- Hardware
Model
- The mathematical model
for the Ball-and-Beam is investigated to determine the
appropriate rules, which govern the assembly. The first
task was to verify the ball dynamics. This was done by
recognizing the basic ball and beam geometry.
- When applying Newtonian
mechanics and the linear property of small angles, the
following equation was verified.
- Thus, the desired
transform was verified in being the following.
- The system dynamics was
found by simple geometry of the beam and motor.
- Applying the liner
property of small angles the following equation was
determined and verified.
- Finally, the motor
dynamics of the system were verified referencing control
system textbooks to lookup the electrical characteristics
of the motor and the torque equations.
Motor Dynamics
- Armature
Resistance(Rm) = 9.0 Ohms
- Motor Torque
Constant(Km) = 00.75 N-m/Amp
- Total Load
Inertia(J) = 7.35 x 10-4 N-m-sec2/rad
- Total Load
Friction(B) = 1.6 x 10-3 N-m-sec/rad
- Gear Ratio(Kg)
= 75
-
- The output is the angle
q measured at the large gear. It is measured using a 10k
Ohm mechanically continous turn potentiometer. The
electrical rotation however is 352 degrees.
-
- The total track length
is 16 inches (40.5 cm). The total track resistance is
approximately 300 Ohms. The two bias resistors are 210
Ohms each.
-
- Control Model
- With the ball and motor
dynamics determined, the block diagram below can be
implemented to control the Ball and Beam. One closed loop
controls the ball position through the beam angle a, while the other controls a by controlling the motor
angle output q as a function of the input
voltage Vin.
-
-
- The subscripted signals
represent the desired positions while the others
represent the measured positions. The control laws to be
implemented are shown in Equations 1 and 2.
a = Px (xd
– x) + Dxx¢ (1)
Vin
= Kp (qd - q) +Kdq¢ (2)
