Challenge LOP-G to Brighton Beach and Back Again

[MontBlanc2012]

Here's a challenge that might appeal:

NASA has proposed parking its planned LOP-G (Lunar Orbital Platform Gateway) station in a Near Rectilinear Halo Orbit (NRHO) and, from this station, allow lunar landers to target landing at any site on the Moon's surface. This raises interesting questions about the most fuel efficient strategies to depart LOP-G and land on the surface; and, equally, to depart the Moon and rendezvous with LOP-G.

Following on from recent postings in relation to halo orbits (See here), I've constructed a scenario with two Delta-Gliders. The first Delta-Glider is parked in a LOP-G style NRHO; and the second is at Brighton Beach. One can either try to find the most fuel-efficient for the first Delta-Glider to land at Brighton Beach from its LOP-G orbit; or one can take the Brighton Bach Delta-Glider and find a fuel efficient rendezvous strategy for rendezvousing with the Delta-Glider.

The Near Rectilinear Halo Orbit that the first Delta-Glider is placed in is a fair proximities for the 'real' proposed LOP-G orbit: it's a L2 halo orbit with a near 4:1 synodic orbital resonance - so this challenge will do a fair job of simulating the type of problems that one might encounter in landing and rendezvousing.

The scenario is as follows:

BEGIN_ENVIRONMENT
  System Sol
  Date MJD 52013.754909351
  Help CurrentState_img
END_ENVIRONMENT

BEGIN_FOCUS
  Ship GL-02
END_FOCUS

BEGIN_CAMERA
  TARGET GL-02
  MODE Cockpit
  FOV 40.00
END_CAMERA

BEGIN_SHIPS
GL-01:Deltaglider
  STATUS Landed Moon
  POS -33.4450800 41.1217030
  HEADING 66.59
  ALT 2.553
  AROT 18.270 -16.773 41.004
  AFCMODE 7
  PRPLEVEL 0:1.000000 1:1.000000
  NAVFREQ 0 0 0 0
  XPDR 0
  HOVERHOLD 0 1 0.0000e+000 0.0000e+000
  GEAR 1.0000 0.0000
  AAP 0:0 0:0 0:0
END
GL-02:DeltaGlider
  STATUS Orbiting Moon
  RPOS -16083384.788 -3158071.948 -5516819.356
  RVEL 382.8380 536.2102 146.3039
  AROT 74.634 -34.764 -157.159
  AFCMODE 7
  PRPLEVEL 0:1.000000 1:1.0000000
  NAVFREQ 586 466 0 0
  XPDR 0
  HOVERHOLD 0 1 0.0000e+000 0.0000e+000
  AAP 0:0 0:0 0:0
  SKIN BLUE
END
END_SHIPS

Just cut and paste this into a .scn text file and place in your scenario directory. Then load and run.

There are no 'rules' for this scenario other than that for the Brighton-Beach to LOP-G rendezvous variant, the orbit of the LOP-G vessel must not be changed. It's on a carefully calculated ballistic trajectory and it's orbit cannot change.

And don't take too long: the orbiting Delta-Glider isn't subject to active station-keeping and after one orbit will start to deviate significantly from the NRHO track.

 

[Ajaja]

Where is "Date MJD" ?

 

[MontBlanc2012]

Ajaja - yep, sorry about that: I accidentally omitted the ENVIRONMENT section of the scenario. I've now amended the scenario in the above.

Cheers.

 

[Ajaja]

Something odd. GL-02 escapes Moon after two revolutions. Is this really NRHO? Can you recheck, please?

 

[MontBlanc2012]

Something odd. GL-02 escapes Moon after two revolutions. Is this really NRHO? Can you recheck, please?

Ajaja: yes, this is a NRHO - at least for the CR3BP. I have previously checked my analysis that the calculated halo orbits preserve the Jacobi constant. And they do to high precision. Halo orbits with perilunes of circa 7,000 km as per the scenario are inherently unstable and are expected to 'leave' the NRHO orbit after about one revolution without active station-keeping.

The fact of the matter is that the halo orbit used as a basis in this scenario is a numerically precise for the CR3BP but it does not (and is not intended to) take full account of gravitational perturbations - principally those due to the Moon's orbital eccentricity and solar tidal forces. Moreover,

Also, note that the because (in two-body terms), the orbital eccentricity is high. This means that the Delta-Glider is only weakly bound to the Moon. After two revolutions, it seems, perturbations kick the Delta-Glider up to an energy state where it can 'escape' the Moon. This weakly bound state is confirmed by calculating the Jacobi constant for the orbit which is close to 3.

This scenario was set up so that readers could get a feel for the kind of orbit that LOP-G will be placed - i.e., a high inclination and high elliptical orbit - and was intended to highlight to orbital rendezvous problems that this orbit creates for going to or leaving LOP-G.

Is it possible to create a high fidelity LOP-G orbit for Orbiter? Yes, it is - but it requires a robust framework for performing active station-keeping over weeks or months; and it requires splicing together a trajectory solution using a full ephemeris gravity model over multiple lunar orbits.

 

[Ajaja]

It has to be more than 2-3 revolutions. For example try this values (for the same Date MJD 52013.754909351):
RPOS -12142686.086 -14397704.633 -5091606.013
RVEL 176.6108 590.1609 100.3061
I'm not sure if it is exactly a NRHO, I didn't do the math, just did a simple "brute-force" optimization in GMAT to find a stable LOP-G-like orbit, and it's definitely much more stable in Orbiter too.

 

[MontBlanc2012]

Ajaja:

it's a case, really, that not all NRHOs are made equal. The scenario I put forward has an initial perilune of around 6,700 km whereas yours appears to have a perilune that is much lower at around 2,500 km. The stability of NRHOs is strongly dependent on this perilune radius.

According to Kathleen Howell (Professor of Aeronautics and Astronautics in the College of Engineering at Purdue University and advisor to NASA on all things halo), the stability of orbits in the CR3BP varies according to perilune as shown the following figure:

This figure was extracted from her paper: Near rectilinear halo orbits and their application in cis-lunar space which is a 'good read' if one is interested in the subject.

Basically this shows that there are regions of stability of NRHOs (in the CR3BP model) and these occur when the perilune radius is very low; or when it is around 15,000 km. But for NRHOs with perilunes of circa 7,000 km as per my scenario, the number of revolutions before departure is expected to be about one. And that's exactly what we observe. On the other hand, you have selected an orbit with a much lower perilune and, according to the graph, it has a higher stability and may survive a few more orbits before departure. So the fact that your halo orbit survives longer that mine before departure is a bit of a blessing - but not unexpected. But it doesn't say anything else.

Congratulations, though, on using GMAT to construct a NRHO.

 

[Ajaja]

Yes, the periapsis was redused by GMAT.
BTW, they want to use a low perilune for LOP-G too:
https://en.wikipedia.org/wiki/Lunar_Orbital_Platform-Gateway
The LOP-G would be placed in a highly elliptical near-rectilinear halo orbit (NRHO) around the Moon which will bring the station within 1,500 km (930 mi) of the lunar surface at closest approach and as far away as 70,000 km (43,000 mi) on a six-day orbit.[15] This orbit would allow lunar expeditions from the Gateway to reach a polar low lunar orbit using 730 m/s of delta-v in half a day. Orbital station-keeping would require less than 10 m/s of delta-v per year.[16]

 

[MontBlanc2012]

I've seen various figures for LOP-G perilune orbital period. As I understand it, the goal is to have a synodic resonance of either 9:2 or 4:1 to avoid occultation of the solar panels. Assuming that the synodic period of the Moon is around 29.5 days, a 4:1 resonance has a period of around 7.35 days; and a 9:2 resonance has a period of 6.5 days. The 4:1 resonance has quite a high perilune; whereas the 9:2 resonance has a substantially lower one.

 

[Ajaja]

Yes, with your periapsis it is less stable, but still, it should be possible to do more than 2-3 revolutions:
RPOS -15969338.210 -3884420.146 -5698604.470
RVEL 368.6800 556.7144 165.8636

 

[MontBlanc2012]

Ajaja:

Thanks for recalculating with GMAT with a more comparable perilune. I guess the question is: where and why is there a difference?

As a bit of preamble, let's just compare the osculating orbital elements of the two orbits:

                             MB                GMAT
------------------------------------------------------------
Perilune radius:           6,770 km          6,779 km
Eccentricity:              0.8460            0.8751
Inclination:               91.76 deg         93.98 deg
LAN:                      198.60 deg        198.73 deg
Argument of Periapsis:     98.61 deg         94.99 deg

Basically, the osculating orbits are essentially of the same shape and size and have more or less the same orientation. The differences reflect the way that one maps the numerical solutions of the CR3BP to an inertial reference frame. GMAT uses a solver that takes into account all of the perturbations acting on the Moon. On the other hand I use a simple mapping from CR3BP to an ER3BP reference frame. Both solutions are bona fide halo orbits - but your GMAT solution is better 'tailored' to long-term stability.

Is the GMAT solution 'better'. Yes, in that it takes into account perturbations in a more accurate and more systematic fashion. But it is also a 'black box' solution and generally I prefer to work with tools that I've built. That's just me.

Can I build a GMAT-like full ephemeris solver? Yes, but not just yet. That's work in progress.

Should one use the GMAT solution in this scenario instead? Well that all depends on what you want to do with scenario? The original intention of the scenario was:

* to illustrate a LOP-G-like NRHO;

* to highlight the practical implications of leaving the quasi halo orbit and landing a manned lander at 'an arbitrary point' (Brighton Beach); and

* to highlight the difficulties of achieving rendezvous with LOP-G where one assumes that one doesn't want to have a crew trapped in a cramped lunar lander for more than perhaps a day or two.

 

[Ajaja]

The original intention of the scenario was:
* to illustrate a LOP-G-like NRHO;

* to highlight the practical implications of leaving the quasi halo orbit and landing a manned lander at 'an arbitrary point' (Brighton Beach); and

* to highlight the difficulties of achieving rendezvous with LOP-G where one assumes that one doesn't want to have a crew trapped in a cramped lunar lander for more than perhaps a day or two.

Thanks again. To watch graphs and videos it's very different from "to touch" those orbits in Orbiter or GMAT. And without your scenario I wouldn't know where to start from and what an initial guess to choose searching for NRHO-like orbits.
Currently for this scenario I see two opposing solutions. The first (time-critical and fuel wasteful) is to catch the ship near the perilune. The second (economical) is to meet our LOP-G in the apoapsis. The first one, I think, is possible to do in Orbiter with standard instruments, but the second one is a real challenge especially without using some third-party tools like GMAT. Although LTMFD or IMFD could help here.
The issues with orbit stability appeared when I tried to apply simple approach to solve the challenge. I waited when the NRHO plane aligns with the longitude of Brighton Beach. Looks like for regular cargo/crew transfers between LOP-G and a non-polar base at Moon it is the best way. The NRHO plane makes whole turn in a month, so we should have 2 windows per month. It doesn't work in our case, but a more stable NRHO would be suitable for simulating such transfer in Orbiter.

 

[MontBlanc2012]

Currently for this scenario I see two opposing solutions. The first (time-critical and fuel wasteful) is to catch the ship near the perilune. The second (economical) is to meet our LOP-G in the apoapsis. The first one, I think, is possible to do in Orbiter with standard instruments, but the second one is a real challenge especially without using some third-party tools like GMAT. Although LTMFD or IMFD could help here.

Yes, I agree. Orbiter isn't well equipped for dealing with these orbits given the readily available tool set. Using GMAT sounds promising, though.

Rendezvous at apoapsis would have astronauts in the lunar lander for three-four days. Let's hope that whatever design for the lunar lander is used, it is a little bit more than a glorified tin can.

The issues with orbit stability appeared when I tried to apply simple approach to solve the challenge. I waited when the NRHO plane aligns with the longitude of Brighton Beach. Looks like for regular cargo/crew transfers between LOP-G and a non-polar base at Moon it is the best way. The NRHO plane makes whole turn in a month, so we should have 2 windows per month. It doesn't work in our case, but a more stable NRHO would be suitable for simulating such transfer in Orbiter.

My understanding is that the orbital plane of the NRHO orbit (or any halo orbit, in fact) co-rotates with the Moon so that on each perilune pass, it crosses over the same patch of the Moon. [As viewed from Earth, the orbit always appears to 'circle' the Moon.] Not sure, then, that the concept of '2 windows per month' works for halo orbits. Keplerian orbits, yes (where the Moon rotates under the orbital plane); halo orbits (where it doesn't and the two co-rotate), no. But maybe I'm thinking about this incorrectly.

('my' NRHO doesn't simulate this co-rotation character of NRHOs very well - although it achieves it to a limited degree and for a limited time. But having looked at the IMFD projection of your GMAT solutions, your solutions should simulate this behaviour much better.)

 

[Ajaja]

My understanding is that the orbital plane of the NRHO orbit (or any halo orbit, in fact) co-rotates with the Moon so that on each perilune pass, it crosses over the same patch of the Moon.

You are right, Moon turns with NRHO plane , my mistake.

 

[Ajaja]

GMAT gives an interesting solution:

The green line is the trajectory to the LOP-G from a low circular orbit above Brighton-Beach and the yellow is the return trajectory. The most interesting fact is that we need less than 200 m/s at the apoapsis to align speed with LOP-G, dock, and then to make a burn to return back.
dV = 2343 m/s + 86 m/s + 99 m/s + 2344 m/s

 

[Ajaja]

I've checked it in Orbiler, and it works.
There is a "lodestar" (for the green trajectory) that can be used to target LOP-G from Brighton-Beach:

GL-03:DeltaGlider
  STATUS Orbiting Moon
  RPOS -3111927.892 -26616079.085 -14832641.036
  RVEL 8.0341 435.9435 99.7092

 

[MontBlanc2012]

The green line is the trajectory to the LOP-G from a low circular orbit above Brighton-Beach and the yellow is the return trajectory. The most interesting fact is that we need less than 200 m/s at the apoapsis to align speed with LOP-G, dock, and then to make a burn to return back.
dV = 2343 m/s + 86 m/s + 99 m/s + 2344 m/s

I'm really quite impressed with what you've managed to achieve here with GMAT.

The dV numbers that you quote make intuitive sense: if one thinks of a NRHO as glorified highly eccentric Keplerian orbit (albeit one subject to string tidal forces from the Earth), the periapsis speed needed to achieve an apoapsis of about LOP-G's orbital apoapsis is around 2330 to 2340 m/s. Moreover, apoapis speed is going to be around 45 m/s so, taking into account the possibility of an up to 90 degree plane change to align planes with LOP-G, means that the maximum rendezvous speed is going to be approximately [MATH]\sqrt{2}[/MATH] of that or approximately 65 m/s. And although this is 'back of the envelope' stuff, it aligns pretty well with your considerably more accurate GMAT results.

There are a couple of takeaways from this:

* GMAT seems to be a useful adjunct to working with three body orbits in Orbiter. In the absence of tools developed specifically for Orbiter, trajectory planing using third party tools such as GMAT is clearly advantageous.

* In a fuel efficiency sense, targeting an arbitrary point on the lunar surface with a lunar lander is beset achieved at LOP-G orbital apoapsis for both departure and return.

* However, this fuel efficient strategy comes at the price of having the crew of a lunar lander being fully dependent on that vehicle for 3-4 days of the descent to the Moon's surface from LOP-G; up to two weeks on the lunar surface; and another 3-4 days for the return ascent back to LOP-G. The lunar lander is going ti have to provide food, water and oxygen (and radiation shielding) for two or more people for at least three weeks. That, I would imagine, is quite a design challenge.

Ajaja, many thanks for your contribution on this subject.