# A knotty problem: Boffins working on fuel-efficient trajectories for space travel

Boffins at the UK's Surrey Space Centre have devised a way of determining the optimal route for spacecraft that doesn't require the engines to burn precious fuel. The concept of shifting from one orbit to another without using fuel is not a new one. Engineers have turned to heteroclinic connections as a way of transferring a …

1. #### Top Boffinery

The kind of Boffinery that I struggle to understand, yet admire greatly

2. #### heteroclinic connection

I had to look this one up, it means:

"In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end of the orbit are the same, the orbit is a homoclinic orbit."*

So, as I understand the article, you put the spacecraft on a trajectory which links two points of equilibrium, so that when it reaches the second point it just kind of stays in that orbit.

*https://en.wikipedia.org/wiki/Heteroclinic_orbit#:~:text=In%20mathematics%2C%20in%20the%20phase,orbit%20is%20a%20homoclinic%20orbit. (Warning, proper mathematics.)

1. #### Re: heteroclinic connection*

《 So, as I understand the article, you put the spacecraft on a trajectory which links two points of equilibrium, so that when it reaches the second point it just kind of stays in that orbit.》

I checked out the wiki article - proper mathematics giving greek alphabet a decent seeing to. 40 years ago I might have had half a clue but pretty clueless now. :(

I imagine it is like throwing a ball right next to a wall on a slighty parabolic trajectory with just enough impetus that just as it reaches the top of its trajectory it is just over the top of the wall and comes to rest on top of the the wall just as it begins its descent. (Assumes the we can treat the horizontal component of the ball's velocity as neglible.)

The time varying position of all three bodies in the orbital case must make the analytic solution close to intractable.

As the first post opined - proper boffinery of the first water.

* apparently doesn't mean bonking one's spouse.

1. #### Re: heteroclinic connection*

I remember finding a paper by some SpaceX guys on the booster landing guidance. My meager maths abilities instantly noped out of there and was last seen running for the hills.

In 50+ years of being a space geek, and 10 years of playing KSP, this is the first I've heard the term "heteroclinic connection"

I do like how "heteroclinic connection" in Wikipedia redirects to "heteroclinic orbit" which has a link to "heteroclinic connection", which actually goes to "homoclinic connection"

Out-of-context statement of the week: "Homoclinic tangles are always accompanied by a Smale horseshoe"

1. #### Re: heteroclinic connection*

KSP is great and I understand their decision for using it in the game but the discrete spheres of influence model mean you miss out on some advanced orbital mechanics. That said, I think its the best compromise to make the game accessible.

4. #### The important questions

So when is this going to be added to MechJeb?

1. #### Re: The important questions

I am not sure about KSP2 but KSP1 physics does not model three body problems. You orbit the body that causes the greatest gravitational field at you current location. When you pass a boundary surface you switch to orbiting a different body. You and MechJeb cannot take advantage of this mathematics without an upgrade to the model.

1. #### Re: The important questions

You and MechJeb cannot take advantage of this mathematics without an upgrade to the model.

And probably a major upgrade in computing power I would think.

5. #### No, yeah, no

I'm not seeing how this gets around the theory of TANSTAAFL. Being a licensed pilot, I know all about trades between airspeed and altitude. The Grand Tour by the Voyager probes was a really special set of circumstances and was limited since if they needed to steal more of Jupiter's gravity/momentum, it would have required propellant since getting closer would mean dipping into the atmosphere and would have been fatal. Lucky they were able to input all the energy into the system at launch that they needed.

1. #### Re: No, yeah, no

The free lunch comes from the middle body of the three. For Earth/Moon/Satelite a tiny change in the Moon's orbital velocity+altitude makes a huge change in the satellite's velocity+altitude because the masses are so different.

2. #### Re: No, yeah, no

"Lucky they were able to input all the energy into the system at launch that they needed."

That wasn't luck, it was maths.

3. #### Re: TANSTAAFL

1) These complicated trajectories tend to take a long time to get where you want to go, at the speed you want to be going at. For example, look at those comet rendezvous missions involving sling-shotting off the inner planets multiple times, taking several years in the process.

2) Tell that to the quantum computing folks. If an O(n) increase in processing elements is supposed to give you an O(e**n) increase in processing power, and that’s not some kind of “something for nothing”, then I don’t know what it is.

1. #### Re: TANSTAAFL

As a too-simplistic to be useful analogy, you can think of it as a high tide in your Earth orbit.

1. #### Re: TANSTAAFL

Maybe it's not too simplistic. As your skiff is bobbing in the harbour waves you can use motion inside the boat to increase or dampen the bobbing. The spacecraft orbit also bobs at ultra low frequency between low and high tide. You should be able to use a really tiny impulse between those points to amplify the bobbing, repeat as necessary until your highest high tide actually crosses the line into the next orbit. The caveats are the least propellant use requires a large number of orbits, and you have to make sure low tide doesn't encounter significant atmosphere. Just like in the ocean there are two high tides per orbit, so there might be two solutions, or if many orbits are needed then they might merge into one solution.

6. XKCD has been there.

7. #### Astrolabe

May be computationally expensive the first time, but I assume they will be re-using the same craft. They will certainly be re-using the same planet and moon. How many starting and ending orbital paths are we talking about? The point is the equations should converge rapidly *if* the inputs are already close to the correct solution. Every time you make the trip(1), you save the computer's triple of orbit-mass-orbit in a larger matrix of positions of Sun-Earth-Moon (lots of those but due to periodicity not infinite). After enough solutions with the computer you have a lookup table. Now you can balance the cost of waiting for one of the previously known configurations versus the cost of computing a new one. Quite likely there will be some spatial arrangements where you just don't make the trip, others where close enough(2) is good enough, and still others where you simply must use the computer.

(1) Actually the computer will generate lots of unused solutions for each made trip, but let's not go crazy and try to pre-compute the entire solution space ahead of time.

(2) I know the maths are fussy, but at heart it's a flying machine and if normal attitude and altitude adjustments drown out the optimizations that's what I mean by close enough.

8. #### Pocket rockets

Real cool math! Nonlinear dynamics, equilibrium points, isoclines, nullclines, homoclinic orbits, heteroclines, separatrixes ... so many inspirational words and concepts to explore. Love it (and Fig. 7 of the linked OpenAccess paper by Owen and Baresi)!

But beware, the manifold of your quasi-periodic halo orbit might separate when passing x = 1 − μ! (eh-eh-eh! Fig. 25). Likely less painful than an Earth → Jupiter → Uranus Mission though (Fig. 8 of the 1966 Grand Tour [PDF]) for anything larger than a suppository! (gnak-gnak-ouch!)

(great article!)

9. #### What does good enough look like?

Some years back I was working on some bleeding edge communications project, run by a project manager who had a background in mechanical engineering (car gearboxes, brakes and the like). He forced me to explain everything to him in layman's terms (always a good filter) and constantly asked "what does good enough look like?", because in our case you could either compute the heck out of our problem or use some fairly rudimentary approximation, which saved a lot of energy and yielded much the same results, especially as all of our nodes were constantly on the move.

When at some point in the future a craft simply has to move (eg some X-wing fighter is on your tail), would you be able to somehow do this by eye if you had enough experience with the maths-generated paths?

1. #### Re: What does good enough look like?

If some X-wing fighter is on your tail and you are trying to figure out how to minimize fuel consumption, you are either very talented or very dead.

1. #### Re: What does good enough look like?

"If some X-wing fighter is on your tail and you are trying to figure out how to minimize fuel consumption, you are either very talented or very dead."

But what if you are on the Kessel run, with only enough fuel for 11 parsecs, eh?

2. #### Re: What does good enough look like?

"If some X-wing fighter is on your tail"

JUST WHO'S SIDE ARE YOU ON, PAL?

10. #### They've created a Tube map for space?

Does it takes into account the everyday experiences of the London Tube lines? Leaves on the track (space debris), bad weather (solar storms and CMEs), the lovely British Euphemism "People on the tracks" (I guess other satellites in your desired path), a driver's strike (failed propulsion systems, or lost communication systems)?

Handle all that, and you've got a winner!

11. The real question is: how many parsecs will it take to do the Kessel run?

1. < 12

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