AE554 Applied Orbital Mechanics

Orbit Estimation BasicsEgemen Đmre

203/01/2008

Orbit Estimation/Determination

� So far we looked into setting up a mathematical model to simulate the trajectory (propagation)

� This fits in the estimation framework, where noisy measurement data is coupled with a semi-decent mathematical model.

� Propagation: Project the orbit to a given time� Given initial conditions (good or bad? GIGO?)

� Estimation: Determine the orbit at a given time� Given set of past measurement data

� Post-processing / on-the-fly

303/01/2008

Orbit Estimation

� The idea is to “estimate” the orbit (or position and velocity) at any given time � Current

� Past

� This is really a trajectory fit to the measurements

measurements Fitted trajectory

Estimated initial orbit Estimated final orbit

403/01/2008

Measure what, exactly?

� Range (between station and satellite) : � time-of-flight principle� One-way, two-way and three-way measurement

� GPS pseudorange � Radar: ranging or altimetry� Laser (extremely accurate): ground-based ranging or satellite-borne

altimetry

� Range rate (between station and satellite)� Repeated pulse transmission (discrete)� Transmitter beacon (continuous i.e., fixed frequency)

� Doppler effect

� Angles� Photo of satellite streak against stationary satellites

503/01/2008

Measurement noise

� Noise: Stuff that makes the measurements deviate from our mathematical models

� Atmospheric effects� Frequency dependent (laser and RF different)� Tropospheric delay� Ionospheric delay

� Total electron content (TEC) along the path� Spatial and temporal changes� Dual frequency measurements to cancel some terms� Meters to tens of meters for GPS frequencies

� Relativistic effects

603/01/2008

Estimation Basics

� Linearised orbit model and state transition matrix (STM)� STM relates initial state to state at any given time

� Observation model: � Relates observed quantities to the state

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703/01/2008

Estimation Solutions - LSQ

� Least squares solution:� State estimation that minimises the sum of the squares of

the “observation residuals” (i.e., minimise ε)

� Problems:� Equal weight to each observation

� Uncorrelated observation errors� Statistical nature of the problem ignored (usually Gaussian

assumption)

� Other solutions exist (weighted least squares, iterative methods etc.)

803/01/2008

Estimation Solutions – Min Variance

� Minimum Variance Estimate (“Batch” or “Sequential”Estimation)� Statistical properties of the estimation

� Mean and covariance matrix associated with the random (Gaussian)error

� Covariance matrix: The “cloud” of possible locations of the state

� Pendulum example!� Needs to be propagated with time� Use State Transition Matrix! (linear estimate) (i.e., Jacobian)

� Find the estimate of the state:� Linear: linear combinations of measurements � Unbiased: zero mean� Minimum variance: Estimation error covariance matrix to be

minimised

903/01/2008

Estimation Solutions - KF

� Kalman Filter (Sequential Estimation)� In control theory, the Kalman filter is most commonly

referred to as linear quadratic estimation (LQE).� Also a “recursive estimator” as it requires previous state estimate

and current measurement to generate current state estimate� Unlike batch estimators

� Estimate the current state using the available measurements

� Add process noise - assumed to be zero mean normal distribution with covariance Qk

111 −−− ++= kkkkkk wuBxFx

Previous state

State transition model

Control vector

Control-input model

Process noise

1003/01/2008

Estimation Solutions - KF

� Predict� Predicted state

� Predicted estimate covariance

� Update� Innovation or measurement residual

� Innovation (or residual) covariance

� Optimal Kalman gain

� Updated state estimate

� Updated estimate covariance

Error covariance matrix (a measure of the estimated accuracy of the state estimate)

1103/01/2008

Estimation Solutions – KF Types

� Classical KF (1960): � Covariance matrix propagation via linearised equations� Good for linear-like and well-behaving problems

� Extended KF (1966?): � Reinitialise reference trajectory after each observation� State transition and observation models are differentiable non-linear

functions� Have to calculate Jacobian for (linear) covariance updates� Essentially linearisation of the non-linear function around the current

estimate� Handles non-linearities better

� Quicker convergence� Problems:

� Not an optimal estimator� Sensitive to mismodelling

1203/01/2008

Estimation Solutions – UKF

� Unscented KF (1997):� EKF performs poorly on highly non-linear problems due to

the linearisation� Only “mean” is propagated non-linearly

� Can we use non-linear functions to propagate the covariance matrix?

� Yes, but only if we take sample points around the mean (2n+1 sigma points for n element state)

� No need to calculate Jacobian!!

1303/01/2008

Estimation Solutions – KF Issues

� User provided estimates for the model or observation “reliability” (or 3s values of modeling or measurement errors)

� Too big a difference and model or observation will be ignored!

� Filters have a mind of their own; parameters to tune!

� Convergence as new observations flow in.

� Numerical stability and convergence is a problem!

� Zero mean Gaussian noise is handled well

� Bias is not!

� State transition and observation models (in classical KF) are linear

� Real world is not linear!

1403/01/2008

Estimation Solutions – KF Issues

� As more measurements flow in, covar matrix and Kalman gain approach zero.� Filter is “self-confident” or can be “smug”� No improvements to state estimate possible� Small errors such as round-off, non-linearities and

simplfications accumulate and lead to divergence� Solution: add process noise that stops the covariance

matrix getting smaller

� KF does NOT need to store past data but batch filter does � On-board or groundstation (off-line) implementation

1503/01/2008

Orbit Estimation Accuracy

� 1960s: hundreds of meters� 1970s: few tens of meters to meters

� Better RF and optical measurement systems� Much better computational power and mathematical models

� 1980s: tens of centimeters� Better gravity field models; solid body and ocean tides� Better computational power

� 1990s: centimeters� Oceanographic satellites and other dedicated missions

(TOPEX/Poseidon, CHAMP etc.)

� EKF is the de facto standard for GPS navigation solutions!

1603/01/2008

Orbit Determination/Estimation

� High-accuracy orbit determination usually limited by dynamic model errors� Centimeter level accuracy (particularly with ranging) is

possible

� But how do we know?� Real orbit is never known� The size of the estimation error covariance is an indicator

� Can be overly optimistic� A function of user-defined process noise

� Laser tracking is cm-level accurate� Only if we can afford it!

1703/01/2008

Orbit Prediction

� Needs good mathematical model (propagation)

� Needs good estimate of initial conditions (estimation)

� Lageos prediction accuracy: 200m after two months

� TOPEX/Poseidon prediction accuracy (laser or GPS): 0.5km after a week

1803/01/2008

Error Sources and Accuracy

� Error sources have chracteristics � Random (gaussian? bias?)

� Systemmatic (periodic? secularly growing?)

� Depends on the orbit in question!

� Accuracy depends on a number of parameters� Truncation errors

� Round-off errors – finite precision problem

� Mathematical model simplifications

� Errors in mathematical and measurement models

� Amount, type and accuracy of tracking data

1903/01/2008

Error Sources - Satellite Force Model� Gravitation parameters

� Mass of the Earth� Geopotential coefficients� Solid Earth and ocean tide perturbations� Mass and position of the Moon and planets� General relativistic perturbation

� Nongravitational parameters� Drag: CD, atmospheric density� Solar and Earth radiation pressure� Thrust� Other (magnetic etc.)

2003/01/2008

Error Sources - Measurement Model

� Inertial and terrestrial coordinate systems� Precession and nutation

� Polar motion

� Ground-based measurements� Coordinates of tracking station

� Atmospheric effects (tropospheric and ionospheric refraction)

� Instrument modeling

� Clock accuracy

� Tectonic plate motion

2103/01/2008

2203/01/2008

� http://en.wikipedia.org/wiki/Kalman_filter