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02042019, 08:36 AM  #1 
Orbinaut

An update on all things Lissajous and Halo
Over the last six months or so, I've posted a number of threads relating to general Lissajous and Halo orbits with a view to developing simple tools that can lock vessel onto a prescribed 'closed loop' orbit around the L1/L2 Lagrange points of the EarthMoon, SunEarth and other restricted threebody systems.
This effort is nearing (successful) completion, and I've begin testing a Lua script toos that allow one to 'lock' a vessel onto one of these close loop orbits and keep on the prescribed indefinitely (or, at least, until stationkeeping fuel is exhausted). At the moment, my stationkeeping Lua script testing is limited to maintaining a vessel at an L1 or L2 Lagrange point and estimating the dV required to keep the vessel at that Lagrange point. If the underlying datasets are good, the total stationkeeping dV requirements should be low since the stationkeeping is primarily limited to offsetting minor gravitational perturbations. For those interested, and in terms of monthly stationkeeping dV requirements, my preliminary estimates of stationkeeping dV requirements are as follows: EarthMoon L1: 1.9 m/s EarthMoon L2 : 2.7 m/s SunEarth L1: 0.6 m/s SunEarth L2: 0.6 m/s SunMars L1: 0.2 m/s SunMars L2: 0.2 m/s SunVesta L1: 0.2 m/s SunVesta L2: 0.2 m/s These dV estimates are not large. However, The my stationkeeping logic is very simple: it just 'locks' a vessel onto the closed orbit trajectory and keeps it there. More sophisticated stationkeeping could easily reduce these dV requirements by upwards of a factor of 10. However, at this time, there is no real need to adopt more sophisticated approaches to stationkeeping since these dV requirements are already very low and there is little to be gained by improving the algorithm further. For those interested, I've just about written up and tested some Orbiter scenarios that illustrate this Lagrange point stationkeeping. When I've finished testing in the next few days, I'll post these scenarios here on OF. In essence, for each of the eight Lagrange points listed above, the scenario will consist of two docked (stock) DeltaGliders. One of the DeltaGliders will be 'locked' to the Lagrange point; and the other will initially be docked to the first DeltaGlider. One can then undock the two DeltaGliders. The first will continue to be locked to the Lagrange point (and perform the required stationkeeping to keep the vessel at the Lagrange point); and the second can manoeuvre freely. These scenarios are designed to test the dynamics in the vicinity of Lagrange points; and offer 'target practice' for those wishing to rendezvous with a vessel 'parked' at a Lagrange point. Most of my recent posts have considered a general description of closed loop (Lissajous) orbits around the Lagrange points  not just the Lagrange points themselves. The next step after this, then, is to extend my stationkeeping algorithm so that it can deal with general Lissajous orbits. Technically, the last remaining problem that I need to solve before I can do this is to properly incorporate the effect of gravity perturbations on the core stationkeeping algorithm. In essence, I know how to deal with this  I just now need to formally work through the maths; write and test an updated version of the stationkeeping algorithm; and then post it. This should take a couple of weeks. Again, I'll write a few more scenarios that illustrate stationkeeping for planar and vertical Lyapunov orbits; Halo orbits; and general Lissajous orbits. So, after considerable effort, the delivery of useful tools with which to perform stationkeeping on closedloop orbits in the vicinity of Lagrange points; and enable efficient transfer to/from/between the closed loop orbits around Lagrange points is now in sight. Yay. 
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02152019, 12:05 PM  #2 
Orbinaut

So excellent!
Can we use the Lagrange MFD to create a halo or a Quasihalo orbit ?I really like the manifold. 
02162019, 01:28 PM  #3 
Orbinaut

Quote:
The point of all of my longwinded notes on Lissajous, Lyapunov and Halo orbits is to go this extra step that Lagrange MFD doesn’t  namely to map those orbits; and to provide a basic method of automated stationkeeping on those orbits . The maths of the centre manifold orbits is significantly more painful than the usual Keplerian, twobody orbits and so it has taken a fair bit of effort to get this far. But my laboured mapping exercise is nearly over and is on the brink of producing something mildly useful. So, although one can’t do much with Halo orbits with Lagrange MFD, in principle we will shortly have a method for ‘dialing up’ an accurate representation of (quasi) Halo orbit; placing a vessel on that orbit; and keeping it there indefinitely.  Post added at 01:28 PM  Previous post was at 11:31 AM  As an quick example of the utility of this mapping exercise, here is an example video of the view of the Earth and Moon as seen from a vessel 'locked onto' a vertical Lyapunov orbit around the L2 Lagrange point. Although not shown in the video, a DeltaGlider is locked onto this orbit in Orbiter 2016 and will stay on this orbit indefinitely. (The algorithm still has a few minor glitches that I need to iron out  but, basically, it's working.) Last edited by MontBlanc2012; 02162019 at 02:08 PM. 
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02192019, 03:05 AM  #4 
Orbinaut

As another quick example of the utility of the Lagrange point centre manifold mapping exercise, here is an example video of the view of the Earth and Moon as seen from a vessel 'locked onto' a horizontal Lyapunov orbit around the L2 Lagrange point.
 Post added 021919 at 03:05 AM  Previous post was 021819 at 12:47 PM  And to complete the trilogy, here is a short video of an arbitrary (quasi) Halo Orbit example around EarthMoon L2. As with all of these videos, a vessel is 'locked' onto a centremanifold (in this case, a Halo Orbit) as described by a 9th order LindstedtPoincaré perturbative solution of the Elliptical Restricted Three Body Problem (EP3BP). In effect, LP solution 'presolves' the equations of motion so that what you see is an accurate representation of the true ballistic dynamics of a spacecraft on such an orbit. And, as with all of the centremanifold orbits, stationkeeping is needed to keep the vessel 'on track' so a Lua script provides a simple stationkeeping method that will keep the vessel on the quasi Halo orbit indefinitely. 
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