1. ASEN 5070: Statistical Orbit Determination I Fall 2015
Professor Brandon A. Jones
Lecture 7: Linearization and the State Transition Matrix

2. Announcements Homework 2 – Due September 11
Lecture Quiz 2 – Due 5pm
Future Lectures
Lecture 8 – Friday 9am
Lecture 9 – Monday 9am
Lecture 10 – Monday 4pm
Lecture 11 – Monday 9am
Lecture 12 – Monday 4pm

3. Today’s Lecture Lecture Quiz 1 Results
Linearization – How we do it? (wrap-up)
State Transition Matrix (STM)
Derivation
Solution Methods

4. Lecture Quiz 1

5. Question 1
Correct – 98% 2%
98% 0% 0%

6. Three Levels of OD: Low accuracy (~1 km) Medium accuracy (~100 m)
Low resolution imaging, space surveillance
Medium accuracy (~100 m)
Medium resolution imaging, orbit prediction, laser tracking
High accuracy (<10 cm)
Relative motion/formation flying
Scientific studies of the Earth

7. Question 2
Correct – 80%
We are modeling the forces acting on a spacecraft using only the two- body force and gravity perturbations (e.g., spherical harmonics model).  Our ECF/ECI coordinate system transformation is the full model, i.e., it includes the Earth’s rotation, nutation/precession, bias, etc. The spherical harmonic model we use has degree and order 70. We propagate a test orbit, and see that the energy
is not constant.  This means that our orbit propagator has an error.
True
False

8. New Integration Constant

9. What about HW 2? Derived from a time-varying potential
In the inertial frame, the potential varies due to the Earth’s rotation
If we assume pure rotation and only zonal harmonics (Jn terms), there is no change in time

10. Question 3
Correct – 76%
Which of these time systems uses the definition of the second based on atomic time?
Atomic Time (TAI)
Terrestrial Time (TT)
GPS Time
UT1
96%
36%
89%
16%

11. Time Systems: Time Scales

12. Question 4
Correct – 27%
Which of the following are assumed when using the Newton-Raphson method of solving a nonlinear system
The observations are nonlinear, but the dynamics are linear
The equations describing the dynamics and the observations are nonlinear
The observations (or known data) have no errors
The number of observations equals the number of unknowns
27%
36%
60%
64%

13. Question 5 Correct – 98% Atmospheric drag is a conservative force True
False

14. Linearization – Why do we need it? (review)

15. General Estimation Problem
How do we estimate X ?
How do we estimate the errors εi?
How do we account for force and observation model errors?

16. Linear Problem
For now, let’s consider a linear problem:

17. Normal Form of Least Squares Estimator
This is the “normal equation” for the least squares estimator
“hat” notation indicates vector that minimizes J(x)
We treat this as the estimated state
We assumed that the state-observation relationship was linear, but the orbit determination problems is nonlinear
We will linearize the formulation of the problem

18. Linearize About Reference

19. Linearization – How do we do it? (continued)

20. Example Linearization: Planar Orbit

21. Now, what is the A matrix?

22. And the answer is:
Which terms are non-zero?

23. And the answer is: Which terms equal 1?
What are the partials w.r.t. μ?

24. And the answer is:
What is the final answer?

25. Linearize the Obs. Model
Computed, not measured values!

26. Linearization – State Transition Matrix

27. State Transition Matrix
When x is linear (note lower case!) then there exists a solution to the linear, first order system of differential equations:
The solution is of the form:
Φ(t,ti) is the state transition matrix (STM) that maps x(ti) to the state x(t) at time t.

28. STM Differential Equation
Constant!
Why?
What is the differential equation for the STM?

29. STM Identities

30. Methods to Generate the STM
There are four methods to generate the STM:
Solve from the direct Taylor expansion
If A is constant, use the Laplace Transform or eigenvector/value analysis
Analytically integrate the differential equation directly
Numerically integrate the equations (ode45)

31. State Transition Matrix – Alternative Derivation

32. STM – Alternative Derivation
Expand X(t) in a Taylor series about X*(t):

33. STM – Alternative Derivation

34. Flat Earth Problem (FEP) STM – Alternative Derivation

35. State Transition Matrix – Laplace Transform

36. Laplace Transforms
Laplace Transforms are useful for analysis of linear time-invariant systems:
electrical circuits,
harmonic oscillators,
optical devices,
mechanical systems,
even some orbit problems.
Transformation from the time domain into the Laplace domain.
Inverse Laplace Transform converts the system back.

37. Laplace Transform Tables

38. Example
Solve the ODE
We can solve this using “traditional” calculus:

39. Example
Solve the ODE
Or, we can solve this using Laplace Transforms:

40. Applied to Stat OD
Solve the ODE:

41. FEP STM – Laplace Transform

42. FEP STM – Laplace Transform