I started to work on optimum maneuver strategy in 1992. I found an
optimum manuver strategy for North-South Station Keeping Manuvers. I gave a name
to this strategy "pipe strategy" (Boru Metodu in Turkish). But When I searched
literature , I noticed that this strategy was already found in 1986 :).

First we will develop a geometrical representation of maneuvers. As it is seen
in Fig-1, icnlination starts from (ko,ho) and reaches to (k1,h1) point after T1
period of time. A maneuver with Dv has been performed at this point to keep
inclination inside of circle. Maneuver causes shift of point (k1,h1) to (k2,h2).
Inclination vector is traces again on T2 curve and reaches to (k3,h3) after T2
time. 


fig-1

This general geometry can be better drawn in Fig-2 to understand optimum
strategy. Instead of shifting (k1,h1) to (k2,h2) we can shift frame (k,h) with
the same amount of Dv but in the reverse direction. With this representation
inclination curve becomes free evolution during T1+T2 period of time. Distance
between centers of two circles gives nothing but Dv, maneuver size. 


Fig-2

If we go further to perform maneuvers during lifetime of satellite , we would
get a picture like in Fig-3. Here we can state the optimum problem :

Minimize R1+R2+R3...Rn by changing centers of circles and with a constrain such
that each beginning and end of cycle points will remain in the corresponding
circle.


Fig-3

Any one can easily see that if we can put all the centers of circles to the line
connecting the first and the last circle centers, R1+R2+R3..+Rn will be the
minimum. It seems like to put all the cycle points inside a "pipe".  


Fig-4

When we are positioning the last circle, we must locate it such as the end of
life point will be on the circle and optimum line must intersect bith the center
of last circle and the end of life point.This allows us to leave inclination
free without making any maneuver till the end of life for the last cycle, as 
seen in Fig-5.


Fig-5

Once the optimum line determined , other center of circles can be located 
freely. But circle must include both cycle points. We can choose the closest 
point to the optimum line (ofcourse this is not unique solution). If we align 
center of circle in this way , inclination at the start of maneuver and end of 
maneuver must be equal.


Fig-6

You can send your comments to Senol Gulgonul