Wheeled Motion

Four Basic Wheel Types

Standard Wheel
Spherical Wheel - A ball with omnidirectional movement.
Castor Wheel - A wheel that can rotate around a castor and it’s own axle.
Swedish Wheel - A wheel made of 45 degree rollers. This allows passive motion in the direction of the rollers.

Stability is guaranteed with 3 wheels
- This is improved by using 4 or more wheels.
Bigger wheels allow the robot to overcome higher obstacles.
Most wheel arrangements are non-holonomic
- There are constraints in the direction that the vehicle can move.
Combining actuation and steering on one wheel complicates design and adds odometry errors.

The velocity of a wheel is determined by the wheel diameter and the change in rotation over time ($\omega=\frac{\Delta\theta}{\Delta t}$)

\[v = \omega r\]

This is an ideal scenario where there are no losses or deformation.

Wheels have two degrees of freedom. They can rotate in the x-axis and translate in the y-axis.

The manoeuvrability of a vehicle depends on the degree of mobility:

$\delta_m$ quantifies the degrees of controllable freedom based on changes to the wheel’s velocity.

\[\delta_m = 3 - N_k\]

where $N_k$ is the number independent kinematic constraints.

Independent means that they are on different axes.

$\delta_m$ quantifies the degrees of controllable freedom based on changes in wheel velocity.

A tricycle has:

Two wheels in the same axis as the back.
- Even if they can turn independently they are on the same axis, therefore dependant.
A front steering wheel.

This gives two kinematic constraints therefore the degree of mobility is:

\[\delta_m = 3 - 2 = 1\]

The degree of steerability ($\delta_s$) is the number of independent steerable wheels (where $0\leq\delta_s\leq2$).

$\delta_s$ quantifies the degrees of controllable freedom based on changes in wheel orientation.

The degree of manoeuvrability $\delta_M$: The overall degrees of freedom that a robot can manipulate by changing the wheel’s speed and orientation

\[\delta_M = \delta_m+\delta_s\]

This is an indicator as to how easy it is for a robot to move around.

Vehicles with non-holonomic constraints are unable to move in all directions of a 2D plane:

A Bike has non-holonomic contraints as it has to rotate in order to go left or right.
A ball is holonomic as it can move in any direction.