I'm aware of the Ackermann Steering Angle, however, I still struggle to come up with an equation that model the relationship between $\dot{x}, \dot{y}$ (the horizontal and vertical velocity of the centre of mass, indicated by yellow dot in the diagram) with the car's velocity($v$) and the steering angle of the wheels (indicated by $\theta L,\:\theta R$ in the diagram)
I want to be able to come up with an expression, assuming that $\theta L=\theta R$. In other words, how do I come up with an equation that describe the relationship between $\theta$ and $\beta$?
I suppose that the Ackerman model is based on the fact that $\theta L\neq \theta R$, thus being able to draw a point of rotation outside the car. However, in the real world, when turning, do actual car have different $\theta$ for its left and right tire?
What my beliefs are:
- The car's tire would be constantly slipping in the Ackermann model.
- The Axle Width (AW), is irrelevant, meaning that the relationship of $\theta$ and $\beta$ is independent of AW, in the equal left and right $\theta$ case.
I come across a website that suggest the relationship to be this for the equal $\theta$ case: $$ tan(\beta) = \frac{FL}{FL+RL}tan(\theta) $$ I have no idea how to derive that relationship. I would really appreciate if someone can please tell me how to come up with that equation or proof that it is correct? I tried using similar triangles to think in "reverse" for coming with that equation, but I can't seem to get it.