I'm not sure whether this is the correct community to post this, so pardon me.

I was studying the Bicycle Kinematic Model and came across 3 possible reference points for analysis - the rear tire, front tire and the centre of gravity. For all these reference points, Instantaneous Centre of Rotation was applied and velocities were derived. Instantaneous Centre of Rotation

I have depicted the velocities of the wheels, the steering angle and the angular velocity of the system.

Under no-slip condition, velocity of the rear wheel can be expressed as -

The front wheel velocity can also be expressed similarly,

For both these equations to hold, R must be equal to R', which is not true. Where am I going wrong? Either both wheels must have different velocities or the no-slip condition must be false, both of which the Bicycle Kinematic Model refutes. I believe I'm missing something right under my nose. Any help will be appreciated. Thanks!

  • $\begingroup$ if v=R*w for a wheel then R is the radius of the wheel, not the radius to a center around wich the bicycle drives. $\endgroup$
    – trula
    Jul 24 '20 at 14:06
  • $\begingroup$ R and R' are not equal, when a bicycle turns like this the front wheel turns faster than the rear because the front wheel has more ground to cover. $\endgroup$ Jul 24 '20 at 18:53
  • $\begingroup$ @trula What you are saying is also correct. If the radius of the wheel is R1 and the angular speed of the wheel is w1, then the velocity of the wheel can be expressed as v=R1*w1 under no-slip condition. This same velocity (v) can also be expressed as v=R*w, where R is the distance from the ICR and w is the angular speed of the system. $\endgroup$ Jul 24 '20 at 22:50
  • $\begingroup$ @AdrianHoward What exactly do you mean by "the front wheel turns faster"? Are you referring to the rate of change of delta? If yes, I don't understand how it affects the equations I have mentioned in the question. If you are suggesting that the longitudinal velocity of the front wheel (v) must be greater than that of the rear wheel, it makes sense and I too believe that it solves the problem. But the Kinematic Bicycle Model assumes the longitudinal velocity for both the wheels to be same. I will link an article in the next comment. $\endgroup$ Jul 24 '20 at 23:02
  • $\begingroup$ @AdrianHoward This article uses the assumption I mentioned before. $\endgroup$ Jul 24 '20 at 23:06

The assumption that the wheels have to have the same velocity is wrong (and in general is not true for two points on a rotating object).

It helps here to think about the extreme case where the front wheel $\delta$ angle is 90 degrees, so that the car pivots about its rear wheel. The front wheel is rolling and the rear wheel is not, so clearly they are rolling at different speeds.


If the radius of the wheels are different, the angular velocities are also different. So both can have the same linear velocity and fulfill the condition of no-slip.

  • $\begingroup$ The wheels don’t have to have the same linear velocity, because the whole vehicle is rotating. $\endgroup$
    – RLH
    Jul 27 '20 at 2:31
  • $\begingroup$ yes. I thought it was a bicycle. $\endgroup$ Jul 27 '20 at 13:43

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