insensitive to changing supply voltage as the batteries run
down. The 49 ohm resistor in series with the motor tends
to make it respond with torque vs. voltage rather than
speed vs. voltage.
The position control has two inverting amplifiers in a
loop that — by itself — is completely unstable. Without the
rest of the system, the position loop will run away to
saturation in about five seconds.
The band suppression filter can be intuitively explained
as follows: At very low frequencies,
C1 and C2 can be ignored. The input
and feedback resistors are both 0.27
+1.6 = 1.8 MW, so the gain is 1. As
the frequency rises, C2 (the larger
capacitor) will begin to short out R2,
reducing the feedback impedance
causing the gain to drop.
As frequency continues to
increase, C1 will begin shorting out
R1, reducing the input impedance
causing the gain to rise. At high
frequencies, R1 and R2 are both
shorted out by C1 and C2, so the gain
comes back to 1.
Component values are chosen to
put the low point at wp = 4. 4 rad/s.
The 555 timer at the upper left is an
optional position control that causes
the command position to move left
and right about 20 cm at 10 second
The realization of the stabilization
algorithm described here has no
accelerometers, no gyros, no bits, no
code, and no processor. It’s a purely
analog solution with just three op-amps and a few resistors and
It illustrates the concepts of
stabilization of the inverted pendulum
without wading through the math
and all of the many lines of code to
be found in a microprocessor solution.
Over the years, I have built four
versions of the stabilized inverted
pendulum, from the 60 pound
vacuum tube version of 1965 to the
two-wheel 12 ounce version of Figure
2 — all using the same math, but each
taking advantage of improved
technology to produce a simpler and
lower cost design.
If this problem sounds like fun,
harvest a motor and some gears from
a junked CD player and give it a try.
Stabilizing an inverted pendulum is a favorite and much
discussed control theory problem for roboticists. The basic
idea is illustrated in Figures 1 and 2. Simply put, the
problem is to drive the robot base forward or backward in
such a manner as to prevent the inverted pendulum design
from falling over.
As you can now see, the math and science behind this
task is not so bad after all. SV
SERVO 12.2017 51