**Omni-wheeled system motion**

We now consider the motion of a simple omni-wheeled system,

where rotation is fixed. Consider Figure 2.3.1, showing the vectors

acting on one driving wheel.

where,

**Vw**is the velocity of the wheel,

**θ**is the reference wheelangle,

**Vin**is the induced velocity on wheel,

**φ**is the reference body

velocity angle, and

**Vb**is the body velocity of robot.

Now Vin and Vw are always orthogonal:

Vb2 = Vw2 + Vin2 (1)

Also:

Vin2 = Vb2 + Vw2 - 2 Vw Vb cos(θ - φ)

= Vb2 + Vw2 - 2 Vw Vb (cosθ cosφ + sinθ sinφ) (2)

Substituting (2) into (1), we may obtain:

Vw = Vb(cosθ cosφ + sinθ sinφ ) (3)For a given rotational velocity of the centre of mass, w, each

wheel must apply velocity:

Vw= Rw (4)

where

**R**is the distance of the wheel from the centre of mass.

Thus, for each wheel:

Vw = Vb(cosθ cosφ + sinθ sinφ ) + Rw (5)

This is a general equation that is independent of the number

of wheels. Consider a three wheeled omni-directional vehicle

with wheels arranged at angles of 0°, 120° and 240°, equation (5)

yields:

Wheel 1 (θ = 0°): Vw 1 = Vb cosθ +Rw

Wheel 2 (θ = 120°): Vw 2= Vb (-0.5 cosθ+0.866 sinθ)+Rw

Wheel 3 (θ = 240°): Vw 3= Vb (-0.5 cosθ-0.866 sinθ)+Rw

from

Mark Ashmore and Nick Barnes, Omni-drive robot motion on curved

paths: The fastest path between two points is not a straight-line

**Omni-wheeled**