION

Project ReportUnscented kalman Filter

Information FilterCombining Estimates

Concept Exam

ASEN 5070

LECTURE 38, 39

12/07, 09/09

2Copyright 2006

Requirements/Suggestions for Term Project Report • **1. General description of the OD problem and the batch and sequential algorithms • **2. Discussion of the results - contrasting the batch processor, and sequential filter.

Discuss the relative advantages, shortcomings, applications, etc. of the algorithms. • **3. Show plots of residuals for all algorithms. Plot the trace of the covariance for

position and velocity for the sequential filter for the first iteration. You may want to use a log scale.

• **4. When plotting the trace of P for the position and velocity, do any numerical problems show up? If so discuss briefly how they may be avoided.

• **5. Contrast the relative strengths of the range and range rate data. Generate solutions with both data types alone for the batch and discuss the solutions. How do the final covariances differ? You could plot the two error ellipsoids for position. What does this tell you about the solutions and the relative data strength?

• **6. Why did you fix one of the stations? Would the same result be obtained by not solving for one of the stations i.e., leaving it out of the solution list? Does it matter which station is fixed?

• **7. A discussion of what you learned from the term project and suggestions for improving it.

• **Required items for the final report

3Copyright 2006

Requirements/Suggestions for Term Project Report• Extra Credit Items

• 8. Code the Extended Kalman Filter• 9. How does varying the a priori covariance and data noise covariance affect

the solution? What would happen if we used an a priori more compatible with the actual errors in the initial conditions, i. e. a few meters in position etc.

• 10. Do an overlap study (see lecture 34).• 11. Code the Potter algorithm and compare results to the conventional Kalman

filter. • 12. Solve for the state deviation vector using the Givens square root free

algorithm. Compare solution and RMS residuals for range and range rate from Givens solution with results from conventional Kalman and Potter filters.

• 13. Plot pre and post fit residuals for the Kalman filter. Include the 1 sigma pre-fit standard deviations on the plot (See Eqn. 4.7.34 of text).

• 14. Convert the estimation error covariance matrix into classical orbit element and nonsingular orbit element space. What elements are most in error? Can you see a reason for this. The coordinate transformation code is being mailed to the class.

• 15. Examine the effects of fixing various parameters such as J2 and seeing the results on the solution and residuals. They will no longer be Gaussian.

4Copyright 2006

4

Unscented Kalman Filter

• Introduced by Julier & Uhlmann circa 1997

• Minimal set of samples (sigma points) are deterministically drawn from the square root decomposition of prior covariance, using a specific algorithm– Monte Carlo draws samples randomly (based upon a priori

statistics)

• Sigma points are propagated through the modeled non-linear function– No Jacobians need be computed

• State estimate and covariance are accurate to 2nd order for all non-linearities– 3rd order accuracy is expected for Gaussian distributions

5Copyright 2006

5


(1) Compute weights: LW m /0

20 1/ LW c

LiLWW ci

mi 2,........,12/1

where:

LL 2

and

(2) Initialize the UKF given:

ttt RXP ,ˆ, 11

(3) Compute Sigma Points (matrix) for t-1

)12(111111

ˆˆˆ LxLtttttt PXPXX

(4) Propagate Sigma Points through non-linear systemnote: this implies integrating 2L+1 state vectors at each time step (i.e. each column)

111/ , tttt F u

141 e

= 3 - L

Use =1

2

L

Use matlab fn.Sqrtm to get squareRoot of P

6Copyright 2006

6


and

(5) Perform a state and covariance time update

L

itti

mit WX

2

01/,

Tttti

L

ittti

cit XXWQP

1/,

2

01/,

where: Q is the process noise covariance

(6) Re-compute Sigma Points to incorporate effects of process noise

)12(1/

LxLttttttt PXPXX

(7) Compute the measurements associated with each Sigma Point vector and their weighted average

)( 1/1/ tttt GY

L

itti

mit YWy

2

01/,

7Copyright 2006

7


and

(8) Compute the innovation and cross-correlation covariances

Tttti

L

ittti

ciyy yYyYWRP

1/,

2

01/,

Tttti

L

ittti

cixy yYXWP

1/,

2

01/,

(9) Compute the Kalman Gain, the best estimate of the state and the a posteriori state covariance

1 yyxyPPtK

ttttt yyKXX ˆ

Ttyyttt KPKPP

where: ty

(10) Replace t with t-1 and return to (3) until all the observations have been processed.

are the observations

Information Filter


Recall that the equations for the standard sequential algorithm(also called the covariance filter) are:


The equations for the information filter are as follows.

Combining Estimates


• Treat estimates as observations.• Write observation equations.• Then use Linear Unbiased Minimum

Variance Estimate Equation, as follows.


112

11

2

^1

21

^1

111

21

1

111^

2

1

2

1

2

^1

^

2

1

2

^1

^

][

][][

)(

0

0

,,

PPP

xPxPPP

yRHHRHx

P

PR

Hx

xy

Hxy

xx

x

TT


:

colorado center for astrodynamics research the university of colorado statistical orbit...

Documents

sequential filter

conventional kalman

extended kalman filter

givens solution

range rate data

data noise covariance

final report

fit residuals