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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 7: Generating the STM. Announcements. Lecture Quiz – Due Wednesday Homework 2 – Due September 13 Bring pen and paper on Friday!. Today’s Lecture. Previous Lecture

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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

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  1. ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 7: Generating the STM

  2. Announcements • Lecture Quiz – Due Wednesday • Homework 2 – Due September 13 • Bring pen and paper on Friday!

  3. Today’s Lecture • Previous Lecture • Linearization – State Transition Matrix (STM) • Alternative Derivation • Solution Methods

  4. Last Lecture – A Couple of Clarifications

  5. Least Squares: Derivation of Normal Form For now, assume G is a linear function • Differentiate with respect to x. What is the answer?

  6. Linearize About Reference

  7. Linearization – State Transition Matrix

  8. State Transition Matrix • Since 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.

  9. STM Differential Equation Constant! • What is the differential equation?

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

  11. State Transition Matrix – Alternative Derivation

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

  13. STM – Alternative Derivation

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

  15. State Transition Matrix – Laplace Transform

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

  17. Laplace Transform Tables

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

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

  20. Applied to Stat OD • Solve the ODE

  21. FEP STM – Laplace Transform

  22. FEP STM – Laplace Transform

  23. State Transition Matrix – Direct Approach

  24. ‘Direct Approach’ to Solving STM • Leverage the differential equation and combine it with analytic diff. eq. methods • Compatible with simple equations, but not with larger estimated state vectors or complicated dynamics

  25. Example Direct Approach

  26. Solution

  27. State Transition Matrix – Numeric Integration

  28. Numeric Integration • For more complicated dynamics, must integrate X*(t) and Φ(t,t0) simultaneously in propagator • Up to n+n2 propagated states • Derivative function must include the evaluation of the [A(t)]* matrix in addition to F(X,t)

  29. Example Implementation • Use the MATLAB reshape() command to turn matrix into a vector • v = reshape( V, nrows*ncols, 1 ); • MATLAB Demo…

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