KINEMATICS AND DYNAMICS OF CONFIGURABLE WHEEL-LEG

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30 Οκτ 2013 (πριν από 3 χρόνια και 7 μήνες)

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345


8th International DAAAM Baltic Conference

"
INDUSTRIAL ENGINEERING

19
-
21 April 2012, Tallinn, Estonia


KINEMATICS AND DYNAMICS OF CONFIGURABLE WHEEL
-
LEG



Sell, R; Aryassov, G; Petrits
h
enko, A
;
Kaeeli, M



Abstract:
In this paper, the kinematics and
dynamics of the configurable wheel
-
leg is
considered. The configurable wheel
-
leg
has been invented by TUT researchers to
increase the mobility of the wheeled
vehicles, mostly unmanned ground
vehicles, on different terrains

and to
overcome obstacles that are unobtainable
to the conventional wheels e.g. climbing on
stairs. The paper presents an overview of
similar wheels and focuses mostly on the
kinematics and dynamics of invented
wheel. The result is a mathematical model
wh
ich can be
used

for further analysis and
simulations.


Key words: wheel dynamics, wheel
-
leg,
variable diameter wheel


1. INTRODUCTION


Today’s world several different types of
mobile robots are in use, both civilian and
military domain. Wheeled robots have

advantages of good performance when
moving on smooth roads. However when
obstacle is on the road or robot needs to
turn off
-
road the good performance is gone
and different type of locomotion principle
have advantages to overcome of the
obstacle or move on

rough terrain. In
rescue robots, it is often case that robot
have to run relatively long on smooth
terrain but then needs to go up to stairs. It
is clear that wheels are not suitable
climbing up to stairs. Therefore these
robots have usually two locomotio
n
options mounted on the robot. Tracks are
mounted with combination of wheel. This
solution is not efficient in terms of energy
consumption, complexity and stability.



Fig.
1
. Wheel
-
leg (Wheg) in closed and
open configuration


In this paper novel patented [1] solution
wheel
-
leg (Wheg), shown in Fig. 1.
i
s

introduced. The solution combines the
advantages of different locomotion
principles by changing the geometry of the
end actuator on the fly. By changing the
wheel geometry kine
matic and dynamic
parameters of the wheel
-
leg are also
changed considerably. The study gives
essential understanding of kinematic and
dynamic properties of the non
-
conventional rigid wheel
-
leg and provides
necessary information to estimate the
minimum torq
ue of wheel
-
leg actuators.

The study is a part of wider research area,
which is connected to the study of
mobility,
maneuverability

and trajectory
control of wheeled mobile robotics
equipped with wheel
-
legs.


2. STATE
-
OF
-
ART


Inventing the wheel has been attracted
mankind already for thousands of years.
Even conventional wheel has been in use
without of major modification a long time
several inventions have been registered
346


during last century. The main driving force
to invent n
ew versions of wheel is to
improve the passability on rough terrains
and climb over the obstacles or holes.
Several inventions have been developed in
Japan [2],
in
US [3],
in
Russia [4] and
other countries. Several dynamically
configurable wheels are also
invented very
recently, party driven by space missions
where the rover has to be energy efficient
and uneven terrain capability on the same
time. Zheng et.al have been introduced the
diameter
-
variable wheel [5] for out
-
door
rover which deploys planar polyg
onal
mechanism. Another concept is presented
by Xinbo et.al. [6]
w
here

wheel is
segmented and by expanding the segments
the diameter is also extended. Well known
application

is so called Galileo
wheel

[7]


which uses elastic long scale expandable
tracks a
s a
tire
. In addition to variable
diameter wheel construction several fixed
sized but different shape end actuators are
developed which can be placed between
wheel and leg. Most
well
-
known

is a
Boston Dynamics RHex robot which uses
elastic half
-
circular wh
eel
-
leg as end
actuator [8]. All described solutions have
their benefits and drawbacks. In next
section we are presenting the solution
integrating the wheel and leg locomotion
principles very tightly by increasing the
energy efficiency and construction
dim
ension by offering the wide scale of
geometry and dynamic performance
change.


3. WHEEL
-
LEG


Wheel
-
leg is a mechanism, invented by
paper authors, that includes good qualities
of both wheels and legs. The result of that
is a good passing ability in differen
t terrain
including stairs and steps. In the smooth
terrain the wheel regime is used. When
terrain changes to hardly passable the
wheel
-
leg adjusts its configuration by
opening the wheel segments so that the
passing ability increases drastically. In Fig.
1

two different regime of wheel
-
leg is
shown, where the change from one regime
to other can be done even during the
normal operation.

Due to design parameters, the wheel
-
leg
can operate in different mode of operation
and change its configuration dynamicall
y
(Fig. 1.). When blades of the wheel
-
leg are
closed, it operates nearly as conventional
rigid wheel. In case of the blades of the
wheel
-
leg are open, the wheel
-
leg operates
as non
-
circular wheel and its kinematic and
dynamic properties vary within its pos
ition.


4. KINEMATICS OF WHEEL
-
LEG


4.1 Kinematics of wheel
-
leg in
the
Cartesian

reference frame


In
Fig. 2

the physical and mathematical
model of the fully open variable diameter
rigid wheel
-
leg constrained to move in
plane and on the rigid surface. The
wheel
-
leg is
modeled

as regular hexagon with
corners rounded. The rounded corners are
represented in Fig. 3 as circles with
continuous lines. The sides of the hexagon
correspond to the tangencies to the rounded
corners of physical model with constant
radiu
s.


Fig.
2
. The physical and mathematical
model of whe
el
-
leg


The position of the rigid body in plane is
completely defined if it is defined the
position of the two points of the rigid body
at any instant time of its motion. To define
the position of the
center

of the wheel
-
leg
the
two

reference frames in Fig. 3 are
introduced: ground
-
fixed reference frame
Oxy

and

reference frame
Bx
1
y
1

that moves
translatory with respect of the ground
-
fixed
reference frame.

347



Fig.
3
. Rotation of half of the hexagon


The position of the wheel
-
leg is chosen so
that at the beginning of the motion the
center

of wheel
-
leg coincides with
coordinate axis
y

of ground reference
frame and angle
θ

of

the tangency to the
rounded corners of wheel
-
leg and
coordinate axis

x

is
θ=0.


Through the rotation of the 1/6
th

of one
revolution of the hexagon, the angle
θ
changes from
0

to
π/3
that corresponds to
the changes of the angle
φ

from
-
π/6

to
π/6

measured from the
y

coordinate axes. The
position vector of the
center

C

of the
hexagon can then be expressed by it
components through the generalized
coordinate
φ

as



(1)


where

-

the component of the vector
r
B

with respect of the coordinate axis
O
x

and
correspond
s
to the initial
position

of
instantaneous center of rotation
;
-

the
component of the vector
r
B

with respect of
the coordinate axis
O
y
;

-

length of the
vector
r
BC

;

-

rotation angle of vector

r
BC

with respect of reference frame
B
x
1
y
1
. The
first time derivative from the both sides of
Eq. (1) yields to velocity expression as



(2)


where



the angular velocity vector of
the vector
r
BC

respect to the reference frame
B
x
1
y
1
.

The second time derivative from the
both sides of the Eq. (2) yields to the
acceleration of the
center

C as



(3)



where

angular acceleration of the
vector
r
BC

with respect to the reference
frame
B
x
1
y
1
.


4.2 General kinematics of wheel
-
leg


In this section the
alternative

method to
establish the kinematic
characteristics

of
the wheel
-
leg is
considered
. It is based on
the Euler
-
Savary formulation of the
moving and fixed cen
troids of wheel
-
leg
.

This method
can have advantage
s in
some
cases when wheel
-
leg moves on the curved
or other surfaces in space.

To derive the
kinematic characteristics of

the wheel
-
leg
the Fig.
4

is used.


Fig.
4
. Rotation of the blade of the wheel
-
leg.


Let the
α
-
α
will be

the trajectory of the
center of the wheel
-
leg, when blade
L

rolls
on rigid surface. The motion of the blade
represents the motion of the moving
centroid on fixed centroid with angular
velocity
.
At the moment, when the blade
is in contact with fixed centroid, the vector
348


r
BC
,
which connects

the center B of the
moving centroid
L
with center of the
wheel
-
leg, forms the angle

-

/6


φ




/6

with respect of the vertical axis
NN
. Thus
,
the rolling time of the wheel blade
is
.

To find the radius of curvature

of the
trajectory of the center
C

of the wheel
-
leg,
we draw the line from point
C

and through
instantaneous center of velocity
C
v
(in the
following for instantaneous center of
rotation the notation ICR will be used).
The center of the curvature
K

of trajec
tory
of the point
C

has to lie on this line. The
angle between the straight lines

СC
v

and
N
N
is

β.
The

angles
β
and
φ

are related to
each other according the theorem of sine as
follows


(4)

Through the very small time interval

dt
,
when blade
L

rolls on the horizontal line,
the center
C

moves to the position
C’

and
ICR rolls to the position
C
v
'
. Normal
C
'
K

to
the trajectory
α
-
α

has to intersect the ICR
of
C
v
'
. From the point
C
v

we draw the
normal to the line
CK’

and denote the
angle
CKC’

as

.


From the angle Δ
СKC
'
and Δ
KC
v
M

we
receive



(5)


Further



(6)


(7)


Substituting the Eq. (6) and (7) to the
Eq.
(5) we receive the
Euler
-
Savary's
equation
[9
-
11] as




(8)

From the Eq. (8) we receive



(9)


and radius of curvature of the trajectory of
the center
C

according to the Fig. 4 is



(10)


w
here

.

The length of the vector

we receive
from the vector equation as



(11)


The length of the vector

through the
components of fixed reference frame can
be written then in the form





(12)


w
here

.


Parametrical representation of the
trajectory of the center
C

according to the
Eq. (12) can be written as follows






(13)




(14)


where
x
0


is initial position of the ICR of
the
C
v
.

The velocity of the center
C

become
s

then
according to the Eq. (12)

as




(15)


Taking the time derivative from the Eq.
(15) we receive the tangential acceleration
as


349





(16)


where

.

Normal acceleration of the point
C

taken
into account the
Eq.(10) and Eq. (15) can
be obtained as follows





(17)

where angle
β

according to the Eq. (4) is
defined as





(18)

Absolute acceleration is defined through
Eq. (16) and Eq. (17) as



(19)


5.
DYNAMICS OF WHEEL
-
LEG


To evaluate the maximum torque that can
be applied to wheel
-
leg in motion on
rigid
surface the Lagrange’s equation with
undetermined multipliers are introduced
[10] in the form



(20)


where
T



kinetic energy of the wheel
-
leg,
Q


generalized forces of wheel
-
leg,
q

and


ge
neralized coordinates and their time
derivatives respectively;
λ



Lagrangian
undetermined multipliers;
f



the equation
of constraint.

As the generalized coordinates the
translation of the center of the rounded tip
B of the blade in the direction of the
x

coordinate axis and rotation about this
center through the angle
φ

are chosen.
According to the chosen generalized
coordinates the position of center C can be
written as



(21)


where
-

is constan
t through the rotation
of the point C with respect to the
y
-
coordinate axis by angle
-
π/6

to
π/6
.

Taking the time derivative from the Eq.
(21) we receive expression of the kinetic
energy as follows



(2
2
)


where
I
z


the moment of inertia of the
wheel
-
leg.

The constraint equation
f

compliant with
generalized coordinates is





(2
3
)


where

and
-

determines the
initial position

of ICR.

According to Fig.
3

the
active forces that do work as
weight
P

of the wheel
-
leg and the torque
M

are

applied to the center of the wheel
-
leg.
Applying the method of virtual work we
receive the generalized forces
corresponding to the virtual displac
ement
of generalized coordinates as follows



(2
4
)


Taking the partial and time derivatives of
Eq. (2
2
) and Eq. (2
3
) according to the Eq.
(20) we receive Lagrangian system of
equation as




(2
5
)


The first equation in Eq. (2
5
) represents the
maximum friction force between the
surface and the wheel
-
leg blade that is
constituent with constraint equation Eq.
(2
3
), i.e motion of wheel
-
leg without
350


slipping. From the second equation of the
Eq. (2
5
) it i
s possible to get the expression
for the maximum torque that can be
applied of the wheel
-
leg.

The generalized coordinates in Eq.
(21)
are
coupled.
Dropping out the term
x
0

that
represents the initial position of ICR, then
the
Eq. (2
5
) can be rewritten as



(2
6
)


The Eq. (2
6
) can be solved numerically for
both equations to determine the values of
the friction forces and applied torques.


6
. CONCLUSION


In this paper the mathematical formulation
of the planar kinematics and dynamics of
the wheel
-
leg is derived. The kinematic
properties of wheel
-
leg were derived by
method of rigid body mechanics and by
alternative

method
based on

the Euler
-
Savary formulati
on of the moving and
fixed centroids. The

use of

Euler
-
Savary
method
can have ad
vantageous
when
wheel
-
leg moves in curved or other
surfaces in space.
The Lagrange’s equation
with undetermined multipliers has been
used to establish relationship between the
maximum applied torque and the friction
force of the wheel
-
leg.

The study of the
kinematic and dynamic properties of the
wheel
-
leg has covered only the basic
theoretical aspects of the wheel
-
leg motion
in horizontal plane. Thus, to evaluate more
capabiliti
es of the wheel
-
leg the
comprehensive analysis of the wheel
-
leg
will be performed in future.


7.
ACKNOWLEDGEMENT


This research was supported by funding of
the Estonian Science Foundation grant No.
8652

8. REFERENCES


1
.

Sell, R.,

Kaeeli, M.
,

Wheel
-
Leg
(Wheg)
.

Patent no EE05283B1, 2009
.

2
.

Tsunasawa M.,
Moving Equipment
,
Patent no 23.

JP60148780, 1985
.

3
.

Holmes, M.,
Vehicle traction assist
device
, Patent no US2006131948, 2006
.

4
.

Denisenko, G,
Wheel Propulsion Unit
,
Patent no RU280562 C2, 2006

5
.

Zheng
L., et.al,
A Novel High
Adaptability Out
-
door Mobile Robot with
Diameter
-
variable Wheels
.

Proc
. of the
IEEE Int.

Conference on Information and
Automation Shenzhen, China
,

2011
.

6
.

Xinbo C., et. al,
Mechanism principle
and dynamics simulation on variable
diameter walking wheel
.

Second
International Conference on Digital
Manufacturing & Automation
,
2011
.

7
.
Galileo Mobility Capabilities,
http://www.galileomobility.com

8
.

Campbell D. and Buehler M.,
Stair
descent in the simple hexapod 'RHex'
,
Proceedings of
the IEEE Int
.

Conference
on Robotics & Automation, Taipei,
Taiwan, September 14
-
19, 2003
.

9
.

Angus, R., W.
Theory of Machines
Including t
he Principles of Mechanisms
.

NABU Press, 2011.

10
.

Cleghom, W., L.
Mechanics of
Machines
.
Oxford University Press,
2005.


Raivo Sell, Ph.D, senior researcher,

TUT,
Ehitajate tee 5, Tallinn
, raivo.sell@ttu.ee


Gennady Aryassov, PhD, Ass. Prof.,
TUT
Ehitajate tee 5, Tallinn
,
gennadi.arjassov@ttu.ee


Andres
Petrits
h
enko, PhD, researcher,
TUT
Ehitajate tee 5, Tallinn
,
andres.petritsenko@ttu.ee


Mati Kaeeli, PhD
S
tudent, Tallinn
TUT
Ehitajate tee 5, Tallinn
,

mati141@gmail.co