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!!
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