Re: [Orekit Developers] status on second order derivatives

[Evan Ward <evan.ward@nrl.navy.mil>]

On 10/30/2014 07:10 AM, Luc Maisonobe wrote:
> Hi Evan,
>
> Le 29/10/2014 21:34, Evan Ward a écrit :
>> On 10/29/2014 03:14 PM, MAISONOBE Luc wrote:
>>> Hi Evan,
>>> Evan Ward <evan.ward@nrl.navy.mil> a écrit :
>>>
>>>> Hi Luc, Paul,
>>>>
>>>> If I understand correctly this issue only affects propagators that do
>>>> not use Cartesian representation of the state. When a non-Cartesian
>>>> representation is used the propagator's position is combined with the
>>>> Keplerian velocity and the resulting combination is not locally tangent
>>>> (osculating) to the propagator's orbit. Am I understanding the issue
>>>> correctly?
>>> Yes, this is exactly what happens.
>>>
>>>> Could this issue be solved by using the propagator's complete state to
>>>> determine the instantaneous P/V and use that to build a CartesianOrbit?
>>> Not exactly. The numericalPropagator once only integrated
>>> CartesianOrbit and it was considered a limitation. Since a few years,
>>> we have added the possibility to directly integrate the orbit type
>>> specifed by user. Moving back to CartesianOrbit only would be a
>>> regression IMHO.
>>>
>>> However what you suggest is also almost what I have in mind since we
>>> really use the propagator complete state. In fact, for the numerical
>>> propagator the conversion between acceleration and orbit is not done
>>> at orbit level after integration, but rather at force level as we use
>>> the Jacobian of the (P,V) vs orbital parameters conversion to directly
>>> get parameters derivatives. When you look at it, the Gauss equations
>>> are really the last columns of the Jacobian. This is done with the
>>> TimeDerivatives interface, which for NumericalPropagator is
>>> implemented using the Jacobian.
>>>
>>> So as far as N?umericalPropagator is involved, I think consistency is
>>> automatically preserved, since we do integrated something that is
>>> computed from the Jacobian.
>>>
>>> The DSST propagator on the other hand propagated directly in
>>> equinoctial parameters and uses another dedicated implementation of
>>> the TimeDerivatives interface.
>> I guess I don't understand why we need more than 6 parameters to specify
>> the osculating Keplerian orbit.
> For Keplerian orbit, 6 parameters are enough and the relationship
> between (a, e, i, pa, raan, M) and (x, y, z, vx, vy, vz) is a clear
> bijection. If we restrict ourselves to Keplerian motion, this
> relationship holds not only at t0 but also at t0+dt for every value we
> want. If we use Keplerian parameters evolution (i.e. only M evolving)
> and recompute at several steps the corresponding Cartesian, we will see
> vx, vy, vz are really the derivatives or x, y, z, because the
> relationship does take Kepler laws into account.
>
> The problem arise when we introduce non-Keplerian parts. Then if we
> blindly apply the same relationship for converting between the two sets
> of parameters, we see that they are not consistent anymore.
>
> What happens is that the mapping:
>
>   (a, e, i, pa, raan, M)     <---> (x, y, z, vx, vy, vz)
>
> is such that when da/dt = 0, de/dt = 0, ... dM/dt = n, then the computed
> vx is dx/dt, vy is dy/dt and vz is dz/dt.
>
> If now we decide to set arbitrarily da/dt to some made up non null value
> and so on for all parameters, we still compute the same 6 values (x, y,
> z, vx, vy, vz) as before (the initial value of a, e, i ... did not
> change), but now differentiating the mapping shows that vx will not be
> equal to dx/dt anymore, vy will not be dy/dt anymore and vz will not be
> dz/dt anymore. The made up da/dt, de/dt will contribute to the results.
> This is what happens for Eckstein-Hechler as it computes the left part
> of the mapping and we (Orekit team) compute the right part using the
> Keplerian-based mapping. If we want proper computation of vx, vy, vz, we
> need the additional parameters.

I agree that we need to account for the element rate terms to correctly
compute the velocity. I think we could do the (a, e, i, pa, raan, M,
rates, other propagator state) --> (p, v) mapping within the Propagator
and then return a Orbit built from the correctly computed PV. This way
the returned orbit would be the osculating Keplerian orbit that matches
the propagator's position and velocity. If the user then translates the
osculating PV to osculating Keplerian elements they will be different
from the propagator's elements, but that is to be expected since the
propagator includes perturbations and osculating elements don't. (Maybe
this is the source of my confusion; when hear "osculating orbit" I think
of a conic orbit that matches the instantaneous PV at a single instant
in time.)

> The problem may not hold for NumericalPropagator (I still have to
> check), because basically we do the computation the other way round. We
> start from x, y, z, vx, vy, vz and deduce a, e, ... from the mapping. So
> as long as our vx=dx/dt, vy=dy/dt, vz=dz/dt, the initial mapping will
> still hold.
>
>> It seems that if we add rates to all the elements then the resulting
>> class could be classified as a Propagator.
> It is only a way to get a consistent (P, V), and as a propagator it
> would be really limited as it is a Taylor expansion. I would simply
> qualify it as a perturbed orbit allowing local expansion.

Extending the Taylor series expansion to higher order derivatives makes
sense if we need to return the propagator's internal state, and the
satellite's PV in one consistent object. Is there a use case for
providing the propagator's internal state? I don't know, but I can't
think of any.

If we need to provide the user with acceleration as well as PV then I
agree that we need a bigger container class to hold the extra
information. Maybe the new PVACoordinates would be the right choice
instead of a PerturbedOrbit class since I don't think we would get the
same extrapolation from a PerturbedOrbit stored in Keplerian vs
Cartesian elements. To put it another way, If I converted a
PerturbedKeplerianOrbit to a PerturbedCartesianOrbit would they both
follow the same path?

I'm not against the PerturbedOrbit classes; I just want to make sure we
don't choose a complex solution to a simple problem.

>> There might be a case for
>> this intermediate level of fidelity/speed, but is it worth the added
>> complexity? Especially since we already have one that is fast (the Orbit
>> class) and one that is precise (the Propagator classes). Maybe I'm still
>> not understanding your proposal.
> I detected the problem when checking some pointing attitude modes that
> are related to spacecraft velocity (alignment of the spacecraft axis
> with ground drift for Earth observation). This mode transforms something
> that is a derivative (velocity) into a regular non-differentiated value
> (an angle). So we go up one order of derivation. For such modes, a wrong
> velocity induces a wrong angle and a wrong acceleration induces a wrong
> angular rate. As the angular rate is used for time shift, I needed it to
> be accurate.
>
> You also mentioned issues with Doppler which would also occur.
>
> We do need accurate velocity, and we do need a velocity that is
> consistent with the derivative of the position.

Definitely agree that getting the velocity correct is important. Thanks
again for all the work you've put into solving this bug.

Best Regards,
Evan

>
> best regards,
> Luc
>
>> Best Regards,
>> Evan
>>
>>>> Then we wouldn't have to modify the existing set of Orbit classes, and
>>>> the user would see the correct osculating P/V. (This might be equivalent
>>>> to your second approach.)
>>>>
>>>> As far as where to put the code, it seems like the conversion code would
>>>> be specific to the internal state representation used by the propagator,
>>>> so I think it makes sense to keep the code as private to the propagator.
>>>> Though if you think there would be other uses for the conversion, I
>>>> think a public static factory method would work well.
>>> Yes, the conversion code is propagator dependent. The first
>>> implementation I played with is Eckstein-Hechler propagator, and it
>>> definitely is Eckstein-Hechler specific (it's a simple derivation of
>>> the original equations, so each time we did compute something like e =
>>> a * c + b * d, now we also have another statement to compute eDot =
>>> aDot * c + a * dCot + bDot * d + b * dDot). For numerical propagator
>>> we already have the derivatives since we start from the derivatives
>>> and integrate them, so its even simpler. For DSST, this will be a mix
>>> as the mean elements are integrated and the short periodics terms are
>>> computed from Fourier coefficients which are straightforward to
>>> differentiate. For ephemeris-based propagator, we will need to compute
>>> the derivatives of the underlying polynomials, which is also
>>> straightforward.
>>>
>>>> Thanks Luc for finding this issue and doing the analysis. I can see how
>>>> this would be an issue when computing the Doppler as well as time
>>>> shifting.
>>> your welcome
>>>
>>> best regards,
>>> Luc
>>>
>>>> Best Regards,
>>>> Evan
>>>>
>>>> On 10/29/2014 06:57 AM, MAISONOBE Luc wrote:
>>>>> Hi Paul,
>>>>>
>>>>> paulcefo <paulcefo@buffalo.edu> a écrit :
>>>>>
>>>>>> Luc,
>>>>>>
>>>>>> Do I correctly understand that your concern is that Keplerian
>>>>>> transformations do apply outside the osculating space?
>>>>> The problem I had was that we did use Keplerian-only expression
>>>>> to set up local Taylor expansions around the current point (a few
>>>>> seconds away). This was slightly wrong when all the parameters were
>>>>> time-dependent and not only the anomaly was time-dependent. Of course,
>>>>> the error increasing with the time offset with respect to the central
>>>>> date at which the Taylor expansion is built. The fix was simply to
>>>>> not forget the derivatives of these other parameters.
>>>>>
>>>>> This Taylor expansion feature is a built-in feature available in all
>>>>> Orekit orbits, it typically allows to do computation in the vicinity of
>>>>> an already computed point without needed to trigger a complete
>>>>> propagator.
>>>>> It can even be used for some computation inside the run of a
>>>>> propagator,
>>>>> as for example when the higher level propagator takes care of the long
>>>>> term propagation and at each step we need some additional points
>>>>> surrounding the current step to compute attitude evolution in some
>>>>> specific modes.
>>>>>
>>>>> My concern was how to implement this fix in our current architecture,
>>>>> and more precisely were to put the code: in an existing class or in
>>>>> a dedicated class which would be used only by propagators.
>>>>>
>>>>> best regards,
>>>>> Luc
>>>>>
>>>>>> Paul
>>>>>>
>>>>>> -- 
>>>>>> Dr. Paul J. Cefola
>>>>>> Consultant in Aerospace Systems, Spaceflight Mechanics, &
>>>>>> Astrodynamics
>>>>>> Adjunct Faculty, Dept. of Mechanical and Aerospace Engineering,
>>>>>> University at Buffalo (SUNY)
>>>>>>
>>>>>> 4 Moonstone Way
>>>>>> Vineyard Haven, MA 02568
>>>>>> USA
>>>>>>
>>>>>> 508-696-1884 (phone on Martha's Vineyard)
>>>>>> 978-201-1393 (cell)
>>>>>>
>>>>>> paulcefo@buffalo.edu
>>>>>> paul.cefola@gmail.com
>>>>>>
>>>>>> On 10/29/2014 6:02 am, MAISONOBE Luc wrote:
>>>>>>> Hello,
>>>>>>>
>>>>>>> As some of you may be aware, I have been working for a few months on
>>>>>>> second order derivatives in the git branch
>>>>>>> position-velocity-acceleration. This work is still ongoing but I hope
>>>>>>> to finish it for 7.0 and merge the branch back to master soon. For
>>>>>>> now, there are still failing tests so I can't do it.
>>>>>>>
>>>>>>> This change should allow us to reach several goals :
>>>>>>>
>>>>>>> - improved accuracy in shiftedBy methods
>>>>>>> - improved accuracy in interpolators (with user-defined
>>>>>>>   choices to use or not first and second derivatives
>>>>>>>   from the sample)
>>>>>>> - improved accuracy in attitude
>>>>>>> - removal of ugly hidden finite differences in some classes
>>>>>>>   (most notably attitude modes) with hard-coded steps
>>>>>>> - hopefully faster Earth transforms, by replacing Hermite
>>>>>>>   interpolation with single point extrapolation
>>>>>>> - availability of non-Keplerian acceleration everywhere
>>>>>>> - availability of angular acceleration in attitude and frames
>>>>>>> - proper composition of dynamics in frames
>>>>>>> - possibility to propagate orbits in non-inertial frames
>>>>>>> - possibility to propagate orbits without a central body
>>>>>>>   (interplanetary missions, Lagrange point missions, ...)
>>>>>>>
>>>>>>> There is one point that bothers me right now. As I removed some of
>>>>>>> the
>>>>>>> ugly finite differences, some non-regression tests started to fail. I
>>>>>>> finally found the raw cause of these failures and was surprised to
>>>>>>> discover an old bug in the way we use the osculating orbits produced
>>>>>>> by the Eckstein-Hechler analytical propagator. This propagator takes
>>>>>>> zonal terms into account, and produces directly circular
>>>>>>> parameters a,
>>>>>>> ex, ey, ... When we compute anything related to geometry, we compute
>>>>>>> Cartesian coordinates using the Orbit getPVCoordinates method. As the
>>>>>>> Orbit classes do not know anything about the perturbation, the (P, V)
>>>>>>> pair does in fact implicitly relies on Keplerian-only expressions. So
>>>>>>> the velocity part is *not* consistent with the derivative of the
>>>>>>> position. The real derivative of the position takes the non-Keplerian
>>>>>>> effects into account which are ignored by getPVCoordinates. The
>>>>>>> difference is small, but as the tests threshold were deliberatly very
>>>>>>> tight, the tests started to fail when the various pointing directions
>>>>>>> were not computed anymore from finite differences mainly involving
>>>>>>> position and when they relied on the computed velocity. So the
>>>>>>> problem
>>>>>>> already happens in the master branch, it is not specific to the
>>>>>>> introduction of acceleration (it was just detected here during
>>>>>>> testing).
>>>>>>>
>>>>>>> The solution is in fact quite simple. If an orbit has been
>>>>>>> produced by
>>>>>>> a non-Keplerian propagator, the propagator already knows about the
>>>>>>> derivatives of the orbital elements (which are circular in the
>>>>>>> Eckstein-Hechler model case but can be any kind of parameters for
>>>>>>> other propagators). The propagator should therefore provide these
>>>>>>> derivatives to the orbit so they can be used in the PVCoordinates
>>>>>>> conversion. The code is very simple and straightforward. I have
>>>>>>> checked this and got very interesting results with
>>>>>>> Eckstein-Hechler/Circular, as for example a simple interpolation over
>>>>>>> a 900s arc with proper velocity/acceleration has a 88m error with two
>>>>>>> base points now whereas it was 5162 m before (and 0.02m vs 650m for 3
>>>>>>> points, 1.0e-5m vs 259m for 4 points).
>>>>>>>
>>>>>>> Here is what bothers me:
>>>>>>>
>>>>>>> Should we create specialized classes for perturbed orbits or
>>>>>>> should we
>>>>>>> simply add a constructor to the existing orbits with the parameters
>>>>>>> derivatives and set them to 0 when they are not known?
>>>>>>>
>>>>>>> For my tests, I created PerturbedCircularOrbit which extends
>>>>>>> CircularOrbit and override the protected initPVCoordinates method and
>>>>>>> the public shiftedBy and interpolate methods. I could also have
>>>>>>> simply
>>>>>>> moved everything into CircularOrbit with a new constructor.
>>>>>>>
>>>>>>> I do not like much the PerturbedXxxxOrbit approach, as it forces to
>>>>>>> create also additional entries in the OrbitType enum with additional
>>>>>>> converters and it becomes awkward if for example a user configures a
>>>>>>> NumericalPropagator to generate XxxxOrbit, despite this propagator
>>>>>>> will in fact really generate PerturbedXxxOrbit because it is what a
>>>>>>> Numerical propagator is for. So there should be either an internal
>>>>>>> modification of the user setting from OrbitType.XXXX to
>>>>>>> OrbitType.PERTURBED_XXXX or an error triggered which would invalidate
>>>>>>> *all* current user code as it would become forbiddent to generate
>>>>>>> XXXX
>>>>>>> orbits now.
>>>>>>>
>>>>>>> On the other hand, the drawback of modifying the existing classes to
>>>>>>> hold the non-Keplerian derivatives is that they will consume more
>>>>>>> memory. I don't think it is a problem with current computers.
>>>>>>>
>>>>>>> In any case, initial orbits created directly from user code or by
>>>>>>> reading files would not include the derivatives and therefore will be
>>>>>>> built as usual (by calling the unmodified classes in the first
>>>>>>> approach, or by using the already existing constructors in the second
>>>>>>> approach, assuming these constructors will automatically set the
>>>>>>> derivatives to Keplerian-only values). In any case, full-blown
>>>>>>> perturbed orbits will be created internally by Orekit propagators,
>>>>>>> which can easily be modified to provide the derivatives they know (by
>>>>>>> creating instances of the new derived classes in the first approach,
>>>>>>> or by using new constructors with additional parameters in the second
>>>>>>> approach).
>>>>>>>
>>>>>>> My humble opinion would be to use the second approach to solve this
>>>>>>> bug. I will probably do this in the position-velocity-acceleration
>>>>>>> branch so it will include accelerations right from the start and will
>>>>>>> be merged to master at the same time as the rest of the branch. Of
>>>>>>> course, this will be a dedicated commits (Git branches are great!).
>>>>>>>
>>>>>>> What do you think ?
>>>>>>>
>>>>>>> best regards,
>>>>>>> Luc
>>>>>>>
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