Exploring the Physics of a Self-Balancing Electric Unicycle

[rcgldr]

A self-balancing electric unicycle consists of a frame+pedals that the rider stands on, a motor where the stator is attached to the frame+pedals, and a rotor which is attached to the wheel+tire.

An EUC is similar to an inverted pendulum: the EUC frame is free to rotate about its axis, limited by motor torque. EUC plus rider is similar to a double inverted pendulum, with the rider free to rotate relative to EUC frame, limited by frame|rider torque. Any torque exerted by the motor onto the EUC frame has to be countered by an opposing torque from the rider to end up with zero net torque on the EUC frame so that the EUC frame does not rotate.

The pedals are long enough (8 to 10 inches or so front to back) for a rider to be able to generate or experience adequate torque with the frame. There may also be other contact points for the rider to be able to generate or experience a torque with the frame. To simplify this question, assume that only the pedals are being used (some riders actually ride this way). Self-balancing components and algorithm will make corrections as needed to keep a leaned rider balanced, holding the rider's lean angle steady and preventing rider from falling forwards or backwards.

Assumptions: During constant acceleration with zero net torque on the frame, the motor is exerting a forwards torque onto the wheel+tire, coexistent with the motor exerting an equal in magnitude backwards torque onto the frame+pedals. Splitting this up into Newton third law like pairs, the forwards torque exerted by the motor onto the wheel coexists with a backwards torque exerted by the wheel onto the motor. The backwards torque exerted by the motor onto the frame coexists with a forwards torque exerted by the frame onto the motor. The backwards torque exerted by the frame onto the rider coexists with a forwards torque exerted by the rider onto the frame.

Leaning related to linear acceleration: The rider has to lean forwards in order to be balanced due to the forward force exerted by the pedals related to linear acceleration.

Question 1: Does the rider need to be further leaned forwards due to the backwards torque exerted by frame onto rider which coexists with a forwards torque exerted by rider onto frame?

Question 2: Assuming the rider does need to lean further forwards, and assuming zero net torque on the frame, is the forwards torque the rider exerts onto the frame exactly equal to the forwards torque that the motor exerts onto the wheel+tire?

 

[Lnewqban]

You can disregard internal torques.
A rigid inverted pendulum needs to counteract any horizontal acceleration of the pivot point with the torque induced by gravity.
Otherwise, the inertia of the mass of the rigid pendulum will remain behind the pivot point (in the horizontal direction), resulting in a non-desired gravity induced torque.

Please, see:
http://large.stanford.edu/courses/2007/ph210/lee2/

 

[rcgldr]

A rigid inverted pendulum

The Stanford link shows an inverted pendulum that pivots on the upper surface of cart that can only exert a linear force. In the case of an electric unicycle, the pivot point of the pendulum is offset above the point of contact with the pavement, attached to a motor that generates a Newton third law like pair of torques at the pivot point. If accelerating forwards, the motor exerts a forwards torque onto the wheel, coexistent with the motor exerting a backwards torque onto the pendulum and vice versa. At the pivot point of the pendulum, the pendulum experiences both an external linear force and an external torque.

That external torque applied at the pivot point should require the inverted pendulum to be leaned further than the Stanford example because in the case of the Stanford example, there's no external torque in addition to the linear force applied at the pivot point of the pendulum.

For a balanced acceleration, the combined center of mass of the EUC and elevated rider needs to be at some angle ahead or behind the contact patch of the tire, which requires the rider to be leaned further forwards (with respect to the pedals the rider is standing on).

An extreme example of this is 10 year old EUC girl accelerating on an EUC that weighs as much or more than her. It's easier to see this if you pause it.

 

[Baluncore]

When the rider places their weight differentially on the two foot rests, does it sense that, and how does it turn to steer left or right ?

 

[rcgldr]

When the rider places their weight deferentially on the two foot rests, does it sense that, and how does it turn to steer left or right ?

There are no sensors in the pedals. The EUC uses a very sensitive tilt sensor mounted to the frame to determine rider inputs, and transitions to | from inclines or declines. The sensor includes a 3 axis accelerometer and 3 axis gyro, which is enough for an algorithm to determine tilt angle, regardless of acceleration.

An EUC can be steered by twisting it and | or by tilting it. Tilting an EUC causes it to steer due to camber effect. Twisting requires some type of angular momentum, such as twisting the upper body and arms left to twist EUC right and vice versa. Angular momentum can be built up by tilting the EUC and twisting the upper body in the same direction of tilt, or tapping the pavement with the inside foot. Tilting is the primary method for steering at normal speeds, and twisting is mostly used for slow speeds. If a rider is weaving left | right by tilting the EUC left | right, the rider can also allow the EUC to twist left | right while the rider keeps the upper body facing forwards.

A rider has to lean for balance in order to turn, using the same counter-steering principle needed for any uni-track vehicle such as a motorcycle. A lean is initiated by steering the EUC out from under the rider to lean the rider inwards. One leaned, a rider steers inwards more to lean less, or steers inwards less to lean more. It takes a while to learn to coordinate lean and tilt depending on speed and turning radius. At slow speeds, the rider leans less than the EUC is tilted, and at around 15 mph or faster, the rider leans less than the EUC is tilted.

 

[Baluncore]

However, similar to a bike but due to different physics, at sufficient speed, usually 6 to 8 mph. Tilting the EUC causes it to steer due to camber effect of the rounded profile tire. The response to tilt depends on tire parameters: wider tires respond more than thinner tires, street tires respond more than off-road knobby tires.

I cannot see how tire contact patch camber can control the change of tyre direction.
Does gyroscopic precession play a significant part in the steering, or is that denied?

 

[rcgldr]

Does gyroscopic precession play a significant part in the steering, or is that denied?

Precession may play a minor role at higher speed, but compared to camber effect, it's too small of an effect for a rider to notice any difference to tilt response. I'm not sure if precession helps or hinders camber response to tilt, depending on speed and tire parameters (the second video below explains tire parameter effects). For this situation, precession is a yaw reaction to an inwards imbalance (roll torque). If the turn is coordinated, there's no imbalance and no precession.

This is a video of a rider making very tight turns at a speed too slow for precession to have any detectable effect. Near the end of the video, your hear a pedal scrape due to the large amount of tilt:

I cannot see how tire contact patch camber can control the change of tyre direction.

When tilted, the inner edge of the contact patch rolls about a smaller radius than the outer edge of the contact patch so the inner edge is moving at a slower speed than the outer edge. This creates a twisting | yaw torque due to friction. The video below doesn't explain the physics well, but it does a good job in showing how tire parameters affect the camber response to tilt. I ride a V8F with a narrow 2.125 inch wide tire (a similar V8S is shown in the video). It turns well, but a wider tire would require less tilting for the same radius turn.

 

[rcgldr]

You can disregard internal torques.

Assume an EUC with a mass e. Assume the center of mass of the EUC is at the axle of a tire with radius r. This means leaning (tilting) the EUC does not offset the center of mass with respect to the contact patch. An inverted pendulum is directly attached to the EUC frame. The inverted pendulum consists of a massless rod supporting a point mass with mass p at some distance h above the pavement. h decreases as pendulum lean increases, but r does not. During acceleration of amount a, there is a total torque about the contact patch: -a * (e * r + p * h). The pendulum has to be leaned forward so that the point mass is offset o ahead of the contact patch to counter the total torque. Let g = acceleration of gravity, then g * (p * o) - a * (e * r + p * h) = 0, for the system to be balanced. If e (mass of EUC) is increased, then o needs to be increased, so an increase in the mass of the EUC requires a greater lean angle of the pendulum.

Getting back to my prior post, motor torque onto wheel: a * (e + p) * r , motor torque onto frame: -a * (e + p) * r. Torque from pendulum: g * (p * o) = a * (e * r + p * h). Assume base of pendulum is at the axle, then the linear acceleration at the axle accounts for a * p * (h - r). Rewriting prior equation: g * (p * o) = a * (e + p) * r + a * p * (h - r). So at the axle, the pendulum experiences a linear force: a * p, and a torque: -a * (e + p) (the backwards torque the motor exerts onto the frame). This is balanced by the pull of gravity | lean angle of the pendulum.

 

[Lnewqban]

I believe that there is the lean forward to command the motor and control system to accelerate.
Once that happens, a further lean forward is needed to compensate for the forward acceleration.
The rider should align his/her (weight + inertia) force vector with the contact patch to keep balance.
Once a desired constant speed is reached, less balancing lean is needed.

 

[rcgldr]

I believe that there is the lean forward to command the motor and control system to accelerate.

As I posted above, similar to counter-steering, there's what I call counter-lean (forwards | backwards). Consider the case when standing on solid ground. In order to initiate a forwards lean from a vertical stance, there needs to be a forwards torque, and this is induced by pressing with the heels. Once leaned, pressure on the toes controls the lean angle.

On an EUC (electric unicycle), that same pressure on the heels commands the EUC to decelerate from under the rider, leaning the rider forwards. Once leaned forwards, the rider presses with the toes to control acceleration and lean angle. More pressure causes the EUC to accelerate a bit under the rider, reducing the lean (or returning to vertical) and less pressure allows the rider to lean more. However, these control motions to lean forwards | backwards are instinctive (since it is how people balance when standing), and a rider only needs to focus on leaning forwards | backwards to accelerate | decelerate without having to think about toe versus heel pressure.

The balancing algorithm uses the sensors and sensing of motor torque making acceleration | deceleration corrections to prevent the rider from falling forwards or backwards despite acceleration, deceleration, head or tail wind, incline or decline, ... , holding a rider's lean angle until the rider commands the EUC to change lean angle.

There are also cases where the EUC overrides a rider's input. For example, an EUC has a limit for speed (well less than it's actual no load max speed), and if the rider reaches or exceeds that limit, the EUC will accelerate under the rider to lean the rider backwards as well as tilt the EUC backwards a bit so that the pedals are tilted back a bit. This is called tilt-back.

In the case of Inmotion EUCs, when transitioning onto an incline or decline, the EUC will auto-lean the rider as needed to maintain speed, and also tilt the EUC a bit forwards | backwards for incline | decline, without requiring any rider input.

Turning is a bit trickier on an EUC. The steering response to tilt is mostly independent of speed, while the amount of lean is related to angular acceleration = speed^2 / radius. This requires riders to coordinate the amount an EUC is tilted, and how much the rider leans. For a low speed, low-g tight turn, the rider tilts the EUC a lot with very little rider lean. At higher speed, the rider tilts the EUC less than the rider leans, and there's a range of speed where tilt and lean are about the same.

Riders can also twist an EUC (relative to rider) as well as tilt it to steer, such as carving, or if trying to keep facing forwards, such as when holding a selfie stick to make videos.

 

[rcgldr]

This video of is a good example of how little movement is needed to ride and turn on an EUC.



Video of a rider (Dawn Champion) taking turns at 30 to 50 mph: