Four-Wheel Differential Drive Robot

Given a four-wheel differential drive robot, what are the kinematics for wheel control? What are the wheel speeds $(u_r,u_{\ell })$ to move the G.C with $v$ and $\omega$?

Similar to the 2-wheel differential drive:

$$\begin{array}{rl}v& =R\omega \\ R_r& =R+\frac{T}{2}\\ \ell & =\frac{L}{2}\\ \tan {\lambda }_r& =\frac{\ell }{R_r}\end{array}$$

Then, we write $v_r$ as:

$$v_r=\omega \sqrt{R_r^2+{\ell }^2}=v_r^w\cos {\lambda }_r$$

We have:

$$\begin{array}{rl}{v}_r^w& =\frac{\omega \sqrt{R_r^2+{\ell }^2}}{\cos {\lambda }_r}\\ & =\frac{\omega \sqrt{R_r^2+{\ell }^2}}{1/\sqrt{1+{\tan }^2{\lambda }_r}}\\ & =\frac{\omega \sqrt{R_r^2+{\ell }^2}}{1/\sqrt{1+\frac{{\ell }^2}{R_r^2}}}\\ & =\omega \left(R_r+\frac{{\ell }^2}{R_r}\right)\\ & =v+\frac{T\omega }{2}+\frac{{\left(\frac{L\omega }{2}\right)}^2}{v+T\omega }2\end{array}$$

Thus, we have

$$u_r=\frac{v_r^w}{r}=\frac{1}{r}\left(v+\frac{T\omega }{2}+\frac{{\left(\frac{L\omega }{2}\right)}^2}{v+T\omega }2\right)$$

The example below shows the derivation process to get the equation for the left wheel to drive the G.C at $v$ and $\omega$.

Note that because of friction, the actual motion becomes different from the ideal case, so we usually use approximate motion model, e.g.:

$$\begin{array}{r}v\approx \frac{r}{2}(u_r+u_{\ell })\\ w\approx \frac{r}{T}(u_r-u_{\ell })\end{array}$$