Omnidirectional and Mecanum Wheels

Mecanum wheels have small rollers set at 45 degrees around their rim. Each wheel pushes the robot partly forward and partly sideways. By spinning the four wheels at the right speeds, the robot can move in any direction while facing any way, which makes it holonomic.

Forward speed Sideways speed (left) Spin rate

Inverse kinematics

For a body velocity $(v_x, v_y, \omega)$, with wheel radius $r$ and half-length plus half-width $l_x + l_y$, the four wheel speeds are

$$\begin{bmatrix} \omega_{FL} \\ \omega_{FR} \\ \omega_{RL} \\ \omega_{RR} \end{bmatrix} = \frac{1}{r}\begin{bmatrix} 1 & -1 & -(l_x + l_y) \\ 1 & 1 & (l_x + l_y) \\ 1 & 1 & -(l_x + l_y) \\ 1 & -1 & (l_x + l_y) \end{bmatrix}\begin{bmatrix} v_x \\ v_y \\ \omega \end{bmatrix}$$

What to notice

Forward motion spins all four wheels the same way. Sideways motion spins diagonal pairs in opposite directions, so the sideways pushes from the rollers add up while the forward pushes cancel. Diagonal motion uses only two of the wheels. The robot can also drive in a straight line while spinning.

The cost is efficiency and traction: rollers slip by design, so mecanum robots push less hard, wear faster, and struggle on rough ground. They shine on smooth factory and warehouse floors.