Ackermann steering: why cars turn the way they do

TL;DR:

• In a turn, every wheel must roll around the same center point, so the inner front wheel turns more than the outer one.

• For wheelbase L, track T and a turn of radius R (measured at the rear axle center), the angles are tan δ_in = L / (R − T/2) and tan δ_out = L / (R + T/2).

• Skid steering has no such geometry, so wheels drag sideways on the ground, wasting energy and wearing tires.

Look at a car from above as it turns left. The two front wheels do not point in the same direction. The one on the inside of the turn is turned more sharply than the one on the outside. This is not a flaw in the steering. It is geometry, and it is the reason a car can turn without its tires fighting the ground.

What it does on a robot

Many outdoor robots, rovers and model-car platforms steer like a car: the rear wheels are fixed, and the front wheels pivot. Compared with differential drive, this is more stable at speed and easier on tires, but it cannot spin on the spot, and it needs room to turn.

The geometry: one center for every wheel

When a vehicle drives round a curve without sliding, each wheel rolls along a circle, and all those circles share a single center, the instantaneous center of rotation. The rear axle is fixed, so the center must lie on the line through the rear axle. Each front wheel must point at right angles to the line joining it to that center.

Let L be the wheelbase (distance between front and rear axles) and T the track width (distance between the left and right wheels). Let R be the turning radius, measured from the center to the middle of the rear axle. Then the inner front wheel is at horizontal distance R − T/2 from the center, and the outer at R + T/2. With the front axle L ahead of the rear axle, each front wheel's angle δ from straight ahead satisfies:

tan δ_in = L / (R − T/2) tan δ_out = L / (R + T/2)

The inner wheel is closer to the center, so its tangent is larger, and it turns more.

Worked example: inner and outer wheel angles

A model car has L = 0.30 m and T = 0.20 m, and takes a turn with R = 0.60 m.

1. Inner distance: R − T/2 = 0.60 − 0.10 = 0.50 m. So tan δ_in = 0.30 / 0.50 = 0.60, and δ_in = 31.0°.

2. Outer distance: R + T/2 = 0.60 + 0.10 = 0.70 m. So tan δ_out = 0.30 / 0.70 = 0.4286, and δ_out = 23.2°.

3. Difference: the inner wheel is turned about 31.0 − 23.2 = 7.8° more than the outer.

Check using the Ackermann condition, cot δ_out − cot δ_in = T / L. We get 1 / 0.4286 − 1 / 0.60 = 2.333 − 1.667 = 0.667, and T / L = 0.20 / 0.30 = 0.667. The two sides match.

The turning circle depends on the largest steering angle the wheels can reach. If the inner wheel can turn at most 35°, then the smallest R satisfies tan 35° = L / (R − T/2), so R − 0.10 = 0.30 / 0.7002 = 0.4285, and R = 0.5285 m. The car-like robot cannot turn tighter than a circle of about 0.53 m radius at the rear axle center, so the full turning circle is about 2 × 0.53 = 1.06 m wide, a little more at the outer front wheel.

If the front wheels were instead set parallel (both at the same angle), their circles would not share a center. One wheel or both would have to slide sideways. That is called scrub.

Why skid steering scrubs

A tracked vehicle or a four-wheel robot with fixed wheels turns by driving the left and right sides at different speeds, like differential drive. The wheels on one side cannot point toward the center of the turn, because they are fixed. As the robot turns, they are dragged sideways across the ground. This costs extra motor power, wears the tires, tears up soft ground, and makes wheel odometry unreliable. The long wheelbase makes it worse. That is one reason a four-wheeled skid-steer robot with a long body turns poorly on grippy floors, and why cars do not do it.

Where it goes wrong

• Ideal versus real linkages. Real cars use a linkage that approximates the Ackermann angles, and often only partly, because tires also deform and slip at speed.

• Speed. At higher speeds, tires develop slip angles, and the real path differs from the low-speed geometry above. The formulas are for slow, no-slip driving, which is what most small robots do.

• Servo limits. A hobby servo steering both wheels with a simple linkage has limited angle and slop, which limits the turning circle.

• Rolling over. Sharp turns at speed can tip a tall robot. A lateral acceleration v² / R that is too high, compared with g × (T/2) / h for a center-of-mass height h, will roll it.

• Cannot spin. If the task needs turning in place, choose a different drive.

Safety: A car-like robot that does not have a way to stop quickly can be dangerous at speed. Test at low speed on an open area, keep spectators away from the path, and have a remote stop. Take care with the steering linkage, which can pinch fingers when it moves.

How real robots use it

Autonomous cars, rovers, farm vehicles and robot-racing cars use Ackermann steering. A planner has to respect the minimum turning radius, which is why paths for these vehicles are smooth arcs and not sharp corners. Motion planning for such vehicles must also deal with the fact that they cannot move sideways, so parallel parking needs a sequence of forward and reverse arcs. Orientation sensing from an IMU helps hold the heading.

Try it yourself

Measure L and T of a toy car or a robot platform. Choose a turning radius R, compute δ_in and δ_out, and set the front wheels to those angles with a protractor. Roll the car along a straight line on paper marked with the arc, and watch whether it follows without sliding. Then set both wheels to the same angle and see how it behaves.

Frequently asked questions

Why does the inner wheel turn more?

It is closer to the center of the turn, so the line from it to the center is steeper. The wheel must point at right angles to that line to roll without sliding.

Can an Ackermann robot turn on the spot?

No. It has a minimum turning radius set by its wheelbase and its maximum steering angle, so it needs room to turn.