An update on all things Lissajous and Halo

MontBlanc2012

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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 Earth-Moon, Sun-Earth and other restricted three-body 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 station-keeping fuel is exhausted). At the moment, my station-keeping 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 station-keeping dV requirements should be low since the station-keeping is primarily limited to offsetting minor gravitational perturbations. For those interested, and in terms of monthly station-keeping dV requirements, my preliminary estimates of station-keeping dV requirements are as follows:

Earth-Moon L1: 1.9 m/s
Earth-Moon L2 : 2.7 m/s

Sun-Earth L1: 0.6 m/s
Sun-Earth L2: 0.6 m/s

Sun-Mars L1: 0.2 m/s
Sun-Mars L2: 0.2 m/s

Sun-Vesta L1: 0.2 m/s
Sun-Vesta L2: 0.2 m/s

These dV estimates are not large. However, The my station-keeping logic is very simple: it just 'locks' a vessel onto the closed orbit trajectory and keeps it there. More sophisticated station-keeping 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 station-keeping 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 station-keeping. When I've finished testing in the next few days, I'll post these scenarios here on O-F. In essence, for each of the eight Lagrange points listed above, the scenario will consist of two docked (stock) Delta-Gliders. One of the Delta-Gliders will be 'locked' to the Lagrange point; and the other will initially be docked to the first Delta-Glider. One can then undock the two Delta-Gliders. The first will continue to be locked to the Lagrange point (and perform the required station-keeping 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 station-keeping 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 station-keeping 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 station-keeping algorithm; and then post it. This should take a couple of weeks. Again, I'll write a few more scenarios that illustrate station-keeping 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 station-keeping on closed-loop 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.
 

Uramogi

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So excellent!
Can we use the Lagrange MFD to create a halo or a Quasi-halo orbit ?I really like the manifold.
 

MontBlanc2012

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So excellent!
Can we use the Lagrange MFD to create a halo or a Quasi-halo orbit ?I really like the manifold.

From what I know of Lagrange MFD, it’s a great tool for mapping the Lagrange points themselves - i.e., in particular L1 and L2. It also has a Interplanetary style n-body integrator so that one can predict the future of one’s current trajectory. However, Lagrange MFD doesn’t map Halo orbits and the like (the center manifold orbits), so it provides little useful information about where these orbits are - or how to stay on them.

The point of all of my long-winded 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 station-keeping on those orbits . The maths of the centre manifold orbits is significantly more painful than the usual Keplerian, two-body 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 Delta-Glider 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.)
 
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MontBlanc2012

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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 02-19-19 at 03:05 AM ---------- Previous post was 02-18-19 at 12:47 PM ----------

And to complete the trilogy, here is a short video of an arbitrary (quasi-) Halo Orbit example around Earth-Moon L2.

[ame="https://youtu.be/-YbpODhSzRE"]https://youtu.be/-YbpODhSzRE[/ame]

As with all of these videos, a vessel is 'locked' onto a centre-manifold (in this case, a Halo Orbit) as described by a 9th order Lindstedt-Poincaré perturbative solution of the Elliptical Restricted Three Body Problem (EP3BP). In effect, LP solution 'pre-solves' 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 centre-manifold orbits, station-keeping is needed to keep the vessel 'on track' so a Lua script provides a simple station-keeping method that will keep the vessel on the quasi Halo orbit indefinitely.
 

MontBlanc2012

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Dear Sir -

Are you still engaged in the study of Lissajous orbits?

Lori
The short answer is: yes.

However, a bunch of more practical issues (associated with macro-economic concerns and, of course, COVID-19) have stolen much of my attention for the last year or so.

At some point, I'm sure, I'll find the time to focus on all things Lissajous once again.

Regards
 
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