ASEN 5070 Statistical Orbit Determination I Fall 2012 Professor Jeffrey S. Parker

CCARColorado Center for

Astrodynamics Research

University of ColoradoBoulder 1

ASEN 5070Statistical Orbit Determination I

Fall 2012

Professor Jeffrey S. ParkerProfessor George H. Born

Lecture 18: CKF, Numerical Compensations




Homework 6 and 7 graded asap Homework 8 announced today.

Exam 2 on the horizon. Actually, it’s scheduled for 11/8! A week from Thursday.

We still have quite a bit of material to cover. Exam 2 will cover:

◦ Batch vs. CKF vs. EKF◦ Probability and statistics (good to keep this up!)◦ Observability◦ Numerical compensation techniques, such as the Joseph and Potter

formulation.◦ No calculators should be necessary◦ Open Book, Open Notes

Announcements




Quiz 13 Review




Quiz 13 Review




Quiz 13 Review




Quiz 13 Review

Only 46% of the class got this right




Quiz 13 Review




Quiz 13 Review




Quiz 13 Review




Quiz 13 Review

0% of the class got this right!




Quiz 13 Review

Observable quantities:• Semimajor axis of both orbits• Eccentricity of both orbits• True anomaly of both satellites• Inclination difference between the orbits• Node difference between the orbits• Arg Peri difference between the orbits




Quiz 14 Review




Quiz 14 Review

100% of the class got this right!




Quiz 14 Review




Quiz 14 Review




Quiz 14 Review




Quiz 14 Review




Quiz 14 Review




Quiz 14 Review




Quiz 14 Review




Quiz 14 Review

Probably the hardest question on this quiz and 78% got it right.




Homework 8

Due in 9 days

Assignment #8



University of ColoradoBoulder

CKF vs. Batch 1 1

, , , , ,k k k k kn n n p n p n nn nx P y K P

The Kalman (sequential) filter differs from the batch filter as follows:

1. It uses in place of . 2. It uses in place of

H 1,k kt t 0,kt t

Or if we don’t reinitialize, then

10 1 0, ,k kt t t t

1,k kt t i.e., we must invert at each stage

This means that is reinitialized at each observation time.

1 0,kt t




Processing Observations One at a TimeGiven a vector of observations at we may process these as scalar observations by performing the measurement update times and skipping the time update.

1p pp

kt

Algorithm1. Do the time update at kt

2. Measurement update

1 1 1, ,Tk k k k k kP t t P t t

1 1ˆ,k k k kx t t x




Processing Observations One at a Time

2a. Process the 1st element of . ComputekY *1 1 1 ,ky Y G

where





2 1

2 1

ˆHere k k

k k

x x

P P

2. Do not do a time update but do a measurement update by processing the 2nd element of (i.e. and are not mapped to )

kY 1ˆkx 1k

P1kt

Compute

etc.

Process pth element of . kY Compute





let ˆ ˆ ,pk kx x

pk kP P

Time update to and repeat this procedure

Note: must be a diagonal matrix. Why?

If not we would apply a “Whitening Transformation” as described in the next slides.

kR

1kt




Whitening Transformation

0E TE R

(1)Where is any sort of observation error covariance

R





0E TE R

(1)

Factor where is the square root of TR SS S R

is chosen to be uppertriangularS

multiply Eq. (1) by 1S

Where is any sort of observation error covariance

R





0E TE R

(1)

Factor where is the square root of TR SS S R

is chosen to be uppertriangularS

multiply Eq. (1) by 1S

let 1 ,y S y 1S

Where is any sort of observation error covariance

R





Now 1 0E S E

1 1T T T TE S E S S RS

1 ,T TS SS S

Hence, the new observation has an error with zero mean and unit variance. We would now process the new observation and use .

1y S yy




Notice the decomposition in the whitening algorithm. ◦ S is a “square root” of R.

Several ways of computing S. A good way is the Cholesky Decomposition

(Section 5.2).

Major benefit of S vs. R: if the condition number of R is large, say 16, the condition number of S is half as large, say 8.

It’s much more accurate to invert S than R.

Cholesky DecompositionTR SS




Let M be a symmetric positive definite matrix, and let R be an upper triangular matrix computed such that

By inspection,

Cholesky Decomposition

Etc. See 5.2.1




M:

Use Matlab’s “chol” function to compute R:





Consider the case where we have the least squares condition

If M is poorly conditioned, then inverting M will not produce good results.

If we decompose M first, then the accuracy of the solution will improve.

Let Then

Easy to forward-solve for z and then we can backward-solve for x-hat.





Notice that the Cholesky Decomposition includes square roots. These are very expensive operations!

A square root free algorithm is given in 5.2.2.

Idea:





Processing an observation vector one element at a time.

Whitening Cholesky

Quick Break

Next topic: Joseph, Potter, …

Questions?




Least squares estimation began with Gauss

1963: Kalman’s sequential approach◦ Introduced minimum variance◦ Introduced process noise◦ Permitted covariance analyses without data

Schmidt proposed a linearization method that would work for OD problems◦ Supposed that linearizing around the best estimate trajectory is better than linearizing

around the nominal trajectory

1970: Extended Kalman Filter

Gradually, researchers identified problems.◦ (a) Divergence due to the use of incorrect a priori statistics and unmodeled parameters.◦ (b) Divergence due to the presence of nonlinearities.◦ (c) Divergence due to the effects of computer round-off.

Kalman Filter History




Numerical issues cause the covariance matrix to lose their symmetry and nonnegativity

Possible corrections:◦ (a) Compute only the upper (or lower) triangular entries and force symmetry◦ (b) Compute the entire matrix and then average the upper and lower fields◦ (c) Periodically test and reset the matrix◦ (d) Replace the optimal Kalman measurement update by other expressions

(Joseph, Potter, etc)◦ (e) Use larger process noise and measurement noise covariances.





Potter is credited with introducing square root factorization.◦ Worked for the Apollo missions!

1968: Andrews extended Potter’s algorithms to include process noise and correlated measurements.

1965 – 1969: Development of the Householder transformation◦ Worked for Mariner 9 in 1971!

1969: Dyer-McReynolds filter added additional process noise effects.◦ Worked for Mariner 10 in 1973 for Venus and Mercury!





Batch:◦ Large matrix inversion w/ potential poorly-conditioned matrix.

Sequential:◦ Lots and lots of matrix inversions w/ potentially poorly-conditioned

matrices.

EKF:◦ Divergence: If the observation data noise is too large.◦ Saturation: If covariance gets too small, new data stops influencing the

solution.

Numerical Issues




Replace

with

This formulation will always retain a symmetric matrix, but it may still lose positive definiteness.

Joseph Formulation




Equivalence of Joseph and Conventional Formulation of the Measurement Update of P

Need to show that

TKH P KH KRK KH P

First derive the Joseph formulation

x̂ x K y Hx subst. y Hx

x̂ x K Hx Hx

x̂ x KH x x K

subtract from both sides and rearrangex

x̂ x x x KH x x K

KH x x K




form ˆ ˆ TP E x x x x

T TP KH P KH KRK

where

,TP E x x x x TR E

0,TE x x since is independent of ix t it





Next, show that this is the same as the conventional Kalman

T T T T TKH P KH KRK KH P KH PH K KRK

T T T T TKH P PH K KHPH K KRK

T T T TKH P PH K K HPH R K





Next, show that this is the same as the conventional Kalman

T T T T TKH P KH KRK KH P KH PH K KRK

T T T T TKH P PH K KHPH K KRK

T T T TKH P PH K K HPH R K

subst. for 1T TK PH HPH R

1T T T T T TKH P PH K PH HPH R HPH R K

T T T TKH P PH K PH K

KH P





Question on Joseph?

Next: Demo!




Example Illustrating Numerical Instability of Sequential (Kalman) Filter (see 4.7.1)The following example problem from Bierman (1977) illustrates the numerical characteristics of several algorithms we have studied to date. Consider the problem of estimating and from scalar measurements and 1x 2x 1z 2z

1 1 2 1z x x

2 1 2 2z x x

Where and are uncorrelated zero mean random variables with unit variance. If they did not have the above traits we could perform a whitening transformation so that they do. In matrix form

1 2

1 1 1

2 2 2

14.7.20

1 1z xz x




The a priori covariance associated with our knowledge of is assumed to be

1

2

xX

x

2 20

1 00 1

P I

Where and . The quantity is assumed to be

small enough such that computer round-off produces

1 0 1

21 1 4.7.21 This estimation problem is well posed. The observation provides an estimate of which, when coupled with should accurately determine . However, when the various data processing algorithms are applied several diverse results are obtained.

1z2z1x 2x

Example Illustrating Numerical Instability of Sequential (Kalman) Filter (see 4.7.1)




Let the gain and estimation error covariance associated with be denoted as and respectively. Similarly the measurement is associated with and respectively. Note that this is a linear parameter (constant) estimation problem.

Hence,

1z 1k1P 2z 2k 2P

, kt t I

1k kP P

1k k k kP I k h P 1

1 1 1 1,2T Tk k k k k kk P h h P h k

We will process the observation one at a time. Hence,





Note: Since we will process the observations one at a time, the matrix inversion in previous equation is a scalar inversion. The exact solution is

1

1 0 1 1 0 1T Tk P h h P h

1

2 21 11 1

1 2

11 4.7.221 2

k

where 2 2 1





1 2

21 4.7.231

P

where 21 2


The estimation error covariance associated with processing the first data point is

21 2

11 11 2

P I I

1 1 1 0P I k h P




2 2 2 1P I k h P

The exact solution for is given by:2P

2

2

1 2 11 (4.7.24)1 2

Example Illustrating Numerical Instability of Sequential (Kalman) Filter (see 4.7.1)Processing the second observation:

1

2 1 2 2 1 2 1T Tk Ph h Ph

where 2 21 2 2 2

2

1 211




The conventional Kalman filter yields (using )21 1

1

1k

21

11P I I

1 2

0(4.7.25)P


Note that is no longer positive definite. The diagonals of a matrix must be

positive. However does exist since .1P PD

11P

1 0P




1

2 1 2 2 1 2 1T Tk Ph h Ph

1

11 2

2 2 2 1P I k h P 1 11 (4.7.26)1 11 2

Compute P2 using the Conventional Kalman

Now is not positive definite ( ) nor does it have positive terms on the diagonal. In fact, the conventional Kalman filter has failed for these observations.

2P 2 0P





The Joseph formulation yields

2 2 2 1 2 2 2 2T TP I k h P I k h k k





The Batch Processor yields

21 11 1 1

1I

12

2

2 1

1 2 1

12

1

101

ii

Ti PhhP


P2 for the batch

2

1 2 (1 3 )1 3 2 4

P




P2 for the batch

2

1 2 (1 3 )1 3 2 4

P

To order , the exact solution for is2P

2

1 1 1 2 1 31 2

1 2 1 3 2 4P





Summary of Results2P

Exact to order 1 2 (1 3 )

(1 3 ) 2 4

Conventional Kalman

1 111 11 2

Joseph

1 2 (1 3 )(1 3 ) 2

Batch

1 2 (1 3 )(1 3 ) 2 4





Homework 6 and 7 graded asap Homework 8 announced today.

Exam 2 on the horizon. Actually, it’s scheduled for 11/8! A week from Thursday.

We still have quite a bit of material to cover. Exam 2 will cover:

◦ Batch vs. CKF vs. EKF◦ Probability and statistics (good to keep this up!)◦ Observability◦ Numerical compensation techniques, such as the Joseph and Potter

formulation.◦ No calculators should be necessary◦ Open Book, Open Notes

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ASEN 5070 Statistical Orbit Determination I Fall 2012 Professor Jeffrey S. Parker

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