# Chapter the center of the robot and is

Chapter 1
Three wheels
omnidirectional mobile robot kinematics

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

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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.