Self Balancing Unicycle Here is a link to a video of a self-balencing unicycle. I'm curious as to how this works. In particular:


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*What provides the torque to keep the rider from falling over? I know it's somehow provided by the motor, but this isn't an external torque (if we define the unicycle as our system), and I assume we need an external torque to change the angular momentum of the unicycle. 

*If you were to try to ride this thing on ice, I assume that it could not prevent the rider from falling over. Is this correct? 

*What would the free body diagrams look like for this unicycle when it is in the act of falling over? When it is righting itself? 
 A: Basically it works on the following principle.
There is a gyroscope used to determine what is the current angle between the vertical and the seat of the monocycle. This can be done because the gyroscope will tend to maintain the same position no matter what the orientation of the monocycle. 
The gyroscope is connected to the engine, so that when it detects torque towards the front, the engine will generate more torque oriented backwards correspondingly (spin up); when it detects torque towards the back, the engine will spin down accordingly.
The engine torque mentioned is generated by increasing (or the decreasing) the angular velocity of the wheel.
So this has two purposes:


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*It corrects any imbalances of the passenger and keeps the wheel directed up.

*The corrections have a net effect of speeding up or slowing down the linear motion (i.e. how fast the monocycle goes), so it's an effective way of controlling the monocycle. If you want to brake you lean back.

A: Sklivvz has the mechanics down. To answer your questions:


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*You're right, if the system were only the rider and the unicycle, there would be nothing to prevent it from falling over. However, there's another component of the system that you're missing: the ground.

*Yes. If the wheel were unable to "push against" the ground, as on a frictionless surface, there would be nothing preventing the rider falling over.

*The FBD for falling over would show the force of gravity on the rider/unicycle system's COG 
pointing down, and the force of the ground on the wheel pointing up, creating a moment couple. (Note that if there was no ground in the system, there would be no moment, and hence no falling.)

*The FBD for self-righting would show the same system, with an additional force (friction) tangential to the wheel on the ground. This force is caused by the wheel's motion in the same direction as the falling.


You could also draw FBD's for the unicycle itself, with the force of the rider bearing down on the seat and the weight of the unicycle on the COG, but unless you need to account for the rider bending back & forth and waving his arms wildly, it's safe to assume that the rider/unicycle are a single body.
A: It is in principle possible for a unicycle to prevent the rider from falling over, even on a frictionless surface, by spinning up the (hopefully massive) wheel enough for the resulting torque to pull the axis under the rider.  Naturally, locomotion on a frictionless surface has its own set of problems.
If the rider is falling sideways, the unicycle would need an additional motor to twist the wheel in an appropriate direction relative to the seat to compensate.  Such twisting could be a potential problem for people whose legs aren't tentacles.
