I Acceleration on an electric unicycle, how much does the rider have to lean?

[rcgldr]

Can someone here check my math?

On an euc (electric unicycle), the motor has to exact the same amount of torque onto the wheel+tire that the rider exerts onto the euc, otherwise the frame would rotate (due to the motor exerting an equal in magnitude but opposite direction onto the frame than it does onto the wheel, a Newton third law like pair of torques).

Choosing some arbitrary numbers:

rider = 200 lb
euc = 100 lb
rider + euc = 300 lb
tire radius = .83333 foot (10 inches)
rider center of mass 3 feet from riders feet

If rider is standing on a plaform accelerating at 1/2 g (32.174 feet / second^2) rider center of mass is 3 x sin(arctan(0.5)) ~= 1.342 feet ahead of feet and 3 x cos(arctan(0.5)) = 2.683 above feet. The external torques onto the rider cancel: torque from gravity = 200 x 1.342 ~= 268.328 lb ft, opposing torque due to acceleration = 100 x 2.683 = 268.328 lb ft. Rider exerts a linear force and zero torque onto the platform.

For the 300 lb of rider + euc, 1/2 g acceleration requires 150 lb force. With a tire radius of .83333 foot, this translates into 125 lb ft of torque. The rider needs to lean forwards enough so gravity generates a 268.328 lb ft + 125 lb ft = 393.328 lb ft of torque. Rider center of mass is ~1.96664 ahead of contact patch.

For 1 g of platform acceleration, rider center of mass is 1.5 feet ahead and above contact patch. For euc, force = 300 lb, torque on wheel = 300 x .83333 = 250 lb ft. Total torque = 300 + 250 = 550 lb ft. Rider center of mass = 550 / 200 = 2.75 feet ahead of contact patch.

 

[jedishrfu]

It's interesting that you worked your problem in English units and not SI metric units.

Doing that makes it harder for many folks to check due to the conversion factors, constants like g, and due to our science training in the MKS system that makes us familiar with the numbers.

 

[A.T.]

It's interesting that you worked your problem in English units and not SI metric units.

Doing that makes it harder for many folks to check due to the conversion factors ...

Using numbers instead of symbols is the first mistake, if you want someone to be bothered to look at it.

 

[berkeman]

On an euc (electric unicycle), the motor has to exact the same amount of torque onto the wheel+tire that the rider exerts onto the euc,

Are you interested in the dynamics of maintaining the balance too, or just the overall torque balance numbers?

I've never tried riding an EUC, but have seen some riders on them. They look fun, but probably not for me. The reason I ask about the dynamics is because on a dirtbike (and especially on a motocross bike), you spend most of your time standing and balancing your body weight on the footpegs. That gives you the best control while riding, and also is the least tiring for you if you do it right.

When you are about to accelerate, you first lean forward (almost like falling forward a few inches), and then apply throttle to catch you at the angle that balances the acceleration that you are shooting for. Similarly, when you are about to brake, you lean back first and then apply the brakes hard enough to balance your lean angle. So the dynamics of leaning forward and backward and when the acceleration/deceleration torque is applied is not such a simple thing. At least not on a MX bike; I don't know if you have to take the same thing into account for your EUC dynamic control...

 

[Baluncore]

A unicycle is an example of an inverted pendulum.
https://en.wikipedia.org/wiki/Inverted_pendulum

 

[jbriggs444]

When you are about to accelerate, you first lean forward

I can ride a real unicycle. One cannot spontaneously lean forward. Instead, one pedals slightly back. This moves the contact point rearward and creates the desired lean angle.

This is entirely similar to the way that one steers a bicycle. One cannot simply lean to the right or left. Instead, one counter-steers in the opposite direction, moving the contact patch out from under. This creates the desired lean angle.

Steering a unicycle is difficult to describe. It is a skill that is learned. One can swivel the hips to create a temporary steering input. This allows one to steer the contact patch out from under. If one accelerates (or decelerates) while the contact patch is offset, that generates a yaw torque to steer with.

My brother, my sister and myself spent a few hours one Saturday morning about 60 years ago learning this all for ourselves. One unicycle. Taking turns. No instructor.

 

[berkeman]

I can ride a real unicycle. One cannot spontaneously lean forward. Instead, one pedals slightly back. This moves the contact point rearward and creates the desired lean angle.

Ah, that makes sense. Do you have any idea how that translates into the OP's EUC? I think EUCs have platforms for the rider's feet (only my impression from seeing some riders), so maybe tilting the platforms is used somehow to initiate the accel/decel?

My brother, my sister and myself spent a few hours one Saturday morning about 60 years ago learning this all for ourselves. One unicycle. Taking turns. No instructor.

Any first aid issues that morning? LOL. Some buddies and I went to an ice skating rink to try to teach ourselves how to skate, and we were bloody messes when we walked out the front door afterwards (I was the worst).

 

[berkeman]

I think EUCs have platforms for the rider's feet (only my impression from seeing some riders), so maybe tilting the platforms is used somehow to initiate the accel/decel?

BTW, EUCs don't have seats, so I guess the feet tilt thing is a big part of controlling the accel/decel. I need to watch this video...

 

[rcgldr]

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EUCs (Electric unicycles are programmed to remain vertical (or at some user set fixed angle), using accelerometers and gyros, similar to a Segway. If the EUC is tilted forwards, it applies or increases forwards motor torque to accelerate and return the EUC back to vertical. If tilted backwards, it applies or increases backwards motor torque to decelerate and return the EUC back to vertical. At constant speed, the rider balances as if standing on a level surface. If accelerating or decelerating, the rider balances as if standing on an inclined surface.

As mentioned by jbriggs444, rider's can't spontaneously lean forwards or backwards. The EUC has to be accelerated or decelerated from under the rider in order to lean. However, this is very similar to how a rider would lean forwards or backwards on a level surface, without realizing what is actually happening. It becomes more obvious in more extreme cases, such as this video where the EUC is accelerated ahead of the rider to setup heavy braking, then accelerated backwards to lean the rider forwards for heavy acceleration:

leaning backwards then forwards

At normal speeds, EUCs turn due to camber effect of a tire with a round profile contact patch. When the EUC is tilted sideways, the camber effect causes the tire to follow a circular path. Tire parameters affect this. At the same time the rider is counter-steering (maybe unaware of it) to lean for balance. In general, for low speed tight turns, the rider barely leans, while the EUC is tilted a lot:

slow speed tight turn

In a high speed turn, the rider leans inwards as needed for balance, while the EUC is barely tilted. Since the centrifugal reaction force is outwards at the pedals and generates an outwards torque onto the EUC, the rider has to pull inwards on the outside upper pad to force the EUC to tilt inwards.

high speed turn

 

[rcgldr]

It's interesting that you worked your problem in English units and not SI metric units.

Using numbers instead of symbols is the first mistake

The reason for this is this is a response some making claims about braking distances at another forum, in English units. However, there is an phone based app that reports peak acceleration and deceleration in mph / sec or kph / sec, so I suggested he use the app to determine the rate of acceleration or deceleration.

 

[rcgldr]

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EUC stability. When riding at sufficient speed, about 8 mph | 12 kph for most EUCs, they become self-balancing left + right. An imbalance generally causes the EUC to tilt and turn into the direction of imbalance, and at sufficient speed, it becomes a self-correcting response. A rider can literally stand still while riding at sufficient speed if riding in a straight line on smooth pavement. Riding an EUC is very similar to riding a bike with no hands on the handlebars, the rider leans for balance, and tilts the bike or EUC to steer. Video of a girl on a Kingsong S18 at about 15 mph showing how little motion is needed to ride an EUC, no twisting, just leaning for balance and tilting EUC to steer.

Girl riding S18

EUC turning - this takes the longest to learn well, since rider has to lean for balance and tilt the EUC to steer, both angles in the same direction, but different magnitudes depending on speed an turning radius.

EUC - speed wobble - the self-correcting response can become excessive and result in a speed wobble. Tire parameters affect this. Wider tires are more responsive than narrower tires. Street tires are more responsive than knobby tires. Smaller diameter tires are more prone to wobbles than larger diameter tires. The currently fastest EUC is an Inmotion P6 with a top speed of 93 mph, and uses a 20 inch diameter tire.

EUC - twisting to turn - this is done for slow speeds. Example of a 3 year old arm flailing, flail arms left to twist EUC right and vice versa. I did something similar over 50 years ago when I was loaned a pedal unicycle for about 2 hours.

3 year old twisting EUC

Video of me riding an EUC back in 2022 when I was only 70 years old:

Me riding EUC

 

[rcgldr]

Are you interested in the dynamics of maintaining the balance too, or just the overall torque balance numbers?

Just the rate of acceleration or deceleration versus how far offset the riders center of mass from the tire contact patch has to be versus 1 g of acceleration or deceleration. I used numbers to simplify the math. This was in response to someone making a claim about braking distance (in English units). However there is an app that reports peak acceleration and deceleration in terms of mph / sec or kph / sec, and I suggest he use that to get an idea of how hard he is braking or accelerating.

Although peak shows 1.41 g, that's acceleration of the phone, which is on the riders moving arm so not accurate. 0 to 30 mph took 2.04 seconds, which is about 0.67 g. The goal was to get just under 2.0 seconds which would be about 0.684 g. The pedals are spiked and there are pads at the top of the sides of the EUC to allow the rider to exert more torque.

Extreme acceleration on an Imotion P6

 

[rcgldr]

EUCs don't have seats

The early ones did

Old EUC - SBU

With the low foot pegs, the EUC couldn't be tilted much, and combined with the narrow tire couldn't turn as responsively. The advantage was it was much easier to learn on, as a rider could just step along with one foot or the other. Top speed was around 12 mph (the SBU v3 may have gotten close to 15 mph). Despite the name, only the EUC was self-balancing, programmed to remain vertical. If the rider used muscles to stay nearly rigid with the SBU, then in that sense it was self balancing. If the rider flexed forwards or backwards at the waist, then technically the rider was doing the balancing of their upper body.

Around 2011, the "Solowheel" switched to using foot pedals instead of foot pegs, which allowed the rider to exert a torque onto the EUC via the pedals, eliminating the need for a seat. As EUCs became more powerful, power pads were added: these are protruding front and back pads on the upper surface that combined with spiked pedals, (no foot slipping) allowed a rider to exert more torque onto the EUC.

My 18XL EUC has small power pads. I don't use the front ones, but I do use the back ones for hard braking (if needed).

18XL top view

18XL front view

 

[A.T.]

Although peak shows 1.41 g, that's acceleration of the phone, which is on the riders moving arm so not accurate. 0 to 30 mph took 2.04 seconds, which is about 0.67 g.

Does the app show only the horizontal acceleration or the total magnitude, including 1g vertical acceleration that you also get when the phone is resting the ground? 1g vertical and 1g horizontal peak will give 1.41g total magnitude peak. The 0.67g horizontal 2s average would not be inconsistent with that.

It also depends how that peak is determined. If it's simply the sample of highest magnitude in the high frequency raw accelerometer output, then it's not reliable, even if it was attached to the vehicle. Does the app alow to adjust the smoothing of the signal?

An alternative is to draw some marks on the ground, and use the slow-mo mode many phone and consumer grade cameras have nowadays.

 

[rcgldr]

Does the app show only the horizontal acceleration

I'm not sure about the app in this case, but suspect that the acceleration is based on change in GPS speed versus time, as opposed to using an internal accelerometer, so it only calculates horizontal acceleration. My guess is the peak of the 1.41 g was due to the fact that the phone was strapped onto the rider's forearm, and that's how fast the rider accelerated his forearm forward when launching on the EUC.

 

[A.T.]

suspect that the acceleration is based on change in GPS speed versus time, as opposed to using an internal accelerometer

Seriously? I don't think this would be accurate for short term accelerations.

The better options are likely:

- Accelerometer with vertical 1g removed.

- Markings on the ground and slowmo video.

 

[rcgldr]

Seriously? I don't think this would be accurate for short term accelerations.
The better options are likely:
- Accelerometer with vertical 1g removed.
- Markings on the ground and slowmo video.

The app could be using the 3 axis accelerometer and gyro MPU | MEMS chip in a phone, which can determine the direction of gravity and compensate.

Again, that peak of 1.41 g was most likely due to the rider shoving his forearm forward at launch (the phone was strapped to rider's forearm).

 

[A.T.]

The app could be using ...

What's the point of using an app, where you are not sure what it actually measures?

(the phone was strapped to rider's forearm).

What's the point in attaching it to the arm, if you want the acceleration of the vehicle?

 

[rcgldr]

What's the point in attaching it to the arm, if you want the acceleration of the vehicle?

Riders do that so they can monitor speed while riding. The monitored speed can be GPS or as reported by the EUC. The app has a correction factor for EUC reported speed that the rider calibrates by comparing GPS speed versus EUC reported speed and adjusts correction factor until they match. The EUC reports speed at 10 times per second or faster.

Example of wrist mounted Garmin watch used as a GPS speedometer and odometer, and what appears to be a Ziren horn. She modded the horn button strap for easier use:

video of watch + horn

I only ride up to 25 mph, so I can hear my phone's speaker well enough with the speaker end pointed upwards, with phone strapped to belt or in an inside jacket pocket. I have the app set to call out speed, battery %, distance, and time every 15 seconds. I taped over the ear holes on my helmet to reduce wind noise.

 

[berkeman]

Riders do that so they can monitor speed while riding. The monitored speed can be GPS or as reported by the EUC. The app has a correction factor for EUC reported speed that the rider calibrates by comparing GPS speed versus EUC reported speed and adjusts correction factor until they match.

What could possibly make those mismatch? Is there that much wheelspin when they are riding their EUCs? What kind of riding are they doing?!

 

[rcgldr]

What could possibly make those mismatch?

My guess is the manufacturers deliberately over-report speed, often around 10%.

 

[A.T.]

Riders do that so they can monitor speed while riding.

OK, but that is useless to get reliable short term / peak acceleration of the vehicle.

You need a accelerometer on the vehicle, or slowmo video and marks on the ground/wheel.

 

[rcgldr]

You need a accelerometer on the vehicle, or slowmo video and marks on the ground/wheel.

EUCs also have 3 axis accelerometers and gyros, and use a algorithm to determine the direction of gravity so they can maintain a vertical orientation (relative to gravity). EUC reported acceleration isn't going to show a peak of 1.41 g though, so that had to be the phone. The zero to 30 mph in about 2 seconds is a much more reliable indicator.

 

[A.T.]

The zero to 30 mph in about 2 seconds is a much more reliable indicator.

If 2s average is a sufficient time resolution, sure. The max. acceleration during those 2s might still be higher.

 

[rcgldr]

If 2s average is a sufficient time resolution, sure. The max. acceleration during those 2s might still be higher.

The time was 2.04 seconds, but I don't know the resolution of the timer.

On an EUC, the motor torque has to match the torque exerted by the rider onto the EUC. This particular EUC has a top speed ~93 mph, so at the test speeds, the acceleration is a function of how much torque the rider can exert onto the EUC and not limited by the motor. Maximum acceleration occurs on the initial part of the transition from acceleration to constant speed. The rider could move his extended arms back to his waist and|or exert more torque onto the EUC in order to decrease the rider's lean angle.

 

[A.T.]

... the acceleration is a function of how much torque the rider can exert onto the EUC and not limited by the motor.

What is the maximal power / torque output of the motor?

 

[jbriggs444]

What is the maximal power / torque output of the motor?

Kingsong 18XL is rated at 2200 W and 104 Nm.

One finds online that rated power and peak power differ by roughly a factor of two. One also finds that there is a constant torque regime up to a certain speed where the constant power regime kicks in. Apparently the controller will throttle up to bring your lean angle back to where the motor can keep you upright in the constant power regime.

 

[rcgldr]

What is the maximal power / torque output of the motor?

As posted by jbriggs444, 18XL nominal (continuous) power is 2200 watts, peak power is 4000 watts. S18 Pro+ is nominal 2200 watts, peak power 5000 watts (higher peak power, hollow core motor). The EUC in question is the Inmotion P6, 6000 watts nominal (my guess this is a thermal limit), 20000 watts peak, with peak torque about 300 Nm. Claimed acceleration is 0 to 50 kph in 1.9 seconds, about 0.7454 g. Best video so far was 0 to 48.3 kph (30 mph) in 2.0 seconds, about .6840 g. In both cases rider weight was not stated. For the P6, the limit is the amount of torque the rider can exert onto the EUC. For the 18XL, the limit is the amount of torque the motor (and battery) can deliver.

 

[A.T.]

For the P6, the limit is the amount of torque the rider can exert onto the EUC.

I don't want give people ideas, on how to kill themselves. But one way to increase that torque would be to attach a kind of railing to the EUC, that the rider can lean into. Then add some weights to the riders shoulders and the railing itself.

 

[Baluncore]

Then add some weights to the riders shoulders and the railing itself.

I don't want give people ideas, on how to kill themselves. But one way to increase that torque would be to attach a kind of railing to the EUC, that the rider can lean into. The EUC and rider form an inverted pendulum. It is the deflection angle from the vertical, of the line from the contact patch to the centre of mass, that decides the acceleration necessary to maintain or reduce that angle of deflection.

Increasing the mass will increase the power required from the motors to stabilise the situation. More mass will require a greater force, but the acceleration will always be balanced against the acceleration due to gravity and the trigonometric ratio of that deflection.