Constant Bearing Intercepts in a Rotating Frame

[Hlynkacg]

So I'm trying to write some code that will mimic the behavior of the Apollo Spacecraft's rendezvous guidance and instrumentation. could use some clarification.

I know from reading NASA's documentation that the system used the Constant Bearing Method but I’m having trouble visualizing how this works in a rotating reference frame (IE when both the target and the interceptor are in orbit). I know what to do with the bearings once I have them but how to calculate in the first place them, is less clear.

My first impulse is to measure the angle between the target’s position and the interceptor’s velocity, but I’m not sure how accurate this is, and whether it would hold true for off-plane intercepts.

 

[ADSWNJ]

Fellow RV guidance programmer ... I salute you!

 

[Hlynkacg]

Forget the salute. I work for a living. Any ideas?

---------- Post added at 21:07 ---------- Previous post was at 20:33 ----------

After some experimentation, using the angle between the interceptor's Orbit Normal and the target as the "lateral bearing" gets me half the solution.

I can accurately predict my range/time to intercept in the horizontal frame, but matching altitude is proving more difficult. Even on a pre-loaded "correct" intercept trajectory the angles between target, interceptor's velocity and interceptor's local vertical vary greatly.

 

[indy91]

I should know a thing or two about Apollo rendezvous, I've been working on that for NASSP. I'm not 100% sure what your goal is, the terms you used are a little bit unclear to me.

If I understand it correctly, your are referring to a concept used during the last part of an Apollo style rendezvous, also used during Gemini. If you are not using a Hohmann transfer (180°) for the terminal rendezvous, but instead a shorter transfer (130°-140°) a very helpful effect occurs, your inertial line-of-sight to the target vessel becomes very close to constant. So you can manually rotate to look at the target, then go into attitude hold (relative to the stars of course) and do all your corrections with translational thrusters, without the need to do any rotations, and keep the target in the window. That way it kind of becomes a 3-dimensional problem instead of a 6-dimensional one for the astronauts to do the final, manual approach to the target. Actually, Buzz Aldrin was one of the first to explore these kind of trajectories in his PhD thesis, in the early 60s.

Anyway, your goal is to calculate this line-of-sight rate that should be close to zero? I could probably come up with a few equations, if that is the goal. I can give you a few primary sources, e.g. the astronauts even had a chart for the line-of-sight in the final part of the rendezvous: http://history.nasa.gov/ap12fj/pdf/a12_lmtb.pdf on page 14. Note that this all only really works, if your starting orbit is concentric (or rather coelliptic when we are talking about orbits) to the target orbit.

Was any of this helpful?

 

[ADSWNJ]

Forget the salute. I work for a living. Any ideas?

Can you show me what NASA documentation you were looking at? I'll take a look and see if I can help.

 

[C3PO]

The only place I can see how constant bearing would work in orbit is for plane alignment.

 

[Urwumpe]

The only place I can see how constant bearing would work in orbit is for plane alignment.

For docking as well, if you have a flight time to the target, that is significant shorter than the orbit period.

http://ijass.org/On_line/admin/files/2)(154-169)20120430.pdf

 

[Hlynkacg]

@ indy91.
At the moment I'm writing a MFD that replicates the LM's Range/Rate and X-Pointer instruments along with a portion of the ascent guidance AP. The goal is to be able to practice Apollo style ascent/rendezvous in (and exercise my code) in a DG or other generic Orbiter vessel before porting it over to my own LM addon.

I am aware of Aldrin's thesis, but the "LOS angle" is where I get lost. Angle relative to what? You need at least two vectors to form an angle.

As I said above using the angle between the rendezvous target and my orbit normal to match orbital planes is pretty straight forward. But completing the intercept is proving a bit more difficult.

Can you show me what NASA documentation you were looking at? I'll take a look and see if I can help.

I downloaded the Delco manual for the Apollo Guidance Computer along with the LM14 Pilot's Manual, and Operation's Handbook off of NTRS but the links don't seem to be working anymore.

---------- Post added at 13:45 ---------- Previous post was at 13:44 ----------

For docking as well, if you have a flight time to the target, that is significant shorter than the orbit period.

http://ijass.org/On_line/admin/files/2)(154-169)20120430.pdf

That looks promising.

 

[indy91]

As the absolute LOS angle the elevation angle of the target was used, so between local horizon and target. Here a neat set of equations used by the AGC for this:

[MATH]U_L = unit(R_P-R_A)[/MATH][MATH]U = unit(R_A \times V_A)[/MATH][MATH] U_P = unit[U_L-(U_L \cdot R_A) R_A/r_A^2][/MATH][MATH]E_A = \arccos[U_L \cdot U_P sign (U_P \cdot U \times R_A)][/MATH]
My LaTeX syntax is rusty, but R_A and V_A are the active vehicle position and velocity vectors, r_A the length of R_A, and R_P and V_P the position and velocity vectors of the passive vehicle (target). E_A is the elevation angle.

The Terminal Phase Initiate (TPI) maneuver was calculated by the AGC to occur at a specific elevation angle (26.6°), which would allow the burn attitude to be close to boresight on the target. After the last midcourse correction the target was again boresighted, i.e. aligned with the x-axis of the LM. As I explained above this attitude was then held throughout the rest of the final rendezvous phase. This is a manual technique and it doesn't actually matter what specific attitude is used, as long as the x-axis is pointed at the target. Then, because the line-of-sight rate becomes close to zero with this rendezvous concept, this attitude doesn't have to be changed anymore.

 

[C3PO]

For docking as well, if you have a flight time to the target, that is significant shorter than the orbit period.

:thumbup:

I assumed we were talking about rendezvous. My bad. :embarrassed:

 

[Hlynkacg]

@Indy91
Thank you for that. though I'm a little confused by what's going on in that fourth equation.

Is that last term supposed to be the dot product of UP and U multiplied by the Vector R_A? or is it the dot product of Up and the cross product of U and R_A?

Likewise we are multiplying the sine (not "sign") of the above by the dot product of Ul and Up to get our input for the inverse (arc) cosine function yes?

Thank you again.

 

[indy91]

Well, mostly the fourth equation is supposed to be the angle between U_P and U_L. The sign, and it is sign, not sine, has to do with the definition of the elevation angle. I think it is supposed to have a consistent definition of the angle, even if the target vessel is above and/or behind the active vessel.

And finally, there is only one order of operations that works with the part of the equation in the sign, so it first is the cross product and then the dot product. The other way around, and you would have to calculate a cross product from a vector and a scalar, doesn't work.

 

[Hlynkacg]

Well, mostly the fourth equation is supposed to be the angle between U_P and U_L. The sign, and it is sign, not sine, has to do with the definition of the elevation angle. I think it is supposed to have a consistent definition of the angle, even if the target vessel is above and/or behind the active vessel.

In essence then, it is the angle between local vertical (unit vector from the orbit's focal point to the vessel) and the relative position offset by 90 degrees. That makes sense actually.

And finally, there is only one order of operations that works with the part of the equation in the sign

I realized that after I got up to get lunch :facepalm: