PEG 4 - meaning of c1 and c2

[Thorsten]

I wonder if someone can clarify this for me (or point me to the right document) - I'm wondering what c1 and c2 in the PEG 4 target really parametrize.

I've found an explanation giving the relation between v_h (horizontal) and v_v (vertical) as v_v = c1 * v_h + c2, but... it doesn't seem to make any sense.

Mathematically the PEG 4 target must be a 2-dim state vector, and I can readily see how HT and theta-t translate into (x,y) given a current position, so c1 and c2 *should* somehow translate into (vx, vy) or (v_h, v_v) to specify the velocity part of that state vector.

Yet if I actually assume v_h and v_v as given, the relation is (unless v_h =0) redundant, since I can always find a c1 that satisfies the relation above and would never need c2. So there doesn't seem to be a mapping to (vx, vy), it must stand for something else, and I wonder what that is.

Thanks in advance.

 

[Urwumpe]

These are energy constants, there are three of them known, C3 is the most famous and also known as characteristic energy (which is twice the orbit energy)


Here is some mathematical background about them:

http://brane-space.blogspot.de/2010/08/celestial-mechanics-ii-energy-constants.html

 

[fred18]

Can I ask you where did you find a detailed explanation of PEG4?

I could only find the old PEG

Thank you very much

 

[Urwumpe]

Can I ask you where did you find a detailed explanation of PEG4?

I could only find the old PEG

Thank you very much

What old PEG?

PEG4 and PEG7 are just different entry points for the same algorithm. It is used literally everywhere in the Shuttle GNC software.

 

[fred18]

What old PEG?

I just mean this:

 

[Urwumpe]

I just mean this:

That is just the basic algorithm. The Space Shuttle uses a more complicated algorithm wrapped around this. The PEG equation is for used for every kind of powered flight, but there are different modes defined (0 - 7 AFAIR) which change the constraints of the PEG algorithm.

 

[indy91]

The original Shuttle Unified Powered Flight Guidance document (UPFG) can be found here: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19740004402.pdf

C1 and C2 are constants to create a linear relationship between vertical and horizontal velocity at Entry Interface. C1 and C2 are chosen so that the Shuttle has the correct entry angle as a function of the inertial velocity. The higher your velocity, the steeper you want to be at EI. And the linear relationship helps with convergence I guess.

EDIT: I wonder why they didn't use the same C1 and C2 for every flight then, because I know they didn't. I guess they wanted to be somewhat flexible with the reentry while still having an onboard capability to calculate the deorbit maneuver.

 

[fred18]

That is just the basic algorithm. The Space Shuttle uses a more complicated algorithm wrapped around this. The PEG equation is for used for every kind of powered flight, but there are different modes defined (0 - 7 AFAIR) which change the constraints of the PEG algorithm.

That was my question: where can i find a bit of documentation of PEG 0-7 modes

---------- Post added at 20:57 ---------- Previous post was at 20:52 ----------

The original Shuttle Unified Powered Flight Guidance document (UPFG) can be found here: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19740004402.pdf

:tiphat:

 

[Thorsten]

These are energy constants, there are three of them known, C3 is the most famous and also known as characteristic energy (which is twice the orbit energy)

Well, the blog entry defines them as (r cross r') = (c1/h, c2/h, c3/h), which, yes, in a suitable coordinate system picks out a large component (the characteristic energy as you say) and two perturbations. But they'd all have the same dimension in this formula.

Whereas the description of the MNVR display where they're supposed to be entered is pretty clear on them not having the same dimension:

(C1 in fps and C2 a dimensionless value),

(JSC-48017, 3-8).

So again I hit the step where there's some information missing :-/

C1 and C2 are constants to create a linear relationship between vertical and horizontal velocity at Entry Interface. C1 and C2 are chosen so that the Shuttle has the correct entry angle as a function of the inertial velocity.

Yes, but see, that's my problem above: The linear relationship

v_v = v_h * c2 + c1

does in fact not determine v_h and v_v uniquely. Or, given v_h and c1, I could already specify any angle unless (v_h = 0) by just giving c2 and never talking about c1 at all. If I set v_h, I can find an infinite number of pairs (c1, c1) which all give me the same v_v.

So there must be a second defining equation somewhere.

Which is why I am asking this question - the various bits and pieces of information I found don't add up to a consistent whole, there's some piece missing which ties all together.

Edit: Ah, wait - I think now I get it: the (c1, c2) target doesn't actually specify a target state vector, it specifies a whole acceptable set of them, so in fact any (v_h, v_v) which lies on the line defined by the linear relationship is acceptable - so the thing not being unique is by design - that correct?

 

[indy91]

Edit: Ah, wait - I think now I get it: the (c1, c2) target doesn't actually specify a target state vector, it specifies a whole acceptable set of them, so in fact any (v_h, v_v) which lies on the line defined by the linear relationship is acceptable - so the thing not being unique is by design - that correct?

Yes, v_h and v_v are calculated from the velocity vector at EI and have a solution basically depending on the inertial velocity magnitude. This velocity vector is a result of the "Required Velocity Routine", which solves Lambert's problem. An iteration is started to adjust the time of flight between deorbit and EI until the equation with C1 and C2 is satisfied. The document I linked above of course goes into much more detail.

 

[Thorsten]

Thanks, that was pretty much what I was looking for!