guidance, flight mechanics and trajectory optimization

only line-of-sight thrusting is used, the velocity perpendicular to the line of
sight will become 103v Thus, unless v is zero (initially), rendezvous
is not possible with &%-of-sight thrusti R g. For.the more accurate approximation
of linear terms in the equations of motion, the same situation pertains
except that some sets of initial conditions would reduce the effect and others
would magnify it. In either case, it is clear that a technique for managing
the relative velocity perpendicular to the line of sight and for reducing it to
zero as range and range-rate are reduced to zero is essential for rendezvous.
Further, though an astronaut could learn to make the necessary corrections by
trial and error, techniques for optimal and for automatic control are needed.
The rendezvous operation may be described as the overall solution to the
following set of interrelated problems:
a. The state determination problem.
b. The trajectory determination and prediction problem.
C. The trajectory control problem.
Each of these problems is discussed briefly here:
a. The state determination problem - Before making a course correction,
the current conditions (i.e., the orbit of the target vehicle and the
relative position and velocity of the active vehicle), must be deter-
mined. It is assumed in this monograph that this information, which
is taken to include error estimation, is available at the start of the
problem, as well as at later times as it may be needed.
b. The trajectory determination and prediction problem - the future sepa-
ration of the two vehicles must be predictable in some fashion in order
that changes of velocity can be determined which will cause the
separation to be reduced to zero at the same time the velocity differ-
ence is nulled. The degree of sophistication required in the equations
of motion will depend on the time to make the maneuver and the levels
of thrust that may be used. For times which are short compared to the
period of the motion and for thrust accelerations which are consider-
ably larger than the differences in the gravitational or other
perturbation accelerations between the two satellites, the motion of
the two vehicles approximates completely the field free space problem.
On the other hand, if the time to rendezvous is of the order of a
quarter of a revolution or longer, the equations must contain peri-
odic effects and secular effects produced by the dynamics of the two
bodies. However, since the object of the maneuver is to effect a
reduction of the relative motion to zero, it is to be expected that
approximate representations will be satisfactory as long as errors in
the rendezvous caused by poor representation at large values of the
relative coordinates can be corrected by subsequent thrusting as the
rendezvous is approached. In fact, this capability for error compen-
sation is required since noise and measurement errors in the data
sensed must be taken into consideration. Both the model errors and