On Fri, 2003-10-24 at 00:11, David Weinshenker wrote:
> This would suggest that for easiest steering, the orbital vehicle
> should in fact make its "horizon turn" as a maneuver on either the
> pitch or roll axis (depending on which way came out "easy") after
> selecting the desired azimuth in a vertical roll.
I think it's only a problem for arrangements with small numbers of
engines. If you have more engines, you should be able to be clever about
it. I don't know what the magic number is, but I have hunches. One thing
that came out of my reasoning that is interesting is that the problem
goes away if the engines are on the same plane as the Cg. Observe:
First, take the thrusts to be represented as a 4-vector, (t1,t2,t3,t4),
where the engines are numbered by taking the one on the positive X
direction as 1 and going clockwise. Take the engine cant as an angle a.
Further, assume t1+t2+t3+t4=c, where c is a constant. Therefore, the
total engine thrusts are represented by the following vectors:
[0,t1sin(a),t1cos(a)]
[t2sin(a),0,t2cos(a)]
[0,-t3sin(a),t3cos(a)]
[-t4sin(a),0,t4cos(a)]
and their positions relative to the Cg by:
[r,0,0]
[0,r,0]
[-r,0,0]
[0,-r,0]
Therefore, the resulting moment is:
[(t2-t4)rcos(a),(t3-t1)rcos(a),(t1-t2+t3-t4)rsin(a)]
When t1=t2=t3=t4, then this is of course zero. To pitch the vehicle
(rotate around the Y axis), set t2=t4, t1=(1.1)t2, t3=(0.9)t2. The
resulting torque is:
[0,-0.2cos(a),0]
No cross-coupling. Now, say we want to pitch and yaw at the same time
(rotate around Y and X axes at the same time. So, we set t1=(1.1)c/4,
t2=(1.2)c/4, t3=(0.9)c/4, t4=(0.8)c/4. The resulting torque is:
[0.4cos(a),-0.2cos(a),0]
This produces no cross coupling as long as the engines are on the same
plane as the Cg. However, if we assume that the engines are all at some
negative z-coordinate, then things get a little more interesting. The
moment becomes:
[(t2-t4)rcos(a)+(t3-t1)rzsin(a),
(t3-t1)rcos(a)+(t2-t4)rzsin(a),
(t1-t2+t3-t4)rsin(a)]
Clearly, this leads to cross-coupling if the Cg is not on the same
plane as the engines. It's also not fixable by rotating orthogonal axes,
because the cant angles are in opposite directions.
This should also explain why John has gotten away with ignoring the
issue -- the Cg of the vehicles he has flown to date has been close to
the plane of the engines compared to the radius, and he has used small
cant angles. As long as:
rzsin(a) << rcos(a)
Then there should be no problem. However, as the vehicles get longer and
skinnier, it becomes more and more of a serious issue. Where precisely
it becomes a problem is dependent on the specifics of the vehicle and
the control system, and that's what accurate software simulations are
for.
-p
--
No science without fancy, no art without facts.
-Vladimir Nabokov
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