University of ColoradoBoulder

ASEN 5070: Statistical Orbit Determination I

Fall 2014

Professor Brandon A. Jones

Lecture 9: Batch Estimator and Weighted LS

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Homework 3 – Due September 19◦ When exporting MATLAB figures, please use a high-quality

image format, e.g., PNG, EPS, etc.◦ Screen captures and JPEGs are typically not the best

option!

Lecture Quiz◦ Covers Lectures 6-8◦ Due Friday at 5pm

Friday and Next Week:◦ Probability and Statistics◦ Book Appendix A

Announcements

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Batch Estimator

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Linearized Equations

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Straightforward way to estimate the state at a time that matches the observations

What about when the observations cover multiple points in time?

LS with Linearized System

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What can we do to estimate the state when we have observations at multiple points in time?

Observations at multiple times?

◦ What tool(s) do we have available to alter the formulation?

◦ Given result from above, how might we alter the formulation to use a single relationship of the form:

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Reformulation for Epoch State

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Reformulation for Epoch State

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Process all observations over a given time span in a single batch◦ The alternative sequential methods will be

discussed later

What are the shortcomings of such a formulation?

The Batch Estimator

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Example Least Squares Problem

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Example Least Squares Problem

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Example Least Squares Problem

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Process all observations over a given time span in a single batch

The Batch Estimator

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No weighting of observations◦ How do we account for different sensors with

different accuracies?

No incorporation of previous information◦ Known a a priori state information◦ How do we include this in the filter?

Shortcomings of Basic LS

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Weighted Least Squares Estimation

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For each yi, we have some weight wi

We define a set of weights

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Effects of Weights in J(x)

Consider the case with two observations (m=2)

If w2 > w1, which εi will have a larger influence on J(x) ? Why?

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Derivation of Weighted LS Estimator

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For each yi, we have some weight wi

We define a set of weights

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Derivation of Weighted LS Estimator

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For the weighted LS estimator:

Weighted Least Squares Estimator

How do we find W ?

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What is the effect of W on the solution?

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What is the effect of W on the solution?

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Weighted Least Squares w/ A Priori

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A priori◦ Relating to or denoting reasoning or knowledge

that proceeds from theoretical deduction rather than from observation or experience

We have:

LS w/ A Priori Formulation

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As you will show in the homework:

LS w/ A Priori Solution

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Computation Method

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Concept Exercises

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Sent via e-mail shortly before lecture

Take a look between now and Friday◦ Feel free to work in groups!

Be ready to answer the questions at the start of lecture

The concept quiz will not be turned in for a grade

Directions