When a vehicle makes a turn, the two front wheels trace out two arcs as shown in the figure below. The wheel facing towards the inside of the turn has a steering angle that is greater than that of the outer wheel. This is necessary to ensure that both front wheels smoothly trace out two arcs, which have the same center, otherwise the front wheels will skid on the ground during the turn.

During a turn, do the rear wheels necessarily trace out the same arcs as the front wheels?


No, in general they do not. You can work this out from the geometry of Ackermann steering, discussed on my website article "Parallel Parking a Car".

In summary: look at the defining geometry for Ackermann steering, which I have sketched below:

Ackermann Steering Geometry

Ackermann steering is defined by the intersection of the central unit normals to (axes of rotational symmetry of) all wheels at a common point $C$ in the diagram. It should therefore be very clear from the diagram that the radius of curvature for each wheel is different: the curvature radiusses for the forward wheel paths are a little bit bigger than those of the hinder wheel paths.

You can also see that if the forward and hinder wheels steer symmetrically as in some four wheel steering systems, corresponding wheel pairs do follow the same arc as the curvature radiusses would then be equal.

Note that the Argand plane co-ordinates are simply for my website, where I want to compute the group of transformations available to the driver and show that these transformations do indeed generate the whole of the proper Euclidean group $E^+(2)$- i.e. any simultaneous translation and planar orientation rotation can be realized as a finite sequence of steering operations.

Real cars always deviate from the Ackermann condition, however slightly. Indeed, if the tyre has a nonzero width, some part of it must either deform cyclically as it passes through the point of contact with the ground and practically there is always some slipping, which you can hear if you drive the car very slowly on a polished concrete surface (e.g. in a undercover carpark) and open your window.


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