Lean Into the Turn

Stability Modeling for a Rowing Tadpole Trike

December 15, 2025 · John Peach

Rowing provides great whole-body exercise, but not everyone lives near the water or has a rowboat. People have tried to combine rowing motion with bicycling, which certainly makes exercising more convenient, but since rowing involves arm and leg muscle groups, trying to maintain balance and steering becomes more complicated than simply riding a bike.

Better stability may be attained by using a three-wheel design with two wheels in the front for steering and braking, and one rear wheel to power the vehicle. To make it feel more like a bicycle, the back portion where the rider sits can be made to pivot about the longitudinal axis so the rider leans into a turn.

In this article, we will examine the leaning steering dynamics of a tadpole rowing trike, setting aside for the moment details of structural design and power transmission. We’ll focus specifically on the relationship between rider lean angle, vehicle speed, and rollover stability during steady-state turns. Using geometric analysis and barycentric stability methods, we’ll map out safe operating regions and determine optimal lean-to-steering ratios. We’ll consider:

How lean angle translates to steering input,
Which combinations of speed and lean keep the trike stable,
Whether the front or rear wheel lifts first in extreme conditions,
How much lateral acceleration the rider experiences.

In future articles, we’ll look at structural design and frame construction, power transmission and rowing mechanisms, braking and suspension, and real-world effects of uneven or loose surfaces.

Introduction: Why Lean at All?

Modern rowing tricycles often use direct steering, but a rowing tadpole A tadpole trike has two wheels at the front and a driving wheel in the back. Delta trikes use the opposite configuration with a single steering wheel in the front and two rear driving wheels. trike requires that your hands operate a rowing lever to simulate the motion of oars in a boat. A lean-steer system is appealing because the rider naturally leans into a turn, shifting the center of gravity (CG) inward, and the lean angle can be mapped mechanically to the steering angle.

If you’ve ever ridden an adult-sized tricycle, you immediately notice that the dynamics are very different from a bicycle. As you make a turn, the tricycle doesn’t lean the way a bike does, so you feel as if you are being thrown outward from the direction of the turn.

If we could adapt leaning into the design of a tricycle, some questions need to be answered. How much lean is needed for stability? How does lean interact with speed? Does a wider track or lower center of gravity help? We’ll look at how leaning and steering interact with stability by considering:

Geometric kinematics
Support-polygon stability analysis
Numerical simulations

The central question of this article is how much lean is useful and safe at realistic speeds.

Modeling the Trike Geometry

To begin the analysis, we need to define relevant geometric parameters:

Wheelbase $L$ is the distance from the rear wheel contact patch Contact Patch: The small area where a tire touches the ground. For a bicycle or trike, this is typically an oval-shaped region a few inches long. All forces between the vehicle and ground must pass through these contact patches. to the line joining the two front wheel contact patches.
Track width $T$ is the distance between the points of contact of the front wheels.
CG location $(x_{cg}, y_{cg},z_{cg}).$ We’ll use the trike centerline as the $x-$ axis, so without leaning, $y_{cg} = 0.$
Roll axis height $h_{axis}$ is the height above ground where the trike pivots to induce lean.
The maximum lean angle will be constrained to be less than $\pm 15 \degree.$

Support Polygon Stability

As the trike makes a turn, the front wheels steer along a circular path with radius $R.$ Isaac Newton formulated three laws of motion, which are:

An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction, unless acted on by a force.
The acceleration $\overrightarrow{a}$ of an object is directly proportional to the net force $\overrightarrow{F}$ applied to it and inversely proportional to its mass: $\overrightarrow{F} = m \overrightarrow{a}.$
For every action, there is an equal and opposite reaction. When one object exerts a force on a second object, the second object simultaneously exerts a force equal in magnitude and opposite in direction on the first object.

The first law says that the trike will continue in a straight line unless the turned wheels impart a force towards the center of the turn, called the centripetal force. Centripetal Force: The inward force required to keep an object moving in a circular path. For a turning vehicle, this force comes from friction between the tires and the road surface. Gravity also acts on the combined mass of the trike and rider in a downward direction, $-z$ , and the trike will be stable if the resultant force

\overrightarrow{F_{res}} = - m \overrightarrow{g} - m \overrightarrow{a}

lies inside the triangle, or support polygon Support Polygon: The geometric shape formed by connecting all ground contact points of a vehicle. A three-wheeled vehicle has a triangular support polygon. The vehicle remains stable as long as the vertical projection of its center of mass falls inside this polygon—just like how you can balance a book on a table as long as its center of mass is over the table surface., formed by the contact points of the three wheels. If the force lies outside the triangle, then the trike is in danger of rolling over.

In a frame fixed to the trike, we can think in terms of an apparent inertial force $-m\vec a$ added to the gravitational force $-m\vec g$ . The resultant $\vec F_{\text{res}} = -m(\vec g + \vec a)$ is what we project to the ground. Since we’re interested in the point that the resultant force vector makes contact with the ground, we only need to consider the vector addition of gravity and acceleration.

Barycentric coordinates provide a convenient method for determining if the resultant force vector is inside, outside, or on the perimeter of the triangle. Suppose $\triangle ABC$ contains a point $P$ and $P$ forms the vertex of three smaller triangles, using the vertices of the original triangle.

The areas of the new, smaller triangles are given in terms of ratios to the original triangle $\triangle ABC$ , so $\alpha$ is the ratio of the green area to the triangle, $\beta$ is the red area ratio, $\gamma$ is the blue area ratio, and $P$ is the point such that $\alpha + \beta + \gamma = 1.$

Alternatively, construct a Cartesian coordinate system anywhere in the $(\mathscr{x},\mathscr{y})$ and draw the vectors, $\overrightarrow{A}, \overrightarrow{B}, \overrightarrow{C}$ from the origin to the three vertices of the triangle. Then, $P = \alpha \overrightarrow{A} + \beta \overrightarrow{B} + \gamma \overrightarrow{C},$ with the constraint that $\alpha + \beta + \gamma = 1.$

The area ratios $\alpha, \beta, \gamma$ are all greater than zero if $P$ lies inside the triangle, but one will be equal to zero when $P$ is on one of the edges, and negative if $P$ is outside the triangle. This provides a quick stability test by setting $S = \min(\alpha,\beta,\gamma)$ , and noting that

$S > 0$ indicates stability,
$S = 0$ means the trike is at the edge of tipping,
$S < 0$ results in a rollover.

Lean Steering

Now, let’s consider the relationship between the steering angle, lateral acceleration, and the lean angle. Suppose the front wheels are turned at an angle $\delta_{in}$ relative to straight ahead. Then, the trike follows the contour of a circle of radius $R,$

R = \frac{L}{\tan \delta_{in}}

where $L$ is the wheelbase length. The lateral acceleration depends on the trike speed $v$ and the radius of curvature,

a_y = \frac{v^2}{R}.

Unlike a typical tricycle, the leaning tadpole allows the rider to tilt into the turn at some angle $\theta.$ By leaning, the rider can shift the resultant force vector contact point $P$ from a dangerous rollover condition outside the triangle to stability inside the triangle. For this analysis, we consider three mechanisms to control steer.

The first is that the rider leans towards the center of the turn, the second method is to control the steering angle through foot pedal action, and the third is to adjust how much lean angle affects steering angle depending on the trike’s speed. At lower speeds, we allow a greater turn angle for each degree of lean, but at higher speeds, we want to reduce the effect to minimize overreaction. In practice, adjusting the lean by speed would be mechanically difficult, so we’ll look at the effects of two different lean factors.

Heatmap: Stability Margin vs Speed & Lean

We can analyze stability by plotting a heat map of rider lean angle against trike speed. This assumes that the rider tilts at some angle $\theta$ relative to vertical, causing the trike’s front wheels to turn at an angle $\delta_{in} = k_{\text{eff}} \; \theta,$ where $k_{\text{eff}}$ Lean-to-Steer Gain $k_{\text{eff}}$ : This coefficient determines how aggressively the front wheels turn in response to leaning. For example, a value of 0.3 could mean that a 10° lean produces a 3° steering angle, while 0.1 produces only 1° of steering. Lower values make the trike more stable but require larger lean angles to navigate turns. is a gain factor applied to the lean angle. At lower speeds, the rider may safely lean more for a sharper turn, but as speed increases, we want to limit the effect by reducing $k_{\text{eff}}.$

At low speeds, we might accept $k_{\text{eff}} = 0.3,$ but at higher speeds, setting $k_{\text{eff}} = 0.1$ might be safer. Another factor that will affect stability is the position of the seat during the stroke. Since the purpose of the trike is to mimic rowing a boat, the seat is allowed to slide forward and backward during each stroke. At the back end of the stroke, the CG has shifted, which could impact the chance of a rollover.

These four plots show the effect of rider lean angle and speed on trike stability. Positive lean angles indicate the rider leans into the turn, while negative angles indicate cases when the rider leans out of the turn. Leaning out might happen if the rider entered a sharp turn at high speed and was “thrown” outwards. In general, though, the rider should anticipate the turn and lean into the curve.

The black dotted lines indicate the margins of stability where the resultant force $F_{\text{res}}$ is on the edge of the triangle, and the color bars indicate the minimum values of $(\alpha,\beta,\gamma)$ from the barycentric triangles. Keeping the lean amplification factor $k_{\text{eff}} = 0.1$ provides safety everywhere except at large outward lean angles and high speeds. As expected, a lean angle of $0 \degree$ is very stable because the bike is traveling in a straight line and the rider is sitting upright in the seat. This is shown by the faint yellow horizontal line in the middle of each figure.

Another region of high stability appears on the “safe” side of the plots where the lean angles are non-negative. You might notice a faint vertical yellow line between speeds of 2.5 and 5.0 m/s when $k_{\text{eff}} = 0.3$ and between 7.5 and 10.0 m/s $k_{\text{eff}} = 0.1.$ This is due to nonlinearities in the chain from lean angle $\theta$ to steering angle $\delta_{in}$ , turning radius $R$ , and lateral acceleration $a_y = \frac{v^2}{R}$ affecting the CG projection Center of Gravity Projection: Imagine dropping a plumb line from the combined center of mass of the trike and rider straight down to the ground. Where that line hits the ground is the CG projection. If this point moves outside the triangle formed by the three wheels, the trike will tip over. and the barycentric margin. Fortunately, this extra margin of safety happens at relatively high speed where we need it.

Which Wheel Tips First?

If you go tearing around a tight turn at high speed, which wheel comes off the ground first? Surprisingly, it’s always the rear wheel that is most likely to lose contact with the ground. This doesn’t mean that the back of the trike will suddenly lift, but the downward force on the rear wheel is reduced in some cases, which could cause the rear wheel to slide sideways. These tipping maps assume steady turning (no braking) and look at whether the projection of gravity plus lateral acceleration leaves the support triangle.

For $k_{\text{eff}} = 0.3,$ it’s possible to slide in a turn during high-speed turns and larger lean angles,

Figure 4.
Limiting edge, $k_{\text{eff}}=0.3$

The interpretation of this plot is:

Green → tipping over front axle line (rear wheel lifts first)
Blue → tipping about the front-right/rear edge
Red → tipping about front-left/rear edge (inner wheel lift)

This plot indicates that neither of the front wheels is likely to lift during a sharp turn, since no areas of the plot are blue or red.

The greater danger of tipping comes from leaning in the wrong direction during a turn. Leaning out by $10 \degree - 15 \degree$ could easily induce the rear wheel to begin a slide. By reducing the lean coefficient to $k_{\text{eff}} = 0.1,$ the effect can be significantly reduced, and eliminated when leaning into the turn,

Figure 5.
Limiting edge, $k_{\text{eff}}=0.1$

From this analysis, it’s clear that keeping $k_{\text{eff}}$ low minimizes the chance of losing steering control.

Lateral Acceleration

The next consideration is to look at how much lateral force the rider will experience during a turn. A turn that induces 0.25 g will feel fairly normal, but getting much above 0.5 g could feel uncomfortable. For both cases of $k_{\text{eff}}$ , the lateral acceleration remains comfortably low.

Figure 6.
Lateral g’s, $k_{\text{eff}}=0.3$ .

Figure 7.
Lateral g’s, $k_{\text{eff}}=0.1$ .

Under most conditions, the rider will feel little lateral acceleration (black) and even in extreme cases, only reaches 0.25 g’s (yellow).

These plots show lateral acceleration as a function of lean angle and speed, but they only account for how lean changes the steering angle. They do not include the stabilizing effect of the rider’s torso moving relative to the frame; we’ll treat that separately in the stability margin plots.

Even if lateral acceleration exceeds 0.5 g’s, the rider may be able to overcome the trike’s built-in mechanical limit of $15 \degree$ by applying $20 \degree - 30 \degree$ of effective body lean through tilting towards the center of the turn. Riders will naturally lean into the turn and dramatically reduce the side load, even if they can’t achieve a mathematically perfect neutral lean.

We can also look at the relationship between turning radius and lean angle at various speeds. At the highest speed of $15 \; \text{m/s,}$ a $20 \; \text{m}$ turn radius is achieved at a lean angle of $15 \degree$ , which we’ve assumed is the mechanical limit for the trike. This is a very reasonable turn radius at speed, and could be enhanced if the rider shifts his or her CG. A 20 m radius is roughly like taking a small neighborhood roundabout at ~54 km/h (33 mph).

Figure 8.
Turning radius and lean angles at speeds.

Conclusion

The analysis of a rowing lean-steer tadpole tricycle shows that:

Lean-steer strongly enhances stability for a rowing tadpole trike.
±15° lean gives large benefits at normal speeds; beyond that, torso lean fills the gap.
Lowering CG is more effective than widening the track, but both help.
A properly tuned steer-lean ratio is key to good feel and safety.
We haven’t yet included hard braking, bumps, or suspension dynamics; those will be added in future iterations. The model does not predict subjective “feel,” dynamic wobble, or braking instability.

This modeling framework will be used as we add suspension, braking force distribution, and dynamic weight transfer in future articles.

Glossary

Barycentric Coordinates: A coordinate system that expresses the position of a point inside a triangle as weighted proportions of the triangle’s three vertices. If the three proportions ( $\alpha, \beta, \gamma$ ) are all positive, the point lies inside the triangle; if any proportion is negative, the point lies outside.

Center of Gravity (CG): The point where the entire weight of an object (trike plus rider) can be considered to act. Also called the center of mass. The location is typically given as coordinates ( $x_{cg}, y_{cg}, z_{cg}$ ) relative to a reference frame.

Centripetal Force: The inward force required to keep an object moving in a circular path. For a turning vehicle, this force comes from friction between the tires and the road surface. Without centripetal force, an object would continue in a straight line (Newton’s first law).

Contact Patch: The small area where a tire touches the ground. For bicycle and tricycle wheels, this is typically an oval-shaped region a few inches long. All forces between the vehicle and ground must pass through these contact patches.

Delta Trike: A three-wheeled vehicle configuration with a single steering wheel in the front and two rear driving wheels. The opposite of a tadpole trike.

Lateral Acceleration: Sideways acceleration experienced during a turn, directed toward the center of the circular path. Measured in “g’s” (multiples of Earth’s gravitational acceleration, where 1 g = 9.81 m/s²). Values above 0.5 g can feel uncomfortable for most riders.

Lean Angle ( $\theta$ ): The angle at which the trike frame tilts from vertical, measured in degrees. Positive lean angles indicate leaning into the turn (toward the center of the curve); negative angles indicate leaning away from the turn.

Lean-to-Steer Gain ( $k_{\text{eff}}$ ): A coefficient that determines how aggressively the front wheels turn in response to leaning. For example, $k_{\text{eff}} = 0.3$ means a 10° lean produces a 3° steering angle, while $k_{\text{eff}} = 0.1$ produces only 1° of steering. Lower values make the trike more stable but require larger lean angles to navigate turns.

Resultant Force ( $F_{res}$ ): The vector sum of all forces acting on the trike, primarily gravity (pulling downward) and centripetal force (pulling toward the turn center). The ground projection of this force determines stability.

Roll Axis: An imaginary line running lengthwise through the trike about which the frame pivots when leaning. The height of this axis ( $h_{axis}$ ) affects how lean angle translates to center-of-gravity shift.

Steering Angle ( $\delta_{in}$ ): The angle at which the inner front wheel is turned relative to straight ahead, measured in degrees. This angle determines the turning radius of the trike.

Support Polygon: The geometric shape formed by connecting all ground contact points of a vehicle. For a tadpole trike, this is a triangle connecting the two front wheels and the single rear wheel. The vehicle remains stable as long as the vertical projection of its center of mass falls inside this polygon.

Tadpole Trike: A three-wheeled vehicle configuration with two wheels at the front (for steering) and a single driving wheel in the back. Named for its resemblance to a tadpole’s body shape.

Track Width (T): The distance between the contact points of the two front wheels, measured perpendicular to the direction of travel. Wider track widths generally improve stability.

Turning Radius (R): The radius of the circular path followed by the trike during a turn. Related to wheelbase and steering angle by $R = L/\tan(\delta_{in})$ . Smaller radii represent tighter turns.

Wheelbase (L): The distance from the rear wheel contact patch to the line connecting the two front wheel contact patches, measured parallel to the direction of travel. One of the fundamental geometric parameters affecting turning behavior.

Code for this article

stability_2x2_bounds.jl generates stability heatmaps.
stability_explorer.jl explores static rollover stability of the rowing tadpole trike.
trike_animations.py ManimCE visualizations (static images + short animations).

Software

Julia - The Julia Project as a whole is about bringing usable, scalable technical computing to a greater audience: allowing scientists and researchers to use computation more rapidly and effectively; letting businesses do harder and more interesting analyses more easily and cheaply.
Manim - Manim (short for Mathematical Animation Engine) is a cross-platform, free and open-source animation engine initially developed by Grant Sanderson in early 2015. Manim is a Python library for creating precise, programmatic animations of mathematical concepts. It allows users to define graphical scenes with mathematical objects, transformations, text, etc., and render them into videos.

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Image credits

Hero: Gemini Nano Banana Pro. (Couldn’t follow directions to generate an image of a tricycle.)
Dimensions and Force Diagram: Eco-Tad Supplemental Owner’s Manual, Design and Stability of Recumbent Tricycle.

Appendix

Appendix A. Barycentric Stability Margin

The stability analysis in this article is based on the idea that the trike remains upright as long as the ground projection of the resultant force lies inside the support polygon formed by the three wheel contact points.

For a tadpole trike, this support polygon is a triangle defined by:

$A$ : front-left wheel contact point
$B$ : front-right wheel contact point
$C$ : rear wheel contact point

A.1 Support triangle geometry

Let the wheel contact points be given in ground coordinates as

A = (x_A, y_A), \quad B = (x_B, y_B), \quad C = (x_C, y_C).

Let $P = (x_P, y_P)$ be the ground projection of the resultant force vector acting through the trike’s center of mass.

A.2 Barycentric coordinates

Any point $P$ in the plane of triangle $ABC$ can be written as a barycentric combination of the vertices:

P = \alpha A + \beta B + \gamma C,

with the constraint

\alpha + \beta + \gamma = 1.

The coefficients $\alpha, \beta, \gamma$ are the barycentric coordinates of $P$ with respect to the triangle.

A key geometric fact is:

$P$ lies inside the triangle if and only if $\alpha \ge 0,\quad \beta \ge 0,\quad \gamma \ge 0.$

A.3 Stability margin definition

We define the stability margin

S = \min(\alpha, \beta, \gamma).

This single scalar summarizes stability:

$S > 0$ : resultant projection lies strictly inside the support triangle
$S = 0$ : projection lies exactly on an edge (incipient tipping)
$S < 0$ : projection has crossed an edge (static instability)

The wheel pair associated with the smallest barycentric coordinate identifies the limiting tipping edge:

$\alpha$ smallest → tipping about edge $BC$ (front-right + rear)
$\beta$ smallest → tipping about edge $AC$ (front-left + rear)
$\gamma$ smallest → tipping about edge $AB$ (front axle line)

A.4 Simple numerical example

Consider a triangular support with vertices

A=(0,1),\quad B=(0,-1),\quad C=(-2,0).

If a force projects to

P=(-0.5,0),

the barycentric coordinates are all positive, giving $S > 0$ and a stable configuration.

As $P$ moves forward toward the line $AB$ , the coordinate $\gamma$ decreases toward zero. When $\gamma = 0$ , the rear wheel is just unloading, and the trike is on the verge of tipping about the front axle line.

Appendix B. Lean-to-Steering Model

This article uses a deliberately simple mapping from lean angle to steering angle to explore stability trends rather than detailed control dynamics.

B.1 Lean angle definitions

$\theta$ : frame lean angle (positive when leaning into the turn)
$\delta_{\text{in}}$ : inner front wheel steering angle

The model assumes the trike frame can lean up to a fixed mechanical limit, while the rider may add additional body lean that shifts the center of mass relative to the frame. Only the frame lean enters the steering model.

B.2 Fixed-gain steering law

For most of the stability analysis, we use a fixed gain relationship:

\delta_{\text{in}} = \operatorname{clamp}\!\left( k_{\text{eff}}\,\theta,\; -\delta_{\max},\; +\delta_{\max} \right),

where:

$k_{\text{eff}}$ is the lean-to-steering gain,
$\delta_{\max}$ is a steering saturation limit.

Two representative gains are explored:

$k_{\text{low}}$ : aggressive steering (tight turns at small lean)
$k_{\text{high}}$ : gentler steering (larger turning radius)

B.3 Turning radius and lateral acceleration

Using standard bicycle geometry,

R = \frac{L}{\tan \delta_{\text{in}}},

where $L$ is the wheelbase.

The resulting lateral (centripetal) acceleration is

a_y = \frac{v^2}{R},

with $v$ the forward speed.

This chain

\theta \;\rightarrow\; \delta_{\text{in}} \;\rightarrow\; R \;\rightarrow\; a_y

is responsible for the curved bands seen in the lateral-acceleration and stability heatmaps.

More sophisticated models could allow $k_{\text{eff}}$ to vary smoothly with speed or include direct coupling to rider input. These extensions are not required to capture the main stability trends discussed here.

Appendix C. Validation and Sanity Checks

Several simple checks were used to validate the model and implementation.

C.1 Straight-line motion

With zero lean and zero steering:

\theta = 0,\quad \delta_{\text{in}} = 0,

the turning radius tends to infinity and $a_y = 0$ . The resultant force projects vertically downward through the center of mass, well inside the support triangle.

C.2 Low-speed lean

At small lean angles and low speeds, lateral acceleration is small, and the stability margin remains positive across all seat positions. This confirms that the trike behaves benignly during gentle maneuvers.

C.3 High-speed cornering

At high speed and large lean, the stability margin approaches zero. The barycentric analysis correctly identifies which wheel pair becomes the tipping axis, matching geometric intuition.

C.4 Symmetry checks

Lateral acceleration magnitude is symmetric in the sign of the lean angle.
Stability margin changes sign appropriately when leaning into vs. away from the turn.

These symmetries provide strong consistency checks on the calculations.

Appendix D. Animations and Visualizations

Several animated diagrams were created using Manim Community Edition to support the analysis:

Top-view turning motion showing radius and centripetal acceleration
Front-view lean showing center-of-mass shift and roll axis
Motion of the resultant force projection within the support triangle

Instructions for running the animations are contained in trike_animations.py. For example, at a command prompt, execute manim -pqh trike_animations.py SupportTriangleProjection.