Modeling of the BOBB System
Figure 1 shows schematic representations
Lagrangian approach was used to model
system relates to the relative movement
the location of the pivot corresponds to
total potential energy as
of the ball on beam balancer system. The
the system. The potential energy of the
of the ball on the beam. Assuming that
zero potential energy, we can write the
where m is the ball mass, x is the translational position (with positive direction
pointing downward) and θ is the angle of the beam shaft. The kinetic energy, T, can
also be computed from
where, v is the translational velocity of the ball, ω is the angular velocity of the
ball, Ia is the beam inertia, and Ib is the ball inertia.
Figure 1: Schematics of the ball on beam balancer system (courtesy of [1])
Since the sensors in this system measure ball translational position, x, and shaft
angular position, θ, we rewrite the kinetic-energy equation in terms of these two
variables by finding the relationship between r and ω, and x. Note that the distance
traveled by the ball is given by
where φ is the rotational angle of the ball with respect to the shaft with respect to
the shaft and r is the rolling, or effective, radius of the ball. The total angle of the
ball is the sum of the angle of the ball with respect to the shaft, φ, and the angle of
the shaft, θ. The rotational velocity of the ball, therefore, is given by
The translational velocity of the ball, v, is given by (see Fig. 1B)
Replacing the rotational and translational velocities in the kinetic-energy equation
yields,
The equations of motion can now be obtained using the Euler-Lagrange equation:
where L is the Lagrangian given by
Replacing L in the Euler-Lagrange equation, we get
We ignore the derivatives of the shaft angle assuming the shaft movements remain
relatively small. The simplified equation of motion is then given by
Applying the Laplace transform to the above equation yields to the following
transfer function
In the above equation, the inertia of the ball is given by
Plugging in Ib in the transfer function G(s), we get
System Simulation and Controller Design
Figure 2 shows the Simulink model of the system. The first part (with the feedback
loop) is representative of the DC motor dynamics and the next block corresponds to
the ball-on-beam balancer transfer function.
Figure 2: Simulink Model of the open loop system
Figure 3 shows the step response of the open-loop system. As expected, the
system is open-loop unstable.
Figure 3: Step response of the open loop system
Figure 4 display the root-locus plot of the system corresponding to the Simulink
model in Fig. 2.
Figure 4: Root Locus of the BOBB System
To make the system stable, the poles in the root locus are moved to the left half
plane by adding two zeros in the form of K(1+αs)(1+βs) to the system transfer
function. Figure 5 shows the root-locus plot of the system with the added PD
compensator. The step response of the new system is plotted in Fig. 6 for K = 1, α
= 0.2857 and β = 1.1111.
Figure 5: Root locus of the BOBB system with the added PD compensator
Figure 6: Step response of the closed-loop system
Fig 7 shows the electronic circuit board that was fabricated to use in this project. It
is essentially made of three parts. the first part is the DC motor driver circuit. The
second part is designed to compare the position of the angular potentiometer
versus the reference position and then amplify the signal. which is later fed into the
computer. And the last part is to amplify the output voltage from the linear
potentiometer (it includes a low pass filter to filter some of the noise added due to
the amplification). The board also features a switch that allows the manual control
of the ball on beam balancer system!
Figure 7: Electronic circuit board
References
[1]. Robert Hirsch, Ball on Beam System, 1999