university of colorado boulder asen 6008 interplanetary mission design statistical orbit...

CCARColorado Center for

Astrodynamics Research

University of ColoradoBoulder 1

ASEN 6008Interplanetary Mission Design

Statistical Orbit DeterminationA brief overview




Before we start Stat OD, let’s talk about homework.

Eduardo is angry.

But first…..




Watch your significant figures! We don’t need to know velocities to the

pm/s. For most situations (especially HW in this

class)◦ Positions: meter level accuracy ◦ Velocities: m/s or cm/s accuracy

Significant Figure




Dawn spacecraft is nearing dwarf planet Ceres

Ceres orbit is between Mars and Jupiter in the main asteroid belt

Ceres was discovered by Father Giuseppe Piazzi 1801. ◦ Ceres was initially classified as a planet and later

demoted an asteroid◦ Ceres was upgraded a dwarf planet in 2006, along

with Pluto (downgraded) and Eris.

Interesting IMD news




Ceres will enter into orbit on March 6, 2015 First mission to visit a dwarf planet

◦ Barely beats New Horizons.

Interesting IMD news




Which missions require spacecraft navigation?

Orbit Determination




Which missions require spacecraft navigation?

ALL OF THEM.

Orbit Determination




Orbit determination is an essential part of any mission

There are (multiple) courses here at CU devoted to the art of OD◦ ASEN 5050, ASEN 6080

If you have any future interest in trajectory design, orbit determination is a highly useful (i.e., essential) skill to have

Orbit Determination




Do we actually know exactly where a spacecraft is?◦ No, there are many sources of error◦ Modeling errors◦ Launch errors◦ Spacecraft performance◦ Observation errors

Orbit Determination




Tracking Data Tracking data may include many types of data – and often should

include many types of data:◦ Ground observations:

Doppler Range 1-Way, 2-Way, 3-Way Angles when very near the Earth Delta-DOR when further from Earth

◦ Relative to other spacecraft, vehicles, bodies GPS Autonav LiAISON Formation Flying

◦ Spacecraft measurements: Accelerations, including drag-free corrections, thrust, etc. Measured mass-flow Attitude measurements




We have observations of a spacecraft at different points in time. How can we estimate its state?

Orbit Determination

X*Measurements




Estimate the state using a filter

Orbit Determination

Observed RangeComputed Range

ε = O-C = “Residual”

X*




What really happens◦ Satellite travels according to the real forces in the

universe◦ We model the motion to the best of our ability, but our

force models contain errors

Overview of the Stat OD Process




Setup.◦ Given: an initial state◦ Optional: an initial covariance

Overview of the Stat OD Process




Setup.◦ Given: an initial state◦ Optional: an initial covariance

◦ The satellite will not be there, but will (hopefully) be nearby True state =

Review of the Stat OD Process




What really happens◦ Of course, we don’t know this!





Model reality as best as possible Propagate our initial guess of the state





Goal: Determine how to modify to match





Goal: Determine how to modify to match


Define

Want




Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best

fit the observations


Define

Want




Process:1. Track satellite


Perfect Observations







Computed Observations




Process:1. Track satellite2. Map observations to state deviation






fit the observations


Least Squares





fit the observations4. Apply and repeat


Least Squares






Imperfect Observations







Same process, but the best estimate trajectory will never quite match the truth, since the observations have noise.







Least Squares




Batch◦ Using Least-Squares or a variant

Sequential◦ CKF◦ EKF◦ UKF (others)

Filter Options




How do we best fit the data?

A good solution, and one easy to code up, is the least-squares solution

Fitting the data




Different ways to get a best estimate

Least Squares

Weighted Least Squares

Least Squares with a priori

Min Variance

Min Variance with a priori




How can we map state deviation errors?

State Deviation and Linearization

Final State:(xf, yf, zf, vxf, vyf, vzf)

Example: Propagating a state in the presence of NO forces

Initial State:(x0, y0, z0, vx0, vy0, vz0)




Perturb the initial state in the x direction◦ x0 = x0 + Dx


37



Initial State:(x0+Δx, y0, z0, vx0, vy0, vz0)

Force model: 0




Propagate the deviated state


38



Force model: 0


Final State:(xf+Δx, yf, zf, vxf, vyf, vzf)




Propagate the deviated state How is the final state altered?


39



Force model: 0






Propagate the deviated state How is the final state altered?


40



Force model: 0






Perturb the position in all 3 directions Now we have a matrix of partials relating the initial

state to the final state


41


Initial State:(x0+Δx, y0+Δy, z0+Δz,

vx0, vy0, vz0)

Final State:(xf+Δx, yf+Δy, zf+Δz,

vxf, vyf, vzf)

Force model: 0




Perturb the x velocity


42



Initial State:(x0, y0, z0, vx0-Δvx, vy0, vz0)

Force model: 0




Perturb the x velocity


43

Force model: 0



Final State:(xf+tΔvx, yf, zf, vxf+Δvx, vyf, vzf)

Initial State:(x0, y0, z0, vx0+Δvx, vy0, vz0)




The relationships describing the change to the final state based on deviations to the initial state are simple because we assumed no forces

The relationships are more complicated with non-linear dynamics


44




How can we map state deviation errors?


45




Linearization Introduce the state deviation vector

If the reference/nominal trajectory is close to the truth trajectory, then a linear approximation is reasonable.

Taylor Series Expansion





The state transition matrix maps a deviation in the state from one epoch to another.

State Transition Matrix




The state transition matrix is constructed via numerical integration, in parallel with the trajectory itself.





Computing the individual partials of the A matrix

You can do it by hand if you enjoy that sort of thing

Alternatively, use MATLAB’s symbolic toolbox◦ A = jacobian(F,X)◦ See MATLAB help file for the jacobian function





Sample MATLAB code

syms mu x y r xdot ydot xddot yddot r = sqrt(x^2 + y^2); xddot = -mu*x/r^3;yddot = -mu*y/r^3; X = [x, y, xdot, ydot];Xdot = [xdot, ydot, xddot yddot]; A = jacobian(Xdot, X); A = subs(A,'(x^2+y^2)^(5/2)','r^5');A = subs(A,'(x^2+y^2)^(3/2)','r^3');A = subs(A,'(x^2+y^2)^(1/2)','r');A = subs(A,'(x^2+y^2)','r^2');

Use the symbolic toolbox to compute the partials for you




Sample MATLAB code output

A = [ 0, 0, 1, 0][ 0, 0, 0, 1][ (3*mu*x^2)/r^5 - mu/r^3, (3*mu*x*y)/r^5, 0, 0][ (3*mu*x*y)/r^5, (3*mu*y^2)/r^5 - mu/r^3, 0, 0]




If you haven’t taken ASEN 5070, it’s a good idea to do so.

For more information on Stat OD visit: http://ccar.colorado.edu/asen5070/

ASEN 5070

http://ccar.colorado.edu/asen5070/

http://ccar.colorado.edu/asen5070/

university of colorado boulder asen 6008 interplanetary mission design statistical orbit...

Documents