Joining two orbiting space vehicles

[danielandpenn]

I'm curious to know how the engineers of NASA determine how to join two moving, or orbiting objects in space. What type of mathematics helps them determine their trajectories? Thanks!

 

[D H]

https://www.amazon.com/dp/0521824923/?tag=pfamazon01-20

A used paperback that costs $90!

A university library will probably have this book.

 

[danielandpenn]

Thanks D H, but I was just curious. I don't want to buy a book. I really just wanted to be lazy and have someone give me a clue. (haha) It fascinates me but I'm in school and raising a family so I don't have the time to add another book to my "to-do" list. I appreciate your giving me the info tho.

 

[A.T.]

https://www.amazon.com/dp/0521824923/?tag=pfamazon01-20
A used paperback that costs $90!

Well it's a book for someone who already owns at least two spacecraft , you know.

 

[DaveC426913]

I'm curious to know how the engineers of NASA determine how to join two moving, or orbiting objects in space. What type of mathematics helps them determine their trajectories? Thanks!

I do believe it's straight orbital mechanics.
Forward (thrust) means out.
Out means back.
Backward (braking) means in.
In means forward.

 

[rcgldr]

Numerical integration is used to calculate orbital paths, to compensate for the variation in gravity. The approaching vehicle uses a lower orbital path, which will have both a faster linear and angular speed. Once it's within range, a burst of thrust is used to initially increase speed and/or alitude, with the ultimate goal being the nearly the same averate orbital radius and speed as the target vehicle. A second burst is needed to correct the shape of the orbital path. Still the approaching vehicle will have a slightly lower and more elliptical path, in order to intercept the target vehicle. Once it's close, small control rockets are used to control rotation and lateral movments. There is a targeting window with cross hairs or something similar on the active vehicle, and a painted target on the passive vehicle. The pilot has to adjust any rotation and lateral movment to get it very close, and approach at very slow speed. The vehicles then collide at some slow speed, and the docking system has some type of shock absorption to deal with this minor collison. The docking mechanism then locks the vehicles together, and if needed forms an air tight seal to allow passengers to move back and forth between vehicles.

 

[D H]

Things proceed in stages. Suppose the approaching vehicle is launched from the ground. The launch has to be very well timed or the rendezvous will not happen. For example, the Shuttle has a 2.5 to 10 minute long window during which the launch must occur if the Shuttle is to later successfully dock with the Space Station. Once on orbit, the approaching vehicle climbs, in stages, to close to the target vehicle's orbit. The goal here is to get close enough to the target vehicle so that sensors can see it, but not so close as to be a hazard. Up until this point, where the target vehicle will be involves a bit of guesswork.

Note that I said "will be". If you've ever gone duck hunting, you know what I mean. To hit the duck you need to aim at the spot where you think the duck will be when the shot crosses its path. Aim directly at the duck and you will miss. Don't take this analogy too far. A duck flies at about 20 m/s. The target vehicle is moving at about 7.7 km/s. The goal in duck hunting is a rather violent collision between the duck and the shot, and exactly where the contact occurs isn't all that important. The goal in rendezvous is a very gentle collision between the vehicles, and exactly where the contact occurs is extremely important.

So, now the two vehicles are about 20 kilometers apart. The approaching vehicle switches from absolute guidance to relative guidance. Relative guidance means the vehicle is using sensors to detect the relative position, velocity, orientation, and rotation of the target vehicle. The approaching vehicle slowly closes the gap between the vehicles down to a few kilometers and then a hundred meters or so. This is the near field rendezvous phase.

Those last hundred meters or so are very delicate. The last thing one wants is an uncontrolled collision. Closing the gap to ten meters or so is done slowly, often with stops. This is the proximity operations phase. How it proceeds depends a lot on how and where the final act of contact is to occur. Some vehicles dock directly with the target vehicle. Some reach a hold point and are grabbed by a robotic arm on the target vehicle. Others might grab the target vehicle with their own robotic arm. The vehicle can approach from behind, in front of, or below the target vehicle. The target vehicle can be cooperative or uncooperative. These decisions are made well before launch and often dictate the design of the approaching and target vehicles.

The final ten meters are the most delicate of all. A docking vehicle will close the final gap with relative speeds in the centimeters per second range. A vehicle that berths will simply stop (not so simply, actually; orbital mechanics can be cruel) and wait to be grappled.

 

[D H]

The underlying mathematics are the Clohessy-Wiltshire equations (aka Hill's equations).

Reference: http://ccar.colorado.edu/asen5050/lecture12.pdf

 

[mbunds]

...Once it's within range, a burst of thrust is used to initially increase speed and/or alitude, with the ultimate goal being the nearly the same averate orbital radius and speed as the target vehicle...

So, you have to speed up in order to slow down!?

 

[rcgldr]

A burst of thrust is used to initially increase speed ... A second burst is needed to correct the shape of the orbital path.

So, you have to speed up in order to slow down!?

The intial burst increases speed, resulting in a "higher energy" eliptical orbit. As the vehicle follows the elliptical orbit, it's speed decreases as it's altitude increases. The second burst is made near the peak (and slowest speed) of the elliptical orbit, with the goal typically being a nearly circular orbit. This process is describe in this wiki article:

http://en.wikipedia.org/wiki/Hohmann_transfer

The point I was trying to get at before, is that the approaching vehicle approaches via a lower orbital path, where both angular and linear speed are faster. I don't know how many bursts are used to transition from the lower orbit to an orbit within the 20 km range that DH mentioned, but the minimum is 2. The idea here is to launch the intercepting vehicle so that it's "normal" climb into an orbital path process results with it being about 20 km behind and below the target vehicle, so that's it's approaching, but at a relatively slow closing speed.

If you watch an Apollo video, such as the DVD, "For All Mankind" (recently released, although it's really old), there are some long sequences where you can see the lunar landing module orbiting just under the lunar orbiting module.

 

[mikelepore]

https://www.amazon.com/dp/0521824923/?tag=pfamazon01-20
A used paperback that costs $90!

We can read part of that book online for free here:
http://books.google.com/books?id=8I2IfZH7PtAC

Lots for free at books.google.com.
They have three kinds of books there.
1. Full view,and you can download it.
2. Full view, but you can't download it
3. Partial preview only, and you can't download it.