Building an Inverted Pendulum System The Ups and Downs of Defying Gravity…

After thinking about it for a while, I realized the force to move the cable carrier would be much lower if I added a spacer to the cart to increase the bend radius of the cable carrier.  With Bill’s help, I made a 1 1/2″ spacer from a piece of extruded aluminum.  I think the piece was a 1 1/2″ square cross section of 80/20 brand extruded aluminum framing material.  This stuff is very light and very strong.  We cut a 1 3/4″ inch piece of it, then drilled two holes 1 1/4″ apart for the existing bolts to pass through.  I added it between the T bar and the cross bar on the cart.  I had to remove the old cable ties at the ends of the cable carrier.  I also had to add a couple of sections to the cable carrier, which was a nuisance because I had to remove the cable connector at the end of my encoder cable (again!) so it would fit through the cable carrier.  The finished cart is shown below.  You can see the spacer directly behind the red encoder.  With the larger bend radius, the cable carrier moves very smoothly and the cart has no noticeable resistance due to the cable carrier anywhere along the track.  I believe this problem is solved!

Cable Carrier with Spacer

The cable carrier arrived a couple of days after I ordered it.  I was a little too busy with “real work” to deal with it for a while, though.  With help from Bill at Shaltz, I finally got it installed tonight.  It was pretty easy to install.  First, I removed the T bar from the top of the cart and attached the moving end of the carrier to the T bar (actually Bill did this, since it involved drilling and tapping holes in metal).  I put the T bar back on the cart and started to figure out where to connect the stationary end.  I held it where I thought it should go, then moved the cart from end to end.  I realized I had about 6 extra links that I didn’t need, so I removed 5 of them.  I double checked the new position and drilled a couple of tiny holes in the shelf, then used a couple of small screws to mount the stationary end.  At this point, everything moved very smoothly with minimal effort.  I disconnected the last link from the stationary end and threaded each cable through one at a time.  I had to cut the connectors off to get a nice clean end for threading.  After threading all three cables through, I used a cable tie at the cart end to secure the cables and keep them from moving around.  I reattached the stationary end and put a cable tie on that end as well.  It looks really nice.

Now the down side.  The cables I used seem to be fairly stiff and don’t like bending as much as the cable carrier wants them to.  When I move the cart all the way to the left (smallest carrier radius), there is much more effort required to move the cart either direction.  When I move it all the way to the right, it is much easier to move because the carrier makes a bigger radius.  I am kind of wishing I had selected carrier with a bigger radius.  It would be sticking out above the cart, but the motion would be a little easier.  I just hope the difference in effort doesn’t totally mess up my cart dynamics.  Hopefully it will be a simple force that is just a function of position and velocity and can easily be corrected for.

After testing the encoders while running the motor and not having any noise issues, I decided it was safe to run all the cables in the same carrier.  I ended up picking the 0202.20-KR28 carrier from KabelSchlepp.  It is wide enough to handle three quarter inch diameter cables, and it is thin enough to fit underneath my cart assembly.  The price for 30 inches of carrier (38 links) is $22.23 plus shipping.  The folks at KabelSchlepp were kind enough to waive the usual minimum order of $50.  I placed the order today and should get it in a few days.  When I install it, I will probably have to remove all the connectors on my cables, then reattach the connectors after threading the cables through the carrier.  Hopefully I will do a better job soldering the second time around.

Today I picked up the finished hardware, and it looks really good.  Other than the cable carrier and the limit switches, everything is assembled and ready to go.

Bill cut off the end of the bearing shaft for me.  It turns out it isn’t a rubber saw at all, although it does kind of look like it.  It is a composite blade (hardware stores call them abrasive blades) that wears away as you cut.  Apparently they work really well for cutting hard metals like the shaft.

Dana mounted the bearing shaft on the shelf for me.  He ended up using a solid block underneath the brackets and just removed the right amount of material to get them to the thickness needed.  Dana also drilled a hole in the pendulum and added a set screw at the end so the pendulum is easy to mount on the encoder.  Bill cut off the end of the pendulum so it is about 30 inches long.  That is about as long as I could make it without hitting the floor when it is sitting on a standard table.

Many thanks again to the guys at Shaltz Automation for helping me build this contraption.  I couldn’t have done it without their help.

I wired up the motor to the 9505 module for preliminary testing.  At this point, the programming begins, so I am going to start a new category called “software” where I will discuss writing and testing software.  I will probably add a few more items in the hardware category when I add cable carrier and limit switches.

Finally had a chance to test the limit switches tonight.  Wired the first one up to the DAQ and didn’t get a signal.  Wired the second one up with the same result.  Figured maybe the connections were bad in the connector, since I didn’t have the special tools to push the wires into the sockets.  Checked continuity with a meter and pushed them in some more just to be sure.  Reviewed the limit switch documentation and verified all the wires were in the right positions.  Everything looked good.  Wired one back up to the DAQ, still no signal.  Started wondering what to try next.  Reviewed the limit switch documentation again, and noticed the sample circuits have a pull up resistor between the voltage source and the signal.  Added a 1K resistor to the circuit, and everything started working great.  The sensor is actually much more sensitive than advertised.  On a flat black surface (my computer keyboard) it works as advertised 2-20mm.  On any white or reflective surface (envelope, etc.) the range is actually between 2 and 3 inches depending on the surface.  I will have to test them on the actual surface they will be used with, but I expect to be able to mount them directly to the shelf.  Originally I expected to need some spacer blocks to get them closer to the surface, but it will be much easier to mount them if that is not necessary.

Other good news – I found out the sbRIO board has been repaired (finally) and has been shipped to me.  I should have it early next week.  That means I will have my choice of RIO chassis to work with.

The limit switches came in today.  They are even smaller than I expected.  I knew what the measurements were, but it didn’t really connect until I had one in my hand.  These would fit in a tight space fairly easily.

I pulled off a strip of ribbon cable that is three conductors wide.  I colored one edge black and the other edge red with permanent markers.  I separated about half an inch of conductors at one end and pressed them into the connector using a small screwdriver.  I think they are in far enough to cut through the insulation and make contact with the wires.  If they don’t make good contact when I test them in a week or so, I will have to find something else to press them in further.  The holes are pretty small.  Perhaps a straightened out paper clip would work well.

While searching the digikey website for photo detectors for another purpose, I came across some optical sensors that should work very well as limit switches.  They reflect infrared light off a diffusing surface with a range of 2 to 22 mm.  I assume the optimum distance is about 11 mm.  Right now I plan on positioning them so they reflect off the bottom surface of the bearing block.  They are small and flat and should fit nicely under the bearing block.  I ordered two sensors (425-2042-ND), three connectors (one spare) (A98801-ND), and ten feet of ribbon cable (AE09G-10-ND) for about $13 plus shipping.  The sensors use three wires: GND, 5V, and signal.  I plan on pulling off three wire strips of ribbon cable and coloring one edge black (GND) and the other edge red (5V) with permanent markers.  I can separate the wires at each end and keep a nice clean cable in between.  My original thought was to buy the individual components and design my own circuit, but it would have cost more than buying the prebuilt sensor.

When I picked out all the hardware during the design phase, I didn’t really explain any of my calculations.  Here are a few numbers of interest:

Encoder calculations

I decided I wanted encoders with a minimum resolution of 1000 pulses/rev.  With quadrature  counting, that gives me 4000 counts/rev.  For the pendulum, this makes the resolution 0.09 degrees/count (or 0.00157 radians/count for those that care).  For position measurement, the 1.25″ diameter gear on the encoder gives a position resolution of 0.00098 inches/count or 0.0249 mm/count.  These resolutions should give adequate feedback even when the speeds are fairly slow.

Motor calculations

IG32P Motor Performance Chart

I started selecting a motor by figuring out what the power limitations were.  From the beginning I planned on using the NI 9505 module, which has a limit of 30V and 5A.  I decided to use a 24V motor since that is a standard voltage and there are a large number of robot motors available that use 24V.  The motor I picked is 24V with a stall current of 5A.  The motor (without gearbox) has a maximum speed of 6000 RPM and a stall torque of about 1900 g-cm (0.186 N-m) according to the chart above.  There were several gearbox combinations available, so I picked the 19:1 ratio that will give me fairly high stall torque of 3.54 N-m.  With an approximate total cart and pendulum mass of 1.4 kg and a moment arm of 1″ (gear radius), the max acceleration of the cart will be 99 m/s^2.  However, this drops off quickly with speed.  Other factors (cable carrier, support platform) will probably limit the acceleration to 50 m/s^2 or so.  With the 2″ gear, the maximum speed is 33 inches/sec (84 cm/sec) and will be fast enough to travel the length of the track in about 1.5 seconds.  If the max speed isn’t high enough, I will look into getting the 14:1 gearbox, which would reduce the torque a little bit and increase the speed.