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ASEN 6008Interplanetary Mission Design
Statistical Orbit DeterminationA brief overview
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Before we start Stat OD, let’s talk about homework.
Eduardo is angry.
But first…..
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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
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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
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Ceres will enter into orbit on March 6, 2015 First mission to visit a dwarf planet
◦ Barely beats New Horizons.
Interesting IMD news
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Which missions require spacecraft navigation?
Orbit Determination
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Which missions require spacecraft navigation?
ALL OF THEM.
Orbit Determination
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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
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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
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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
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We have observations of a spacecraft at different points in time. How can we estimate its state?
Orbit Determination
X*Measurements
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Estimate the state using a filter
Orbit Determination
Observed RangeComputed Range
ε = O-C = “Residual”
X*
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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
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Setup.◦ Given: an initial state◦ Optional: an initial covariance
Overview of the Stat OD Process
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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
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What really happens◦ Of course, we don’t know this!
Review of the Stat OD Process
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Model reality as best as possible Propagate our initial guess of the state
Review of the Stat OD Process
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Goal: Determine how to modify to match
Review of the Stat OD Process
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Goal: Determine how to modify to match
Review of the Stat OD Process
Define
Want
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations
Review of the Stat OD Process
Define
Want
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Process:1. Track satellite
Review of the Stat OD Process
Perfect Observations
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Process:1. Track satellite
Review of the Stat OD Process
Perfect Observations
Computed Observations
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Process:1. Track satellite
Review of the Stat OD Process
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Process:1. Track satellite2. Map observations to state deviation
Review of the Stat OD Process
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations
Review of the Stat OD Process
Least Squares
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations4. Apply and repeat
Review of the Stat OD Process
Least Squares
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations4. Apply and repeat
Review of the Stat OD Process
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Astrodynamics Research
University of ColoradoBoulder 28
Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations4. Apply and repeat
Review of the Stat OD Process
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Astrodynamics Research
University of ColoradoBoulder 29
Process:1. Track satellite
Review of the Stat OD Process
Perfect Observations
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Process:1. Track satellite
Review of the Stat OD Process
Imperfect Observations
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations4. Apply and repeat
Review of the Stat OD Process
Same process, but the best estimate trajectory will never quite match the truth, since the observations have noise.
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Process:1. Track satellite2. Map observations to state deviation3. Determine how to adjust the state to best
fit the observations4. Apply and repeat
Review of the Stat OD Process
Least Squares
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Batch◦ Using Least-Squares or a variant
Sequential◦ CKF◦ EKF◦ UKF (others)
Filter Options
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How do we best fit the data?
A good solution, and one easy to code up, is the least-squares solution
Fitting the data
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Different ways to get a best estimate
Least Squares
Weighted Least Squares
Least Squares with a priori
Min Variance
Min Variance with a priori
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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)
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Perturb the initial state in the x direction◦ x0 = x0 + Dx
State Deviation and Linearization
37
Initial State:(x0, y0, z0, vx0, vy0, vz0)
Final State:(xf, yf, zf, vxf, vyf, vzf)
Initial State:(x0+Δx, y0, z0, vx0, vy0, vz0)
Force model: 0
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Propagate the deviated state
State Deviation and Linearization
38
Initial State:(x0, y0, z0, vx0, vy0, vz0)
Final State:(xf, yf, zf, vxf, vyf, vzf)
Force model: 0
Initial State:(x0+Δx, y0, z0, vx0, vy0, vz0)
Final State:(xf+Δx, yf, zf, vxf, vyf, vzf)
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Propagate the deviated state How is the final state altered?
State Deviation and Linearization
39
Initial State:(x0, y0, z0, vx0, vy0, vz0)
Final State:(xf, yf, zf, vxf, vyf, vzf)
Force model: 0
Initial State:(x0+Δx, y0, z0, vx0, vy0, vz0)
Final State:(xf+Δx, yf, zf, vxf, vyf, vzf)
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Propagate the deviated state How is the final state altered?
State Deviation and Linearization
40
Initial State:(x0, y0, z0, vx0, vy0, vz0)
Final State:(xf, yf, zf, vxf, vyf, vzf)
Force model: 0
Initial State:(x0+Δx, y0, z0, vx0, vy0, vz0)
Final State:(xf+Δx, yf, zf, vxf, vyf, vzf)
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Perturb the position in all 3 directions Now we have a matrix of partials relating the initial
state to the final state
State Deviation and Linearization
41
Initial State:(x0, y0, z0, vx0, vy0, vz0)
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
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Perturb the x velocity
State Deviation and Linearization
42
Initial State:(x0, y0, z0, vx0, vy0, vz0)
Final State:(xf, yf, zf, vxf, vyf, vzf)
Initial State:(x0, y0, z0, vx0-Δvx, vy0, vz0)
Force model: 0
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Perturb the x velocity
State Deviation and Linearization
43
Force model: 0
Initial State:(x0, y0, z0, vx0, vy0, vz0)
Final State:(xf, yf, zf, vxf, vyf, vzf)
Final State:(xf+tΔvx, yf, zf, vxf+Δvx, vyf, vzf)
Initial State:(x0, y0, z0, vx0+Δvx, vy0, vz0)
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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
State Deviation and Linearization
44
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How can we map state deviation errors?
State Deviation and Linearization
45
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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
State Deviation and Linearization
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The state transition matrix maps a deviation in the state from one epoch to another.
State Transition Matrix
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The state transition matrix is constructed via numerical integration, in parallel with the trajectory itself.
State Transition Matrix
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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
State Transition Matrix
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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
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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]
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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
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