Wheels, Speed & Turning: The Math of Movement

Two wheels. Different speeds. Infinite paths.

1

Which End Is the Front?

Before we talk about speed, let's settle something fundamental. This trips up almost everyone at first. Look at this robot and ask yourself: which end is the front?

Use arrow keys or buttons below to drive:

The Wheels PULL the Robot

The drive wheels are the front. They lead. The small caster wheel in the back just drags along for balance - like a shopping cart's back wheels.

Think About Walking

Your feet step forward first, and your body follows. You don't walk backwards and call it "forward." The drive wheels are the robot's feet.

Push vs Pull

Try pushing a broomstick across the floor by the end. It flops everywhere! Now pull it - perfectly stable. Drive wheels PULL the robot forward. That's why they're in front.

2

Equal Speed = Straight

A differential drive robot has two wheels. When both spin at the same speed, the robot drives perfectly straight. Try it!

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Same Speed

Both wheels cover the same distance per second, so the robot moves in a straight line.

Speed = Distance

Higher speed means more ground covered. The direction stays the same when both are equal.

3

Different Speeds = Arcs

When one wheel goes faster than the other, the robot curves. The slower wheel is on the inside of the turn.

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40
Turn Radius: --

The Slower Wheel

The slower wheel traces a smaller circle. The robot pivots around the slower side.

Bigger Difference = Tighter Turn

A large speed difference creates a tight arc. A small difference makes a gentle curve.

Think About It
If the right wheel is faster, which way does the robot turn? Left! It always turns toward the slower wheel.
4

Opposite Speeds = Spin

When wheels spin at equal speeds but in opposite directions, the robot pivots in place with zero turning radius!

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Zero-Radius Turn

The pivot point is exactly between the two wheels. The robot spins without moving forward.

Why It's Useful

This lets robots turn to face any direction without needing space to curve around.

Real Robots
Tank treads and wheelchair drives use this exact principle. Each side can be independently controlled for precise turning.
5

The Math Behind the Motion

One formula predicts every path. Adjust the sliders and watch the radius calculation update in real-time.

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R = (L / 2) × (vL + vR) / (vR - vL)
R = turn radius, L = wheelbase (distance between wheels), vL/vR = wheel speeds
vL = vR
R = Infinite
(Straight line)
vL = -vR
R = 0
(Spin in place)
Current
--
Adjust sliders

The Wheelbase (L)

The distance between the two wheels. A wider wheelbase makes the same speed difference produce gentler turns.

Linear Velocity

The robot's forward speed is the average of both wheels: v = (vL + vR) / 2

Angular Velocity

How fast it turns: omega = (vR - vL) / L. Bigger difference = faster rotation.

6

Path Challenges

Real robots don't just use one speed setting - they follow a program: a sequence of steps. Build your program below to trace the target shapes!

Circle
Radius = 120
Not attempted
Square
Side = 160
Not attempted
Figure-8
Loop radius = 80
Not attempted
Your Program
1.
Step Math: Speed: 90/s Radius = 120 Full circle: 8.4s
Hint for Circle: One step with different L and R speeds, Long duration
Real-World Connection
This is exactly how real robots are programmed: a sequence of drive commands, each with different wheel speeds and durations. Roomba vacuums, Mars rovers, and warehouse robots all work this way.

Master of Motion!

You've discovered how two simple wheels create infinite paths. Real robots use this exact math to navigate the world!

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Paths Drawn
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Challenges Completed
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Time Exploring

Equal = Straight

Same wheel speeds produce straight-line motion.

Speed Ratio = Curve

The ratio between wheel speeds determines the turn radius.

Opposite = Spin

Equal but opposite speeds create zero-radius pivot turns.

It's All Math

A simple formula predicts every possible path.

Real Robots Use This

Differential drive is how most wheeled robots navigate.

Ready to Create?

Put your new knowledge into practice!

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