Trikke Magic: Leveraging the Invisible

Draft: 23 November 2022

Most body-powered vehicles employ cranks, gears and chains to transfer rider generated momentum; none is as direct as Trikke locomotion. An invisible yet powerful mechanism, the Magic Lever, converts good rider technique into carving a great Trikke ride. Mathematical models of the dynamics and kinematics of a Trikke T78 Air were derived and incorporated into computer simulations. Typical Trikke responses to rider input were analyzed with the goal of improving riding techniques. Indeed, knowing how a Trikke works reveals several opportunities to leverage the magic.


The information in this paper is true and complete to the best of my knowledge. All recommendations are made without guarantee on the part of the author. The author disclaims any liability in connection with the use of this information. Further, I am not affiliated, associated, authorized, endorsed by, or in any way officially connected with Trikke, or any of its subsidiaries or its affiliates. The official Trikke website can be found at The name “Trikke” as well as related names, marks, emblems and images are registered trademarks of Trikke.

How I Became a Daily Trikke Rider


Watching this new way to exercise elegantly flow by on its TV infomercial just looked like so much fun! Owning one was inevitable but I held off as long as I could. I enjoyed my daily dose of endorphins and thanked God every day that I could still run fast. But in recent years my calves were cramping worse from deep vein damage and swelling due to varicose veins. Discovering calf wraps and cross-training with a bike helped, but it was time to buy that Trikke!

Author carving
Michael carving
(normally wears a helmet)

My first ride was preceded by watching lots of "how to" videos and scouring the web for scholarly articles on Trikke locomotion. Being a computer "geek" with a bachelor's degree in physics makes that kind of prep mandatory. If there is any serious information about Trikke propulsion, it has obfuscated search terms or it doesn't exist. With a couple theories brewing in my head, I test rode a T78 and Sport 8 while passing through Atlanta in June 2017.

A bit like paddling at the front of a raft using a wheel on on a pole rather than a paddle - that's how I thought it should work! Making it "go" wasn't too difficult but it was slow. The test-drive parking lot was not flat. I was able to "carve" slowly where it was slightly down hill or flat. Climbing a slight incline stressed my arms, though I made it work. How did the people in the videos make it look so casual at decent speed? A bit disappointed that my physical instincts failed me, there was plenty of evidence that those casual riders were doing something I wasn't.

A week or so later, I began riding my own T78 Air every day at an almost-no-where-flat linear park near my work place. As theory and experience advanced, I made progress every day. By three weeks my rides began reaching my running speed. Progress continued via a combination of gaining upper body strength from the Trikke, a growing understanding of how it works and "listening to" and learning the responses of the Trikke to my moves. Thin water shoes enabled a better feel of those shifting decks! The relationship is very much like that described in one of the "dragon rider" novels we read as children; dragon and rider bonded psychologically, anticipating each other's actions. In a word, "magic". Yes, I gave my T78 and T8 names.

Not a Trikke Rider (Yet)?

Trikke Anatomy - What is a Trikke?

All body-powered Trikke vehicles consist of three long tubes attached to wheels with a shorter tube (handlebar) used to provide comfortable leverage for turning and cambering. The three main tubes articulate via a cambering joint that constrains all three wheels to camber at about the same angle, at the same time. The guide-wheel in front is mounted as a caster wheel. Deceptively simple design. Pure genius, synergistic engineering.

For ease of illustration, most depictions of a Trikke have been stripped down to a stick figure. Parts of the model Trikke are referred to by the names indicated in the image on the left in Figure 1. On the right, the main spans and articulations are illustrated in Figure 2.
Parts of a Trikke
Figure 1 - Parts of a Trikke
Main spans and articulations
Figure 2 - Main spans and articulations

A Magic Lever

Magic became even more magical when I realized what the main power producing mechanism of a fitness or body-powered Trikke is. Of course, the rider supplies the power, but the mechanism has to be realized through levers, gears, pulleys and such - right? It took a good, long look at the Trikke frame and the progress I made riding it to analyze the [physics] of the ride. Four levers hide in that three-tube frame and 3 mechanisms bring it to life. That wasn't enough. Which mechanisms work best in which road conditions? A forward-feed, discrete, [calibrated] Trikke/Rider [simulation] demonstrated the typical [behaviors] needed to answer those questions and more. Over three years after my first ride, about an hour a day riding and an hour "hobbying", at last I was able to make a contribution to the Trikke Community. But I still wasn't perfectly clear about how I was generating the torque that feeds the physical mechanisms; until now [Quick Start], over 5 years from that first ride.

Principle 1: All driving motion feeds into a Turn-Center Arc.

Corollary: When its handlebars are pointed straight ahead, a Trikke is at rest or coasting.

What is the main mechanism that drives a Trikke? Look at the illustration on the right to find out; Figure 3. It shows the workings of the most important of the 4 levers; the black dashed line. What makes this lever so magic is that it is not a "solid" lever as one might expect. It arises out of coordinated friction at each wheel contact patch on the road. The same friction forces that constrain wheels to roll forward, make them turn around a turn-center just like a lever rotates on a fulcrum (black dot at bottom). Every part of the Trikke must also rotate around the turn-center on a turn-center arc (i.e., orbit). Parts farther to the "outside" rotate faster than those closer to the fulcrum on the "inside". Parts in the same orbit or when the Trikke is not turning have the same speed.

One of the farthest parts is the front-wheel contact patch (center of black wheel in the upper left). This guide-wheel receives the action (purple ray). The farther out it sweeps, the faster it moves relative to the turn-center fulcrum. Closer in we find the center of mass of both the rider and Trikke (brown dot). They combine creating a system center of mass that constantly changes position; shifting with both the rider and the configuration of the Trikke via steering and cambering. This is the load (orange dot and ray) on the lever (black dashes); the stuff that gets pushed by the action in orbit around the fulcrum. While the load is not actually located on the lever, it acts as if it is on the lever simply because both the Trikke and rider revolve around the turn-center as a unit. The result is the same: give the guide-wheel a push, via a jetting action, and the center of mass follows it in its own orbit (brown ray).

Jetting turns the rider's angular momentum along the guide-wheel path (purple ray) into linear momentum at the system center of mass (brown ray). Like most systems, there is loss in this momentum transfer. One source of that loss is the ratio of the system center of mass turn-radius to the guide-wheel contact turn-radius. If it is less than one, the usual case, there is a proportional loss. At one, there is no loss due to this source. Greater than one, there is a proportional gain; this can only happen near the center-line when steering turns from a straight line. There is always a loss, just not due to this particular geometric factor. Can the rider move the system center of mass toward the guide-wheel orbit to take advantage of this? Not sure - but it would be difficult.

On the other hand, action at a right angle to the guide-wheel dissipates in wheel deformation, heat and other forms of friction as in Figure 4. Unless so strong that the wheel slips or structurally fails, radial momentum along the magic lever does not affect speed. In fact, action (purple double ray) along any turn-radius (brown line) at any point (brown dot) on the Trikke or rider meets the same fate, friction across the wheels, illustrated as red double rays. Therefore the rider's side-to-side momentum generally has no positive effect on speed! However, the weight shift can be a negative. But a rider can still take advantage of this fact to setup for a push by shifting position without affecting speed negatively.

It's not very complicated. It's virtually magic!

Radial Action
Principle 2: Radial action dissipates in wheel deformation, heat and friction. Sideways action has no effect on speed.

Corollary: A Trikke at rest remains at rest if its handlebars are pointed straight ahead; no amount of rider twisting or force in any direction will change that.

Three Ways to Tame a Trikke

"Carving vehicles" earned that epithet by similarities with skating sports. In particular the way that a ski or ice-skate blade cuts or melts into the surface to hold its edge. A Trikke wheel doesn't sink into the road, but its path-constraining effect is the same. This edge-grabbing characteristic is responsible for the "snake-like" paths etched by athletes in these sports. A Trikke is really good at it.

Turning the Magic Lever
Principle 3. Jetting the guide-wheel by foot and/or fist accomplishes Trikke Locomotion.

Corollary: Every handlebar turn triggers the small effect of Castering.

Incremental Speed

You now know three mechanisms that feed the guide-wheel of your Trikke. Accelerating a Trikke using these mechanisms means building up the speed a little at a time; in increments. One drives a Trikke in cycles. Half of a cycle to the right followed by half of a cycle to the left. A cycle can be a bit shorter than a second or restfully long up to a few seconds. Seem short? You'll get there. At 11.5 mph, my cadence falls just short of 1.5 seconds per cycle; 42 cycles per minute. Initially over 12 mph, it gets closer to a second, but lengthens a little maintaining cruise.

From on board the Trikke, you create all the guide-wheel jet-acceleration each cycle in the local frame of reference. Watching from the side of the road, you can see that the guide-wheel travels faster than the rest of the Trikke, certainly faster than the local jet-speed. This is the global frame perspective. So, when the rider stops driving, the Trikke coasts to a stop in the global frame, but motion stops immediately in the local frame. Seems a bit odd, but physically, it's as if the rider in the local frame is blind to the world around the Trikke and can't see it moving by. The observer in the global frame sees everything but at different speed and position. But the rider can "feel" global accelerations locally.

This difference is important because both are right about the speed of the Trikke. The rider creates its local jet-acceleration every cycle, and its speed adds to the Trikke-Rider System's global speed. A graph from a simulated constant acceleration and alternating turn-angle configuration shows this in Figure 8. The average global speed (green graph) builds up as long as two local jet-acceleration (blue) spikes are added every drive-cycle. Each spike represents a jetting action, left then right. Note they are each the same height. So when the Trikke speeds up to its maximum or terminal velocity for the current conditions, the rider must continue to add local jet-acceleration to the guide-wheel. At that point the rider adds exactly as much local jet-speed as friction takes away in the global frame. Note the steps - horizontal lines - added to the average global speed are not evenly spaced vertically. They depend on global friction that changes with global speed. The higher steps occur where the friction is lowest. In the figure, the Trikke increases speed over 9 mph in less than 10 cycles.

Principle 4: Building up speed is incremental; a little more each drive-cycle.

Corollary: At terminal velocity, the rider adds exactly as much jet-speed to the Trikke as is removed by friction, wind and inclines.

As mentioned, the global speed of the Trikke is not the global speed of the guide-wheel. During the first half of the drive-cycle, the rider turns to a certain angle and keeps it there until the next half of the drive-cycle. After steering to the same angle on the other side, the rider holds it until the end of the cycle. So, each drive-cycle has a maximum angle that is held for about half of the cycle to each side. The global speed of the guide-wheel follows this turn-angle along a snaky path, but the Trikke moves forward at a slower speed and turns less. An approximate mathematical relationship for this slithery situation multiplies global guide-wheel speed by the cosine of the turn-angle minus some loss due to friction to get the global Trikke (system center of mass) speed. So, when the turn-angle is zero, straight ahead, the guide-wheel and center of mass speeds are equal and coasting to zero. When steered to a right angle to one side, no jet-speed can be added and the speed must be zero or the Trikke is skidding to a stop or worse.

Trikke Magic

Everyone likes magic especially when it works in their favor. Like most good magic, it's difficult to assess what is actually going on, even when you know the physical principles involved. This article intends to ground the information in the [Quick Start] Trikke riding guide in science. The mechanism of walking a box across a floor invokes the basic magic of riding a Trikke - jetting! Keep reading to understand how that can be.

Box-Trot Arcs
Figure 9: Box-Trot Arcs

If the box is not too heavy and has the right dimensions, placing the closest edges on your toes is the easiest way to walk it. Your feet are then frozen the same distance apart and can't easily move relative to each other throughout the effort. Instead, one foot must rotate arced around the pivot provided by the other as in Figure 9. When you master this "Box-Trot", the pivot foot does not rotate at the ankle, instead it is turned as a unit by the body rotating on the ball of the foot. This motion begins at the pivot-side hip rotating the spine and upper body slightly ahead of the lower body. As the upper body gains angular momentum, the core muscle groups lock the lower body into the rotation increasing the body's angular momentum. The whole body seems to rotate together. Very difficult to see on a video. Synergy follows:

  1. The foot to the outside of the rotation (outside foot) raises ever so slightly with the hip twist to pass over the floor.
  2. Your body and the box's minimal axis of angular inertia are almost parallel and close together making it easy to turn the box with you.
  3. At the end of rotation, your outside foot has moved forward on an arc. Alternating pivot feet and rotations then moves you forward on arcs.

Principle 5: Riding a Trikke is like walking a box across a floor.

Corollary: A Trikke is a much better dance partner.
Turn Center Arcs
Figure 10: Turn-Center Arcs
in a Cambered Left Turn.

Turn-Center Arc: A Trikke is physically constrained to move along a Turn-Center Arc as illustrated in Figure 10. Two gray rays parallel to the wheel axles meet at the Turn-Center off the left edge of the figure trying not to be noticed. Any arch equidistant from the Turn-Center is a Turn-Center Arc. When the Trikke turns right, the Turn-Center moves off the right edge of the figure. Again, your Trikke cannot move on any other path, it is stuck on an arc like an antique record player needle in a vinyl record. Like walking a box across the room.

The dance of riding a Trikke coordinates your generation of angular momentum along a Turn-Center Arc with steering and cambering the Trikke. A turning Trikke given enough angular momentum accelerates; all twisting and force channeled into a Turn-Center Arc builds its angular momentum - this turns the "magic lever".

Motion along any other path, does not contribute to Trikke movement; it only increases tread friction and frustration. All driving motion follows a Turn-Center Arc. Co-rotation - you and your dance partner; this makes the Trikke magic.

Notice that Turn-Center Arcs lie in the plane of the pavement. Vertical motion does not lie in that plane. Except for cambering, explained below, consider it useless. Generate motions parallel to the ground at all times.

Figure 11: Body Co-rotation and
Effort in Orange and Red

Not only does the "Box-Trot" work to move your Trikke, the Trikke is designed to amplify it. After basic body position, cambering and pushing help leverage the angular momentum you create illustrated from a top view in Figure 11. Your feet cannot separate until you camber the steering column. Camber it by lowering your inside hand or raising your outside hand or both, slanting the top of the steering column toward the inside of the turn as in the figure. Cambering adds no motion along any Turn-Center Arc and does not change steering. It allows your feet to separate just enough to straighten the outside knee a little. The outside foot does not move on the deck, but it feels a push across the top of the deck (red arrow); never into it - that is wasted vertical motion. Avoid pushing into it by lifting your heel slightly.

Push the outside foot (red arrow) and hand (orange arrow) much more forward than their inside counterparts. The inside foot supports your upright body and does not push at all. Use the inside hand to keep its side of the handlebar stationary - something like rowing the handlebar, where your inside hand is an oar lock. Synergism again follows:

  1. It forces your body to twist slightly into the outside arm push. When you are used to it, especially if you can feel the power, amp up the body twist.
  2. It helps you throw a harder outside foot-push as your hips twist with it.
  3. Your body moves backward slightly adding to the push into the arch path.

Notice that cambering with your body upright also produces three synergistic benefits:

  1. Allows your feet to separate giving your outside leg the opportunity to push into a Turn-Center Arc.
  2. Cambers all of the wheels adding traction and stability in the turn.
  3. Shifts your weight toward the inside wheel contact point with the ground, decreasing friction on the outside wheel - because you are pushing across the outside deck, not down on it.

Jetting Arcs
Figure 12: Jetting Arcs
in the Global Frame

Note that the pivot in the Box-Trot is a foot or equally, a corner of the box. Co-rotation of the box and body utilizes the same pivot axis. On a Trikke these are separated. The body rotates about its own center of mass, while the Trikke rotates about its Turn Center. Physically, the body rotation translates to the Turn Center via the Parallel Axis Theorem. Thus we can correctly think that the body rotation pushes the green lever in Figure 12 with the force of the green arrow. The green lever is part of the rear-axle turn-lever which is functionally equal to Figure 3 the Magic Lever because they turn as a rigid unit. So body rotation feeds into a push of the Magic Lever; jetting.

Your Trikke is your new dance partner. You will learn to feel the dance from one side to the other; smoothly and quickly. In your mind's eye, see yourself gliding through this dance - videos of others riding a Trikke may help. However, like trying to learn a language from songs, the cues in those videos are likely too subtle or easily misinterpreted. Does the Trikke turn the rider or vice-versa? The rider must always turn the Trikke, unless coasting.

For example, it may feel like you or look like the person in the video is pushing hard on the outside foot and "unloading" the inside foot. That does happen through rotation you can't see or feel easily. However, the center of mass remains closer to the inside wheel and not where the push is directed over the deck - unless the inside arm of the Trikke is broken. A good rider keeps his head stationary, horizontally and vertically with respect to the trail while slinging the Trikke under his swaying body. The diagram in Figure 14 is a simple representation and a bit exaggerated for clarity.

How to Steer and Camber as Quickly as Possible

You may be surprised that leaning back helps turn and camber the handlebars faster! As a physical object, the Trikke has three axes each at a right angle to the others that prescribe how it rotates around its center of mass (where they meet). These are angular inertial axes, two of which are illustrated in Figure 13. One, not shown for clarity, lines up with the y-axis across the Trikke. It is the most difficult to rotate around as one or two wheels would have to leave the ground!. Another difficult axis to spin around points forward and up (orange line).

Your friend, the minimal one (brown line), is three times easier to rotate around than either of the other two. It extends from the Trikke's center of mass through where a seat would be if a Trikke had one. The more aligned your body axis twists and turns are to this axis the easier it is to turn the handlebars. Standing back on the decks and remaining upright is enough to take advantage of this alignment. If you lean back too far you may impress your friends with a wheelie or a shocking tumble.

The minimal axis is not a stationary axis. With cambering it tips from side to side slightly into the turn like an upside-down pendulum. Technically, these axes are the eigenvalue unit vectors of the Trikke's inertial tensor. Note, for orientation, the x-axis (purple) and z-axis (light blue) were drawn as well.

Finesse Steering
Principle 6: Align your body axis with the Trikke's minimal angular inertia axis to ease steering, cambering and therefore jetting.

Corollary: Moving or leaning forward requires increased body rotation to obtain the same jet.

Physics, math and computer based analysis guided by riding experience provided the foregoing explanations of the magic bottled up in a Trikke. Let's take advantage of them to ride a Trikke with grace and conservation of effort.

Starting Out

The easiest way to begin a ride is to grab the handlebars, push the Trikke forward a few steps and hop on. You're rolling! Don't need a simulation for that. A carving purist however, grabs the handlebars, steps up onto the decks, turns the guide-wheel about 80-degrees to a favorite side and torques into a turn at a practiced jet-speed. This won't be easy to accomplish at first and on rough ground it will always be difficult. It requires balance, cultivated body twist and foot thrust. If it's important to learn this right away, start pointed slightly down hill. Otherwise, wait until you've mastered carving and it will come naturally.

Start with your feet touching the back of the decks and your weight close to the inside turn wheel. Keep the steering column light. You want the turn-center and your center of mass to be close to that inside wheel so you can spin around it. This also aligns your body axis better with the minimal rotation inertial axis so the turn can be quick. There are a few body movements to coordinate simultaneously. Perform these movements illustrated in Figure 14 together, as quickly as you dare.

Turn Center Arcs
Figure 14: Swing body,
outside foot and hands around

When pushing the deck with your foot, do so without pushing down into the deck; keep the push across the top as much as you can. This goes for pushing the steering column as well. Push forward, not down. Pushing down diverts your energy into deforming the wheels' contact patches generating heat and friction instead of speed. It may help to raise the handlebars higher; standing straighter levels the foot and hand pushes. This is all part of keeping the guide-wheel light.

Light Guide-Wheel
Principle 7: Keep the guide-wheel light; only allow enough weight for turning traction.

Corollary: The front-wheel does not operate as a drive-wheel or paddle.

Gaining Speed

Once your Trikke is rolling, the best way to gain speed is also the fastest way. When I first rode, I tried what felt like rowing at the bow of a boat to ramp up the speed. I understood rowing. My arms felt like they were going to fall off, but I was riding my Trikke! The tires wore out in a couple of months. Carving properly is so much easier and much less wear on the tires. Carving requires balance, cultivated body twist, outside hand and foot thrust; i.e., jetting. Carving will likely take a few weeks to learn well if you work at it every day.

Starting from a slow roll, the goal for the first couple of cycles is to gain momentum. At this slow speed, carving means keeping the turn-center and your center of mass close to the inside trailing wheel. As the Trikke begins to build momentum, you can move your feet forward toward the front of the decks if that helps. It may be easier to control your center of mass by leaning back and squatting a little. Standing too far back limits you to lean forward, which misaligns with the minimal angular rotation axis. Use similar body movements as in starting out, but begin to camber. These actions are performed together:

Repeat the above movements increasing the cadence up to your normal carving drive-cycle; mine is about 1.5 seconds. Reduce the turn-angle and deepen cambering as you get closer to top speed. At top speed, turns will be very slight, camber and cadence optimized. Jetting feeds directly into the almost straight guide-wheel path. The "magic" lever works near its top efficiency with the turn-center some distance to the side; it lines up with your trailing wheels. Jetting practically straight with almost all camber really feels like magic because it seems like pushing yourself forward. And we all know that kind of perpetual motion can't happen.

Top Speed

As you reach terminal velocity, turns are very slight and cambering is maximized. If you were leaning your body side-to-side, that will decrease; naturally limited by less wide turns. It is only natural at this point to be riding practically vertical, leaning back and squatting slightly. Because you are leaning back, steering is easy and the guide-wheel load is light, which reduces its friction. Cambering feels more like thrusting the handlebars side-to-side away from you. Some might call it punching. The outside foot still pushes forward into the guide-wheel path, but feels like an extension of your body twist. Because you are vertical and the Trikke cambers under you, the total center of mass hovers near the center-line. This increases the geometric turn-radius ratio so that jetting is more efficient and helps the trailing wheels share friction more evenly. When most of the load is on one wheel, its friction can be much greater than the total friction when the load is equally shared.

Understand the [physics] and the technique will come. Rely only on technique and it will eventually fail.

Over Hill and Dale and other assorted topics

What is the "Sweet Spot"?

A "sweet spot" in a physical process is technically a minimum, maximum, inflection or resonance of some characteristic or behavioral quantity of the process. When [calibrating] the Trikke [simulation] the measured friction profile had a minimum at about 4.8 mph, about 2.14 m/s in Figure 15. That means the Trikke "wants to" or is most easily coxed into operating at or near that speed.

Friction actually reduces when the Trikke speeds up from standing still as if it wants to go faster. Accelerating past the minimum friction speed, makes the friction increase. Past about twice the minimum (green zone), friction becomes much more difficult to overcome (red zone). 12 mph (5.3 m/s) is about my terminal velocity on a flat, smooth road with no wind on a warm day. At terminal velocity, the rider's jet-speed matches the speed lost to all forms of friction. Note, the profile was measured for the one Trikke I had at the time. Yours will likely have the same form, but could be shifted or widened.

Another sweet spot is the presumed resonance I feel when jetting with both feet. By cambering and pulling on the handlebars, I provide the restoring force necessary to create the resonance. I can tune the sweet spot to whatever speed I like. Though it doesn't take as much energy as rapidly jetting with one foot at a time, it takes considerably more effort to maintain than normal carving. This may be a sweet spot in a high energy state, like "passing gear" in a car.

In online videos, I don't think "sweet spot" means either of these. I think it simply means a rider has figured out how to carve. The simulation shows that for my Trikke, I have to provide about 2 m/s2 jet acceleration to carve properly. "Properly" means reaching a velocity of 4.2 m/s (9.4 mph) or more. According to the simulation, it is difficult without coasting and resting to carve at a speed between 2.2 and 4.2 m/s because of the friction profile. Carving properly by pushing and jetting one foot at a time (alternating, assisted hemijetting) is an open-ended process like riding a bike; it will propel the Trikke as fast as you are able to coordinate the effort up to a terminal velocity determined by path and weather conditions. I'm pretty sure no one believes that the sweet spot is terminal velocity - you have to exercise pretty hard to achieve that.

What is "Punching"?

Videos mention punching in a few different contexts and seem to explain it as a vertical force on the end of the inside handlebar or a quick thrust forward on both handlebars. The vertical force is quick cambering; forcefully pulling or pushing the guide-wheel across the forward path. By itself, the simulation shows it accomplishes nothing. However, it is part of good technique for reaching top speed with strong carving. Both hands punching forward is a quick, direct push. That is effective at stepping the Trikke forward in a turn, or after one if followed by a turn before resetting position. When directed around with a turn, this punch can also recruit the castering levers to step a little more. So, punch either way; it can only help.

What is "Leaning"?

Like "sweet spot", it is difficult to know what people mean by "lean".

Saga of the Trikke Ballerina

Starting in the "prior art" patents to J. Gildo Beleski Jr. Cambering Vehicle and Mechanism. US Patent 6,220,612 B1. Dec 20, 2005, the idea that the Trikke-like vehicles worked somehow based on the Law of Conservation of Angular Momentum alluringly illustrated by a ballerina drawing her arms in was spun. The poorly explained association continued though each patent indicating to me that they had no idea about how it actually works or that since the appeal worked in the previous patents it was seen as necessary to obtain following patents. I suppose had they explained the "magic lever", it would not have been seen as so mysterious a mechanism as to deserve a patent.

Unfortunately, this myth has been propagated in Trikke literature, not just owner's manuals but training and "educational materials" targeting schools as well.

Why can't the Trikke Ballerina explain Trikke movement? An elegantly spinning ballerina does illustrate conservation of angular momentum and conservation of energy beautifully. But precisely because of that, she can never make a Trikke move! The problem is - she has constant energy. She spins faster as she pulls her arms inward, but in order to do that she cannot give away any energy; she has to keep it, conserve it. The Trikke needs energy to offset that lost to friction and to produce kinetic, forward motion. Energy the ballerina is not able to give to it.

Now, consider the ballerina alternating spinning directions, like a Trikke rider. To change directions, she cannot conserve energy because she must stop spinning at some point to change directions - her energy becomes zero! Suppose she manages to transfer that rotation to the Trikke, like a real rider. When she changes spin direction, to conserve the energy she gave away, she must take it back to spin up again - all of it! The Trikke ends up with no net energy from the ballerina.

A Trikke must gain energy from its rider, not conserve it. The rider is not a ballerina. The rider is more like a washing machine agitator motor accelerating to spin up the Trikke in alternating directions. Conservation of angular momentum is essential in constraining various aspects of the physics models representing Trikke locomotion, but it is not its principle of propulsion. Conservation can't explain that; Trikke propulsion is an open system. Like riding a bike, it will go faster if you work harder - up to external, terminal conditions. The principle of Trikke propulsion can be analyzed as alternating co-rotation from an angular viewpoint or pushes of an alternating "magic lever" from a linear one. Neither is a "conservative" process.

Why Steering Angle Decreases with Speed

Suppose you generate the same jet-speed each drive-cycle. That speed reduced by friction increases the global guide-wheel speed. If the maximum turn-angle remains the same each drive-cycle, then your Trikke will want to make wider and wider curves across the road. It travels further each half of the drive-cycle as simulated in Figure 16. Guide-wheel (brown trace), left (red trace), right (green trace) and Trikke center of mass (blue trace) all grow width-wise.

The natural rider response is to shorten each subsequent drive-cycle to keep the Trikke closer to the center of the road. In normal travel, the rider tends to keep the Trikke's path-width fairly constant. So, he ends up ping-ponging across the path more and more quickly. However, there comes a point where the handlebars can't be turned fast enough, even with the body aligned with the minimal inertial rotation axis. Shortening the drive-cycle and not changing the steering angle, unintentionally caps Trikke speed. There is a faster way!

To reach terminal velocity, about 12 mph, comfortably, a rider reduces the turn-angle as the Trikke increases speed. At terminal velocity the turn-angle will be almost zero and you will mostly be cambering. It's amazing that jetting still works at such small turn-angles! Your feet and hands are essentially pushing straight ahead - at least it feels that way. If the wheels were as large as bicycle wheels, I might believe camber thrust has something to do with it. Maybe. When the body axis aligns with the Trikke's minimal angular inertia axis, drive-cycle times can become lower than a second.

How does the steering angle change with speed? Holding a constant path-width as the Trikke accelerates, leads to a simple relationship among turn angle θ, path-width w, global speed V and drive-cycle time τ. As global guide-wheel speed increases, the steering angle is approximately inversely proportional to that speed and the drive-cycle time. But is also proportional to twice the path-width. For low speed, this relationship becomes the argument of an arcsine function; θ = sin-1((2 w)/(V τ)). This leads to the conclusion that reaching maximum speed while keeping path-width and drive-cycle time constant, a rider must reduce the steering angle as in Figure 17.

Speed and Steering
Principle 9: As speed builds, decrease steering angle and camber wide.

Corollary: Increase steering cadence to accelerate.

Experiment on Your Trikke

Experience Static Friction Like a Ninja

Goal: Step the Trikke forward by pushing then reset your position by pulling without losing ground.

Principle: Static friction and low impulse resist motion.

Try this experiment with your Trikke on a flat surface (works best on a rough or soft surface).

  1. Position yourself on your Trikke, with feet toward the front of the decks, hands gripping the handlebars and body close to them. Steer straight with no camber. Use the breaks to hold still.
  2. When it is still, release the breaks. It should remain halted. Carefully and simultaneously, push the handlebars forward, pop up on your toes and arch your back as far back as you can without falling off. Your center of mass may have moved a foot backward.
  3. Hold that position and let your Trikke come to a stop. It will probably stop almost immediately and only travel about a foot. Don't use the breaks.
  4. Now very slowly, settle forward to your initial position close to the handlebars. This is the "Like a ninja" part - letting static wheel friction hold the Trikke still for you.
  5. Unless your surface is fairly smooth, you should eventually be able to get back to your original position without moving the Trikke backward. You get ninja points! That is static friction in the wheels at work. A weak impulse cannot overcome it, but a normal one can.
  6. Repeat to understand that the slower impulse provides a weaker force. How fast can you make the slow impulse? Try this on carpet or a rough road.
This is possible to do on a Trikke, but it is not useful for carving.

Two steps to nowhere

Goal: Step the Trikke forward by pushing then step it back by pulling.

Principle: Conservation of linear momentum prevents system center of mass from moving.

Find a flat, smooth surface. Ice can be used for this one. Helmet please.

  1. Position yourself on your Trikke, with feet toward the front of the decks, hands gripping the handlebars and body close to them. Steer straight with no camber. Use the breaks to hold still.
  2. When it is still, release the breaks. It should remain halted. Carefully and simultaneously, push the handlebars forward, pop up on your toes and arch your back as far back as you can without falling off. Your center of mass may have moved a foot backward.
  3. Hold that position and let your Trikke come to a stop. It will probably stop almost immediately and only travel about a foot, even on ice. Don't use the breaks.
  4. It should remain halted. Carefully and simultaneously, pull the handlebars to you, drop your heels to the decks and stand up straight the way you started out.
  5. The Trikke should be back where it started. Trikke wheel bearings are pretty easy to move, so it shouldn't take much effort to return to the original position. Even a relatively small impulse can overcome its static friction.
  6. Repeat to understand that both the push and the pull impulse overcome the static friction force.
This is why it is not generally a good idea to push on a Trikke that's going straight - it's step will be lost to the reset pull.

Two turning steps to nowhere

Goal: In a turn, step the Trikke forward by pushing then step it back by pulling.

Principle: On an arch conservation of angular momentum allows the system center of mass to take a step. Opposite steps cancel.

Find a flat, smooth surface. Ice can't be used for this one. Helmet please, anyway.

  1. Position yourself on your Trikke, with feet toward the front of the decks, hands gripping the handlebars and body close to them. Steer to one side with no camber. Use the breaks to hold still.
  2. When it is still, release the breaks. It should remain halted. Carefully and simultaneously, push the handlebars forward into the turn, pop up on your toes and arch your back as far back as you can without falling off. Your center of mass may have moved a foot backward. Your outside arm should be extended and the other bent.
  3. Hold that position and let your Trikke come to a stop. It will probably stop almost immediately and only travel about a foot. Don't use the breaks.
  4. It should remain halted. Carefully and simultaneously, without changing the turn-angle, pull the handlebars to you, drop your heels to the decks and stand up straight the way you started out.
  5. The Trikke should be back where it started. Trikke wheel bearings are pretty easy to move, so it shouldn't take much effort to return to the original position. Even a relatively small impulse can overcome its static friction.
  6. Repeat to understand that both the push and the pull impulse overcome the static friction force even when the handlebars are turned.
This is why it is not generally a good idea to push on a Trikke that's turned if you are not going to turn the other way before pulling - it's step will be lost to the reset pull.

Two turns, two steps, only one counts

Goal: In a turn, step the Trikke forward by pushing then turn and pull to lock in the step.

Principle: On an arch conservation of angular momentum allows the system center of mass to take a step. A turn can dampen a step if the action is perpendicular to it.

Find a flat, smooth surface. Ice can't be used for this one either. Helmet please.

  1. Position yourself on your Trikke, with feet toward the front of the decks, hands gripping the handlebars and body close to them. Steer to one side with no camber. Use the breaks to hold still.
  2. When it is still, release the breaks. It should remain halted. Carefully and simultaneously, push the handlebars forward into the turn, pop up on your toes and arch your back as far back as you can without falling off. Your center of mass may have moved a foot backward. Your outside arm should be extended and the other bent.
  3. Hold that position and let your Trikke come to a stop. It will probably stop almost immediately and only travel about a foot. Don't use the breaks.
  4. It should remain halted. Carefully and simultaneously, turn the other direction as you pull the handlebars to you, drop your heels to the decks and stand up straight the way you started out. Keep the pull at right angles to the turn. Do this more slowly at first.
  5. The Trikke should remain where the push put it.
  6. Repeat to understand that the push impulse overcomes the static friction force, but dumping the pull radially against the turn increases the wheel's friction to hold it in place.
Reseting by pulling against a turn is a nice talent to learn, but not necessary. Using Trikke geometry, one push sets you up for the next to the other side.

Fred Walsh, the National Trikke Trainer, showed me these exercises that follow. I practiced the first one every day before my regular rides for months afterward. It was really helpful. Only played the second one with Fred so far; and that was a bit one-sided. If you ever get a chance to take a lesson from Fred, do it. Even if you're a pro, you'll learn a few things.

Pulse-Turn Exercise

Goal: Keep the Trikke moving in a circle.

Principle: Carving in a circle requires more movement of the outside arm and leg.

Find a flat, smooth surface. No ice. Helmet please.

  1. Position yourself on your Trikke, with feet toward the front of the decks, hands gripping the handlebars with body straight or leaned back some. Carve up to a normal speed.
  2. Turn to one side while simultaneously pushing into the turn with both hands and the outside foot. This means pushing the outside handlebar and deck more than the inner ones. Your body will naturally lean into the turn. As you practice, lean less until you remain upright to maximize twist.
  3. Hold the turn, or turn slower if you want a larger circle.
  4. Relax the turn toward straight. Regain your initial position by moving radially. Radial movement won't cause a step backward.
  5. Repeat, continuing around the circle indefinitely.
This teaches you good jetting technique and reseting your push position by using Trikke geometry; one push sets you up for the next. Experiment with quick turns and push. Expand the circle as you go faster.

Trikke Tag

Goal: Tag another player's back wheel with your front wheel.

Principle: Agility and speed.

Helmet please. Knee and elbow pads if you get really competitive.

  1. Chose a tagger - "it".
  2. Form a distant circle around the tagger. The distance depends on skill levels and ground space.
  3. The tagger counts down from 3 to "go".
  4. The tagger tries to tag another player's back wheel with his front wheel.
  5. Everyone else avoids this state.
  6. When the tagger succeeds, the player tagged is "it", becoming the tagger.
Challenges each player to be more agile and faster.


Camber, cambering Wheel tilt angle from vertical rotated around a wheel's forward path direction. Tilting a Trikke's steering column serving as the Camber-Lever to the left or right side creates a nearly equivalent vertical tilt in all three wheels. Foot-deck tops also angle synergistically to increase grip in a turn. For Trikkes, cambering generally refers to tilting the steering column from side-to-side coordinated with steering. Cambering enhances the effects of carving and castering.

Camber-Lever A second-class lever of the stem with fulcrum on the road and action from cambering that moves the yoke. It also interacts with the two levers involved in castering.

Camber thrust When a cambered wheel rotates, tread-particle elliptical trajectories are constrained to run straight when contacting the ground. This asymmetry causes a change in their momentum (a force) at right angles to the inside of the camber angle. This force and increasing change in camber pull the wheels into the turn avoiding dangerous wheel slips to the outside.

Carve, carving Most dictionaries don't define a sports sense of this word! [] does for skiing and skateboarding, basically a turn accomplished by leaning to the side and digging an edge or wheel into the path. In the case of "carving vehicles" or "CV"s, carving is a transfer of momentum from the rider to the vehicle synergistically constrained by ground forces on the wheels. The Trikke seems best engineered to claim this action as a primary form of locomotion. This motion mechanism provides action to the guide-wheel contact point around the Turn-Lever.

Caster, castering Most dictionaries don't define a sense of this word as a verb! Here, it is defined as a form of locomotion. To caster is to move by quickly turning a wheel with positive caster and positive trail back and forth to produce forward motion. The offset contact patch causes the vehicle frame head or yoke to move sideways a couple of inches pulling part of the vehicle's trailer mass into the turn. Impulse created by this motion in the direction of the drive wheel path first pulls, then pushes on the vehicle's stem assembly. It is the sideways rotation not the stem pull or push that creates a small jetting impulse. This appears to be the primary means of locomotion for the [EzyRoller], [Flicker], [PlasmaCar] and Wiggle Ride-On Car by Lil' Rider (no web presence) rider toys and one of the modes for a Trikke. Two levers mechanize this motion: the Yoke-Lever and the Trailer-Lever.

Caster angle - positive, negative The angle a steering column makes with the road. It is positive when the angle slopes up toward the back of a vehicle. It is negative otherwise.

Caster pull When the handlebars of a Trikke are turned and tilted quickly toward the center, the geometry of the steering mechanism and constraints on wheel motion, throws the yoke into the turn up to a few inches and pulls the Trikke's Extended-Stem and trailer a quarter inch or so closer together in a fraction of a second. Though the rotation is small, most of the weight of the Trikke and its rider get spun around the transom point. The pull generates no immediate jetting impulse. Steering resistance increases as caster pull builds local angular momentum.

Caster-push When the handlebars of a Trikke are turned and tilted quickly away from the center, the geometry of the steering mechanism and constraints on wheel motion, throws the yoke into the turn up to a few inches and pushes the Trikke's Extended-Stem and trailer a quarter inch or so apart in a fraction of a second. As the turn reaches its limit, the built up angular momentum transforms into a jetting impulse. The rider synergistically leverages this push for speed by synchronizing carving impulses with it.

Citizen scientist An individual who voluntarily contributes time, effort, and resources toward scientific research in collaboration with professional scientists or alone. These individuals don't necessarily have a formal science background. See What is citizen science? [Citizen Science].

Conservative force A force that can be expressed as the gradient of a potential. When this is possible, the work done by the force is not dependent on the path taken. A "round trip" by different paths of the same length requires the same amount of work. Gravity is a familiar conservative force.

Design Of Experiment (DoE) A type of designed experiment contrived to efficiently identify (screen for) the control factors that affect the process being studied the most (main effects). DoE can model and compare the effects of several factors against each other using Yates analysis. It attempts to optimize the differences in effects and minimize the number of trials needed to obtain them. When models are attained they have least-squared, multilinear properties subject to the confounding structure of the experimental factors. When not attained, the true process model is non-linear.

Direct-Push and Pull, slinging Intentional impulse created as the rider quickly moves his or her center of mass in the opposite direction of the handlebars. Lightening the load on the drive wheel and pushing the Trikke forward with arms and legs feels like "slinging" the Trikke forward. In order to retain the ground gained, this maneuver must be concurrent with or followed by steering, cambering or both. During the push or pull, the mechanism produces a jetting impulse; acceleration for the push, deceleration for the pull.

Dynamics Study involving variables related to the generation of an object's motion. Answers questions about how motion is produced.

Equation of motion (EOM) Application of conservation laws or principles like least action and balance of forces and torques to produce global kinematic and other equations for a system. For example, an approach due to Lagrange starts with energy conservation, equivalences between kinetic and potential to derive system velocities, then other kinematic quantities and sometimes Newton's force-based EOMs can be derived for the system. D'Alembert's principle allows various constraints on part velocities and positions to be incorporated in some EOMs. Other approaches create state-based generalized coordinates and use lie algebras to represent system functions. While many EOMs cannot be solved symbolically, they lend themselves to numerical solutions via computer simulation.

Extended-Stem The part of the TRS that is turned by the rider during castering. Composed of the stem, hands and wrists of the rider's body. Does not include body parts considered part of the trailer. In lever systems, one-third of a linkage between two moving parts can be shown to act as if stationary with respect to the closest attachment. So, one-third of a hand and arm acts as if stationary to the stem, another third to the trailer and a third acts with the arm's own momentum.

Extended-Trikke The part of the TRS that slings during carving. Composed of hands, wrists, feet, ankles and parts of the rider's body that sling with the Trikke. The part of the reduced rider's body producing the jetting and Direct-Push impulse is not part of the Extended-Trikke. In lever systems, one-third of a linkage between two moving parts can be shown to act as if stationary with respect to the closest attachment. So, one-third of a foot and lower leg acts as if stationary to the deck, another third to the rider and a third acts with the lower leg's own momentum.

GCF Global Coordinate Frame, "lab" or "observer's" frame may contain an infinite number of LCFs twisting, moving and even changing size everywhere. The Trikke-Rider System LCF is fairly well behaved inside the GCF; it has a common z-axis and doesn't change scale. Observers can see and measure the TRS as it moves and turns and can all obtain the same values. But they can't feel the accelerations of the Trikke like the rider.

Jetting, hemijet The angular component of carving. Beyond the "natural" body twist accompanying slinging, the rider extends the foot outside the turn by lowering his center of mass, leaning back and twisting the hips into the turn. This angular impulse adds angular momentum to the system via the Parallel Axis Theorem through rotation about the TRS center of mass. The word comes from an outdated skiing technique using both feet. Technically, this is "alternating hemijetting" since each foot independently executes half of a "jet" (a hemijet) in one drive-cycle. However, it is possible to jet with both feet.

Kinematics Study involving variables related to the classification and tracking of an object's movement. Answers questions about the form and characterization of motion, not how the motion is produced.

Lateral plane An imaginary plane that separates the front half of an object from its back half. Its normal is parallel to the front-to-back axis of the object.

LCF Local Coordinate Frame, inertial frame or just local frame is the rider's world where the Trikke and everything on it seems relatively stationary compared to the rest of the world. It is "inertial" because the Trikke and rider "feel" centripetal force in a turn and the accelerations of its movement. Yet the Trikke remains close to the rider. "Forward" is pretty much ahead. There is a wheel always under the left foot. Things are local and for the most part in their expected places. When the rider measures things relative to his locality in the GCF, they are usually different than the same things measure by observers in the GFC.

Magic Lever Any Turn-Lever passing through the front-wheel axle or transom used to think about or compute the jetting reactions of the Trikke. Really, any turn-lever will do as they all move as if embedded in a solid, but the two are the simplest to use in mathematical models.

MA, Mechanical Advantage A ratio, percentage or number expressing the force multiplier or efficiency of a lever system. For simple levers, it is the distance of the action to the fulcrum divided by the distance of the load to the fulcrum.

Nonholonomic system (also anholonomic) In physics and mathematics a nonholonomic system is a type of system that ends up in a different state depending on the "path" it takes. In the case of the Trikke, frictional forces and some geometric constraints restricting velocity but not position prevent the system from being represented by a conservative potential function. It is "non-integrable" and not likely to have a closed-form solution.

Normal A "normal" to a plane, is a ray starting at the plane which is at right angles to every ray in the plane that starts at the intersection point. Notice that the normal cannot lie in the plane. Its unit direction is also the "direction" or "orientation" of the plane.

Parallel Axis Theorem One of two theorems by this name. Translates a moment of inertia (moi) tied to a center of mass and rotation axis to an offset but parallel axis. If the moi is expressed as a tensor with no particular rotation axis, then it translates to an arbitrary point, usually on the object. The second namesake does the same for an angular momentum vector. There is also a theorem called the "Second Parallel Axis Theorem" which is simpler to apply in some contexts.

Reduced rider About fifteen percent of the rider's body mass acts as if it is a part of the Trikke. Hands and wrists move largely in unison with respect to the handlebars. Corresponding parts of the legs act as if part of the decks on the arms of the Trikke. This "reduces" the effective mass and distribution of the rider, while increasing that of the Extended-Trikke and Extended-Stem.

Sagittal plane An imaginary plane that separates the left half of an object from its right half. Its normal is parallel to the left-right axis of the object.

Steering column, stem The controlling mechanism surrounding the steering axis. Composed of the steering column, handlebars, rider's hands and parts of his forearms (about 1/3) as well as the yoke, front wheel, axle, bearings and attachments. It is considered part of the Extended-Trikke, but is the part of the TRS not included in the trailer. It is also a lever; the Camber-Lever.

Stride A body in motion has linear and rotational (or angular) velocity. Parts of the body have different velocities depending on their distance from the rotation center. Stride is the difference between a part's velocity and the linear velocity of the whole body at its mass center. Rotational velocity can be expressed as a linear velocity, v̅, at right angles to a radius pointing from the rotation center: (v̅ = ω̅ × r̅). When the rotation center is the turn-center of the TRS, and the radius ends at a wheel, the difference between v̅ and the TRS linear velocity is that wheel's stride.

Swath As a Trikke snakes its way along, the outer edges of the wheels trace a wide ribbon-like path down the road. At terminal velocity, if one connects the outer most edges of the wavy ribbon on both sides to get a long rectangle, the width of that outlined area is the "swath" covered by the Trikke. In simpler terms, it is the width of the path covered by the Trikke. If something is within the swath of the Trikke, there's a decent chance it will be hit depending on Trikke's trajectory.

Trailer The parts of a TRS that are rotated around the transom point and pulled or pushed slightly by the Trailer-Lever while castering. Consists of the parts of a Trikke other than the steering column, front drive wheel, handlebars, "handlebar-stationary" hands and parts of the forearm (about a third of it).

Trailer-Lever Caster action is applied at the yoke by the Yoke-Lever which rotates it and pulls or pushes it. The center of mass of the trailer is its load. Its fulcrum is on the Transom point. The Trailer-Lever is a second-class lever and push-rod.

Transom Used to indicate the points on the line between the rear wheel contact points with the ground or axles. A point along the transom instantaneously acts as the Trailer-Lever fulcrum. Its position is determined by the ratio of instantaneous friction at each wheel. Thus, when the load is nearer the left wheel, the transom point, is nearer the left wheel. For equal friction on both wheels, it is in the center of the line between contacts.

TRS Trikke-Rider System. Refers to the entirety of the physical system composed of the Trikke, all of its relevant parts and behaviors and the rider with all relevant parts and behaviors needed to complete the current investigation. Though the road, air and other parts of the environment are required to operate the system, they are not considered part of it.

Turn-Center Arc As a part of the Trikke turns around the turn-center it traces a circular arc or orbit at a constant radius until the steering angle is changed or a wheel slips. Note, each part has its own orbit, closer to or further from the turn-center. Those further out travel around at a faster speed because their orbits are larger. Consider also that parts of the rider turning with the Trikke have their own turn-center arcs concentric with the parts of the Trikke.

Turn-Center A Trikke orbits this stationary point at a constant radius until the steering angle is changed or a wheel slips. When steering straight, the turn-center is not defined and the turn-radius is considered infinite. Jetting makes use of the turn-ray as a second-class lever acting on the system center of mass.

Turn-Lever A turn-ray is the ray from the turn-center to a point on the Trikke. When the point is the guide-wheel contact point or on the transom, the ray becomes an important motive lever as all TRS motion must move the contact point around the turn-center.

Yoke Idealized point of intersection for the three structural tubes that characterize a Trikke. This genius articulation is fairly complex and precisely manufactured. It is the soul of the Trikke - if there is one. Constant motion from steering, cambering and road vibration are robustly endured, while keeping all the wheels aligned without toe-under or splaying.

Yoke-Lever Constitutes part of the castering mechanism set in motion by turning the stem. The rotation produces an action at the yoke, which is also the load. Its fulcrum is the guide-wheel contact patch. The movement of this yoke-load acts on the Trailer-Lever which rotates it and pulls or pushes it. The Yoke-Lever is a degenerate lever.



[Trikke Patent] J. Gildo Beleski Jr. Cambering Vehicle and Mechanism. US Patent 6,220,612 B1. Dec 20, 2005. Google Patents US6976687B2. Prior: 1999 US09434371 Active, 2000 US09708028 Active, 2002 US10331558 Active, 2004-06-25 US10876497 Expired - Fee Related, 2004-11-18 US20040227318A1 Application, 2005-12-20 US6976687B2 Grant.

Control Theory in Robotics

[RoboTrikke] Sachin Chitta, Peng Cheng, E. Frazzoli, V. Kumar. "RoboTrikke: A Novel Undulatory Locomotion System". In Proc. IEEE Int. Conf. Robotics and Automation, pages 1597-1602, Barcelona, Spain, April 2005. DOI: 10.1109/ROBOT.2005.1570342.

[Roller-Racer] P. S. Krishnaprasad and D. P. Tsakiris. "Oscillations, SE(2)-Snakes and Motion Control: A Study of the Roller Racer." Technical report, Center for Dynamics and Control of Smart Structures (CDCSS), University of Maryland, College Park, 1998.

[Roller-Walker] Gen Endo and Shigeo Hirose. "Study on Roller-Walker: Multimode steering control and self-contained locomotion". In Proc. IEEE Int. Conf. Robotics and Automation, pages 2808-2814, San Francisco, April 2000. DOI: 10.1109/ROBOT.2000.846453


[CarveABike] Sachin Chitta, V. Kumar. "Biking Without Pedaling". November 2006. Chitta, Sachin. “Biking Without Pedaling.” (2006).

[Air Force] Charles E. Clauser, et al., Air Force Systems Command, Wright-Patterson Air Force Base, Ohio. August 1969. Pub AD-710 622.

[Moment Of Inertia] Dr. J. B. Tatum. Chapter 2, 'Moment Of Inertia', section 2.8. University of Victoria. 2017., 2019.

[physics] Michael Lastufka, "Body-Powered Trikke Physics" June 2020.

[calibrated] Michael Lastufka, "Empirical 2nd Degree Friction Model Solution" June 2020.

[simulation] Michael Lastufka, "Dynamic Model of a Trikke T78 Air" June 2020.

[behaviors] Michael Lastufka, "Survey of Simulated Trikke Behaviors" June 2020.

[Quick Start] Michael Lastufka, "Trikke Magic: Start Carving" June 2022.

[co-rotation] Michael Lastufka, "Qualitative Body-Powered Trikke Physics" November 2022.

[George Box] Box, George E. P. (1976), "Science and Statistics" (PDF), Journal of the American Statistical Association, 71: 791–799, DOI: 10.1080/01621459.1976.10480949.

[Nonstandard Analysis] MathWorld: Nonstandard Analysis 2019.

[POV-Ray], 2019.

[], 2019.

[EzyRoller] EzyRoller LLC. 22588 Scenic Loop Rd., San Antonio TX 78255 - USA., 2019. Email:

[Flicker] Yvolution USA Inc. 2200 Amapola Court, Suite: 201, Torrance, CA 90501 – USA., 2019. Email:

[PlasmaCar] PlaSmart Inc. 228-30 Colonnade Road, Nepean, Ontario K2E 7J6 - Canada., 2019. Customer Service 1-877-289-0730 Ext. 214

[Citizen Science] SciStarter., What is citizen science?, 2019. Email: