Kinematics Model of

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Nov 13, 2013 (3 years and 8 months ago)

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Kinematics Model of
Nonholonomic Wheeled Mobile
Robots for Mobile Manipulation
Tasks


Dimitar Chakarov


Institute of Mechanics
-

BAS, 1113 Sofia,
“Acad.G.Bonchev” Str., Block 4

Outline

Introduction

Holonomic and nonholonomic WMR


Direct and reverse kinematics task of
nonhoonomic WMR

Conclusion

1.Introduction(1/2)

The

robotized

technologies

are

very

quickly

spreading

for

domestic,

service

and

entertainment

needs
.

A

big

number

of

scientific

investigations

and

scientific

activities

form

a

new

scientific

field

during

the

recent

years

devoted

to

mutual

interaction

among

robots

and

the

human

being
.

This

is

an

interdisciplinary

scientific

field

that

covers

robotics,

computer

sciences,

and

the

science

of

knowledge,

physiology

and

sociology
.

Robots

are

going

very

soon

to

assist

the

human

being

on

a

wide

range

of

problems,

which

are

not

attractive,

they

are

dangerous,

not

well

paid

or

boring

to

humans
.

Robots

assistants

are

going

to

work

in

the

future

as

patient

sitters,

as

security

guards,

as

rescuers

and

fire

rescuers,

in

surgery

and

rehabilitation,

in

domestics

and

in

offices,

in

mining,

in

building

as

well

as

in

stores

and

museums
.


1.Introduction(2/2)

In order to work together with and to assist and to interact with
people the new robot generation must posses a mechanical
structure that is suitable for this partnership in the human not
organised and unknown surrounding environment.


Wheeled mobile robots (WMR) are long ago invented in areas,
where they interact with humans as service robots for remote
book reading in a library or for serving tea and meals, human
following and guiding robots, entertainment and cleaning
robots.

The compatible with people robots must integrate mobility and
manipulation. Mobile manipulator systems hold promise in
many industrials and service applications including
manipulations, assembly, inspection and work in hazardous
environments. The integration of a manipulator and a mobile
robot base places special demands on the vehicle's mechanical
system.
.

The objective
of the present paper is to evaluate the possible
solutions of mobile robot bases and to build a kinematics
model of WMR suitable for mobile manipulation tasks.

2.
Holonomic and
nonholonomic WMR

(1/5)

The wheeled
mobile robots (WMR) are divided
into two basic
types
-

holonomic and nonholonomic. Theoretically, the
holonomic mechanical systems comprise links, that impose
restrictions on the limb velocities, and after integration these
restrictions can be reduced to restrictions only on the limb
locations. When WMR do not impose restrictions on the motion
velocities in the 2D (planar) solution they are called holonomic.

The holonomic WMR possess maximal number of degrees of
freedom in the 2D (planar) solution h=3. In the field of the
mobile robots the term holonomic is used as an abstract term for
WMR with three degrees of freedom. Thus, every WMR with
three degrees of freedom in plane is called a holonomic one
.


2.
Holonomic WMR

(2/5)

Various mechanisms are used as universal or omni wheels,
orthogonal or ball wheels in order to achieve a holonomic
motion. [5]. The holonomic WMR allow easier motion planning
in a plane. A holonomic WMR is shown on Fig.1.

Fig.1. Holonomic WMR

2.

Nonholonomic WMR

(3/5)

The nonholonomic mechanical systems comprise links restricting the system
velocities; thus these restrictions cannot be integrated. In this way the
nonholonomic WMR impose restrictions on the velocities of the motions in
plane.

Due to this reason the nonholonomic WMR possess less than three degrees
of freedom in plane
h < 3.
They are simpler in construction and thus cheaper,
with less controllable axes and ensure the necessary mobility in plane. Due
to this reason a kinematics model of nonholonomic WMR with two degrees
of freedom
h=2
are derived in the present work

.
A nonholonomic WMR is
shown on Fig.2.

Fig.2. Nonholonomic
WMR.

2.

Nonholonomic WMR

(4/5)

X
1

X

Y
1

Y

V
c
r

V
c
l

V
P

P

b

b

r

r

r






ф

l



O

r

C

V
c

X
1

T

L

Z
1




Fig.3. A general model of non
-
holonomic WMR.

In

Fig
.

3

a

generalised

model

of

a

nonholonomic

WMR

with

h=
2

is

presented
.

It

includes

two

symmetrically

allocated

driving

wheels

with

radii

r
.

The

nonholonomic

WMR

include

a

various

number

universal

wheels

for

keeping

up

the

balance

in

plane
.

This

wheel

is

not

driving

one

and

it

is

not

included

in

the

kinematics

model
..

In

the

robot

centre

P

is

connected

a

local

co
-
ordinate

system

PX
1
Y
1

,

where

X
1

is

along

the

axis

of

symmetry,

and

Y
1

is

along

the

axis

of

the

driving

wheels
.

The

angle

between

the

axis

X
1

and

the

axis

X

of

the

immovable

co
-
ordinate

system

OXY

is

denoted

with

ф
.

The

distance

between

the

driving

wheels

along

the

axis

Y
1

is

2
b
,

and

the

angular

velocities

of

the

left

and

the

right

wheel

are

given

,

.


l
θ

2.

Nonholonomic WMR

(5/5)

X
1

X

Y
1

Y

V
c
r

V
c
l

V
P

P

b

b

r

r

r






ф

l



O

r

C

V
c

X
1

T

L

Z
1




When

one

wheel

is

rolling

on

a

straight

line

without

slipping

with

angular

velocity


,

its

centre

is

moving

with

velocity

Vc
.

The

velocity

of

the

oscillate

point

T

with

the

plane

L

is

0

and

thus

the

equation

(
1
)

is

fulfilled
:


θ

0



r
θ
V
V
c
T

(
1
)

This

equation

can

be

integrated

and

can

be

presented

as

a

link

among

the

angular

and

the

linear

position

of

the

wheel
.





When

the

wheel

is

rolling

along

a

curved

line

the

linear

velocity

of

its

centre

Vc,

in

the

base

co
-
ordinate

system

OXY

depends

on

the

wheel

orientation

in

the

plane

defined

by

the

angle

ф
.

These

equations

(
2
)

can

not

be

integrated

in

order

to

define

relations

only

between

the

wheel

positions
.

In

the

plane

motion

on

the

wheel

velocities

are

imposed

restrictions,

thus

the

mobile

devices

from

the

type

shown

in

Fig
.

3
,

are

called

nonholonomic

WMR

0
sin
0
cos




φ
r
θ
Y
φ
r
θ
X
c
c




(
2
)

l
-

θr = 0

3. Direct kinematics task of
nonhoonomic WMR.
(1/3)

X
1

X

Y
1

Y

V
c
r

V
c
l

V
P

P

b

b

r

r

r






ф

l



O

r

C

V
c

X
1

T

L

Z
1




The

nonholonomic

mobile

devices

include

two

co
-
axial

driving

wheels,

the

velocities

of

their

centres

Vcr

и

Vcl

are

co
-
linear

with

the

axis

X
1

of

the

local

co
-
ordinate

system

PX
1
Y
1
.

The

velocity

Vp

of

the

centre

P

of

the

mobile

platform

is

also

co
-
linear

with

the

axis

X
1
.

The

plan

of

velocities

of

WMR

in

the

plane

OXY

is

presented

in

Fig
.
1
.

The

following

equations

can

be

derived
:





(
4
)

l
c
r
c
V
V
b
2




)
V
V
(
2
1
V
l
c
r
c
p


(3)


(4)


Fig.3. A general model of
non
-
holonomic WMR.

3. Direct kinematics task of
nonhoonomic WMR.
(2/3)

If

we

derive

the

upper

equations

along

the

axes

of

the

base

co
-
ordinate

system

OXY,

where

the

velocity

of

the

centre

Vp

is

presented

by

the

co
-
ordinates

,




and

the

velocities

Vcr

и

Vcl

of

the

right

and

the

left

wheel

is

defined

with

the

help

of

(
2
)



,



thus

equations

are

derived

defining

the

kinematics

of

WMR

in

the

base

co
-
ordinate

system

T
p
p
0
p
]
Y
;
X
[
V



l
r
l
r
p
l
r
p
b
2
r
b
2
r
sin
2
r
sin
2
r
Y
cos
2
r
cos
2
r
X


























(
5
)

0
sin
0
cos




φ
r
θ
Y
φ
r
θ
X
c
c




(
2
)

3. Direct kinematics task of
nonhoonomic WMR.
(3/3)

If the velocity

V
p

of the centre of the mobile platform and
its velocity of rotation


we combine in the vector:







T
p
p
]
;
Y
;
X
[
X







and

the

velocities

of

the

driving

wheels


l


,


we

combine

in

the

vector


T
l
r
]
;
[







then

the

direct

task

of

the

kinematics

of

WMR

is

presented

by

the

vector

equation






S
X
where



















b
2
r
;
b
2
r
sin
2
r
;
sin
2
r
cos
2
r
;
cos
2
r
S
(
6
)

(
7
)

(
8
)

(
9
)

4. Reverse kinematics task
of nonhoonomic WMR.
(1/3)

It

is

necessary

to

solve

the

reverse

task

of

the

kinematics

in

order

to

plan

the

robot

motion

and

the

robot

control
.

In

order

to

find

a

solution

for

the

reverse

task

of

the

kinematics

of

WMR,

the

pseudo

inverse

matrix

S+

can

be

used,

because

matrix

S

is

not

a

quadratic

one
.




It is necessary to use an additional restriction on absolute
parameters defined by the
nonholonomic links of WMR because in this case the number
of the input parameters (6) is bigger than the number of the
output parameters (7):


X
S





0
X
A


T
p
p
]
;
Y
;
X
[
X






(10)


(11)


4. Reverse kinematics task
of nonhoonomic WMR.
(2/3)

Matrixes

S+

and

A

can

be

easily

defined

by

using

the

equalities

(
3
)

and

(
4
)

presented

in

the

form

below
:

l
c
p
r
c
p
V
b
V
V
b
V








(12)


when we define the upper equations along the axes of the local co
-
ordinate system PX1Y1. The velocity of the centre
Vp

is defined
along the axes
X1
and
Y1
by means of consideration of their angle
of rotation
ф
. In this way equations (12) along axis X1 present:

l
p
p
r
p
p
r
b
sin
Y
cos
X
r
b
sin
Y
cos
X






















and

along

axis

Y
1

present
:

0
cos
Y
sin
X
p
p







(13)


(14)


4. Reverse kinematics task
of nonhoonomic WMR.
(3/3)

Equation

(
13
)

express

the

reverse

task

in

kinematics



from

where

for

the

matrix

S+

we

can

derive
:



Equality

(
14
)

expresses

the

restriction

equation




between

the

velocities

of

the

nonholohomic

WMR

including

the

matrix
:





X
S


















b
;
sin
;
cos
b
;
sin
;
cos
r
1
S
0
X
A




0
;
cos
;
sin
A




(15)


(16)


5. Conclusion.

The developed kinematics model of the nonholonomic WMR
can be used for control creation in mobile tasks or in mobile
manipulation tasks.

Control can be based on the direct kinematics task by means
of using equation (5). In this case the equilibrium of the
given velocities of the wheels , guarantee motion along a
straight line, and the difference between them defines robot
rotation. This control mode belongs to a lower level and is
more convenient for derivation of on
-
line tasks of motion.

Control can be build up on the inverse kinematics task by
using equation (10). In this case control models are derived
including tracing a path considering the nonholonomic
restrictions (11). These control schemes are more
sophisticated, they can consider also the dynamics of WMR,
they can include adaptiveness and robustness.

Thanks for your attention!!