Body-Powered Trikke Physics

Draft: 23 November 2022

At first sight a body-powered Trikke appears to possess no mechanism to propel it. No pedals, gears, chains or motion transduction equipment. Three ingeniously connected tubes form two trailing arms terminated over wheels and a tall steering column, called the "stem", articulate at the "yoke" to camber all three wheels at once. The caster-mounted wheel beneath the stem guides the vehicle under power of the rider's motive gyrations. Riders grip the stem handlebars standing with feet on their respective arm decks. Four levers lay hidden among this conceptually simple geometry setting three mechanisms in motion that power the Trikke. First principles guide this study of Trikke locomotion through the derivation of mathematical models for these motion-producing mechanisms. Simple physics shows how riders' angular momentum cranks a virtual class II lever that forms inside a turn to one side, then flips to the other while steering the Trikke's signature sinusoidal path. This provides two opportunities to drive forward every cycle. A turntable on a spinning base best summarizes the mechanism. Drawing it all together, a "simple" equation of motion and two concept demonstration "toys" complete the analysis. These models of local dynamics highlight the Trikke's synergistic design to understand what makes a Trikke move.

Trikke Levers and Mechanisms of Motion

Trikkes coast when not turning. To ride a Trikke is to constantly swivel the handlebars from one side to the other while coordinating the various activities that provide power to the system. Mechanisms that drive mechanical propulsion generally boil down to actions that move levers and push or pull rods. Levers, gears (which are notched, curved levers) and wheels all rotate. There are very few of these elements on a Trikke and fewer linkages between them. Some of the linkages identified are real. Others arise from real system components, like the road, in more conceptual ways.

Two primary types of actions, linear and rotational, propagate motion through mechanical systems. System analysts may describe these actions using "generalized coordinates" that guide forces and torques that shape and obey Equations of Motion, their dynamics and constraints. Solving these equations for energies, local accelerations, velocities, positions of parts, performance envelopes and global kinematics provide a deep understanding only visible in the details. This approach works best when the local dynamics lead seamlessly to global kinematics.

However, the simple mechanics of a Trikke provide formidable obstacles to such an approach, including being a nonholonomic system. Setting aside that machinery, this first principles study derives the local dynamics largely from momentum conservation. As the masses of Trikke components don't change significantly, the word "action" used below often means "momentum" or "velocity" rather than "energy" or "force" and "torque". The goal here is not to provide a global solution for the kinematics by solving equations of motion, but to elucidate the local dynamics that more scholarly approaches like [RoboTrikke] did not.

Organization: This report includes general terminology needed throughout, followed by an introduction of the moving-part "actors". Levers are illustrated and the mechanisms they animate are briefly described at an "executive" level. Physical concepts and some vector analysis required to understand and represent the mechanisms more precisely follow for the benefit of motivated readers. Conceptual derivations of the mechanism models provide the main body of the paper. Most of the interesting and useful insights are in these sections. I tried to make them accessible at some level to readers without a great amount of familiarity with physics or analysis. After conclusions, the appendices contain mathematical proofs, useful physical concepts and more detailed derivations of the physics models. The results of this analysis have been implemented in a forward-feed, discrete, [calibrated] Trikke/Rider [simulation] demonstrating typical Trikke [behaviors] that provide a basis for Trikke [magic].

Terms for Turning

Turns and words related to them form the context of Trikke motion; see the top view illustration in Figure 1. Turns are initiated by rotating the handlebars to the right or left around the steering column. The entire assembly that rotates and the casing that does not make up the stem (blue in diagram). Cambering the stem from side-to-side is not turning and does not result in a turn. It results from leaning the stem to the left or right.

The stem's wheel or guide-wheel (black object near top-left) touches the road on its contact patch, thought of as a point. "Guide" denotes best what the wheel does. Note, it is not a "drive" wheel; it's mainly passive, providing little forward impulse. This term distinguishes it from the trailing-wheels (other black objects) of the left and right arms.

The point around which the Trikke turns or orbits is its turn-center (black dot at bottom). When the Trikke turns left, it is to the left of the Trikke's forward axis (the x-axis along the center-line). Distance measured between the turn-center and a point on the Trikke is that point's turn-radius. A turn-radius is the length of the turn-ray from the turn-center pointing to the referenced point. To find the turn-center, locate the intersection of the line normal (black dashes) to the guide-wheel face at the contact point and the gray line between the trailing wheel contact points, the base-line. A Trikke traveling a straight line has an infinite turn-radius / turn-ray with undefined orientation. As a Trikke moves, its turn-center changes from moment-to-moment; it is instantaneous.

Inside the turn is toward the turn-center from the Trikke center-line, while outside is away from it. At a right angle to a turn-radius, pointing forward at a point on the Trikke is that point's path ray. It is always considered to be a unit vector; it has length 1. The purple ray near the top-left illustrates the guide-wheel contact point path ray.

Every model lives in some "domain". This one inhabits a 3-D mathematical world of local coordinates denoted by the basis vectors <X̑, Ŷ, Z̑>.

• x-axis, X̑: The forward direction of the Trikke is +x; parallel to the center-line
• y-axis, Ŷ: The left side is +y.
• z-axis: Z̑: Up is +z.

A turn to the left increases the steering angle θ (theta). Zero θ points straight ahead toward +x. Cambering the handlebars to the left decreases the camber angle γ (gamma) to negative values; its zero points straight up. These conventions maintain a right-handed coordinate frame when used with the usual vector operations.

Trikke Actors that Push and Pull

Conceptually, three assemblies, each with one wheel, coordinate the motion of a Trikke. They are controlled by a forth, the rider. Together these constitute the Trikke-Rider System (TRS). Two assemblies, the left and right arms, create a platform for the rider. They articulate with each other at the yoke via a cambering mechanism controlled by the third assembly, the stem. Moving the stem across the forward axis causes all three wheels to camber or lean from vertical in the direction of the stem. This action is called cambering. Two push/pull pairs of these assemblies implement the actors in the Trikke motive mechanisms.

In the analysis, some parts of the rider's body are considered to be part of the stem and the decks of the Trikke. For example, hands on the handlebars move with the handle bars, not the rider. Up to 15% of the rider's mass is considered part of the Trikke. This allows inertial effects to be modeled more accurately. In Figure 2 the rider-actor, lacking some mass, is dubbed the Reduced-Rider. Where the context is clear, any construct including the core of the rider is called "rider". The first combination analyzed below, includes most of the rider's body as one actor and the Extended-Trikke including the rest of the rider's mass as the other actor. A second partition of the system in Figure 3 isolates the the Extended-Stem that includes the steering column and forearms of the rider's body as one actor. The trailer includes most of the rider and everything else as the second actor.

 Figure 2: Carving actionsActor 1 Extended-TrikkeActor 2 Reduced-Rider Figure 3: Caster actionsActor 1 Extended-StemActor 2 Trailer

Four Levers

Levers identified in mechanical systems are of three classes based on how their fulcrum, load and action are placed. "Linkages" connect their fulcrums, loads and actions. They can be physical levers minimally shaped like rods or irregular shapes bearing little resemblance to a rod without a line of material passing through its elements; in a sense "virtual".

One of the four levers involves the physical structure of the steering column with linkages extending from the guide-wheel's contact point to the handlebars. This Camber-Lever is a second-class lever with large mechanical advantage that makes moving the handlebars from side-to-side easy.

The most important lever is not physical in the same sense. Anchored at the instantaneous turn-center and extending to the guide-wheel contact point, the Turn-Lever "virtual" lever pulls the entire Trikke around. The physical linkages to the load of this lever are realized in the structure of the Trikke and extend to the road by the constraints of friction on all three wheels. It is also the longest lever, stretching almost to infinity.

Two more levers operate in tandem to rotate, pull and push the trailer into, toward and away from the guide-wheel contact. One is degenerate with action at the intersection of all three tubes, the yoke. This Yoke-Lever, has its fulcrum at the guide-wheel contact with load and action at the yoke. Its mate, the Trailer-Lever, extends from its action at the yoke, to the instantaneous transom point. Mostly, it rotates its load - the center of mass of the trailer.

Figure 4 below illustrates the four Trikke levers. Movement of the dark brown dot, the TRS center of mass, is the goal of these levers. It is constrained to move in the direction of the dark brown ray. Along the way, the trailer's center of mass (orange dot) may also move.

1. Camber-Lever: As the rider steers and cambers a Trikke, the guide-wheel's contact point (green dot under guide-wheel in Figure 4) forms a fulcrum (blue ^) for the handlebar action (blue double-ray) through the Camber-Lever (blue line near circumscribed '1'). The yoke is the load (blue load). It is a second-class lever. Cambering can move the yoke side-to-side a couple decimeters. Because of the castered guide-wheel's negative rake, merely turning the stem also moves the yoke in orbit around the contact point. Yoke movement is the action behind "Castering" a Trikke. When the guide-wheel cambers, the rear-wheels camber with it to very near the same angle. They are permanently cambered about -5°; bottom further out than top.
2. Turn-Lever: Nothing happens in Trikke locomotion without involving this lever (green line near circumscribed '2'). When the stem is set to a particular turn angle, θ, lines (dashed-gray and solid-green) extended through each axle of the three wheels intersect at a point called the turn-center. As the name implies, a Trikke orbits this point at this turn-radius (green line), as long as the turn angle doesn't change. However, the turn angle constantly changes, so technically, the green line is the "instantaneous guide-wheel contact turn-ray". The turn-center is the fulcrum (green ^). Action occurs at the guide-wheel contact point (green patch under wheel with green ray). The TRS center of mass is the load acted on (displaced light brown dashed-line on the Turn-Lever). It can be a second or third-class lever.
3. Yoke-Lever: Extending from its stem contact fulcrum (light blue ^ under wheel) to the yoke, this degenerate lever (light blue line near circumscribed '3') rotates around the Camber-Lever. Due to the rotation, it acts (light blue ray) on the yoke, which is also its load (light blue load). It constitutes one of two levers involved with the castering mechanism.
4. Trailer-Lever: The trailer along with this lever (orange line near circumscribed '4') constitutes the second part of the castering mechanism. Action (orange ray) rotates, pushes and pulls this lever at the yoke. It is anchored at a transom fulcrum (orange ^) which causes it to move its load, the trailer's center of mass. It is a second-class lever and push-rod. The trailer includes the Trikke arms with their wheels and the rider minus body parts that are kinetically part of the handlebars - like hands and wrists. The trailer can be thought of as a distribution of sub-masses with two road contact points.

Three Mechanisms

Three mechanisms Jetting, Carving and Castering combine synergistically via the four levers to propel three-wheeled, body-powered Trikke riders' fun. Trikke locomotion is characterized by the serpentine track carved mostly side-to-side across a road as the rider coordinates steering, cambering and carving. These mechanisms allow riders to conjure the forces to snake out speeds over 4.5 meters per second on a level road. Note that "carving" includes "jetting", but jetting is the most fundamental mechanism of Trikke locomotion, so it is often mentioned outside of the carving context.

• Carving Mechanisms

Jetting, Direct-Push and Direct-Pull are the carving mechanisms driven by two of the four Trikke levers. Jetting is the simplest, employing the virtual Turn-Lever. When activating any of these mechanisms, the rider produces an action at the guide-wheel contact point through rotation. The fulcrum of the Turn-Lever pivots on the turn-center causing the TRS center of mass, the load, to orbit the turn-center. Jetting pries the Turn-Lever like a crowbar dislodging a stone. Constrained to orbit and being solidly connected to the stem via human legs and metal bars, the TRS center of mass must move with the rotation of the lever.

1. Jetting: This term comes from an old skiing technique used until the mid 1980's when ski materials improved dramatically. It involved shoving both feet forward to top small ridges or up-sloping turns. Dropping one's center of mass to power the jet made it difficult to maintain balance.

Trikkes are much more stable and the jet is mainly performed one foot at a time - a hemijet. The physical goal of hemijetting (or just jetting) a Trikke is to accelerate the rotation of the outside arm around the Turn-Ray Lever into the guide-wheel path. When the guide-wheel accelerates, every part of the Trikke is compelled to accelerate with it in proportion to its turn-radius ratio. A part further from the turn-center than the stem orbits faster than the stem. The inside wheel orbits the slowest because it is closest to the turn-center. Model-wise, the guide-wheel pushes the Turn-Ray Lever which pulls the TRS center of mass around the turn-center. This is a consequence of the constraints provided by friction on the wheels.

Unlike linear actions, jetting does not require the rider to "reset" center of mass position; the action is generated without moving the rider's local center of mass. There is NO recoil. When led by a hip twist and a delayed turn to the same side, the rider's angular momentum is first reflected then transferred to the Trikke-Rider pair as co-rotation via The Law of Conservation of Angular Momentum. Pure jetting is a ROTATION. Each of the other mechanisms describe how linear actions between Trikke components also synergistically feed into a jet rotation. A Trikke CANNOT move backward when the rider jets forward, unless slipping down a steep hill. A simple system that embodies momentum transfer through rotation is a turntable on a spinning platform.

However, when turning to the left, jetting by rotating the left side of the body around to the front opposes the turn and reflects the jet into acceleration. The construction of the Trikke makes this difficult to perform from a stop so that a casual rider attempting it as an experiment does not experience much acceleration at all. Some riders may experience this differently and "wind up" before and during a turn, building bodily rotation but not releasing it to the Trikke, like caster does.

2. Direct-Push and Pull

Direct Push-Pull mechanisms feed into jetting, but lose some momentum in the process.

1. Push: Assuming a Trikke is steered straight ahead, pushing it forward by recoiling the rider backward does not change the TRS center of mass location. The center of mass is stationary throughout the motion. However, the rider and Trikke centers of mass do shift position in opposite directions to conserve momentum.

Constrained on an arch as a Trikke turns, the pushing rider and pushed Extended-Trikke agents generally have different turn-radii. When this is the case, conservation of angular momentum demands that a rider's push cause the TRS system center to move! However, the motion stops when the push stops. Never-the-less, the TRS center of mass takes a "step". During the step, a velocity manifests that is proportional to the ratio of the agents' moments of inertia and the ratio of Turn-Lever length to the TRS turn-radius. Motion ends as the rider and Extended-Trikke continue together as one unit in a perfectly inelastic collision. In the detailed derivation, this angular process is treated linearly via components at the guide-wheel contact point. Never-the-less, the Push orbital component becomes the jet and its radial component grounds into Earth's momentum via friction across the wheels. Angular and linear momentum are conserved.

2. Pull: Pull is the opposite of Push. TRS center of mass location is stationary in a no-turn, linear pull. But like the direct-push constrained on an arch, the TRS center of mass is also not stationary. It takes a "step", but in the opposite (wrong) direction. Again, it originates from the conservation of angular momentum, while it ends in a perfectly inelastic collision. Opposite a push, a backward jetting rotation occurs via the Turn-Lever to motivate the step while the pull is in effect. The beauty of Trikke dynamics, one of many synergies, is that a push and turn to one side resets the rider's center of mass for a push and turn to the other side without needing a strong reset pull! It becomes a weak impulse, or better, no impulse.

• Caster Mechanism

A couple of studies have concluded that a Trikke is propelled forward simply by cyclical steering; see [RoboTrikke] and [Roller Racer]. Personal experience shows this is a weak form of locomotion, especially starting from a stop. Note, it is very difficult to perform this task "pure castering" without throwing weight around or knowing for sure that the ground is level. Off you go when unintended jetting or downslope kicks in! When controlled to allow only castering (no cambering, extra jetting or slope) the guide-wheel contact point grinds in place. To emphasize caster-only locomotion, a rider must plant feet in front of the decks on the arm-bars close to the yoke. That moves the rider's center of mass forward enough to be demonstrably thrown into the turn by the Trailer-Lever; generating a decent jet. In normal practice, the rider is too far back to produce a caster-jet strong enough to initiate locomotion without added jetting or a good push. Furthermore, once moving, castering itself does not sustain speed, but crawls to a stop. However with wide, quick turns, direct-push and jet, castering weakly enhances a moving Trikke.

1. Caster-Pull and Push

1. Pull: When changing turn direction in a drive-cycle, the Yoke-Lever rotates the yoke pulling it closer to the guide-wheel contact point. Negative rake changes the guide-wheel contact point as the stem turns. When straight, the contact point is closest to the yoke and the yoke is highest above the wheel. As it turns outward, the contact point moves into the turn away from the descending yoke. The movement can be a few decimeters when cambering, especially sideways. Caster-Pull happens as the stem turns to the neutral, straight forward position.

Pull occurs mostly sideways into the turn powered by the rider's handlebar twist and cambering. This causes a rotation of the Trailer-Lever. When the rider turns the stem very quickly, or moves forward on the deck platforms, turning resistance can be felt. Angular momentum around the transom point increases as the guide-wheel straightens and then passes on to the next stage of the mechanism - Caster-Push.

Forward pull also occurs in this first half of the turn, but fizzles out as the guide-wheel absorbs it geometrically, merely by slowing its spin. However, the rise in yoke height above the wheel shrinks the base-projection distance to the transom causing the Trailer-Lever to shrink as much as one-percent. But this Direct-Pull is very small compared to the rotation and it dissolves completely when the wheel straightens. It is ignored in this Castering Model.

2. Push: As the guide-wheel passes neutral (straight ahead), the Yoke-Lever lengthens due to negative rake as the contact point moves further into the turn and away from the yoke. This time, the wheel spins up covering the extra distance "shorting out" linear push-back on the yoke. Rotation increases the angular momentum of the trailer around the transom. Again, there is a change in Trailer-Lever length, but it is due to the drop in yoke height. It is too small to matter.

When steering slows down near the end of the turn, the trailer's angular momentum releases as jet impulse in a perfectly inelastic collision until the turn stops. The strength of the impulse depends on how fast and how much the wheel is turned and the distance between the rider's center of mass and the yoke. The closer, the stronger. The jetting impulse produced during the end of the turn adds to the jetting rotation around the Turn-Lever that moves the TRS center of mass. Unlike the Direct Push-Pull mechanisms, castering does not result in a "step" and there is no "reset" needed. Like the Jet, it is a rotation.

Actions Constrained to an Arched Path

Wheels constrain a vehicle to travel a straight path or an arched path. Trikkes decelerate on straight paths, coasting to a stop. On a straight path, forward push accomplished by body recoil, does not move the TRS center of mass position under normal conditions. There is simply no sustainable way to accelerate a Trikke on a straight, flat path. Actions of the rider on an arched path, however, do propel the Trikke.

Each time a rider carves a Trikke away from its starting position, it tacks out a slightly different path, ending up in a different state. This nonholonomic behavior complicates the dynamics of the TRS. The normal development of equations of motion must be enhanced using constraints sustained by the three wheels via D'Alembert's principle. However, it is not necessary to develop complete equations of motion to derive useful quantities for understanding the TRS, but to leverage first principles and lots of diagrams.

The following dynamic motive models are based on momentum conservation. Pushing and pulling rely on the mechanics of linear momentum conserved as two masses separate or approach each other and then "rejoin" in an inelastic collision. This equal and opposite law also applies in angular actions. Rotation necessitates counter rotation. However, many of the counter reactions are synergistically consumed by constraints on the wheels grounding to the Earth's huge moment of inertia allowing only path-aligned actions to run free.

Rotation and Vectors

Vector and other various notations used in this study are described in Table 1 on the right. Distinctions among the different contexts in which vectors are commonly used have been made. For example, the free vector V̅ is decorated differently depending on whether it is being used as a position, bound as a ray, needed as a direction unit only or as a magnitude. Three dimensional vectors are used throughout.

Some aspects of rotation are purely geometric. For example, suppose a linear (aka. path, arch) velocity of a rotation is established at one point p͆ at radius r̚p from its turn-center T͆c. Then the path velocity of another point q͆ on the same rotating object is proportional to that at p͆ via vρp/vρq = rp/rq. Note, in the following "ρ" is used as a symbol indicating "path".

For objects constrained to motion on a path, a pattern that determines the amount of vector action imparted along a divergent path is most useful. The dot product gives the magnitude of the vector v̚ anchored at the path origin that aligns with the path ρ̑; a scalar. Multiplying by the path direction unit ρ̑ then gives the result u̚ as a ray with the direction of the path at the path origin. Note the parentheses in the following equation emphasize the order of operations. In many vector equations order does matter, but hints like parentheses may be omitted because there are other conventions at play.

eqn rv1: u̚ = (v̚ · ρ̑) ρ̑

Some vector patterns used in the derivation of the mathematical models of the physical dynamics of the Trikke are derived in Appendix A. For example, it is often necessary to solve a cross-product for one of its arguments. Cross Product Inverse shows one such derivation. Likewise, how to find the lever-arm of an action to a point of rotation is given in the Lever-Arm derivation. Vector Decomposition shows how lever-arms are important in deducing the motion of objects.

Some well known laws and theorems of Physics are also employed like Newton's Third Law and the Parallel Axis Theorem for moment of inertia.

Jetting Path Action Vectors

Gaining and maintaining top speed on a Trikke requires the mastery of alternate hemijetting (aka. jetting). A "jet" is the simplest and most direct motive mechanism of the three explored in this study. A rider generates a jet impulse in various ways involving the body core, legs and feet and/or shoulders and arms. Pinning the rider's center of mass to the inside of the turn and generating quick body rotation while thrusting a well placed foot on the outside of a turn does the trick. Subtle, well-coordinated arm, body tilting and slumping motions can increase effectiveness. However, upper body or lower body techniques can also be effective on their own. Alternating the techniques makes Trikking a total body fitness machine. Each motive mechanism feeds into the Jetting Model to produce locomotion. Jetting is the most basic motive mechanism of the Trikke and easiest to grasp.

Jetting as a Lever

Figure 5 illustrates the simplicity of "Jetting" to move the center of mass (purple dot) of a rotating system. Constrained to turn around its turn center (black dot) by its wheels, a Trikke rider initiates a rotation action (orange ray) by twisting and pushing into the guide-wheel path (brown arc). The entire Trikke-Rider System must turn at the same angular rate as the guide-wheel. Doing so, each part of the Trikke has a velocity at a right angle to its turn-ray, which originates at the turn-center. The distance between the turn-center and the part is its turn-radius. Divide the part's turn-radius by that of the guide-wheel's contact point to get its portion of the guide-wheel's speed. In the figure, the velocity of the Load (purple ray) is B/A times the speed of the guide-wheel contact, but at a right angle to the purple turn-ray. In this case, B/A is less than one, but when the guide-wheel turns in the other direction, B/A can be briefly greater than one.

Since angular momentum is always conserved, what rotates around the turn-center in the opposite direction of the rider's jet? Since the rider produced it, it's not the rider. Since the rider spins opposite to the turn before turning the other way, [Conservation] requires the Trikke-Rider system to co-rotate forward around the Turn-Center.

This "Magic lever" is a 2nd class lever when B/A < 1 and becomes a 3rd class lever when B/A > 1. It is also a virtual lever since there is no actual rod anchored at the ends of the turn-center and action. The road and its friction with the wheels create it. This paper does not consider how the action is created by the rider. Only that the rider has ginned up the action and overcome all the forces of nature to produce the velocity at the guide-wheel contact point. These include various types of friction and inertia. The rotational inertia of the system is best evaluated at the turn-center via application of the Parallel Axis Theorem to its parts. See Figure 14 for a jetting, carving demonstrator concept; perhaps not unlike what the [RoboTrikke] project attempted. Jetting is described in the context of Trikke dynamics in the next paragraphs.

Some of the local dynamic quantities have a local value and a global one. The local value represents the newly generated quantity during the infinitesimal time-span represented in the diagrams. Some of the local ones combine into their global counterparts. These global quantity symbols are not generally shown, but are understood to be there. For example, the TRS already moves with some global velocity v̚ at its position ©͆ in a snap-shot of time. The local value Δv̚ adds to that, becoming part of the global value. However, the local symbols are not generally prefixed with a 'Δ', since this study does not include equations in the global context except for the equations of motion.

Though the TRS does not turn freely around its center of mass, it is the best place to compute the local change in velocity. Its direction is known by constraint, its orbit. Only its magnitude remains, and that's dictated by geometry once the velocity of the guide-wheel contact point is known.

Two main actors appear in the Jet snap-shot diagram, Figure 6: the Extended-Trikke with center of mass at ©͆E (blue dot) and the reduced rider with center of mass at ©͆R (orange dot). In this section, "Trikke" means Extended-Trikke and "rider" means reduced rider. While purely jetting, the rider's position relative to the Trikke is stationary. Jetting is all about adding rotation ω̚σ through the stem contact point C͆σ (center of guide-wheel) around the turn-center T͆c (black dot). The result of the rider's rotation action on the Trikke is linear motion v̚ (brown ray) along the TRS path unit ray ρ̑ (parallel to v̚). The effect of the input rotation ω̚σ (thick black arrow) depends on the guide-wheel contact turn-radius r̚σ (black dashes), and the specifics of the rider's technique that determines ©͆ and therefore r̚.

To avoid constructing a model of how the rider generates ω̚σ (a future paper) consider the rider's jet input to be the locally generated linear velocity produced at the guide-wheel contact v̚σρ (purple ray). From this linear velocity, the stem contact rotation ω̚σ derives. All parts of the Trikke rotate around the turn-center at this angular velocity at the instant in time framed by the diagram. Its linear component at the TRS center of mass (brown dot) derives according to the simple geometry of the diagram. Note the Trikke's moment of inertia need not be accounted for by this use of v̚σρ. Inertial resistance, the various wheel frictions and air resistance are assumed to have been overcome by the rider's effort in establishing this linear velocity.

Once ω̚σ is known, the turn-ray from the TRS center of mass r̚ (thin brown line) factors into the velocity v̚ (thick brown ray) at the TRS center of mass ©͆ along the path ρ̑ (parallel to v̚). Like every part of the Trikke, ©͆ is constrained to move in orbit of the instantaneous turn-center T͆c with rotation ω̚σ due to the constraining power of the wheels at their contact points C͆σ, C͆l and C͆r. The magnitude of v̚σρ is simply related to v by r/rσ.

Jetting can be reversed when a rotation ω̚σ about the turn-center occurs in the opposite direction. The effects are clearly opposite those of a corresponding forward rotation. Though the "pulls" of the following mechanisms can produce such rotations, they are typically over-powered by the global rotation of the Trikke or nixed by the rider's technique. When there is an effect, it reduces the global rotation about the turn-center.

Summary of Derivation

The symbols involved in the Trikke hemijet model derivation are defined in Appendix B.

The purpose of this model is to find an equation for the locally generated instantaneous velocity of the TRS center of mass, v̚ due to rider jetting.

Given: The guide-wheel contact's linear velocity v̚σρ generated by the reduced rider via jetting actions; The guide-wheel contact C͆σ; TRS turn-center T͆c; turn-rays for the guide-wheel contact r̚σ and TRS center of mass r̚.

Find: The TRS velocity v̚ due to the rider's jetting rotation that results in a linear velocity of the guide-wheel contact path v̚σρ.

Main lever-arm is C͆σ - T͆c.

The local instantaneous velocity of the TRS center of mass, v̚ derived for this Trikke hemijet model is:

eqn jp4: v̚ = (ȓσ × v̚σρ) × r̚/rσ

Pushing and Pulling a Trikke

As mentioned above, direct pushing and pulling on a vehicle traveling a straight line does little to promote acceleration. In some cases, a quick impulse can exceed natural frequency processes (as in starting a swing) or friction work functions (hopping a chair forward while seated) to cause system center of mass motion. Such were not found as sustainable mechanisms for riding a Trikke. These diagrams show that pushing and pulling a Trikke produce system acceleration moderated by the inertial properties and geometry of the actors. Computer model runs show that direct-push can contribute a significant momentum change in a carving cycle.

There are two actors (see Figure 2), the Extended-Trikke which moves at the guide-wheel contact point C͆σ and the reduced rider with center of mass at ©͆R. They are treated as semi-independent entities constrained to their arched paths and subject to the conservation of momentum. Combined, these produce a TRS center of mass displacement or "step". Analysis of this mechanism is almost as simple as jetting.

Linear vs. Angular Push/Pull

In terms of mathematical modeling, a Trikke's secondary mode of propulsion Direct-Push and Pull leverages the difference between linear and angular system center of mass motion; see Figure 7. When a linear push or pull is contained in a system, the center of mass of the system does not change (brown dots in left diagram). This is not so when the reaction of one actor (light blue dot in right diagram) is constrained to follow an arch. It also helps when their masses are quite different. The larger one (orange dot) gives more energy to the smaller one (light blue dot), which then travels farther on a circular path bending back toward the larger one on at least one axis (vertical axis in diagram).

In the linear push/pull diagram on the left, the system center of mass (brown dot) is not affected by the actions of the masses being pushed apart and then pulled together. The rays roughly indicate the direction and magnitude of the actions and the solid red and blue lines indicate the tracks of the masses. Conservation of momentum grants larger effects (action and track) to the smaller mass.

In the right diagram of angular push/pull inertial effects also favor the small mass. However, since the smaller actor must round the arc, it cannot be driven on an opposite path to the more massive body as in the linear case. Rather, it bends back forcing the system center of mass to leave its arc (indicated by green signed Δ) as it balances position between the two masses. Any deviation from its arc is an opportunity to drive other mechanisms. Note: an arc push on a Trikke can only contribute orbitally to the reaction, since radial reactions dissipate in friction and dump residual momentum to the Earth at right angles to the wheels.

Direct-Push Path Action Vectors

Movement of the rider generates a linear velocity v̚R (orange ray in Figure 8). This produces a momentum proportional to the rider's mass mR that must be balanced by the amount transferred via the guide-wheel path ρ̑σ to the Trikke and rider plus that lost radially along ȓσ to friction and gained Earth momentum. Though it will ultimately move, ©͆E (blue dot) is not the receiving focus of the rider's action. Rather, the guide-wheel contact point C͆σ receives the separation action via a sequence of linkages during this snap-shot of time. Speed that would be imparted to ©͆E manifests at C͆σ because it is free to move, but it is constrained to align with its path direction unit vector, ρ̑σ (parallel to v̚σρ).

The force of the push acts equal and opposite to both the Extended-Trikke and rider - Newton's Third Law. However, the force is not typically aligned with the guide-wheel path. Therefore only its component aligned with the path (orbital) affects it, the rest (radial) becomes side-ways drag on the wheels. Stated in terms of momentum, before separation local momentum is zero. After the separation stops, the sum of momenta is still zero. But because the Extended-Trikke is constrained to movement in orbit around T͆c (black dot), it can only receive angular momentum. The Rider is free to move anywhere during the separation. This means the two actors transfer equal and opposite angular momenta by conservation of that quantity, but the rider's linear - off orbit - momentum is dumped radially into friction at the wheels. This radial momentum causes the Earth to gain an opposite amount: first transferring some to molecules touching the wheels which alter physically, perhaps some chemically, and alter speed and rotation; secondly, any amount left over to the change the rotation of the Earth itself. Treating as a force or momentum transfer gives the same result when split into orbital and radial components.

This reasoning also shows that if the push (or pull) action v̚R lies along ȓσ, no rotation is produced but the rider moves backward (or forward) at speed vR with no effect on the TRS velocity. On the other extreme, when v̚R aligns with the path ρ̑σ, only rotation is produced which to first order has the same solution as the conservation of linear momentum with no movement of the TRS center of mass at all during the separation event. What happens between these extremes, is examined next.

When a force acts at an angle to a constrained path, it splits into perpendicular components, one along the path and one across the path. Their vector sum must add up to the original force. Such components are found by dotting the force vector by each path. In the push/pull analysis, the force along the path provides thrust for the orbital momentum while the force across the path increases the path-constraint drag at the wheel. For a real wheel, this sideways force may cause the wheel to slip to the outside of the turn on a push and to the inside on a pull. The model assumes no slip at all.

Finding the change in guide-wheel velocity due to a push or pull is straight-forward. See the result in the summary below.

The rider can only recoil backward so much before ©͆ travels behind the trailing wheel axles. At that point, the Trikke ride may end abruptly with a wheelie. Combined with the drive cycle time, this naturally limits how much push a rider can generate. In normal operation, the rider stops rearward movement to "join" the Extended-Trikke inelastically where the two bodies now behave as if stuck together. All the local momentum of the rider returns to the system stopping further ©͆ push-motion. What motion has occurred remains as a "step" in the displacement of the guide-wheel contact.

Direct-Pull Path Action Vectors

When the jetting action is reversed by a pull, the change in guide-wheel velocity also reverses. If the pull can be avoided, so can its displacement loss. When a turn and pull follows a push, the step achieved by the push is where the turn begins grabbing the pavement. The decelerating step of the pull can be "paid" for by the rider's jetting input. However, good riding techniques exist that avoid or greatly reduce a position "reset" pull due to the synergies of Trikke geometry.

Unconstrained, this mechanism converts pull into deceleration around the turn-radius of the guide-wheel which is the core of the jet mechanism. The Jetting input action is the increment in rear-ward linear velocity of rotation induced by the pull. ©͆R and ©͆E can only move until the rider ends the pull or bumps into the handlebars. They stop inelastically as the arms and legs of the rider hold mR and mE in place at the proximal extreme position. At that moment the masses "fuse", momentarily accelerating to the global TRS rotation.

Summary of Derivation

The symbols involved in the Trikke Direct Push-Pull Model derivation are defined in Appendix B.

The purpose of this model is to find an equation for the locally produced velocity of the guide-wheel contact point, v̚σρ. This can then be used to determine TRS v̚ at ©͆ using the jetting model.

Given: TRS center of mass ©͆R and previous ©͆R' (used to determine the derivatives ∂(*)/∂t); guide-wheel path unit ρ̑σ; rider mass mR; guide-wheel contact C͆σ; TRS mass M, orbit unit ρ̑Trs and turn-center T͆c.

Find: The TRS velocity v̚ via the intermediate linear velocity of the guide-wheel contact path v̚σρ around T͆c and the across wheel drag D̚ due to the push or pull.

Note when v̚R · ρ̑σ > 0, the rider is pushing into the guide-wheel path. When v̚R · ρ̑σ < 0, the rider is pulling.

Main lever-arm: C͆σ - T͆c

Jetting velocity input:

eqn dp6: v̚ρσ = - mR (∂©͆R/∂t · ρ̑σ) ρ̑σ/(M ρ̑Trs · ρ̑σ)

This locally generated velocity is treated as input to the jet model to find its contribution to v̚ at the TRS center of mass.

The rider's recoil in terms of drag or friction at the wheels is

eqn dp7: D̚ = mR ∂((∂©͆R/∂t - (∂©͆R/∂t · ρ̑σ) ρ̑σ)/(M ρ̑Trs · ρ̑σ))/∂t

Caster-Pull/Push Path Actions

So far, the motive mechanisms discussed are directly rider controlled. Castering is an indirect rider effort mediated by physical linkages from the guide-wheel contact point to the yoke, the Yoke-Lever and the Trailer-Lever. Computer model runs suggest caster-pull/push contributions at best attain about a seventh of the momentum that direct-push produces in a normal carving cycle. For the Trikke, it is a relatively complex, but small effect.

Caster-Pull is described first then Caster-Push, which is the mechanism with opposite x-axis orientation; both produce momentum primarily across the guide-wheel path. In a drive-cycle, the natural steering sequence invokes Caster-Pull followed by Push. A little jet action results as the Caster-Push transfers the Pull momentum from the trailer to the Trikke and rider. These actors are different from those involved in direct-push/pull. In castering, they are the Extended-Stem and Trailer of Figure 3.

Caster-Pull Action Vectors

A turn toward the center (brown arrow in Figure 10) pulls the yoke Y͆ into the turn via the Yoke-Lever (blue line) and up to several centimeters closer to the contact point C͆σ, especially with cambering. It also raises the yoke vertically up to five-centimeters. This in turn cranks the Trailer-Lever (green line) into the turn and brings it closer to the twisting guide-wheel contact point C͆σ. This contraction mostly slows the guide-wheel rotation, but it also rotates the trailer's center of mass ©͆t slightly into the turn and forward.

As the yoke rises, the trailer center of mass is drawn closer to the yoke by triangle geometry; the base decreases as the hypotenuse tips up; a Direct-Push. This contraction amounts to less than one-percent of the base-distance from yoke to transom T͆p. It is ignored in this simple model.

What cannot be ignored is the yoke being pulled into the turn which causes the trailer to gain angular momentum around the transom. This momentum does not come from nor detract from the TRS. Driven by the rider, both the stem's local increase in angular momentum around the contact point and the trailer's are generated by turning the handlebars. Rotation of the Trailer-Lever (green arrow) moves the trailer's center of mass ©͆t into the turn accounting for most of the momentum change. Note the trailer rotates around a point between the rear wheels closer to the one with greater instantaneous friction - the transom point. This angular momentum continues to build independent of the TRS until the stem turns past the forward direction and the turn slows.

Caster-Push Path Action Vectors

Continuing the turn past center, as illustrated in Figure 11, pushes the guide-wheel contact point C͆σ and yoke Y͆ apart via the Yoke-Lever (blue line) and dips the yoke lower. The tire increases its spin. Especially when the tires have a tread pattern, the rider can hear the guide-wheel slow down with the pull, then speed up with the push. This action continues to crank the Trailer-Lever (green line) around the transom T͆p and further into the turn, pushing it slightly backward as the yoke descends. This very slight Direct-Push is ignored in this model.

The trailer's angular momentum may continue to build as the trailer's center of mass ©͆t continues to move. As the turn completes, the movement begins to slow down and its momentum is released through a jet (v̚σρ) down the guide-wheel path in a perfectly inelastic collision. If the rider's center of mass approaches the handlebars, trailer momentum increases but the handlebars become more difficult to turn. When poised further back near the transom, trailer momentum contributes little, but turning is easy. The rider is also closer to and more aligned with the Trikke's minimum rotational inertia axis. Notice the lever-arm L in Figure 11. The longer it is, the more momentum feeds into the jet. When the wheel is straight, L has zero-length and no momentum is transferred. In summary, as the guide-wheel turns out, the lever-arm lengthens and momentum "bleeds" into the jet until the steering rotation stops.

Summary of Derivation

The symbols involved in the Trikke caster model derivation are defined in Appendix B.

The purpose of this model is to find equations for the contribution to guide-wheel contact point velocity due to Caster-Push. This velocity represents a jet that feeds into the Jetting Model.

Given: The wheel contacts C͆σ, C͆l, C͆r, the absolute value of their path friction Fl, Fr and the center-line length; local yoke position Y͆; guide-wheel contact C͆σ' C͆σ", transom point T̑p', its rotation ω̚Tp' and time duration Δt' from the previous time-step; The guide-wheel path unit ρ̑σ; TRS orbit unit ρ̑Trs, mass M, trailer center of mass position ©͆t, moment of inertia It, mass mt and time duration Δt of the frame.

Find: The TRS velocity v̚ via the intermediate linear velocity of the guide-wheel contact path v̚σρ using the Trailer-Lever rotation ω̚Tp and angular momentum of the trailer around the transom point T͆p.

Note when ∂C͆σ(t)/∂t × T͆p̑ - ω̚Tp' > 0, the rider builds trailer rotation via turning toward or somewhat past straight. When negative, the trailer is transferring momentum to the TRS via the jet whose equation renders below.

Main lever-arms: C͆σ - Y͆, Y͆ - T͆p

Castering rotates the trailer around its transom point located in the local frame at:

eqn cp1 and cp2 combined: T͆p = <-║center-line║, C͆ly - (C͆l - C͆r)y Fr/(Fl + Fr), 0>

During the Caster-Push, the momentum from moving the yoke results in a jet velocity v̚σρ.

eqn cp10: v̚σρ = -(It + mt (T̑p - ©͆t)2) ((T̑p × (C͆σ' - C͆σ)/(Tp Δt)) - (T̑p' × (C͆σ" - C͆σ')/(Tp' Δt'))) × (T͆p - C͆σ) × ρ̑σ × ρ̑σ/(M ρ̑Trs · ρ̑σ)

This locally generated velocity is treated as input to the jet model to find its contribution to v̚ at the TRS center of mass.

Conclusions

Adding angular momentum, jet, to a turning system predictably makes it turn faster! Utilizing a virtual lever to transduce that jet into system velocity makes riding a Trikke mysteriously magic for many. Direct push, typically a more intuitive mechanism, in a Trikke performs in a manner foreign to most people's experience producing cyclic steps or no effect at all. Rotation on a Platform helps illustrate wind-up in castering and jet and their subsequent release.

Stripping the TRS to first-order components and mechanisms exposes a simple toy many of us made as children when beckoned by a freshly waxed floor or new carpet. A piece of corrugated cardboard or carpet remnant (for the waxed floor) sufficed to slide across the floor. Inevitably, we began "walking" it across. Sliding it from side-to-side by rotating around the "planted" foot as much as possible. To this effect, hips swung to power the moving foot around. Then the other way around moving the other foot, each time shifting the weight to the planted foot. This is essentially a Trikke with no stem and no wheels, the Trikke arms reduced topologically to a rectangle of stiff, bottom-slick material as illustrated in Figure 13. As children, we instinctively understood alternate-hemijetting.

Conceptually, to add push and pull, anchor a light vertical frame to the slick rectangle. Shoving it forward and jetting a foot to separate the centers of mass (apparatus and child) causes it to slide. Suppose one kept balanced enough, the friction work function could be overcome to reset to the initial state without sliding back. Ready to push again. Though the behavior is similar to a Trikke, the reason the step is "kept" is not the same. The Trikke keeps it mainly because the rider can use the Trikke's geometry to avoid the pull.

Note this action is not skiing and it is subtly different from skating; it's "jetting"!

This look at Trikke physics does not include how the rider coordinates jet, push and caster. Each action must be coordinated and can be separately emphasized in rich and varied rides. An experienced rider can also produce a type of carving resonance. Quick cambering with the handlebars acts to restore the central stem position while jetting with both feet together to swing the trailer into the guide-wheel path against the stem's vertical rotational inertia. It is a tangible coordination between the two that feels like something children perform on a swing. Remember executing similar motions, hands grasping the chains while standing on a playground swing to achieve resonance against gravity? When quick enough, this type of resonance can help a rider maintain speed against the resistance of wind and slope or provide a fun, alternate way to ride.

Locally Generated Velocities of a Trikke

Table 2 summarizes the equations derived for the local velocities of Trikke motive dynamics.

eqn jp4 eqn dp6 eqn cp10 v̚ Linear TRS path velocity v̚σρ Linear guide-wheel path velocity supplied by rider Δt Time-duration of a time-step r̚{j} Turn-rays: j in {no-sub=TRS_CM, σ=Stem} ρ̑{j} Path unit ray: j in {σ=Stem, Trs=TRS_CM} It Moment of Inertia around trailer center of mass C͆σ Guide-wheel contact point T̑p Transom point between the rear wheels, based on friction M TRS Mass m{j} Mass: j in {R=Rider, t=trailer} ©͆{j} Center of mass: j in {R=Rider, t=trailer} T̑p Transom point between the rear wheels, based on friction v̚ (ȓσ × v̚σρ) × r̚/rσ Locally generated TRS center of mass velocity v̚σρ - mR (∂©͆R/∂t · ρ̑σ) ρ̑σ/(M ρ̑Trs · ρ̑σ) Direct Push-Pull local linear guide-wheel path velocity v̚σρ -(It + mt (T̑p - ©͆t)2) ((T̑p × (C͆σ' - C͆σ)/(Tp Δt)) - (T̑p' × (C͆σ" - C͆σ')/(Tp' Δt'))) × (T͆p - C͆σ) × ρ̑σ × ρ̑σ/(M ρ̑Trs · ρ̑σ) Caster-Push local linear guide-wheel path velocity

Using the velocities from these results, Newtonian equations of motion can be written. As an exercise, the following simple EOM was constructed using a couple of complicated geometric functions. See Appendix B for variable definitions. Much insight was gained deriving these velocities from first principles. Solving this "simple" equation is another matter, if it can be done. It can be programmed in a discrete simulation. This author has programmed such a simulation that uses all of these velocity results including Direct Push-Pull and Castering and is friction [calibrated]. See [simulation]

The guide-wheel contact path unit ray is expressed using a composition of rotation functions with the signature, Raxis(angle) vector-to-rotate. Note, steer angle θ, camber γ, ȃ is the axle center unit, X̑ the forward direction unit ray of the Trikke toward +x. (parallel to the center-line) and Z̑ is the vertical direction unit ray of the Trikke toward +z (up).

eqn eom1: ρ̑σ(θ, γ) = (RR(γ) σ̑(θ) (R(γ) ȃ)) × Z̑

Load on each wheel was solved by satisfying two equations, one for total load and the other balancing the torques of the wheels around the TRS center of mass. The result was an equation for the load on each wheel. Like the equations for the rotation function, these equations are not unique and purely geometric. Such details are easily obtained.

eqn eom2: Lj(t) = Instantaneous Load at time t due to gravity on the Trikke and Rider portioned to each axle j ϵ {σ, r, l}.

After noting simple models covering a Trikke's inertial and friction characteristics, a TRS center of mass EOM was constructed. It takes into account Rider input, angular rotation terms (TRS and wheels), velocity-based friction at the TRS center and velocity for each wheel as a scalar term times the TRS center path direction vector. It does not include Direct Push-Pull or Castering. It demonstrates the usefulness of at least one of this study's velocity equations; the center of mass path velocity for which the other velocity models have alternative expressions.

eqn eom8: M Ɑ̚ = (M ∂(vσρ r/rσ)/∂t + ∂(vσρ (I r + Iw(rσ + rr + rl)/(2 rσ2)))/∂t + vσρ (vσρ A ρair M μT r/2 + μw (Lσ rσ + Lr rr + Ll rl))/rσ) ((ȓσ × ρ̑σ) × ȓ)̑
Trikke Jetting Concept Demonstrators

Analyzing and modeling the dynamics of a mechanical system is often not enough to convince ourselves that the results are meaningful. Simulation data and graphics may seem realistic, but leave doubts about unknown or hidden flaws. Sometimes a physical demonstrator helps to bridge the credulity gap. In the spirit of [RoboTrikke], I offer these demonstrator concepts.

JettingBot

Jetting manifests when the rider's mass hovers on the inside of the turn near a trailing wheel and a rotation is produced into the guide-wheel path. Figure 14 presents a concept using a side-to-side swinging mass to produce both of those effects.

Note the familiar Trikke T-frame used in this paper's diagrams with two trailing wheels and a central spar (light blue). At the tip of the spar, a pair of limiters (purple) keep the guide-wheel from turning too far to the left and right. The caster-mounted wheel has negative rake so that it trails its pivot (orange dot) on the spar. It is free to turn up to the set limits. At mid spar, a pendular arm and weight pivots (orange shape) side-to-side. The larger dot at the back of the mass indicates it is much more massive than the small pivot in the front. Perhaps the motor and cam/gears can be part of this larger swinging mass. Success likely depends on the mass being greater than the rest of the demonstrator. No spring, limit switch, cam or gear design is offered here. That exercise is left to the reader. No batteries included. Your mileage may vary.

Operationally, the swinging mass accomplishes three goals. 1) It moves the center of mass closer to the inside trailing wheel of the turn. 2) The large mass causes the front wheel to turn to the same side when it abruptly stops. 3) While the mass swings across the frame, the frame's center of mass is pushed toward the inside of the turn continuing rotation into the guide-wheel path. Continuing, the mass reaches its oposite limit and rotation in the opposite direction to the current turn is produced. The entire system begins to rotate in the opposite direction. Each cycle, the guide-wheel turns the other way following the kick of the pendular arm and the center of mass moves into the turn due to angular acceleration of the mass-arm. When enough of the system's rotational inertia is engaged, the demonstrator should scoot along the guide-wheel's path in a zig-zag "carving" fashion. One digital robot configuration worked like this; it wasn't the best, but was simple.

RowBot

Figure 15 illustrates a sculling demonstrator concept that seems similar to carving. It's essentially a windshield wiper on wheels. In the rear over a pair of stabilizing wheels, a motor and gear box (black circle) produce the oscillating motion of the forward spar. At the tip of the spar, a pair of limiters (purple) keep the guide-wheel from turning too far to the left and right. The caster-mounted wheel has negative rake so that it trails its pivot (orange dot) on the spar. It is free to turn up to the set limits. The back-end must have more mass than the spar assembly to assure guide-wheel driven locomotion. This might be accomplished by housing batteries near the motor. The front wheel has to have enough weight on it to ensure traction.

In operation, the guide-wheel trails the side-ways motion of the spar, but only to the set limits. Setting the limits so the wheel turns almost perpendicular to the spar makes forward travel easier, but slower. This allows the push on the wheel to one side, but also guides some of that push into forward motion. This drive mechanism is fundamentally different than those of the Trikke. It simulates "sculling at the front of a boat" to pull forward.

Rotation on a Platform

Example of Rotation on a Platform

Jetting and Castering have an underlying similarity. They are both special cases of rotation on a platform. Imagine a stationary Trikke with a large turntable spanning the deck platforms. Its top disk has the same moment of inertia as the Trikke, when spun around the same axis. The Trikke's guide-wheel is neutral and the turntable is not spinning. This system has no angular momentum and no linear momentum. It is on a frictionless surface, perhaps ice.

Turn it on. Conservation of angular momentum implies that the platform gains an equal but opposite amount of angular momentum as the turntable top so that the system total remains zero throughout the spin-up. The turntable mechanism overcomes twice the inertia as when on a stationary platform, so by its relationship to energy, each agent maxes out at 1/\[2] the RPM, say 7.07 RPM. The system gained energy. Note momentum transfers throughout the spin up and ends when "terminal" spin has been reached. Now, the turntable supplies only enough energy to the system to match the energy dissipated by friction.

Turn it off. Friction in the internal workings of the turntable slow both spinning components equally until they stop. There is no way to construct a friction mechanism for the top and another for the platform; there is only one and it is equally shared. Both agents spindown at the same rate. All the energy gained was lost in work done by friction likely at journal or bearing interfaces.

Now, more realistically, put the Trikke with its turntable on a paved road!

Turn the turntable on. The top of the turntable begins to turn, slowly building up spin to 10 rpm. The system has gained angular momentum, it is no longer zero! Was angular momentum NOT conserved? Why did the platform not spin? Friction at the wheel contact points prevented platform spin because the wheels did not slip. Moreover, because the contact points did not move, no work was done to consume the turntable's energy. All of the energy went into spinning only the top. The system gained the same amount of energy as on ice. However, angular momentum WAS conserved; the huge moment of inertia of the planet caused it to spin up an infinitesimal amount transferring almost no energy to it. The ratio of energy imparted to the top vs the Earth is Earth's moment of inertia vs the turntable top's moment of inertia; an inverse relation. No momentum was transferred to the Trikke, only the Earth. That's why we can ignore the Earth energy-wise in these situations and not worry that angular momentum is not conserved.

When the turntable is turned off, the friction in the device does work that dissipates all the energy and it comes to a stop. No momentum is transferred to the Trikke, only the Earth.

Now, turn the guide-wheel to 45-degrees opposite the spin of the top.

Turn the turntable on. Torque created by the reaction to the spin-up acts on the two rear wheels equally in opposite directions and does not move because they must follow their arched paths set by the guide-wheel. However torque action on the guide-wheel is now unbalanced. It pushes the guide-wheel forward with a linear velocity that depends on the torque's lever-arm length. When the lever-arm's length is about the same as the radius of the turntable, its acceleration matches the linear acceleration of the turntable top's edge. With such a push, the Trikke circles, but at a lower angular speed because its turn-radius is much larger than the turntable's. The Trikke accelerates until the top has reached its maximum rotation. Then the Trikke coasts while the top spins, no more input. That's one way to jet a Trikke!

Turn the turntable off. Again there is reaction torque as the top spins down. The two rear wheels are still equally torqued in opposite directions and do not move. Torque action on the guide-wheel still acts through the same lever-arm, but in the opposite direction. The Trikke accelerates backward - it slows down faster than normal coasting and may even roll backward at the end of motion when the turntable stops.

Finally, turn the guide-wheel to 45-degrees in the same direction as the spin of the top. Spinning up the turntable reverses the previous thought experiment. The Trikke accelerates backward into a loop then coasts when the spin-up is complete. Turning it off accelerates it forward causing it to come to a stop and maybe even roll forward a bit until the turntable stops, depending on how much it was allowed to coast.

The moral of these stories is that accelerating rotation, not continuous spin, causes a jet action and only when the guide-wheel is turned. It is also possible to jet backward. The ratio of moments of inertia in a rotation system determines how energy is split between the actors. Lever-arms and friction determine the linear speeds achievable with that energy. Jetting a Trikke means always turning the turntable on, steering the opposite direction and turning it on again - riders never give it a chance to turn off! In castering, momentum transfers into a jet on the "slow-down side" as geometry turns the castering spin off.

Vector Decomposition

Derivation of Vector Decomposition

The fundamental concepts underlying the derivations of Trikke motive mechanisms demonstrates the concept that an action at the surface of a rigid body produces linear action on a ray to or from the center of mass and rotation around it.

A point mass with velocity has momentum, p̚ = m v̚. The velocity is attributed to its instantaneous location and points in the direction of travel. Real objects are not points, but it is useful to think of them as point masses in some contexts. Mathematically, this point mass representation is located at the object's "center of mass", ©͆.

When an object receives a push A (thick light orange ray in Figure Av1) on its surface, the event provides a linear action a (light orange ray) in the direction of ©͆ (brown) and a rotating action ω̚ (light orange arrow) around it. The amount of each depends on the skewness of the interaction with respect to the center of mass ©͆ via its contact ray R̚ (purple and turquoise rays are examples). A push directly toward ©͆, aligned or anti-aligned with the contact ray, produces only a linear action on its "point mass" model. Tangential pushes, perpendicular to the contact ray, result only in rotation of the "point mass" model. It is also possible for an action to pull the object.

Generally, the linear resultant is a̚ = A̚ · (©͆ - A͆)/║©͆ - A͆║, and the rotational one is a̚θ = A̚ - A̚l, which is at a right angle to a̚ in the plane that contains the points ©͆, A͆ and the ray A̚. When the action A̚ is a velocity, (could be a force, momentum, etc.) the linear momentum imparted to m is Δp̚ = m a̚, where p̚ represents the current momentum ray and Δ indicates the "delta" (imparted action) being examined since the point mass might already be moving and spinning. To find the imparted rotation, identify the lever-arm through which a̚θ acts. In this illustration it is: L̚ = (©͆ - A͆) × Ȃ × Ȃ (light gray ray for push, light blue ray for pull), which points away from the rotation center ©͆. The action turns the lever-arm crank against the moment of inertia, I, of the object giving it a rotation of ω̚ in a direction normal to the afore mentioned plane.

When A̚ is a velocity, it is thought of as the actual measured velocity of the interaction. Any resistance intrinsic to the motion (like friction, inertia, etc.) is assumed to have been overcome to achieve the velocity. It is the linear velocity of the rotation around ©͆; a̚θ = ω̚© × L̚ ©. Take the inverse of the cross product to solve for the rotation of the point mass; ω̚© = (L̚© × ȃθ)/a©. The angular momentum gained by the object is L̚© = I ω̚©. I is assumed to be the moment of inertia of the spin axis Is; a positive real number. When using a moment of inertia tensor instead, Is = ║It s̑ I║ where s̑ is the unit spin axis vector, It is the transpose of matrix I and ║║ is the L2 norm using the dot product on the vector argument. Decomposition of vectors into linear and rotational components is fundamental to many analyses.

While the light orange action A is a "push", the orange one, is a "pull". Note how the dot product makes the linear action at the center of mass "pull" it toward the action contact point.

Cross Product Inverse

Derivation of Cross Product Inverse

http://www.euclideanspace.com/maths/algebra/vectors/vecAlgebra/inverse/index.htm

Inverse of cross multiplication using triple product identity

Let A̅ B̅ C̅ be free vectors, × the cross multiplication symbol and · the dot product symbol.

Given C̅ = B̅ × A̅ solve for B̅.

eq q1: A̅ × C̅ = A̅ × (B̅ × A̅)

Using the vector triple product identity gives

eq q2: A̅ × C̅ = B̅ (A̅ · A̅) - A̅ (A̅ · B̅)

Rearrange with B̅ (A̅ · A̅) on the LHS

eq q3: B̅ (A̅ · A̅) = A̅ × C̅ + A̅ (A̅ · B̅)

Note the dot product: A̅ · A̅ = A2 (for example if A̅ has dimension 3 then A̅ · A̅ = ax2 + ay2 + az2)

eq q4: B̅ = (A̅ × C̅ + A̅ (A̅ · B̅))/A2

Can't quite eliminate B̅ on the right hand side unless A̅ and B̅ are mutually perpendicular in which case A̅ · B̅ = 0 so that

If C̅ = B̅ × A̅ with B̅ and A̅ mutually perpendicular then

eq q5: B̅ = (A̅ × C̅)/A2

A more useful form of this result is:

eq q5: B̅ = (Ȃ × C̅)/A

where A̅ has been separated into its magnitude A and direction (or unit) Ȃ.

Lever-Arm

Derivation of Lever-Arm

Let A̚ be an action ray, C͆ a rotation center not on the line containing A̚, × the cross multiplication symbol and · the dot product symbol. Ȃ is the unit vector (direction) of A̚ and A͆ is some point on A̚.

Find L̚ the lever-arm from C͆ perpendicular to Ȃ. This is the arm through which the action A̚ rotates about C such that A̚ = ω̚ × L̚ and so, ω̚ = (L̑ × A̚)/L by inverting the cross product. Likewise, A̚ = -L̚ × ω̚ => L̚ = -(ω̑ × A̚)/ω.

What if ω̚ is not known?

Let C͆ - A͆ and A̚ lie in a plane with normal,

eq a1: n̚ = (C͆ - A͆) × Ȃ

The cross-product of this normal and the action ray unit is perpendicular to the action ray in the plane. Its length is the shortest distance between C͆ and the line containing A̚.

eq a2: n = ║(C͆ - A͆)║ sin θ for θ the acute angle between the vectors.

Since n̚ and A̚ are perpendicular, n̚ × Ȃ has the same magnitude as n̚. Therefore this part of the lever-arm equation serves only to place the lever-arm properly oriented in the plane.

║(C͆ - A͆)║ sin θ is the shortest distance between C͆ and the line containing A̚ by the properties of a right-triangle and definition of the sin() function.

Therefore:

eq a3: L̚ = (C͆ - A͆) × Ȃ × Ȃ

Types of Levers

Each type of lever has a fulcrum (green anvil) about which the lever rotates, an action (blue ray) that moves the lever and a load (orange ray) that is moved by the lever. From the locations of these features on the lever, three types of levers are defined based on their "Mechanical advantage", see Figure ma1.

When two of the elements of a lever operate at the same position, the lever is degenerate and may not do any work. For example, a Class I or II lever with fulcrum and load at one end and action at the other rotates without moving the load. Similarly, a Class II or III lever with fulcrum at one end and load co-located with its action at the other may rotate and do work.

Table 1: Non-Degenerate Mechanical Lever Classes
Class ILoadFulcrumAction0 < (Other End - Interior)/(Interior - End) < ∞
Class IIFulcrumLoadAction(Other End - End)/(Interior - End) > 1
Class IIIFulcrumActionLoad(Interior - End)/(Other End - End) < 1

Appendix B: Model Derivations

Trikke Model Symbols
• A Effective frontal area of TRS for air resistance (m2).
• Ɑ̚ Instantaneous TRS acceleration ray (m/s2).
• i Contact point for each wheel; i ϵ {σ,l,r} (m).
• ©͆j Center of mass for assembly j ϵ {E,R,T} (m).
• E Extended-Trikke symbol and index
• ε̚ Instantaneous linear direct push or pull ray (m/s).
• γ Instantaneous cambering angle γ (rad).
• Ij Moment of inertia of subscripted actor; j ϵ {E,R,T,w} (kg m2).
• Ijtc Moment of inertia of subscripted actor around the turn-center; j ϵ {E,R,T,w} (kg m2).
• Li(t) Instantaneous Load at time t due to gravity on the Trikke and Rider portioned to each axle; i ϵ {σ,l,r}
• l Left index
• M Total mass of the Trikke-Rider System (kg).
• mj Mass of subscripted actor j ϵ {E,R,T} (kg).
• ρ̑i Path unit ray for each wheel; i ϵ {σ,l,r} (m).
• ρ̑j Path unit ray for center of mass; j ϵ {E,R,T,t} (m).
• ρair Mass density of dry air at sea-level (kg/m3).
• r Right index
• R Reduced Rider symbol and index
• Instantaneous TRS center of mass turn-radius ray (m).
• i Turn-radius ray to each wheel; i ϵ {σ,l,r} (m).
• j Turn-radius ray to each center of mass; j ϵ {E,R,T} (m).
• Y Yoke-Lever ray from guide-wheel contact to road projection of yoke (m).
• R(α) v̚ Rotation function around vector "axle" a̚ by angle α of vector v̚.
• σ Stem symbol and index
• σ̚t Ray from guide-wheel contact to road projection of trailer center of mass (m).
• σ̚Y Yoke-Lever ray from guide-wheel contact to road projection of yoke (m).
• TRS Trikke-Rider System symbol; T when used as index.
• c Instantaneous turn-center position vector from yoke (m); tc when used as index.
• θ Instantaneous steering angle θ (rad).
• μf Velocity dependent friction coefficient; f ϵ {T,w} (hz).
• ν0 Natural frequency "nu" of the falling stem Hz.
• Instantaneous TRS center of mass velocity (m/s).
• σρ Instantaneous velocity along guide-wheel path (m/s).
• i Instantaneous velocity ray for each wheel; i ϵ {σ,l,r} (m/s).
• j Instantaneous velocity ray for center of mass; j ϵ {E,R,T} (m/s).
• w Wheel symbol and index
• ω̚ Instantaneous TRS center of mass angular velocity vector (hz).
• ω̚k Instantaneous angular velocity vector; k ϵ {tc,σ,ε,R} (hz).
• ω̚* Possible TRS center of mass angular velocity vector due to slip (hz).
• Y Yoke symbol and index
• The forward direction of the Trikke is +x. (parallel to the center-line)
• Ŷ The left side is +y.
• Up is +z.
• t Trailer-Lever ray from road projection of trailer center of mass to road projection of yoke (m).

Jetting Path Action Vectors

Derivation of Jetting Path Action Vectors

The purpose of this model is to find equations for the locally developed velocity of the TRS center of mass, v̚. A summary is presented above in Jetting Path Action Vectors.

Given: The guide-wheel contact's linear velocity v̚σρ generated by the reduced rider via jetting actions; The guide-wheel contact C͆σ; TRS turn-center T͆c; turn-rays for the guide-wheel contact r̚σ and TRS center of mass r̚.

Find: The TRS velocity v̚ starting with the linear velocity of the guide-wheel contact path v̚σρ.

Jetting results in rotation action around the Trikke turn-center. In an unconstrained situation, as if the Trikke were on ice, the Parallel Axis Theorem for angular momentum would be applied to the TRS center of mass ©͆ to find its rotation. That is not appropriate as a Trikke is constrained to follow an arch. The rider's rotation must be applied to the guide-wheel contact point which only has freedom to move around the current Trikke turn-center. No rider model is assumed. Such a model would no doubt involve consideration of the moment of inertia of some of the rider's body parts and that of the Extended-Trikke with appropriate lever-arms, invocations of the Parallel Axis Theorems, etc.. The end result would be the same, the rider's instantaneous linear jet velocity as a direct increase or decrease at the guide-wheel contact point.

Rotation of the guide-wheel contact C͆σ (center of guide-wheel in Figure Bj1) and therefore all TRS parts around the turn-center T͆c (black dot) is increased (or decreased) by the same amount. The lever-arm is the turn-ray to the guide-wheel contact r̚σ (black dashes).

eqn jp1: r̚σ = C͆σ - T͆c

Solve for the rotation from the equation that computes the linear velocity v̚σρ (purple ray) from the rotation ω̚σby inverting the cross-product v̚ = ω̚ × r̚.

eqn jp2: ω̚σ = (ȓσ × v̚σρ)/rσ

since all the vectors are perpendicular to each other.

Find the linear velocity at the TRS center of mass (brown dot) by noting its turn-radius (brown line).

eqn jp3: r̚ = ©͆ - T͆c

Expanding the expression v̚ = ω̚ × r̚ to the given symbols produces,

eqn jp4: v̚ = (ȓσ × v̚σρ) × r̚/rσ

v̚ is the instantaneous velocity of the TRS about the turn-center accounted for by jetting.

Direct Push-Pull Path Action Vectors

Derivation of Direct Push-Pull Path Action Vectors

The purpose of this model is to find equations for directly pushing or pulling a Trikke, which produces a local velocity v̚ at the TRS center of mass. A summary is presented above in Pushing and Pulling a Trikke.

Given: TRS center of mass ©͆R and previous ©͆R' (used to determine the derivatives ∂(*)/∂t); guide-wheel path unit ρ̑σ; rider mass mR; guide-wheel contact C͆σ; TRS mass M, orbit unit ρ̑Trs and turn-center T͆c.

Find: The TRS velocity v̚ via the intermediate linear velocity of the guide-wheel contact path v̚σρ around T͆c and the across wheel drag D̚ due to the push or pull.

Mass moving with a velocity exhibits momentum. As the push strengthens, its speed increases and so does its change in momentum. Change in momentum is called impulse. Since the masses in this system are constant, changes in object velocities behave like impulses. Change in momentum over time is another way to define force. A force applied over a time is impulse, which is just the amount of change in momentum; in this case velocity change of a mass.

When riding, the rider is not independent of the Trikke. The push will end when the rider rejoins the Trikke in a perfectly inelastic collision attaining the same velocity as the Trikke. Furthermore, depending on the rider's technique, a pull may be required to reset position. In this case, from the physics viewpoint, a pull is merely a push in reverse.

Input for the push or pull comes from the rider's center of mass position ©͆R (orange dot in Figure Bd1). As it moves relative to the Extended-Trikke, a velocity v̚R (orange ray) develops over time at the rider's center of mass ©͆R.

Because the Trikke and rider are a connected system (loosely when necessary), when in direct contact they must conserve momentum together. When traveling in a straight line, a rider's push produces a velocity v̚R component along the Trikke's path ρ̑σ and one across it.

eqn dp2: v̚R = (v̚R · ρ̑σ) ρ̑σ + (v̚R - (v̚R · ρ̑σ) ρ̑σ)

Along the path, the component of rider push increases the Extended-Trikke's momentum.

eqn dp3: 0 = mEE + mR (v̚R · ρ̑σ) ρ̑σ

where the Extended-Trikke's mass is mE, velocity v̚E and those of the rider are mR and v̚R.

This conservation equation is easily solved for the increase in Extended-Trikke velocity v̚E.

eqn dp4: v̚E = - mR (v̚R · ρ̑σ) ρ̑σ/mE

Across the path, the component of rider push dumps into friction D̚ at the wheel axles and contact points.

eqn dp5: D̚ = mR ∂(v̚R - (v̚R · ρ̑σ) ρ̑σ)/∂t

However, to interface properly with the jetting model, a change is needed. The Jetting model assumes a path velocity v̚ρσ that will affect the entire Trikke-rider mass. But the current path velocity was derived only in the context of Extended-Trikke momentum which has much less mass. To reconcile, the velocity must be multiplied by a factor of mE/M. However, the effective TRS mass involved will be a little less than M since the Trikke is turning and the center of mass path ρ̑Trs is not aligned with the Guide-wheel path ρ̑σ. A better estimate for the receiving mass is M ρ̑Trs · ρ̑σ, where the pathing units of the TRS center of mass and the stem orbits are dotted together. This increases v̚σρ a little.

With this adaptation factor the increase in Guide-wheel velocity v̚ρσ is

eqn dp6: v̚ρσ = - mR (∂©͆R/∂t · ρ̑σ) ρ̑σ/(M ρ̑Trs · ρ̑σ)

Across the path, the component of rider push dumps into friction D̚ at the wheel axles and contact points. Technically, to conserve the momentum produced by the rider's motion across path, the molecules at the tread and ground interface absorb some of the momentum in various ways and the Earth reacts to the rest. Such a reaction is small indeed.

eqn dp7: D̚ = mR ∂((∂©͆R/∂t - (∂©͆R/∂t · ρ̑σ) ρ̑σ)/(M ρ̑Trs · ρ̑σ))/∂t

In a turn, this force is distributed to the guide-wheel ΔD̚σ and rear wheels ΔD̚L, D̚R proportionally to the angles subtended by the rider's center of mass position to the guide-wheel and to the wheel-base.

Of spans the angle between the center of mass turn-ray unit tȓcm to the guide-wheel turn-ray unit tȓσ.

eqn dp8: Of = acos(|tȓcm · tȓσ|)

Or spans the angle between the center of mass turn-ray tȓcm to the wheel-base turn-ray, which is parallel to the y-axis in a direction determined by the steering angle.

eqn dp9: Or = acos(|tȓcm · sign(θ) Ŷ|)

The front wheel receives less drag when the angle between the center of mass and the wheel-base is small.

eqn dp10: ΔD̚σ = Or * D̚/(Of + Or)

The rear wheels share the left overs.

eqn dp11: ΔD̚L = D̚R = (D̚ - D̚σ)/2

Caster Pull-Push Path Action Vectors

Derivation of Caster Pull-Push Path Action Vectors

The purpose of this model is to find equations for the contribution to guide-wheel contact point velocity due to Caster-Push. This velocity represents a jet that feeds into the Jetting Model. A summary is presented above in Caster Pull-Push Path Action Vectors.

Given: The wheel contacts C͆σ, C͆l, C͆r, the absolute value of their path friction Fl, Fr and the center-line length; local yoke position Y͆; guide-wheel contact C͆σ' C͆σ", transom point T̑p' and time duration Δt' from the previous time-step; The guide-wheel path unit ρ̑σ; TRS orbit unit ρ̑Trs, mass M, trailer center of mass position ©͆t, moment of inertia It, mass mt and time duration Δt of the frame.

First solve for the transom point defined by the friction on the rear wheels. Then develop the relationships between the guide-wheel contact point and the travel of the yoke and how it turns the Trailer-Lever. Compute the rotation of the Trailer-Lever. Find the differential of this value with time. If it is positive, the momentum is still building independently, so no jet is acting. When it is negative, the angular momentum starts to feed the jet. Determine the lever-arm that feeds the momentum into the jet and then the velocity of that jetting action v̚σρ. Finally, this guide-wheel contact velocity is fed into the jetting model.

Determine the position of the transom point using the rear wheel contact positions and the absolute value of their forward path friction. The absolute values preserve the expected rotation center when one wheel moves in the opposite direction of the other.

eqn cp1: T͆py = C͆ly - Tw Fr/(Fl + Fr)

Where Tw = (C͆l - C͆r)y is the y component of the line between the rear wheel contact points.

The full position at path-level is

eqn cp2: T͆p = <║center-line║, T͆py, 0>

Velocity of the yoke is opposite that of the contact point as it is pulled or pushed.

eqn cp3: v̚Y = -∂C͆σ(t)/∂t ≈ (C͆σ' - C͆σ)/Δt

Which is also the linear velocity of the trailer rotation around the transom point, T͆p

eqn cp4: v̚Y = -ω̚Tp × (Y͆ - T͆p)

Solve for the trailer rotation due to castering using the Cross Product Inverse, while noting that the yoke is the origin of the LCS; Y͆ = <0, 0, 0> in the local coordinate frame.

eqn cp5: ω̚Tp = T̑p × v̚Y/Tp

Compute the Trailer-Lever delta rotation in the Δt time-step where the tick in ω̚Tp' indicates the rotation before the time-step began.

eqn cp6: Δω̚Tp = ω̚Tp - ω̚Tp'

When Δω̚Tp is positive, the momentum of the trailer produced by the rider turning the handlebars builds independently of the TRS. v̚σρ is zero.

When Δω̚Tp is negative, the momentum transfers to the TRS via a jet velocity v̚σρ.

Compute the lever-arm L̚ from the trailer rotation to the jetting action aligned with the guide-wheel path unit vector ρ̑σ using the Lever-Arm equation.

eqn cp7: L̚ = (T͆p - C͆σ) × ρ̑σ × ρ̑σ

Rotation transfers the trailer's change in angular momentum through the lever into the Guide-wheel path to the TRS; a consequence of conservation of momentum. At the contact point, this momentum transfer can be expressed linearly as Δp̚Trs = Δp̚t or with more detail

eqn cp8: M v̚σρ = -It Tp Δω̚Tp × L̚

Note the expression on the left for Δp̚Trs assumes the total TRS mass receives the transferred momentum. Generally the effective mass involved will be a little less since the Trikke is turning and the center of mass path is not aligned with the Guide-wheel path. A better estimate for the receiving mass is M ρ̑Trs · ρ̑σ, where the pathing units of the TRS center of mass and the stem orbits are dotted together. This increases v̚σρ a little.

Expressed as the velocity at the contact point using the better estimate, the jet velocity due to trailer rotation is

eqn cp9: v̚σρ = -It Tp Δω̚Tp × L̚/(M ρ̑Trs · ρ̑σ)

Substituting out to the given variables and simplifying, the instantaneous contribution to the jet by castering is

eqn cp10: v̚σρ = -(It + mt (T̑p - ©͆t)2) ((T̑p × (C͆σ' - C͆σ)/(Tp Δt)) - (T̑p' × (C͆σ" - C͆σ')/(Tp' Δt'))) × (T͆p - C͆σ) × ρ̑σ × ρ̑σ/(M ρ̑Trs · ρ̑σ)

This instantaneous linear velocity at the guide-wheel contact is now treated as a jet input adjusted via the impulse-momentum theorem.

Trikke EOM

Simple Equation of Motion

As indicated in the introduction, Equations of Motion (EOM) can be written for a Trikke, but not solved easily. This very simple EOM only represents jetting; no Direct-Push or Pull and no castering. It models the effects of Trikke input steering angle θ, cambering angle γ and rider jetting vσρ on the mass, inertia and friction of the system and wheels. It relies on two rather complex functions. One uses a rotation operator to find the guide-wheel path direction given θ and γ. Friction requires a load calculation for each Trikke wheel which requires a TRS center of mass calculation and function to emulate the rider's intentional center of mass position, coordinated with θ, γ and vσρ. For this EOM, assume it is constant.

The guide-wheel contact path unit ray is determined by composing rotation functions like this:

eqn eom1: ρ̑σ(θ, γ) = (RR(γ) σ̑(θ) (R(γ) ȃ)) × Z̑

The composition order first finds R(γ) σ̑ the cambered unit stem and R(γ) ȃ and cambered unit axle. Then it rotates the axle around the stem by the turn angle. Finally, the cross-product with the Z-axis makes the path unit point forward at the appropriate angle.

Load on a wheel was determined by solving the equations for total mass and total torque of each wheel around the TRS center of mass simultaneously.

eqn eom2: Li(t) = Instantaneous Load at time t due to gravity on the Trikke and Rider portioned to each axle {σ, l, r}.

The general form of a Newtonian EOM is

eqn eom3: M Ɑ̚ = Forces and torques operating on system

In this example, the forces are split into a scalar factor and a directional one for conceptual simplicity.

Here is a simple rider jetting force model where vσρ is the rider's instantaneous jetting input velocity. vσρ could be a sine or cosine function or something much more complex as long as it matches the frequency of θ and γ. This EOM is a forward-feed model - it does not require feedback for control.

eqn eom4: Jetting force = M ∂(vσρ r/rσ)/∂t

Next consider the torques via moment of inertia for the system and the wheels. Angular velocity can be represented as a linear velocity wrapped around a rotation radius. In this model, all the rotations are related by radii ratios to the main turn-radius and the jet velocity. Conveniently, they factor out. The moments of inertia must be moved to the turn center via the Parallel-Axis Theorem.

eqn eom5: Torques = ∂(vσρ (I r + Iw(rσ + rr + rl)/(2 rσ2)))/∂t

Air resistance, wheel and axle friction are velocity dependent. Wheel friction also requires a gravity load which depends on the instantaneous TRS center of mass, which relies on the rider's motion. Jetting velocity and the radii ratios have been factored out.

eqn eom6: Friction forces = vσρ (vσρ A ρair M μT r/2 + μw (Lσ rσ + Lr rr + Ll rl))/rσ

The direction of the force at the TRS center of mass isolates the vector operations to a single factor. Since the length of the cross-products involve trigonometric functions, not preserving units, it is made into a unit.

eqn eom7: TRS center of mass Force Direction = ((ȓσ × ρ̑σ) × ȓ)̑

Simple TRS center of mass EOM without Direct-Push, Pull or Castering.

eqn eom8: M Ɑ̚ = (M ∂(vσρ r/rσ)/∂t + ∂(vσρ (I r + Iw(rσ + rr + rl)/(2 rσ2)))/∂t + vσρ (vσρ A ρair M μT r/2 + μw (Lσ rσ + Lr rr + Ll rl))/rσ) ((ȓσ × ρ̑σ) × ȓ)̑

Glossary

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! [sportsdefinitions.com] 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. 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.

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 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 the Transom point. The Trailer-Lever is a second-class lever and push-rod.

Transom Used to indicate the point on the line between the rear wheel contact points with the ground acting instantaneously 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, or just transom, 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 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, 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.

References

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.

Papers

[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.

[magic] Michael Lastufka, "Trikke Magic: Leveraging the Invisible" June 2020.

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

Other

[sportsdefinitions.com] SportsDefinitions.com. http://www.sportsdefinitions.com/, 2019.

[EzyRoller] EzyRoller LLC. 22588 Scenic Loop Rd., San Antonio TX 78255 - USA. https://www.ezyroller.com/, 2019. Email: info@ezyroller.com

[Flicker] Yvolution USA Inc. 2200 Amapola Court, Suite: 201, Torrance, CA 90501 – USA. https://yvolution.com/, 2019. Email: support@yvolution.com

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