ASEN 5050
SPACEFLIGHT DYNAMICS
Spacecraft Navigation
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 36: Navigation
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Announcements
• FCQs today! End of class.
• STK Lab 3 due riiiiiight now!
• STK Lab 4 due 12/12
• Final Exam on 12/12, due 12/18
– Take-home, open book open notes
• Final project and exam due 12/18
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Schedule from here out
• 12/5: Spacecraft Navigation
• 12/8: Final Review, part 1
• 12/10: Final Review, part 2
• 12/12: Small Body Missions
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Space News
• Orion’s Exploration Flight Test 1: Friday 12/5 at 7:05 am
Eastern Time (5:05 am Mountain!). Duration: 4.5 hours.
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ASEN 5050
SPACEFLIGHT DYNAMICS
Navigation
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 36: Navigation
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Spacecraft Navigation
• Mission Design
– Define trajectories, error tolerances, navigation strategies, tracking
strategies, science requirements, etc.
• Launch
– Launch targets, guidance, tracking, performance estimation
• Hand-off to the spacecraft operators
– Best estimate of the spacecraft’s state and associated uncertainty
• Tracking and orbit determination
– Ground station tracking and maybe other sources of tracking (GNSS,
TDRSS, LiAISON, SoOp, etc)
– Satellite telemetry
– Estimate the spacecraft’s state and associated uncertainty
• Maneuver design and execution
– Maneuver guidance (attitude, timing, propulsion, execution)
– Maneuver execution estimation
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Spacecraft State
• The “State” ( X ) describes anything that the satellite
operators wish to estimate, with associated
uncertainties. Example state parameters:
– Satellite’s position and velocity (6 parameters)
– Satellite’s mass, Cd, Cr, and other similar parameters
– Maneuver execution parameters
– Science
• Atmospheric density profile (GOCE, DANDE, etc)
• Gravity signatures (GRACE)
– Others as needed
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Spacecraft State Uncertainty
• The state is estimated and an uncertainty is associated
with that estimate.
– State = X
– Uncertainty = P, the variance-covariance matrix
Symmetric
matrix
Useful variation, showing only
the correlations in the lower-left
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Orbit Determination
• The art of estimating the state of a satellite in the
presence of a great deal of uncertainty.
• We start with either no information or an estimate of
the state.
• We model the dynamics (gravity, SRP, drag, etc)
• We acquire tracking data, including noise and
systematic errors
• Filter the data and generate an estimate of the state.
• Take ASEN 5070 for a lot more information.
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Tracking Data
• Ground stations
–
–
–
–
1-way, 2-way, 3-way
Range, Doppler, RA, Dec
Delta-DOR
Other beacons, Laser reflectors
• Satellite tracking
– TDRSS, GNSS, LiAISON
• Telemetry information
– Gyros, accelerometers, propulsion system data
• Other
– OpNav, SoOps, X-Ray Pulsar, Radar, Lidar, etc.
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Methods of Initial Orbit Determination
• Observations of range, azimuth, and elevation
• Angles-only observations
– Laplace’s Method
– Gauss’s Technique
– Double r-iteration
• Mixed Observations
– Range and Range-Rate Processing
– Range-only Processing
• Three Position Vectors and Time
– Gibbs Method
– Herrick-Gibbs
• Two Position Vectors and Time  Lambert’s Problem
–
–
–
–
Minimum Energy
Gauss’s Solution
Universal Variables
Battin Method
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Range, Azimuth, Elevation
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Range, Azimuth, and Elevation
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Angles-Only Observations
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Three Positions and Time
Problem Statement: Given three time-sequential position vectors, determine the
velocity vector corresponding to the 2nd position vector. Note that determining the
position and velocity vectors at the 2nd time essentially completely determines the orbit,
and they may be converted into a complete set of orbital elements.
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Gibbs Method
We have covered a lot of equations here, so lets summarize the
Gibbs Method:
Given: r1, r2 , and r3
Determine: v2
Algorithm:
Z12 = r1 ´ r2 Z23 = r2 ´ r3 Z31 = r3 ´ r1
æ Z ×r ö
acop = 90°- cos-1çç 23 1 ÷÷
è Z23 r1 ø
Check if too close together:
r1 × r2
r2 × r3
cosa12 =
and cosa23 =
r1 r2
r2 r3
Check if coplanar:
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Gibbs Method
N = r1Z23 + r2 Z31 + r3 Z12
D = Z12 + Z23 + Z31
S = (r2 - r3 )r1 +(r3 - r1)r2 +(r1 - r2 )r3
B º D ´ r2 and Lg º
m
ND
Compute velocity:
Lg
v2 =
B + Lg S
r2
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Lambert’s Problem
Given two positions and the time-of-flight between them, determine
the orbit between the two positions.
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Orbit Determination
• If your state has 6 parameters and you get 6 pieces of
information, you can estimate each parameter – but
you are very vulnerable to modeling and
measurement errors!
• Standard OD methods involve collecting as much
data as possible and using a least-squares estimate to
fit the state to the data.
– i.e., 106 pieces of information, including noise and errors!
– Batch processor
– Kalman Filters
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Orbit Determination
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Things to Note
• Not all observations are created equal.
– Range and Doppler data only provides line-of-sight
information.
– Angular data – or data over a long arc – may be very useful
to improve the observability of the state.
• Clean data should be weighted higher than noisy data.
• The satellite is never where you think it is. Pay
attention to the covariance matrix!
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Maneuver Planning
• Once you have an
estimate of your state,
you may require a
maneuver downstream
in order to achieve the
mission’s
requirements.
• What are your
acceptable error
corridors for each
orbital element / state
parameter?
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Maneuver Execution Error
• Maneuvers are never performed perfectly
– Though they’re getting better and better.
• There is typically a minimum expected execution
error – you can’t perform better without switching to
a different system / nozzle / flow rate.
• There is also typically a proportional error that grows
with the maneuver’s duration / magnitude.
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Maneuver Navigation
• Gates Model, describing maneuver execution errors.
– C. R. Gates, “A Simplified Model of Midcourse Maneuver Execution
Errors,” Tech. Rep. 32-504, JPL, Pasadena, CA, October 15, 1963.
• Error in pointing of the burn
• Error in magnitude direction of burn
• Sample these errors and apply them to Monte Carlo
simulations.
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Gates Model
• Error in pointing of the burn
• Error in magnitude direction of burn
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Gates Model
• Now, define a coordinate frame, where z is aligned
with the commanded Delta-V direction:
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Gates Model
• Sample a simple Gaussian distribution
times to generate
three
• Execution errors, in maneuver frame:
• Finally, rotate them into any useful frame.
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Navigation
• Example Gates Model inputs into a navigation study:
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Flight Operations
• Good practices in spaceflight navigation
– Practice! Tune your filter on all kinds of data. Be ready for a wide variety
of data.
– Perform lots of solutions and compare them.
• Solutions over different arcs of time, long and short
• Solutions with different data types
• Solutions with different parameters, consider parameters, process noise, etc.
– Present the information in a manager-friendly format, but understand the
details.
• B-Plane, error ellipses, covariance, residuals
– Look at that covariance in different ways.
– Don’t neglect the numerical details.
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Missions that require navigation
• All of them.
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Spacecraft missions
• All spacecraft require OD
– For communication
• Most spacecraft require OD for other reasons
– For science
– For mission planning
– For maneuver design
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ASEN 5050
SPACEFLIGHT DYNAMICS
FCQs
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 36: Navigation
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