Two-Body Launch Vehicle Problem
Ron Graham

If the launch vehicle is treated as two lumped masses -- one for the main body and one for the engine(s) -- then there are three degrees of freedom:
Parameters
Parameter Description Units
I inertia, main body mass * distance
IE inertia, engine mass * distance
l distance, gimbal to CG distance
lE distance, gimbal to engine mass distance
m mass, main body mass
mE mass, engine mass
T force, engine thrust mass * distance / time2
x axial translation distance
y lateral translation distance
δ rotation, engine radians
θ rotation, rigid body radians
θC attitude command radians

A small-angle approximation applied to the two rotational DOFs leaves the following expression for lateral translation:

two-body equation 1

For rigid body rotation:

two-body equation 2

For engine rotation:

two-body equation 3

Here, the axial acceleration of the vehicle is given by

two-body equation 4

A simpler representation assumes the thrust acts directly on the gimbal point, and that the main body inertia includes the inertia of the engines as well. Then you are left with

two-body equation 5

The engine angle, δ, is the output of the attitude control system and actuator dynamics. If we assume sufficient frequency separation between actuator and control system, we can actually ignore the controller. The resulting controller output is

two-body equation 6

A starting point for control system design would be to choose some bandwidth and damping ratio to be achieved by this equation, ignoring integral action. If we do this we can set values for proportional (KP) and derivative (KD) gains, from

two-body equation 7

...and then we could add a small amount of integral action afterwards, to ensure no attitude bias.

two-body diagram


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