Inverted Pendula
Design, manufacture and stabilize an unteathered inverted pendulum using a hand manufactured drag force actuator with analog and digital control systems.

This is a high bandwidth page by 1999 standards
Click on any of the images to see a larger shot.

The project began with inspiration from a web page I can no longer find.  Some guy built a drag force stabilized pendulum using an RC servo as the actuator and a rate gyro for feedback.  His control law was implemented using a MOT 68HC11 and an emperically derived lookup table for actuator effort vs deflection (as I remember it anyway.)  After a few minutes of looking at his work I set out to build a similar system using linear control and a bit of analysis.  I quickly intoxicated three other engineering geeks with this idea and they hopped on board as well.

I would really like to put a link to his page here, so if anyone has it please e-mail me.

Here is a concept sketch and a free body diagram sketch which we linearized about vertical.

A few days of brain storming and a strong desire to take control of the CNC cutting laser in the student shop yielded this first pass prototype design on the left.

While traveling to Detroit on a plane I came up with these sketches to the right for pendulum aesthetics. (Yeah so I wish I could do Industrial Design some times, what of it?)

Here is our first pass at a limited angle torquer, voice coil actuator.  This actuator works on the Lorentz force principle.  Current through a coil siting in a strong magnetic field generates a force perpendicular to the field and the direction of the current.  The result is a +-60 degree actuator with an output torque proportional to the current applied and the number of turns in the coil.

To the right is a peeled open CAD versoin of our actuator.  The trapezoidal blob trapped in the y-shaped arm is the coil.  The silver and gold magnets represent South and North pointing fields respectively.  Thus we have a push pull actuator.  As the current runs clockwise in the field, the gold side will pull the coil to the left while the silver side pushes to the left, or vice versa.  The arm then rotates about its pivot as the magnets stay fixed.

Above you see the theory, this is what it actually looks like.  The magnets are salvaged out of some old hard disk drives that were laying around.  The steel is 3/16" thick sheet and was cut out with a hack saw by hand and then ground into a circular shape.  The holes were laid out by hand with calipers and drilled 10 thousanths oversized to make sure the whole thing actually goes together.

Currently we have a 100 turn coil with a resistance of 2.2 Ohms.  We will probably re-wind this coil with a lower resistance wire to get more power out at lower voltages.

The large balsa wood fly swatter attached to the steel body is called the fan (ok, so it does not look exactly like the CAD model.)  The drag generated by the fan moving through air generates a reaction force which will right our pendulum. (Yes we are using a 1/8th inch drill bit for our axle.)
Here we are trying to test the controllability of our system.  The pendulum body is starting at a 5 degree lean toward our power supply.  We then actuate the fan and see if we can tip ourselves in the opposite direction. 

It worked at 10V and 4.5A or so.  By rewinding the coil with lower resistance wire we should be able to get 5A out of a 6.0V NiCd battery pack which will be carried on-board the pendulum.

Now that we have the bare minimum of function down we need to iterate.  This includes moving from 1/4 inch acrylic to 3/32 or 1/16 inch acrylic for the body structure to get the weight down.

Here are some sketches for an agressive looking pendulum.  The key difference between these two sketches is the taper of the base structure. 

The control system for this guy is not too exciting yet.  We are using an acclerometer oriented perpindicular to gravity for a tilt sensor for feedback.

This is wired into a lead circuit with a zero at 15 rad/sec and a pole at 60 rad/sec.  This signal is then fed into a darlington push-pull driver which pushes the current through our coil.

It would be nice to stick a root locus 
here for those interested.  But then
the analysis would have to be correct...
The above system was not stabilizeable (by us) for a number of reasons.  The primary reason was the accelerometer had this annoying property of measuring accelerations.  Kinesthetically it went something like this:
  1. Pendulum and fan are perfectly verticle
  2. Small disturbence tilts the pendulum a little bit
  3. Small tilt measured by accelerometer and amplified by lead
  4. Actuator actuated with incredibly high frequency response
  5. Accelerometer measures these small on-axis accelerations
  6. Lead amplifies them and we have a nice 200 Hz oscilator.

The fan was also too small, so we would saturate the actuator at one extreme before generating enough force to right the pendulum.  So we made a bigger fan.

Mike did not like the look of this fan, so he made one too.  It is wicked cool lookin' (and lighter wirght.)

So, we went for another design iteration to integrate all of the suff we have learned so far:
  1. The base now has a cool trapezopid look to it and weighs ~half as much.
  2. The attachment of the actuator to the base now has a cool box-beam structure which is much stronger than before.
  3. We designed a drum drive hall effect sensor system to sense tilt angle.
  4. We got a cool looking new drag area which weighs practically nothing.
  5. The counter mass to place the center of gravity of the fan at the pivot was integrated into the bobbin assembly.
You never know how dependent you are on a CNC cutting laser until it goes down for two weeks.

Black is cool

Here is a look at our sensor detail.  It is a non-inverted pendulum with a bearing pivot placed at the axis of rotation of the pendulum base.

The main idea of this design was to make a large horozontal motion out of a small rotational motion.  We then used this horozontal motion to drag a magnet across the face of a hall effect sensor.  Check out the sketch on the right and compare to the acrylic on the left.

The elegently zip-tied nuts are mass added to help overcome the sticktion of the drum drive.

Here is our new bobbin with a flange for mass ballist.

On the right is a cheep trick to adjust the CG of the system.  The bolts are moved up and down to adjust the relative CG height.  This way we have a way of sneaking up on the poles in the right half plane.

This is a stable configuration.  In the unstable picture below the duct taped bolts are about 3 inches below the base.  I could probably calculate where that puts our poles, but I will save that for later.

We are stable!

The system never really stands perfectly verticle.  The air drafts in then room and hysteresis in the angle sensor keep it oscilitaing about verticle.

Here are some action shots of our system responding to an impulse at its base.