Chapter 1

Three wheels

omnidirectional mobile robot kinematics

The notation

and most background information in this section are cited from 7 unless stated otherwise.

The robot under study in this project is a three-wheel

omnidirectional robot, all of the wheels are driven and are Swedish 90degree

wheels arranged as shown in Figure 2.1.

This section introduce notation used to present the robot motion in both global

frame and local frame of the robot, and develop a kinematic representation of

the whole robot.

?Figure 2.1 Three wheels omnidirectional robot. Adapted from 7.

1.1 Robot position

The model of the robot chassis is referred to as a rigid

body, ignoring all joints and degrees of freedom internal to the robot and its

wheels.

1.1.1

Global and local frames

Describing the position of the robot needs a steady

reference frame for the plane, this frame is called global reference frame, its

two axes are

and

. The frame refereed to the robot itself is

called local reference frame, the origin

of this frame is a reference point

chosen on the robot chassis (P), and the two axes

and

are

relative to this point, in this case (P) is chosen at the center of the robot

and

is

aligned with the axis of the second wheel. For motion description It’s also

important to find a relationship between these two frames, the angular

difference between the local and global frames is given by ?. This notation is shown in Figure

2.2.

?Figure 2.2 Global and local frames. Adapted from 7.

1.1.2

Robot pose

The pose of the robot at any instant is described by three

elements; its x and y coordinates, and the angle between global and local

references. The pose can be written as vector with three elements and it is

referred to the global frame:

And the robot motion at a certain pose

referred to global frame is given by the velocities

,

and

:

1.1.3

Robot motion mapping

To map the motion from a global reference to the robot local

frame, an orthogonal rotation matrix (R(

)) is used:

The robot motion at a certain pose

referred to local frame is given by:

1.2 Forward kinematics

Forward kinematics predicts the overall motion of the robot

on global frame given the speed and geometry of the wheels. The three wheels

omnidirectional robot has three wheels, each radius is r. The distance between

the center (P) and any of the three wheels is

. The wheel rotational position is given by

and the wheels spinning speed is given by

. Figure 2.3 shows the previous

parameters.

?Figure2.3 Robot and wheel geometry and speed. Adapted from 7.

The forward kinematics model represents the motion of

the robot as function of r,

,

,

,

and

:

To map the motion from local frame to

global frame the inverse orthogonal rotation matrix

is used:

1.3 Wheels kinematic constraints

In order to find the kinematic model for the mobile robot,

the wheels motion constraints should be presented. The kinematic properties and

constrains varies for different wheels types. Since the robot in this project

has 90 degrees Swedish wheels, this section will evaluate a presentation for

the Swedish wheels constrains.

At the begging, it’s important to make three assumptions

that simplify wheels motion constraints presentation, and present pure rolling

and rotation for the wheel without sliding content:

1.

The wheel’s plane remains

vertical.

2.

There is only one point of

contact between the wheel and the ground at the single moment.

3.

No sliding occurs at the

point of contact.

Under the previous assumptions,

there are two general motion constrains every wheel type:

1.

The concept of rolling

contact: the wheel should roll in appropriate amount based on the motion along

the direction of wheel plane to achieve pure rolling.

2.

The concept of no lateral

slippage: orthogonal sliding must not happen for the wheel on its plane.

?

Figure2.4 Swedish 90

degrees wheel and its parameters. Adapted from 7.

The Swedish wheel as shown in

Figure 2.8 has a radius r. The position

A is expressed by the distance between the center P and the wheel

and the angle

. The wheel rotational position around the

horizontal axis is

.

is

the angle from the wheel plane to the robot chassis and it is fixed for the

Swedish wheel.

is

the angle between the rollers axes and the wheels main axis, in case of 90

degrees Swedish wheel

.

The rolling constraint for the wheel to

achieve pure rolling along the wheel plane is given by:

But for the Swedish wheel, the small

rollers spin around the zero component of

the velocity at the contact point. Thus, moving in the direction of this

zero velocity component is impossible without sliding. For that, the rolling

constraint formula is represented along the velocity zero component instead of

the wheel plane :

For

the robot in this project,

is zero because the wheels are tangential to

robot chassis ,

for the Swedish 90 wheel is zero the rolling

constraint formula along the velocity zero component:

(*)

The sliding orthogonal to the wheel plane

is not constrained for the Swedish wheel because of the free rotation of the rollers

.

This is presented :

For

the robot in this project :

1.4 Complete kinematic model

A full model for the three wheels omnidirectional robot shown in Figure 2.3 can be built by rewriting the rolling

constrain equation (*):

(**)

Such that

is

a constant diagonal 3×3 matrix that consists the robot wheels radiuses. In this

project all radiuses are equal (r ):

And

is

a 3×3 matrix that relates to the rolling constrains:

Rewriting equation (**):

Representing the robot

motion as linear velocities in x and y direction

and

, and the rotational velocity

.

The robot motion can be described as a linear velocity V and

an angular velocity

as shown in Figure

2.3. The relationship between

,

,

in the global

frame and V ,

is

represented in the following equations:

1.5 The values of the angle

for the three wheels

As a convention

is

along the axis of the first wheel as shown in Figure 2.5. Thus, the angle

between

and the wheel (

) value for each wheel is constant and

their values are (

, (

and (

.

Figure ?2.5 Robot axis along wheel 1 axis and

value

As the robot changes its orientation, the angle between the

global frame and the local frame ?

shown in Figure 2.2 changes. The value of angle

at

the initial position shown in Figure 2.5 is zero.

Chapter 1

Three wheels

omnidirectional mobile robot kinematics

The notation

and most background information in this section are cited from 7 unless stated otherwise.

The robot under study in this project is a three-wheel

omnidirectional robot, all of the wheels are driven and are Swedish 90degree

wheels arranged as shown in Figure 2.1.

This section introduce notation used to present the robot motion in both global

frame and local frame of the robot, and develop a kinematic representation of

the whole robot.

?Figure 2.1 Three wheels omnidirectional robot. Adapted from 7.

1.1 Robot position

The model of the robot chassis is referred to as a rigid

body, ignoring all joints and degrees of freedom internal to the robot and its

wheels.

1.1.1

Global and local frames

Describing the position of the robot needs a steady

reference frame for the plane, this frame is called global reference frame, its

two axes are

and

. The frame refereed to the robot itself is

called local reference frame, the origin

of this frame is a reference point

chosen on the robot chassis (P), and the two axes

and

are

relative to this point, in this case (P) is chosen at the center of the robot

and

is

aligned with the axis of the second wheel. For motion description It’s also

important to find a relationship between these two frames, the angular

difference between the local and global frames is given by ?. This notation is shown in Figure

2.2.

?Figure 2.2 Global and local frames. Adapted from 7.

1.1.2

Robot pose

The pose of the robot at any instant is described by three

elements; its x and y coordinates, and the angle between global and local

references. The pose can be written as vector with three elements and it is

referred to the global frame:

And the robot motion at a certain pose

referred to global frame is given by the velocities

,

and

:

1.1.3

Robot motion mapping

To map the motion from a global reference to the robot local

frame, an orthogonal rotation matrix (R(

)) is used:

The robot motion at a certain pose

referred to local frame is given by:

1.2 Forward kinematics

Forward kinematics predicts the overall motion of the robot

on global frame given the speed and geometry of the wheels. The three wheels

omnidirectional robot has three wheels, each radius is r. The distance between

the center (P) and any of the three wheels is

. The wheel rotational position is given by

and the wheels spinning speed is given by

. Figure 2.3 shows the previous

parameters.

?Figure2.3 Robot and wheel geometry and speed. Adapted from 7.

The forward kinematics model represents the motion of

the robot as function of r,

,

,

,

and

:

To map the motion from local frame to

global frame the inverse orthogonal rotation matrix

is used:

1.3 Wheels kinematic constraints

In order to find the kinematic model for the mobile robot,

the wheels motion constraints should be presented. The kinematic properties and

constrains varies for different wheels types. Since the robot in this project

has 90 degrees Swedish wheels, this section will evaluate a presentation for

the Swedish wheels constrains.

At the begging, it’s important to make three assumptions

that simplify wheels motion constraints presentation, and present pure rolling

and rotation for the wheel without sliding content:

1.

The wheel’s plane remains

vertical.

2.

There is only one point of

contact between the wheel and the ground at the single moment.

3.

No sliding occurs at the

point of contact.

Under the previous assumptions,

there are two general motion constrains every wheel type:

1.

The concept of rolling

contact: the wheel should roll in appropriate amount based on the motion along

the direction of wheel plane to achieve pure rolling.

2.

The concept of no lateral

slippage: orthogonal sliding must not happen for the wheel on its plane.

?

Figure2.4 Swedish 90

degrees wheel and its parameters. Adapted from 7.

The Swedish wheel as shown in

Figure 2.8 has a radius r. The position

A is expressed by the distance between the center P and the wheel

and the angle

. The wheel rotational position around the

horizontal axis is

.

is

the angle from the wheel plane to the robot chassis and it is fixed for the

Swedish wheel.

is

the angle between the rollers axes and the wheels main axis, in case of 90

degrees Swedish wheel

.

The rolling constraint for the wheel to

achieve pure rolling along the wheel plane is given by:

But for the Swedish wheel, the small

rollers spin around the zero component of

the velocity at the contact point. Thus, moving in the direction of this

zero velocity component is impossible without sliding. For that, the rolling

constraint formula is represented along the velocity zero component instead of

the wheel plane :

For

the robot in this project,

is zero because the wheels are tangential to

robot chassis ,

for the Swedish 90 wheel is zero the rolling

constraint formula along the velocity zero component:

(*)

The sliding orthogonal to the wheel plane

is not constrained for the Swedish wheel because of the free rotation of the rollers

.

This is presented :

For

the robot in this project :

1.4 Complete kinematic model

A full model for the three wheels omnidirectional robot shown in Figure 2.3 can be built by rewriting the rolling

constrain equation (*):

(**)

Such that

is

a constant diagonal 3×3 matrix that consists the robot wheels radiuses. In this

project all radiuses are equal (r ):

And

is

a 3×3 matrix that relates to the rolling constrains:

Rewriting equation (**):

Representing the robot

motion as linear velocities in x and y direction

and

, and the rotational velocity

.

The robot motion can be described as a linear velocity V and

an angular velocity

as shown in Figure

2.3. The relationship between

,

,

in the global

frame and V ,

is

represented in the following equations:

1.5 The values of the angle

for the three wheels

As a convention

is

along the axis of the first wheel as shown in Figure 2.5. Thus, the angle

between

and the wheel (

) value for each wheel is constant and

their values are (

, (

and (

.

Figure ?2.5 Robot axis along wheel 1 axis and

value

As the robot changes its orientation, the angle between the

global frame and the local frame ?

shown in Figure 2.2 changes. The value of angle

at

the initial position shown in Figure 2.5 is zero.