Per discussion at the last meeting, here is a simple model for the slew
rate:          /
              / \                         /\
             / (theta)                    ||
            /     \                       ||
   (antenna) <------- (distance) -------> /\ (rocket)
                                             (on pad)

Assume the antenna is located a distance (d) from the launch point and
the rocket leaves the pad vertically with constant acceleration (a)
and altitude (h). If the antenna tracks the rocket by rotating upward
with angle (theta) then

  h = 1/2 a t^2

  theta = ArcTan[ h / d ]

Take the time derivative of (theta) to get the angular (slew) rate

  slew = (a d t) / (d^2 + (1/2 a t^2)^2)

To find the maximum slew rate, take the time derivative of (slew) and
set the result to zero and solve for (t). There is only one positive
real root, and it is a maximum. The result is the time at which the
slew rate is greatest

  tMax = sqrt( 2 d/(a * sqrt(3)))

Substituting (tMax) back into the slew rate formula gives the maximum
slew rate

  slewMax = 3^(3/4)/2 * sqrt(a/(2 d))

Example:  (a) = 100 m/s^2, (d) = 100 m
  tMax    = 1.07457 s
  slewMax = 0.805927 r/s

BTW, i may have missed the discussion on this, but the antenna
patterns aren't very tight, maybe 30 deg 1/2 width, so i think a
10 deg pointing accuracy is a reasonable initial goal. Probably a
little looser is still ok.

psas-avionics mailing list

Reply via email to