guidance, flight mechanics and trajectory optimization

when a = semi major axis
Q = one of Herrick's universal variables
A = matrix
A similar relation can be written for the velocity and variation in the orbit
parameters
The "correctl' velocity required for rendezvous is then given by
2.3 PROPORTIONAL GUIDANCE
xr = 2 +si
4.
= ‘gi ‘&&)&A? i’
Proportional guidance refers to a class of guidance laws which exhibits
smooth thrust functions chosen to be proportional to some function of range
and range rate. The mechanization of these laws requires throttable motors,
this requirement may be undesirable when compared with the relative simplicity
and reliability of constant thrust motors. Also, these methods are ex-
pected to be less optimum than those using constant-thrust motors since
optimization of maneuvers indicates that thrusting should occur in periods
of either full or zero thrust (see Section 2.4). The computation of the
thrust vector, however, is quite simple since the gravity model is not
introduced into the equations.
An example of proportional guidance is presented by Sears and Fellman
(Reference 2.10). In this reference, the rocket thrust is determined from
the equations
c
= s,[p- p I-s,+ St)
c = s3[p-$a+s, (p (ys)
where Sl, S2, S3, S4, S , and S6 are constants and LS is the angular rate
of the line of sight. 'i The subscripts r, c and z refer io the radial,
circumferential and out of plane direction.) If the constants Sl and S3
are made equal and S2 and Sk are set to zero, the thrust direction will be
along the line of sight and the thrust magnitude will be determined by the
difference between the range rate and the square root of the range. For
this case, the trajectory as seen by an observer on the target satellite
will appear as in Figure 2.13.
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