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Re: [Orekit Users] Generic extremum events
Hi Thomas,
Le 08/09/2015 20:43, MAISONOBE Luc a écrit :
>
> Thomas.Fruth@dlr.de a écrit :
>
>> Hello,
>
> Hi Thomas,
>
>>
>> at the DLR GSOC we are presently evaluating whether Orekit could be
>> integrated into our Mission Planning tool suite.
>>
>> One important functionality that we need is the calculation of
>> extremum events, such as for example the time of maximum elevation in
>> a given topocentric frame or the time of minimum range with respect to
>> a given Earth observation target.
>>
>> From investigating Orekit we identified two different approaches to
>> achieve this task:
>>
>>
>> 1) To return the derivative of the value that is to be minimized
>> or maximized in the function g in the specific implementation of
>> AbstractDetector. This approach is for instance chosen by the new
>> Latitude/LongitudeExtremumDetector classes; however, it requires some
>> knowledge about the derivative (as in this example handled by
>> OneAxisEllipsoid.transform(PVCoordinates, Frame, AbsoluteDate)), which
>> might not always be readily available.
>>
>> 2) To extend the present event handling mechanism such that it
>> allows to configure whether the given function g should be searched
>> for its root, or be minimized or maximized. My naïve guess is that an
>> extremum search without passing an explicit derivative might be
>> achieved relatively easily by switching between the solver presently
>> used in EventState.evaluateStep(...) and an optimizer (e.g. the
>> BrentOptimizer from Apache commons-math) as needed.
>>
>> I would be glad if some experienced Orekit users could comment whether
>> my assessment on using Orekit for extremum events is correct,
>
> Yes, your assumptions are correct.
>
>> or if I might have missed some other possibility to achieve this task
>> with Orekit's present functionality. If there yet doesn't exist a
>> generic solution to find extremum events (such as described in option
>> 2 above), are there any plans to include such functionality?
>
> In fact, I think the first approach is the more straightforward one.
> When the g function of your ExtremumElevationDetector will be called,
> it will get a SpacecraftState which does contain the information about
> position, velocity and acceleration which are sufficient to compute
> simply all derivatives. You don't even need to care by yourself about
> the exact expression for the derivatives since DerivativeStructure from
> Apache Commons Math can do it for you. Here is how it can be done:
>
> public double g(final SpacecraftState s) throws OrekitException {
>
> // get position, velocity acceleration of spacecraft in topocentric
> frame
> final Transform inertToTopo = s.getFrame().getTransformTo(topo,
> s.getDate());
> final TimeStampedPVCoordinates pvTopo =
> inertToTopo.transformPVCoordinates(s.getPVCoordinates());
>
> // convert the coordinates to DerivativeStructure based vector
> // instead of having vector position, then vector velocity then
> vector acceleration
> // we get one vector and each coordinate is a DerivativeStructure
> containing
> // value, first time derivative, second time derivative
> final FieldVector3D<DerivativeStructure> pvDS =
> pvTopo.toDerivativeStructureVector(2);
>
> // compute elevation and its first and second derivatives
> final DerivativeStructure elevation =
> pvDS.getZ().divide(pvDS.getNorm()).asin();
>
> // return elevation first time derivative
> return elevation.getPartialDerivative(1);
>
> }
>
> Note that in the method above, we never write explicitely any
> derivative, they are
> computed analytically using chain rule thanks to Apache Commons Math.
>
>
> The second approach as you describe it would lead to some problems as
> when the
> propagator used is a numerical propagator, it is not the Orekit EventState
> class that is used but the one from Apache Commons Math in the ode
> package. This
> means that the same change would have to be done in both libraries. Its not
> impossible, but cumbersome. There may be a better approach in similar
> cases when
> direct differentiation is not possible, it is to use a wrapper function
> that does
> perform the differentiation for you and use the regular event detection
> on the
> wrapped function.
Another problem with the extremum solver is that we would have problem
with the pre-filtering of the search interval.
Currently, the root solver is not triggered at each step, bit only
when a root is known to lie in the step (according to the
maxCheckInterval setting). This two stages search saves a *lot* of
computation.
If we want to search for extremum, we would have to do the same and
identifying an extremum occurs would imply computing the derivative
anyway, so if we have it, we can just use it all the way through,
first to bracket the search interval, then to find the root.
>
> In other words, you could implement an f function that computes the
> function you
> want and then wrap it using a finite differences differentiator that
> would create
> function g by wrapping function f so that each time g(state) is called,
> in fact
> you would get two calls to f. Basically you would end up with somethig
> like:
>
> double g(state) {
> double fPlus = wrapped.f(state.shiftedBy(+step));
> double fMinus = wrapped.f(state.shiftedBy(-step));
> return (fPlus - fMinus) / (2 * step);
> }
>
> With such a wrapper, implementors can easily get extremums without
> worrying about
> difficult derivatives, and no changes to any library is needed.
> The shiftedBy method is well suited for small offsets like the ones used
> in finite
> differences.
>
> As the detector you mention is really simple, I think we may add it in
> the next
> few days, using the code snippet above for the first approach. It is
> clearly in
> line with the new LatitudeExtremum and LongitudeExtremum added recently.
I have just added the ExtremumElevationDetector in the development
version in the git repository, you can give it a try now.
best regards,
Luc
>
> Hope this helps,
> Luc
>
>>
>> Best regards
>>
>> Thomas
>>
>> --------------------------
>> Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR)
>> German Aerospace Center
>> Space Operations and Astronaut Training | Mission Operations |
>> Oberpfaffenhofen | 82234 Wessling | Germany
>>
>> Dr. Thomas Fruth
>> Telephone +49 8153 28-2432 | Telefax +49 8153 28-1456 |
>> thomas.fruth@dlr.de<mailto:thomas.fruth@dlr.de>
>> DLR.de<http://www.dlr.de/>
>