Play with itSet the two wheel speeds
Worked solutionOne step at a time
Both wheels circle the same point
The robot is a rigid body, so while it turns, every part of it goes round one point on the floor (the turning centre) at the same turn rate ω. A point at radius r moves at speed ω·r. The fast wheel is on the outside, the slow wheel on the inside, and their radii differ by exactly the wheel gap W.
Subtract to get the turn rate
Take one equation from the other and R drops out.
Add to get the centre's speed
The centre is halfway between the wheels, so it moves at their average.
Radius is speed over turn rate
In one line: R = (W/2) · (vleft + vright) / (vleft − vright).
Answer: C, 0.75 m
Sanity check: the inner wheel is at 0.75 − 0.25 = 0.5 m and the outer at 1.0 m. The outer circle is twice as long, and the outer wheel is twice as fast, so both finish a lap together. The left wheel is the fast one, so the robot curves to its right.
What the simulation showedA real wheeled robot, not a formula
The reel's robot is a real rigid-body simulation: a 31 kg chassis on two motor-driven wheels (wheel gap 0.50 m, wheel radius 8.5 cm) and two free casters, steered only by commanding wheel speeds.
| Left / right | Formula | Simulated robot |
|---|---|---|
| 1.0 / 0.5 m/s | R = 0.75 m, lap 6.28 s | R = 0.750 m, lap 6.30 s |
| 1.0 / 0.4 m/s | R = 0.583 m | R = 0.583 m |
| 0.9 / 0.3 m/s | R = 0.50 m | R = 0.500 m |
| +0.5 / −0.5 m/s | spins at 114.6°/s | spins at 114°/s, centre still |
- In the quiz case the left wheel's track is a 1.00 m circle and the right wheel's a 0.50 m circle, with the centre exactly between them.
- It only matched after a fix. With flat cylinder wheels the first run drove a 0.99 m circle instead of 0.75 m: a wide wheel touches the floor at both edges, and the two contacts fight each other in a turn. Giving each wheel a single contact point brought it back to the formula. Real robots have the same problem, which is why wide tyres scrub and odometry drifts.
Wrong answersWhy each one is tempting
| Pick | What went wrong |
|---|---|
| A · 0.25 | W/2: the distance from the centre to a wheel. |
| B · 0.5 | The inner wheel's radius, R − W/2. |
| D · 1.5 | Added the speeds and stopped. |
Follow-ups an interviewer asks nextTry these before opening them
Left wheel +0.5 m/s, right wheel −0.5 m/s. What does it do?
The average is zero, so the centre does not move, and ω = (0.5 − (−0.5)) / 0.5 = 2 rad/s. It spins on the spot, one full turn every 3.1 s. This is why these robots can turn around in an aisle barely wider than they are.
You double both wheel speeds. What happens to the radius?
Nothing. Sum and difference both double, so R is unchanged. The robot drives the same circle twice as fast. Radius depends only on the ratio of the wheel speeds and the wheel gap.
The robot's wheels are 10 cm in radius. What wheel spin rates give 1.0 and 0.5 m/s?
Wheel rim speed = spin rate × wheel radius, so 1.0 / 0.1 = 10 rad/s (95 rpm) and 0.5 / 0.1 = 5 rad/s (48 rpm).
Why does the real robot's circle come out slightly different from the formula?
The formula assumes the wheels roll without slipping and the wheel gap is measured between two perfect contact points. Real tyres have a contact patch with width, they scrub sideways a little in a turn, and wheel radius changes with load. That is why wheel odometry drifts, and why robots fuse it with a gyro or lidar.
