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I'm curious how people plan an orbital rendezvous. This question explains the basic principles, but not in enough detail for me to understand how it actually works.

From playing Kerbal Space Program, it seems that this is very hard to get right:

  • If I try to match the orbit (by adjusting my periapsis/apoapsis and inclination), I invariably end up with a large phase angle from my target.
  • If I try to time the launch so that I go straight into my target's orbit, even being a second or two late can mean I end up several kilometers away from the target, but it is very hard to predict how long a launch will take to that accuracy.
  • The process is very counter-intuitive, because if I am near my target and I try to thrust towards it, what happens instead is that I warp my orbit on the other side of the planet.
  • When I try to manipulate the phase angle by going into a slightly higher orbit, it becomes hard to make predictions because the phase angle doesn't stay constant when my orbit is more eccentric than the target.
  • If the phase angle difference is large, I need to either spend a lot of fuel making big adjustments to my orbit, or I need to wait for dozens of orbits to catch up to my target.

All of this seems very error prone and time intensive. KSP doesn't model things like provisions, so the problem is a bit easier. So long as you can create even a tiny difference in phase, you can just crank up the time warp and wait for it to synchronize. But I'm sure that in reality, it wouldn't make sense to send astronauts up and then have the sit in orbit for days or weeks, painstakingly adjusting their phase angle with the ISS.

So, how do space agencies plan their rendezvous missions? Do they have a very precise estimate for the launch, and end up at the right phase when they match orbits? Or is there a good algorithm for optimally synchronizing orbits once you are already in some orbit?

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So, how do space agencies plan their rendezvous missions?

Carefully, and with computers. Computer also plan and perform most rendezvous burns. Rendezvous nowadays is likely to be fully automated.

The first step in planning a rendezvous is to determine the launch opportunities and launch windows. Cooperative rendezvous (e.g., the Soyuz, ATV, HTV, Dragon, or Cygnus rendezvousing with the Space Station) involves an active chaser vehicle and a passive target vehicle. If the chaser vehicle is launched, as are all the vehicles cited above, that cannot occur at any arbitrary time. Launching at any time would necessarily entail extremely expensive plane changes.

The vehicle must instead launch when the launch will make the chaser be in more or less the same orbital plane as the target vehicle. This happens roughly twice a day, near the time when the orbital plane crosses the launch site. The chaser must also launch so that it is moving in the same direction as the target. This coupled with launch constraints (e.g., a vehicle can't launch southwest from the Cape) cuts the twice-a-day launch opportunities down to one.

Those once a day launch opportunities apply only to a rendezvous plan that can handle any phase difference between the chaser and target. The downside is a potentially long rendezvous process. The upside is launch opportunities occur once a day.

There's only one opportunity per year if the rendezvous plan can only accommodate a one degree span of phase angles. That's obviously not good. A rendezvous plan needs some slack in it so it can accommodate a reasonable span of phase changes. A plan that has the chaser transferring direct from orbit insertion to the target is a recipe for a plan with no slack.

Finally, there's the separate concept of a launch window. What happens if there's a fixable problem during the pre-launch countdown? Can the launch occur a few minutes later than planned and still result in a successful rendezvous? Every minute delay results in about 1/8th degree of plane change and 4 degrees of phase change. At some point, the discrepancies become more than the vehicle can accommodate and the window closes.

Do they have a very precise estimate for the launch, and end up at the right phase when they match orbits?

The only way to end up at the right phase is to start somewhere near the right phase. A rendezvous plan must be able to handle some range of phase differences, but expecting a rendezvous plans to handle the full 360° range of phase differences inevitably means a very long rendezvous sequences.

You can't have both a short rendezvous sequence and handle a large swath of phase differences. You can only launch during the launch window that surrounds a launch opportunity.

Your key problem is that you are trying to transfer directly from orbit insertion to the target vehicle. That's a "Doctor! It hurts when I do this! 〈bonk〉" type of situation. Don't do that then. You need to use phasing orbits are the mechanism by which errors are reduced and slack is introduced. Even the fast six hour rendezvous used by the Russians use phasing orbits.

Or is there a good algorithm for optimally synchronizing orbits once you are already in some orbit?

There are lots of good algorithms. At some point, a real spacecraft will need to use Lambert targeting. Lambert's problem asks "I'm here in some orbit at some point in time, but I want to be there in some other orbit at some later time. How do I accomplish that?"

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  • $\begingroup$ KSP has the advantage of launching from the equator. We can force the inclination to be zero merely by being a good pilot. If trying this in the real world, the inclination differences will eat your fuel so fast it's not funny. $\endgroup$ – Joshua Aug 15 '16 at 16:36
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Going from Earth, the most important thing is getting the plane right: both the inclination and the longitude of the ascending node (LAN), since Earth is rotating. Thus, launch windows first and foremost depend on getting directly into the target object's plane (which means timing the launch to a second).

Second, you are correct in ascribing importance to phase angle. Weeding out the "phase angle too large" windows for the desired time-to-rendezvous leaves you with a much smaller set of launch times. You also want to place restrictions on the section of the orbit where the final phases of rendezvous and docking take place (illumination and, often, radio comms visibility from mission control).

Third, you have to understand that there are errors in the rendezvous process that have to be removed with the least expenditure of propellant:

  • launch timing error
  • orbit insertion error
  • thrust and time-of-burn discrepancies for the active spacecraft

Thus, there are pre-defined times when orbit correction maneuvers are scheduled based on the errors estimated with the help of ground tracking and ballistics facilities.

The answer to your question is, accordingly, both: a goodly amount of prior planning and preparation is involved to launch "just right", plus an optimizing algorithm of orbit correction maneuvers is employed.

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According to this article, it can take days to rendezvous with the ISS.

On the other hand, Gemini 6A and 7 rendezvous'd in only about 6 hours from the second launch, and after using a standard 2-day profile for manned flights to ISS for a long time, Soyuz flights have switched to a 6 hour profile.

Modern missions do have better estimates for launch times than the typical KSP player does, but there's always a bit of uncertainty in launcher performance and trajectory, so the initial conditions for the rendezvous are unpredictable.

Going to ISS at 420km altitude, if you start in a lower 320km orbit, you'll gain about a degree of phase every 10 minutes. So the worst case would be 2.5 days for the initial interception, but several more iterations of approach have to occur after that.

In KSP, the usual technique I use is to match planes with craft A in a circular orbit and craft B in an elliptical orbit with apoapsis just touching A's orbit. B is then gaining on A in each orbit. On each apoapsis, examine the next upcoming closest approach in the map view; when the next closest approach is an overshoot, burn prograde at the apoapsis (raising periapsis, making a bigger slower orbit) to reduce the overshoot to zero.

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  • $\begingroup$ Since the ISS's has an orbital period of about 90 minutes, the ATV must be taken over a hundred revolutions to RV - this isn't actually that much different from my worst RVs in KSP. But isn't it wasteful? They say they want a standard RV time that is good for every mission, but they could probably save on launch costs by eliminating the mass of all the food the astronauts would need during those days. $\endgroup$ – Superbest Nov 30 '14 at 5:47
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    $\begingroup$ Standard for the manned Soyuz flights to ISS was 2 days, previously, and they've switched to a 6 hour profile. I'm not sure what the pros and cons are. en.wikipedia.org/wiki/Soyuz_TMA-08M $\endgroup$ – Russell Borogove Nov 30 '14 at 6:24

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