Four-Wheel Front Steer Robot

A 4-wheel front steering robot has the following form:

Often this is simplified to a bicycle model, where two wheels are lumped into a single wheel

• There are a few different forms of the bicycle model depending on the point of interest.

Bicycle model for G.C

For example, the bicycle model for the geometric center/centroid:

$$\begin{array}{rl}\dot{x}& =v\cos (\theta +\beta )\\ \dot{y}& =v\sin (\theta +\beta )\\ \omega & =\dot{\theta }=\frac{v}{R}=\cos \beta \frac{\tan \delta }{L}v\end{array}$$

From the right triangles that connect the axles to the I.C.R:

$$\begin{array}{rl}\tan \beta & =\frac{L/2}{R_r},\tan \delta =\frac{L}{R_r}⟹\tan \delta =2\tan \beta \\ \cos \beta & =\frac{1}{\sqrt{1+{\tan }^2\beta }}=\frac{2}{\sqrt{4+{\tan }^2\delta }}\end{array}$$

Thus, we have

$$\begin{array}{rl}\dot{x}& =v\cos (\theta +\beta )\\ \dot{y}& =v\sin (\theta +\beta )\\ \dot{\theta }& =\frac{2\tan \delta }{L\sqrt{4+{\tan }^2\delta }}v\end{array}$$

• where we have $\beta ={\tan }^{-1}\left(\frac{\tan \delta }{2}\right)$

Kinematics for wheel control

What are the wheel speeds $(u_r,u_f)$ to move the G.C with $v$ if we are doing rear-wheel drive or front-wheel drive?

Rear Wheel Drive

Pure rolling at the real wheel (with wheel radius $r$) gives its linear speed:

$$v_r^w=r{u}_r=R_r\dot{\theta }$$

which results in

$$\begin{array}{r}\dot{\theta }=\frac{v}{R}=\frac{v_r^w}{R_r}⟹v_r^w=\frac{R_r}{R}v\\ \cos \beta =\frac{R_r}{R}=\frac{2}{\sqrt{4+{\tan }^2\delta }}\end{array}$$

Thus, we have:

$$v_r^w=\frac{2}{\sqrt{4+{\tan }^2\delta }}v$$

Equivalently, if the rear wheel is the actuator (rear-wheel drive):

$$v_r^w=\frac{r\sqrt{4+{\tan }^2\delta }}{2}{u}_r$$

Front Wheel Drive

Similarly, for front-wheel drive, we would have

$$v=\frac{r\cos \delta \sqrt{4+{\tan }^2\delta }}{2}{u}_f$$