University of ColoradoBoulder

ASEN 5070: Statistical Orbit Determination I

Fall 2014

Professor Brandon A. Jones

Lecture 15: Statistical Least Squares and

Estimation of Nonlinear System

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Lecture Quiz Due by 5pm

Homework 5 Due Friday

Exam 1 – Friday, October 11

Announcements

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Statistical Least Squares w/ a priori

SLS and Estimation of Nonlinear System

Today’s Topics

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Statistical Interpretation of Least Squares

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Weighted Least Squares

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Observation Errors

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State Estimation Error Description

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State Estimation Error Description

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State Estimation Error Description

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Statistical LS w/ a priori

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Statistical LS w/ a priori

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Statistical LS w/ a priori

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Statistical LS w/ a priori

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State Estimation Error Description

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Still need to know how to map measurements from one time to a state at another time!

Measurement Mapping

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State Update

Since we linearized the formulation, we can still improve accuracy through iteration (more on this in a future lecture)

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Statistical Least Squares Solution for NonlinearSystem

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Computation Algorithm of the Batch Processor

p. 196-197 of textbook (includes corrections)

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Computation Algorithm for the Batch Processor

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Why Reuse A Priori Information?

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The batch filter depends on the assumptions of linearity◦ Violations of this assumption may lead to filter divergence◦ If the reference trajectory is near the truth, this holds just

fine

The batch processor must be iterated 2-3 times to get the best estimate◦ The iteration reduces the linearization error in the

approximation

Continue the process until we “converge”◦ Definition of convergence is an element of filter design

Assumptions with the Iterated Batch

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Post-fit Residuals RMS

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Convergence via Post-fit Residuals

If we know the observation error, why “fit to the noise”?

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No improvement in observation RMS

Other Convergence Tests

No reduction in state deviation vector

Maximum number of iterations

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Instantaneous observation data is taken from three Earth-fixed tracking stations◦ Why is instantaneous important in this context?

LEO Orbit Determination Example

x, y, z – Satellite position in ECI xs, ys, zs are tracking station locations in ECEF

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Effects of Iteration

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Improved Fit to Data

0 100 200 300 400-2000

0

2000Range Residuals (m)

Pas

s 1

observation number0 100 200 300 400

-20

0

20Range Rate Residuals (m/s)

observation number

0 100 200 300 400-1

0

1

Pas

s 2

observation number0 100 200 300 400

-5

0

5x 10

-3

observation number

0 100 200 300 400-0.05

0

0.05

Pas

s 3

observation number0 100 200 300 400

-2

0

2x 10

-3

observation number

RMS Values (Range σ=0.01 m, Range-Rate σ = 0.001 m/s)

Pass 1 Pass 2 Pass 3

Range (m) 732.748

0.319 0.010

Range Rate (m/s)

2.9002 0.0012 0.0010

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Estimated State Uncertainty

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Estimated State Uncertainty

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Estimated State Uncertainty

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Advantage of Different Data Types

Image: Hall and Llinas, “Multisensor Data Fusion”, Handbook of Multisensor Data Fusion: Theory and Practice, 2009.

FLIR – Forward-looking infrared (FLIR) imaging sensor

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Inverting a potentially poorly scaled matrix

Solutions:◦ Matrix Decomposition (e.g., Singular Value Decomposition)◦ Orthogonal Transformations◦ Square-root free Algorithms

Numeric Issues◦ Resulting covariance matrix not symmetric◦ Becomes non-positive definite (bad!)

Batch Processor Issues