# Orbit Determination¶

The `org.orekit.estimation`

package provides classes to manage orbit determination.

## Scope¶

Orbit determination support in Orekit is similar to other space flight dynamics topics

support: the library provides the framework with top level interfaces and classical

implementations (say distance and angular measurements among others). Some hooks are

also provided for expert users who need to supplement the framework with mission-specific

features and implementations (say specific delay models for example). The provided objects

are sufficient for basic orbit determination and can easily be extended to address more

operational needs.

## Organization¶

There are two main sub-packages: `org.orekit.estimation.measurements`

and `org.orekit.estimation.leastsquares`

.

### Measurements¶

The `measurements`

package defines everything that is related to the measurements themselves, both the theoretical

values and the modifications that can be applied to them. All measurements must implement the `ObservedMeasurement`

interface, which is the public API that the engine will use to deal with all measurements. The most important

methods of this interface allow to:

- get the observed value
- estimate the theoretical value of a measurement,
- compute the corresponding partial derivatives (with respect to state and parameters)
- compute the time offset between measurement and spacecraft state

The estimated measurements can be modified by registering one or several `EstimationModifier`

objects. These objects will manage notions like tropospheric delays, biases, ground antennas position offsets ...

A typical operational case from a ground stations network would create distance and angular measurements, create

one bias modifier for the on-board delay for distance measurements, a few modifiers for each ground station

(position offset, delay), modifiers for tropospheric and ionospheric delays and add them to corresponding measurements

(i.e. all distance measurements would share the same on-board delay object, but distance measurements performed

by two difference ground stations would refer to different sets of ground station positions offsets for example).

The classical measurements and modifiers are already provided by Orekit in the same package, but for more advanced

needs, users are expected to implement their own implementations. This ensures the extensibility of this design.

### Least Squares¶

The `leastsquares`

package provides an implementation of a batch least squares estimator engine to perform an orbit

determination. Users will typically create one instance of this object, register all observation data as measurements

with their included modifiers, and run the least squares filter. At the end of the process, a fully configured propagator

is returned, including the estimated orbit as the initial state and the estimated propagator parameters. The estimated

measurement and propagator parameters can also be retrieved by themselves.

The `BatchLSEstimator`

class creates an internal implementation of Hipparchus `LeastSquaresProblem`

interface

to represent the orbit determination problem and passes it to one of the `LeastSquaresOptimizer`

implementations to

solve it. Several choices are possible, among which `LevenbergMarquardtOptimizer`

and `GaussNewtonOptimizer`

. The former

is considered more robust and can start from initial guesses farther than the second one. If `GaussNewtonOptimizer`

is

neverthelesss selected, it should be configured to use `QR`

decomposition rather than `LU`

decomposition for increased

stability in case of poor observability. During the resolution, the selected Hipparchus algorithm will call the `evaluate`

method of the local `LeastSquaresProblem`

model at each algorithm test point. This will trigger one orbit propagation with

some test values for the orbit state and the parameters (for example biases from the measurements modifiers parameters

or drag coefficients from the force models parameters). During the propagation, the Orekit event mechanism is used to

collect the state and its Jacobians at measurements dates. A `MeasurementsHandler`

class performs the binding between the

generic events handling mechanism and the orbit determination framework. At each measurement date, it gets the state

and Jacobians from the propagator side, calls the measurement methods to get the residuals and the partial

derivatives on the measurements side, and fetches the least squares estimator with the combined values, to be

provided back to the Hipparchus least squares solver, thus closing the loop.

### Estimated parameters¶

Users can decide what they want to estimate. The 6 orbital parameters are typically always estimated and are selected

by default, but it is possible to fix some or all of these parameters. Users can also estimate some propagator parameters

(like drag coefficient or radiation pressure coefficient) and measurements parameters (like biases or stations position

offsets). One use case for estimating only a subset of the orbital parameters is when observations are very scarce (say

the first few measurements on a newly detected debris or asteroid). One use case for not estimating any orbital parameters

at all is when calibrating measurements biases from a reference orbit considered to be perfect Selecting which parameters

should be estimates and which parameters should remain fixed is done thanks to the `ParameterDriver`

class. During setup,

the user can retrieve three different `ParametersDriversList`

from the `BatchLSEstimator`

:

- one list containing the 6 orbital parameters, which are estimated by default
- one list containing the propagator parameters, which depends on the force models used and are not estimated by default
- one list containing the measurements parameters, which are not estimated by default

Then, looping on the elements of these lists, the user can change the default settings depending on his/her needs

and for example fix a dew orbital parameters while estimating a few propagation and measurements parameters.

#### Parameters values changes¶

Once everything has been set up, the `estimate`

method of `BatchLSEstimator`

is called. The least squares solver will

then modify the values of the parameters that have been flagged as selected (and hence should be estimated). The

estimator does not know the meaning of any of the parameters, they appear all the same for it. Under the hood,

each parameters was in fact created by an object which knows what the parameter mean, like for example an object

involved in the drag computation. This object uses the observer design pattern to monitor each change attempted by

the optimization algorithm, and it will adapt its computation according to the last change performed. This design

improves the decoupling between the upper layer managing the batch least square estimation and the lower layer to

which force models or biases belong. It therefore allows user to add their own parameters if they create specific

force models, specific measurements or specific measurements modifiers. All they need to do is provide some

`ParameterDriver`

instances and implement the `ParameterObserver`

interface to monitor when the estimator will

change these new parameters.

The class diagram above depicts the parameter update mechanism for the case of ground station position offset. The

`Range`

and `RangeRate`

measurements classes refer to a `GroundStation`

instance (one instance shared by all

measurements using this station) that provides to the upper layer 3 parameters representing the East, North and

Zenith offset for station position. If some station position offset is flagged to be estimated, the `BatchLSEstimator`

will change its value at each new evaluation, without knowing what this change really involves underneath. As the

parameters values are changed, the ground station will be notified of the change thanks to the `ParameterObserver`

it did register to the parameters, and it moves its associated `TopocentricFrame`

according to the updated offsets.

The `Range`

and `RangeRate`

measurements theoretical values will therefore be computed naturally using the updated

station position. Orbital parameters, propagation parameters and measurements parameters are all handled the same

way.

#### Parameters normalization¶

Parameters normalization is used to present a more balanced vector to the least squares algorithm. Without normalization,

the vector component corresponding to the semi-major axis would have an order of magnitude of a few millions whereas

the vector component corresponding to the eccentricity would be 10 orders of magnitude smaller (assuming user decided

to estimate an orbit in Keplerian parameters set). If central attraction coefficient were estimated, the discrepancy

between the largest and smallest component could even reach 20 orders of magnitudes. Least squares optimizers do not

handle such vectors properly. In order to cope with this problem, the mathematical least squares algorithm only sees

normalized values for parameters, while the physical models see real values for the same parameters. The normalized

value is always computed as:

normalized = (physical - reference) / scale

The reference value and the scale are fixed. The scale is related to the expected excursion around reference that

can be expected in a typical problem. It is not really important to have it precisely computed as the goal is only

to avoid huge orders of magnitudes. Any scale that allows the normalized value to be somewhere between 1/1000 and 1000

is good enough. For this reason, the scale is hard-coded for each parameter. In order to increase numerical stability,

the hard-coded values are powers of 2 so sequences of multiplications and divisions when converting between normalized

and physical values do not introduce computation errors. As an example, the scale factor for drag coefficient has

been set to 2⁻³ whereas the scale factor for central attraction coefficient has been set to 2⁺³².

#### Parameters bounds¶

Some parameters values are forbidden and should not be used by the least squares estimator. Unfortunately, as of

early 2016 the Hipparchus library does not support simple bounds constraints for these algorithms. There is

however a workaround with parameters validator. Orekit uses this workaround and set up a validator for the full

set of parameters. This validator checks the test values provided by the least squares solver are within the

parameters bounds, and if not it simply force them at boundary, effectively clipping the values. Just like

scaling factors, the minimum and maximum bounds are currently hard-coded in the library. The limits have been

set to quite loose value, as they are only meant to prevent computation failures (like negative eccentricities

or semi-major axes). If anyway the least squares algorithm tries such extreme values, there is probably a

problem with either the measurements or the propagator configuration.