# What's the exact formula for calculating steering angles of a modern car?

I'm interested in how the steering angles of a modern car are calculated, I'm assuming it should be similar to Ackermann Steering formula.

However, the wikidepia page for Ackermann Steering mentions that "Modern cars do not use pure Ackermann steering, partly because it ignores important dynamic and compliant effects, but the principle is sound for low-speed maneuvers."

• Hi Paul, welcome to engineering. Could you share a reference which explains a bit better "my understanding modern cars don't use exactly that" – NMech Dec 8 '20 at 16:08
• @NMech Hello ! I know wikipedia is not always the most trusted source but the page for Ackerman Steering mentions this "Modern cars do not use pure Ackermann steering, partly because it ignores important dynamic and compliant effects, but the principle is sound for low-speed maneuvers." – Paul Dec 8 '20 at 21:39

I think with this question you are touching a subtle matter. More precisely the conceptual difference between using an equation for design and for control.

IMHO, the Ackermann steering theory is most typically used in the design stage of a vehicle. The idea, is to provide a tool for calculating the steering arms with respect to the axle distance and turning radius of a vehicle.

After these parameters are calculated there was little use to the Ackermann steering theory - at least until recently when embedded computers starting to find their way into cars.

However, recently with the advent of embedded computers, electric cars and stability system there is a need for knowing where will the vehicle be headed if I turn the steering will by x?

A great starting point one would guess is the Ackermann steering theory. However, as wikipedia correctly states, Ackermann's principle is sound for low speed maneuvers. Wikipedia states that because of the slip forces on a tire during a cornering maneuver.

As you can see in the following image, when a vehicle is moving with a velocity v and the tire is turned at a angle $$a$$ which should correspond to turning radius R, the trajectory of the vehicle is not following that turning radius.

The reason is because of longitudinal slip of the tires. I think everyone, who has driven a bit has tried to take a turn just a little bit faster that he/she should have. The result is invariably, that you can feel the car starting to slip, and going of on a tangent (if you are not lucky or fast enough to react).

What you need to take into consideration for that is the following:

• steering angle (Ackermann steering)
• velocity of vehicle
• Turning radius of the material
• mass distribution/center of gravity
• tire/ground interaction (e.g. soil, asphalt, ice)
• temperature
• ....

There are at least a dozen models for this type of work.

So, to conclude, in modern cars there is the need to to calculate the direction a vehicle will be headed given certain conditions. However this need is a subject for vehicle control/dynamic behaviour. While the Ackermann steering theory is an integral part of the answer to the above question, it is most commonly employed autonomously on the design stage of the vehicle.

In my opinion, steering angle can be obtained by the information of lateral forces which are produced by the tires. Force can be calculated by chasis dynamo or any mechanism and any angle can be estimated or obtained by filtering methods or some of calculations. In bicycle model, we ignore some of important things and results from it may be far from correct values. Even so, bicyle model gives us useful informations.