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

Announcements Homework 2 – Due September 11 Lecture Quiz 2 – Due Friday @ 5pm Future Lectures Lecture 8 – Friday 9/11 @ 9am Lecture 9 – Monday 9/14 @ 9am Lecture 10 – Monday 9/14 @ 4pm Lecture 11 – Monday 9/21 @ 9am Lecture 12 – Monday 9/21 @ 4pm

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

Lecture Quiz 1

Question 1 Correct – 98% 2% 98% 0% 0%

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

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

New Integration Constant

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

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%

Time Systems: Time Scales

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%

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

Linearization – Why do we need it? (review)

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?

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

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

Linearize About Reference

Linearization – How do we do it? (continued)

Example Linearization: Planar Orbit

Now, what is the A matrix?

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

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

And the answer is: What is the final answer?

Linearize the Obs. Model Computed, not measured values!

Linearization – State Transition Matrix

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.

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

STM Identities

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)

State Transition Matrix – Alternative Derivation

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

STM – Alternative Derivation

Flat Earth Problem (FEP) STM – Alternative Derivation

State Transition Matrix – Laplace Transform

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.

Laplace Transform Tables

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

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

Applied to Stat OD Solve the ODE:

FEP STM – Laplace Transform

FEP STM – Laplace Transform