The report discusses several model tasks for controlling a platform
with an attached mathematical pendulum or a double pendulum. Small
oscillations are considered. Optimal control is the force acting
horizontally on the platform, or the moment acting on the driving
wheels of the platform. The driving wheel model is discussed
separately. The platform is transferred in a straight line from a
state of rest at a given distance to a state of rest in a given time.
The minimization of energy approach is considered. Vibration damping
occurs. Expressions for optimal control by the Pontryagin maximum
method and using the generalized Gauss principle are found.
Several problems are considered, depending on how the pendulum is
positioned and attached.
a). The initial position of the mathematical
pendulum is vertically upward (unstable position of the pendulum). The
pendulum is pivotally attached to the platform.
b). The balance
position of the pendulum and the initial position are vertically
downwards.
c). The pendulum is facing up, attached to the platform
along with a spiral spring. The equilibrium position of the pendulum
is deflected from the vertical by a given angle. The problem takes
into account the mass of the trolley, the mass of the wheels, the mass
of the motors attached to the wheels, the mass of the pendulum load,
the mass of the pendulum rod.
d). A manipulator consisting of two
weighty rods and a load at the end is attached to the platform. The
lower rod is attached to the platform together with a spiral spring,
the second rod is attached to the first one either rigidly or with a
spiral spring. Unlike the previous model problems, the number of
degrees of freedom increases.