Al-khwarizmi
Engineering
Journal
Al-Khwarizmi Engineering Journal, Vol.4 , No.1 , pp 17- 26 (2008 )
Minimizing error in robot arm based on design optimization for high
stiffness to weight ratio.
Ahmed Abdul Hussain Ali
University of Baghdad
College of Engineering, Mech. department
(Received 8 May 2007; accepted 3 December 2007)
Abstract:
In this work the effect of choosing tri-circular tube section had been addressed to minimize the
end effector’s error, a comparison had been made between the tri-tube section and the traditional
square cross section for a robot arm, the study shows that for the same weight of square section and tri-
tube section the error may be reduced by about 33%.
A program had been built up by the use of MathCAD software to calculate the minimum weight of a
square section robot arm that could with stand a given pay load and gives a minimum deflection. The
second part of the program makes an optimization process for the dimension of the cross section and
gives the dimensions of the tri-circular tube cross section that have the same weight of the
corresponding square section but with less deflection.
Key word: robot arm stiffness, flexible manipulator, robot structure analysis, flexible link robot.
Introduction:
The links of serial manipulators are
usually over designed in order to be able to
support the subsequent links on
the chain and the pay load to be manipulated.
However, increasing the size of the links
unnecessarily requires the use of larger
actuators resulting in higher power
requirements. Optimum robot design has been
addressed by many researchers as found in the
open literature; Shiakolas and koladiye [1]
discuss the application and comparison of the
evolutionary techniques for optimum design of
serial link robot manipulators based on task
specifications. The objective function
minimizes the required torque for a defined
motion subjected to various constraints which
considering kinematics, dynamic and structural
conditions. The design variables examined are
the link parameters and the link cross sectional
characteristics, the developed environment was
employed in optimizing the design variables for
a SCARA and an articulated 3-DOF PUMA
type manipulators. In the work developed by
Marcus Pettersson et al. [2] an optimization
problem are formulated to minimize the weight
of the gearboxes, by choosing different discrete
gear boxes, and changing the lengths of the
arms continuously, subjected to a few
requirements on acceleration capability reach
and pay load capacity. Analysis of stiffness of
manipulator link can be found in Abdel malek,
K. and Paul, B.[3] where aspects of the
structural design of the manipulator arm are
presented. Prismatic joints of manipulator arm
are based upon a cross sectional design of the
links that provides a high stiffness to weight
ratio compared with a hollow round cross-
section.
The case that we study in this work is
the robot that consists of three arms as shown
in fig. (1). Where the first arm is vertical and
the second and third arm are horizontal this
gives the maximum reach (completely stretched
out) for the robot arm and will yield the
maximum deflection for the robot.
Ahmed Abdul Hussain Ali Al-khwarizmi Engineering Journal, Vol.4 , No. 1 PP 17-26 (2008)
18
Prismatic joints:
Most manipulator link cross- section are
either hollow round or hollow rectangular.
Hollow links provide convenient conduits for
electric power and communication cables,
hoses, power transmission members, etc.
Rivin[4] has studied the influence of cross-
sections on the deflections both in bending and
torsion. He had compared hollow square with
hollow circular cross sections. Rivin states that
a square cross section can provide a 69 to 84
percent increase in bending stiffness over a
circular hollow cross section with only a 27
percent increase in weight.
In this paper a different cross-section is
introduced, consisting of three tubes centered
on the vertices of an equilateral triangle. This
cross section is referred to as a tri-tube
configuration the hollow square link will be
referred to as a uni-tube configuration, as
shown in fig. (2).
Deflection due to pure bending:
Links with an open end manipulator are
normally modeled as cantilevers. Consider a
simple cantilever with solid or hollow cross –
section as shown in fig.(3).To study the
proposed cross –section, we use the following
equations for moments of inertia (2nd moment
of area) about any diametrical axis through the
centroid of area.
Uni –tube:
For the uni- tube depicted fig.(2,a) the moment
of inertia about the neutral axis is
tBb
bB
I tubeuni 2,
12
44
Where B, b is the outer and inner sides,
respectively for the uni-tube construction, t is
the thickness of the uni-tube and the area is
22 bBA tubeuni
Tri-tube:
For the tri-tube depicted in fig.(2,b) the
moment of inertia about the neutral axis is
2
22
2244
60sin
3
1
4
2
60sin
3
2
464
3
hdD
hdDdDI tubetri
tDd 2
Where D and d are the out side and inside
diameters, respectively, for each tube on the
equilateral triangle, t is the thickness of each
tube in the tri – tube construction.
The area of each tube is
tDddDA tubetri 2,
4
22
To demonstrate the deflection due to loading,
consider the third arm beam depicted in
fig.(3,c).
83
1
4
33
3
33
3
3
LqLW
EI
gmWA
L
ALg
L
gM
q
,
Where M is the mass of each arm, m is the
mass of the gear box and the mass of the load
to be manipulated at the end of the arm, q is the
weight per unit length of the beam, W is the
load in Newton, g is the gravitational
acceleration and L is the length of the arm, A is
the cross sectional area of the beam, is the
specific density.
To get the reactions (force and moment) at the
fixed end of the third arm we equate the
summation of forces and moment to zero i.e.
33330 LqWFFy
33
2
33
3
2
0 LW
Lq
MOM
The same thing may be said for the second arm
(fig.3-b) taking in to account the effect of
moment in calculating the deflection i.e.
283
1
3
23
4
22
3
223
2
2
LMOLqLWF
EI
The reactions at the fixed end will be
32222 FWLqF
32322
2
22
2
2
MOLFLW
Lq
MO
For the first arm (fig.3-a) we assume that the
deflection at the free end is due to bending
moment only and the effect of compressive
loads on the whole deflection is neglected
therefore
1
2
11
1
2EI
LMO
Ahmed Abdul Hussain Ali Al-khwarizmi Engineering Journal, Vol.4 , No. 1 PP 17-26 (2008)
19
The total deflection at the end effecter of the
robot manipulator arm will be
21
2
23 total
The sequence of analysis in this work is to
calculate the weight of the lightest structure
that has a square hollow section and with stand
the given loading condition this may be
achieved by letting the stress in each arm
reaches the maximum allowable stress to avoid
failure of the structure, the equation for
calculating the stress in the third arm may be
written as
12
2/*
4
3
4
3
33
3
bB
BMO
I
YM
By letting the stress equal the allowable stress
and assuming the thickness of the tube walls to
be 2mm we may found the dimension of the
third arm, this had been done by the aid of a
program built up using MathCAD software.
The stress in the second arm may also be
calculated in the same way i.e.
12
2/*
4
2
4
2
22
2
bB
BMO
The dimension of the first arm fig.(3.a) is
calculated by equating the maximum stress
induced in it with the maximum allowable, this
maximum stress is found by the Rankine-
Gorden formula [5] which is a combination of
the Euler and crushing loads for a strut
ceR FFF
111
For very short strut eF is very large,
eF
1
can
therefore be neglected and cR FF , for very
long struts eF is very small and
eF
1
is very
large so that
cF
1
can be neglected.
Thus eR FF . The Rankine formula is
therefore valid for extreme values of
slenderness ratios. It is also found to be fairly
accurate for the intermediate values. Thus, re –
writing the formula in terms of stresses
)/(1
111
111
eY
Y
Ye
Ye
R
Ye
Ye
YeR
YeR AAA
For a strut with one end free and the other fixed
2
2
4 L
IE
Fe
and
AL
IE
e 2
2
4
The crushing load on the first arm is
121 WFFFc
A
Fc
Y
The final stress 1 on the first arm is thus the
sum of the direct stress calculated by Rankine
formula and that due to bending generated by
the exerted moment 2MO as was explained in
figure (3-a and b)
1
12
1
2/*
/! I
BMO
eY
Y
bendingR
From this equation we may find the dimension
of the first arm. After knowing the dimension
of each arm the weight of each arm may be
found and also the total weight of the
manipulator structure will be determined. The
next step in the analysis is to input those
information to the program to began the
process of changing the dimension of the cross
section to minimize the total error (deflection)
at the end of the robot arm this process gives
many generations of the dimensions of the arm
cross section which satisfies the conditions
specified for the maximum and minimum error
allowed at the end effecter and also the
permissible increase in the weight of the robot
structure specified from us, from between all
those generation the program select the best
generation or probability that gives the lightest
weight and the less deformation. The next step
in the analysis is to input the new weight of the
robot arms to the optimization process for the
tri –tube cross section shown in figure(2-b) and
trying to find the best dimensions that gives the
highest moment of inertia for the cross section
so as to minimize the deflection in each arm
and also the total deflection of the robot at the
end effecter, i.e. the mass of uni –tube section
Ahmed Abdul Hussain Ali Al-khwarizmi Engineering Journal, Vol.4 , No. 1 PP 17-26 (2008)
20
found by the program should be equal to the
mass of the tri –tube section which is equal to
)
4
3
(2)(
4
3
2
222
3
3
D
RSdDL
VM
2/
3
sin
3
2
DhR
Where is the density of robot arm metal, R
is the radius of the two flanges (stiffeners)
welded at each end, S is the thickness of the
flanges and h is the distance between the
vertices of the equilateral triangle.
The optimization problem is defined as follow
tDDh
Iimize
MM
tubetri
tubeunitubetri
2,
)(max
In the optimization problem the thickness of the
tubes )(t and the thickness of the flanges )(S
are assumed to be 2mm.
The results of the optimization problem showed
that the tri –tube section that have the same
weight (mass) of a uni –tube may improve the
stiffness of the robot and minimize the total
deflection in about 33%, this results means that
we may construct a robot having tri –tube
section which is less in weight from that of uni
–tube section and both of them having the same
end effecter deformation.
Results:
In order to verify the analysis of the previous
section a run had been done which has the
following characteristics for the robot arm
29 /10200 mNE
26 /10120 mNall ,
2sec/81.9 mg ,
mLmLmL 4.,45.,5. 321 ,
3/7850 mkg , m0022.0max ,
m0005.0min , mT 002.0 ( tube thickness
), mS 002.0 (stiffener thickness),
kgm 5.91 (mass of the first gear box),
kgm 4.42 (mass of the second gear box),
kgm 503 (manipulated mass). The available
gear boxes for the application are given in the
list of table 1
The results of the program shows that for the
given configuration the minimum weight for
the structure of the uni –tube robot is
(Wmin=19.986 N), the robot with such structural
weight could manipulate the load with out
failure because the stress in each arm is less
than or equal to the allowable stress, but the
deflection of the end point effecter is very
large. The iteration process for increasing the
dimension of the section to minimize the
deflection and letting it be within the range
(0.0005< <0.0022) shows that there are 22
generation all of which has a deflection
(0.0005< <0.0022) and also a weight
(W<Factor * Wmin ) the permissible weight
factor (Fac.) for increasing the weight was
chosen to be (Fac.=1.35). The dimensions of
the inner side of the uni –tube section for the 22
generation are shown in fig.(4). The relation
between the total deflection at the end -effecter
and number of generation is shown in fig.(5).
The relation between the new weight of the
robot structure and its generation is shown in
fig.(6). The program chooses the best
generation which has the less variable
(variable= weight*deflection), the relation
between the variable and the generation is
shown in fig(7) it is obvious that the generation
no. 16 has the minimum value, the dimensions
of the section for that generation are B1
=0.07464m, B2 =0.06397m, B3=0.05657m and
has a deflection total =1.87125*10
-3 m and a
total structural weight Wtotal =26.917N. Those
results are the input for the next step in the
program for calculating the dimensions of the
tri –tube section in which an optimization
problem where solved to maximize the moment
of inertia for the section in terms of the
dimensions h and D, the results of the program
are shown in table (2)
The total deflection for the tri –tube
configuration
21
2
23)( tubeTriTotal =1.31462*10
-3 m
The deflection for the tri –tube configuration is
less than that for the uni –tube which was found
to be ( 310*8725.1 tubeuni ) this results
shows that the tri –tube section reduces the
deflection in a bout 29.7% from that of uni –
tube section, the result may be improved to
reach a value of 33.38% if we change the
weight factor (Fac.) to make it equal to (1.3) on
the other hand if (Fac.) is increased to(1.65) the
Ahmed Abdul Hussain Ali Al-khwarizmi Engineering Journal, Vol.4 , No. 1 PP 17-26 (2008)
21
improvement in deflection would be less and
equal to (6.02%) those results for iteration and
optimization are shown in table(3)
Another interesting feature in the design of the
uni –tube section is that if the weight factor
(Fac.) was chosen to be 1.3 and the range of
deflection is limited to <0.0022 we would
find only five generation which satisfies the
previous mentioned configuration and if (Fac.)
was changed to 1.65 and < 0.00104 we
would find only 4 generation these results
which are shown in table (4) shows the band of
limits of the design of robot in other word we
can not find a robot with a weight factor less
than (Fac.=1.3) and has a deflection less than
0.0022m or we cannot find a robot with a
weight factor less than (Fac.=1.65) and had a
deflection less than 0.00104m those results of
iteration are shown in table(4).
A flow chart of the program used is shown in
fig.(8).
Conclusions:
This paper presents a method for optimization
of robot design in the conceptual design stage.
The robot is modeled in the MathCAD package
and the optimization problem is formulated as
to determine the dimension of robot arm in
order to minimize the weight and maximize
stiffness this formulation can be interpreted as
to design the cheapest possible robot that will
still meets the design demands. The
optimization method showed good capability in
finding an optimum set of dimension of the arm
of robot manipulator with three degree of
freedom.
The optimization method shows that the tri –
tube is superior to uni –tube section in
minimizing deflection in about 33%.
The presented work provides a good support
for conceptual robot design.
Fig. (1) robot configuration
(b) (a)
Fig.(2) (a) uni-tube configuration
(b) tri-tube configuration.
2q
2
3L2L
1
3q
1L
m3
m1 m2
3
Ahmed Abdul Hussain Ali Al-khwarizmi Engineering Journal, Vol.4 , No. 1 PP 17-26 (2008)
22
Fig.(3) modulation of manipulator links
Table 1 torque–mass relation for available gear
boxes [2]
Out put torques (N.m) Mass
(kg)
101
231
572
1088
1499
2176
4361
6135
2.5
4.4
9.5
12.7
18
28
47
69
Table (2) dimension of the tri –tube section
arm
No.
h (m) D (m)
*10-3
Itri –tube(m
4)
*10-7
Deflection
( )
m *10-4
3
2
1
0.10028
0.10440
0.10440
7.52652
9.22085
11.92
1.75056
2.48207
3.42024
8.09087
7.35704
3.0044
Table (3) results for iteration and optimization
problem
Fac.
tubeTri
(m)*10-3
tubeUni
(m)*10-3
Improvement
In deflection
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.39973
1.31462
1.2189
1.16048
1.09686
1.04631
0.985197
.948732
2.10114
1.87125
1.6237
1.48623
1.3437
1.22646
1.08422
1.00955
33.38%
29.7 %
24.9 %
21.9 %
18.37%
14.68%
9.13 %
6.02 %
Table (4) limits of deflection-weight factor for
robot design
Fac. max (m) generations
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
0.0022
0.0019
0.0017
0.0015
0.0014
0.0013
0.0011
0.00104
5
2
3
2
3
0
2
4
Ahmed Abdul Hussain Ali Al-khwarizmi Engineering Journal, Vol.4 , No. 1 PP 17-26 (2008)
23
In
n
er
-s
id
e
d
im
en
si
o
n
(
B
)
fo
r
u
n
i-
tu
be
s
ec
ti
o
n
in
(m
m
)
1 3 5 7 9 11 13 15 17 19 21
0.04
0.05
0.06
0.07
0.08
0.09
B1
B2
B3
B1
B2
B3
generatonsgenerations
fig.(4) correlation between inner side dimension and no. of generations.
T
o
ta
l
de
fl
ec
ti
o
n
total
(m
m
)
0 2 4 6 8 10 12 14 16 18 20 22
0.0017
0.00176
0.00182
0.00188
0.00194
0.002
0.00206
0.00212
0.00218
0.00224
0.0023
generations
generations
Fig.(5) correlation between the deflection and no. of generations.
Ahmed Abdul Hussain Ali Al-khwarizmi Engineering Journal, Vol.4 , No. 1 PP 17-26 (2008)
24
T
o
ta
l
w
ei
g
h
t
(
N
)
0 2 4 6 8 10 12 14 16 18 20 22
25.5
26
26.5
27
generations
generations
Fig.(6) correlation between robot weight and no. of generations.
V
ar
ia
b
le
(
V
A
R
.)
0 2 4 6 8 10 12 14 16 18 20 22
0.05
0.055
0.06
generations
Fig.(7) correlation between (Variable) and no. of generations.
Ahmed Abdul Hussain Ali Al-khwarizmi Engineering Journal, Vol.4 , No. 1 PP 17-26 (2008)
25
References:
[1]. P.S. shiakolas, D. koladiya and J.kebrle
“optimum robot design based on task
specifications using evolutionary techniques
and kinematic, dynamic, and structural
constraints”, international journal of inverse
problems in engineering, volume 10, number 4,
pp 359-375, 2002.
[2]. Marcus P., Peter K., Xiaolong F.A., Johan
A. and doniel W. “industrial robot design
optimization in the conceptual design phase”,
IEEE –mec. Of robot, 2004.
[3]. Abdel –Malek K. and Paul B., “criteria for
the design of manipulator arms for high
stiffness to weight ratio”, SME journal of
manufacturing systems, Vol.17, No.3, pp.209-
220.
[4]. E.I. Rivin, mechanical design of robots,
1988, Mc Grow –Hill, Inc, New York.
[5]. E.j. Hearn “Mechanics of materials”,
Pergamon press, 1977.
Fig.(8) flow chart of the program built-up by the use of MathCAD soft ware
Defining: E, all , m1,m2,m3, ,T,S,
Calculation of B3, B2, B1 and
Wmin=q3*L3+q2*L2+q1*L1
For B1 =B1+0.001
For B2=B2+0.001
For B3=B3+0.001
Calculation of:
321 ,, ,
2
3
2
23 total ,Wrobt
If
maxmin total
If
Wrobot>Fac.*Wmin
Saving
generations
Finding minimum
(VAR=Wrobot total )
1
1
Optimization for tri-tube
section
h>D , Wtii-tube=Wuni-tube, D 2T
Maximize (I), find h,D
Finding: 123 ,,
21
2
23, tubetritotal
No
No
Yes
Yes
Ahmed Abdul Hussain Ali Al-khwarizmi Engineering Journal, Vol.4 , No. 1 PP 17-26 (2008)
26
على أساس التصمیم األمثل ولنسبة جساءة الى وزن ) الروبوت(تقلیل الخطاء في الذراع األلي
.عالیة
أحمد عبد الحسین علي. د
كلیة الھندسة/ جامعة بغداد
قسم المیكانیك
:الخالصة
لطرفي في الذراع األلي ، تم في ھذا البحث تم دراسة تاثیر استخدام المقطع الثالثي األنابیب الدائریھ ألجل تقلیل الخطاء ا
اجراء مقارنھ بین المقطع الثالثي األنابیب و المقطع المربع التقلیدي للذراع األلي ، الدراسھ بینت بانھ لكال الذراعین ذات المقطع
. ٣٣%الثالثي و المربع والذان لھما نفس الوزن ممكن تقلیل الخطاء بحدود
ساب أقل وزن للذراع اآللي ذي المقطع المربع الذي یمكن أن یتحمل األوزان المسلطھ لح MathCADتم كتابة برنامج باستخدام
.ویعطي أقل تشوه
ي الجزء الثاني من البرنامج یقوم بعملیة األمثلیھ ألجل ایجاد ابعاد المقطع ذي األنابیب الثالثیھ الدائریھ والذي لھ نفس وزن الذراع ذ
.المربعالمقطع المربع ولھ تشوه أقل من نضیره
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4
6
8
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12
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Al-khwarizmi
Ahmed abdul hussain ali
Al-khwarizmi Engineering Journal, Vol.4 , No. 1 PP 17-26 (2008)
Al-khwarizmi
Engineering Journal
Al-Khwarizmi Engineering Journal, Vol.4 , No.1 , pp 17- 26 (2008 )
Minimizing error in robot arm based on design optimization for high stiffness to weight ratio.
Ahmed abdul hussain ali
University of Baghdad
College of engineering, Mech. department
(Received 8 May 2007; accepted 3 December 2007)
Abstract:
In this work the effect of choosing tri-circular tube section had been addressed to minimize the end effector’s error, a comparison had been made between the tri-tube section and the traditional square cross section for a robot arm, the study shows that for the same weight of square section and tri-tube section the error may be reduced by about 33%.
A program had been built up by the use of MathCAD software to calculate the minimum weight of a square section robot arm that could with stand a given pay load and gives a minimum deflection. The second part of the program makes an optimization process for the dimension of the cross section and gives the dimensions of the tri-circular tube cross section that have the same weight of the corresponding square section but with less deflection.
Key word: robot arm stiffness, flexible manipulator, robot structure analysis, flexible link robot.
Introduction:
The links of serial manipulators are usually over designed in order to be able to support the subsequent links on
the chain and the pay load to be manipulated. However, increasing the size of the links unnecessarily requires the use of larger actuators resulting in higher power requirements. Optimum robot design has been addressed by many researchers as found in the open literature; Shiakolas and koladiye [1] discuss the application and comparison of the evolutionary techniques for optimum design of serial link robot manipulators based on task specifications. The objective function minimizes the required torque for a defined motion subjected to various constraints which considering kinematics, dynamic and structural conditions. The design variables examined are the link parameters and the link cross sectional characteristics, the developed environment was employed in optimizing the design variables for a SCARA and an articulated 3-DOF PUMA type manipulators. In the work developed by marcus Pettersson et al. [2] an optimization problem are formulated to minimize the weight of the gearboxes, by choosing different discrete gear boxes, and changing the lengths of the arms continuously, subjected to a few requirements on acceleration capability reach and pay load capacity. Analysis of stiffness of manipulator link can be found in Abdel malek, K. and Paul, B.[3] where aspects of the structural design of the manipulator arm are presented. Prismatic joints of manipulator arm are based upon a cross sectional design of the links that provides a high stiffness to weight ratio compared with a hollow round cross-section.
The case that we study in this work is the robot that consists of three arms as shown in fig. (1). Where the first arm is vertical and the second and third arm are horizontal this gives the maximum reach (completely stretched out) for the robot arm and will yield the maximum deflection for the robot.
Prismatic joints:
Most manipulator link cross- section are either hollow round or hollow rectangular. Hollow links provide convenient conduits for electric power and communication cables, hoses, power transmission members, etc. Rivin[4] has studied the influence of cross-sections on the deflections both in bending and torsion. He had compared hollow square with hollow circular cross sections. Rivin states that a square cross section can provide a 69 to 84 percent increase in bending stiffness over a circular hollow cross section with only a 27 percent increase in weight.
In this paper a different cross-section is introduced, consisting of three tubes centered on the vertices of an equilateral triangle. This cross section is referred to as a tri-tube configuration the hollow square link will be referred to as a uni-tube configuration, as shown in fig. (2).
Deflection due to pure bending:
Links with an open end manipulator are normally modeled as cantilevers. Consider a simple cantilever with solid or hollow cross – section as shown in fig.(3).To study the proposed cross –section, we use the following equations for moments of inertia (2nd moment of area) about any diametrical axis through the centroid of area.
Uni –tube:
For the uni- tube depicted fig.(2,a) the moment of inertia about the neutral axis is
Where B, b is the outer and inner sides, respectively for the uni-tube construction, t is the thickness of the uni-tube and the area is
Tri-tube:
For the tri-tube depicted in fig.(2,b) the moment of inertia about the neutral axis is
Where D and d are the out side and inside diameters, respectively, for each tube on the equilateral triangle, t is the thickness of each tube in the tri – tube construction.
The area of each tube is
To demonstrate the deflection due to loading, consider the third arm beam depicted in fig.(3,c).
Where M is the mass of each arm, m is the mass of the gear box and the mass of the load to be manipulated at the end of the arm, q is the weight per unit length of the beam, W is the load in Newton, g is the gravitational acceleration and L is the length of the arm, A is the cross sectional area of the beam,
is the specific density.
To get the reactions (force and moment) at the fixed end of the third arm we equate the summation of forces and moment to zero i.e.
The same thing may be said for the second arm (fig.3-b) taking in to account the effect of moment in calculating the deflection i.e.
The reactions at the fixed end will be
For the first arm (fig.3-a) we assume that the deflection at the free end is due to bending moment only and the effect of compressive loads on the whole deflection is neglected therefore
The total deflection at the end effecter of the robot manipulator arm will be
The sequence of analysis in this work is to calculate the weight of the lightest structure that has a square hollow section and with stand the given loading condition this may be achieved by letting the stress in each arm reaches the maximum allowable stress to avoid failure of the structure, the equation for calculating the stress in the third arm may be written as
By letting the stress equal the allowable stress and assuming the thickness of the tube walls to be 2mm we may found the dimension of the third arm, this had been done by the aid of a program built up using MathCAD software. The stress in the second arm may also be calculated in the same way i.e.
The dimension of the first arm fig.(3.a) is calculated by equating the maximum stress induced in it with the maximum allowable, this maximum stress is found by the Rankine-Gorden formula [5] which is a combination of the Euler and crushing loads for a strut
For very short strut
is very large,
can therefore be neglected and
, for very long struts
is very small and
is very large so that
can be neglected. Thus
. The Rankine formula is therefore valid for extreme values of slenderness ratios. It is also found to be fairly accurate for the intermediate values. Thus, re –writing the formula in terms of stresses
For a strut with one end free and the other fixed
and
The crushing load on the first arm is
The final stress
on the first arm is thus the sum of the direct stress calculated by Rankine formula and that due to bending generated by the exerted moment
as was explained in figure (3-a and b)
From this equation we may find the dimension of the first arm. After knowing the dimension of each arm the weight of each arm may be found and also the total weight of the manipulator structure will be determined. The next step in the analysis is to input those information to the program to began the process of changing the dimension of the cross section to minimize the total error (deflection) at the end of the robot arm this process gives many generations of the dimensions of the arm cross section which satisfies the conditions specified for the maximum and minimum error allowed at the end effecter and also the permissible increase in the weight of the robot structure specified from us, from between all those generation the program select the best generation or probability that gives the lightest weight and the less deformation. The next step in the analysis is to input the new weight of the robot arms to the optimization process for the tri –tube cross section shown in figure(2-b) and trying to find the best dimensions that gives the highest moment of inertia for the cross section so as to minimize the deflection in each arm and also the total deflection of the robot at the end effecter, i.e. the mass of uni –tube section found by the program should be equal to the mass of the tri –tube section which is equal to
Where
is the density of robot arm metal,
is the radius of the two flanges (stiffeners) welded at each end,
is the thickness of the flanges and
is the distance between the vertices of the equilateral triangle.
The optimization problem is defined as follow
In the optimization problem the thickness of the tubes
and the thickness of the flanges
are assumed to be 2mm.
The results of the optimization problem showed that the tri –tube section that have the same weight (mass) of a uni –tube may improve the stiffness of the robot and minimize the total deflection in about 33%, this results means that we may construct a robot having tri –tube section which is less in weight from that of uni –tube section and both of them having the same end effecter deformation.
Results:
In order to verify the analysis of the previous section a run had been done which has the following characteristics for the robot arm
,
,
,
,
,
,
( tube thickness ),
(stiffener thickness),
(mass of the first gear box),
(mass of the second gear box),
(manipulated mass). The available gear boxes for the application are given in the list of table 1
The results of the program shows that for the given configuration the minimum weight for the structure of the uni –tube robot is (Wmin=19.986 N), the robot with such structural weight could manipulate the load with out failure because the stress in each arm is less than or equal to the allowable stress, but the deflection of the end point effecter is very large. The iteration process for increasing the dimension of the section to minimize the deflection and letting it be within the range (0.0005<
<0.0022) shows that there are 22 generation all of which has a deflection (0.0005<
<0.0022) and also a weight (W<Factor * Wmin ) the permissible weight factor (Fac.) for increasing the weight was chosen to be (Fac.=1.35). the dimensions of the inner side of the uni –tube section for the 22 generation are shown in fig.(4). The relation between the total deflection at the end -effecter and number of generation is shown in fig.(5). The relation between the new weight of the robot structure and its generation is shown in fig.(6). The program chooses the best generation which has the less variable (variable= weight*deflection), the relation between the variable and the generation is shown in fig(7) it is obvious that the generation no. 16 has the minimum value, the dimensions of the section for that generation are B1 =0.07464m, B2 =0.06397m, B3=0.05657m and has a deflection
=1.87125*10-3 m and a total structural weight Wtotal =26.917N. Those results are the input for the next step in the program for calculating the dimensions of the tri –tube section in which an optimization problem where solved to maximize the moment of inertia for the section in terms of the dimensions h and D, the results of the program are shown in table (2)
The total deflection for the tri –tube configuration
=1.31462*10-3 m
The deflection for the tri –tube configuration is less than that for the uni –tube which was found to be (
) this results shows that the tri –tube section reduces the deflection in a bout 29.7% from that of uni –tube section, the result may be improved to reach a value of 33.38% if we change the weight factor (Fac.) to make it equal to (1.3) on the other hand if (Fac.) is increased to(1.65) the improvement in deflection would be less and equal to (6.02%) those results for iteration and optimization are shown in table(3)
Another interesting feature in the design of the uni –tube section is that if the weight factor (Fac.) was chosen to be 1.3 and the range of deflection is limited to
<0.0022 we would find only five generation which satisfies the previous mentioned configuration and if (Fac.) was changed to 1.65 and
< 0.00104 we would find only 4 generation these results which are shown in table (4) shows the band of limits of the design of robot in other word we can not find a robot with a weight factor less than (Fac.=1.3) and has a deflection less than 0.0022m or we cannot find a robot with a weight factor less than (Fac.=1.65) and had a deflection less than 0.00104m those results of iteration are shown in table(4).
A flow chart of the program used is shown in fig.(8).
Conclusions:
This paper presents a method for optimization of robot design in the conceptual design stage. The robot is modeled in the MathCAD package and the optimization problem is formulated as to determine the dimension of robot arm in order to minimize the weight and maximize stiffness this formulation can be interpreted as to design the cheapest possible robot that will still meets the design demands. The optimization method showed good capability in finding an optimum set of dimension of the arm of robot manipulator with three degree of freedom.
The optimization method shows that the tri –tube is superior to uni –tube section in minimizing deflection in about 33%.
The presented work provides a good support for conceptual robot design.
Fig. (1) robot configuration
(b)
(a)
Fig.(2) (a) uni-tube configuration
(b) tri-tube configuration.
Fig.(3) modulation of manipulator links
Table 1 torque–mass relation for available gear boxes [2]
Out put torques (N.m)
Mass
(kg)
101
231
572
1088
1499
2176
4361
6135
2.5
4.4
9.5
12.7
18
28
47
69
Table (2) dimension of the tri –tube section
arm No.
h (m)
D (m)
*10-3
Itri –tube(m4)
*10-7
Deflection (
)
m *10-4
3
2
1
0.10028
0.10440
0.10440
7.52652
9.22085
11.92
1.75056
2.48207
3.42024
8.09087
7.35704
3.0044
Table (3) results for iteration and optimization problem
Fac.
(m)*10-3
(m)*10-3
Improvement
In deflection
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.39973
1.31462
1.2189
1.16048
1.09686
1.04631
0.985197
.948732
2.10114
1.87125
1.6237
1.48623
1.3437
1.22646
1.08422
1.00955
33.38%
29.7 %
24.9 %
21.9 %
18.37%
14.68%
9.13 %
6.02 %
Table (4) limits of deflection-weight factor for robot design
Fac.
max (m)
generations
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
0.0022
0.0019
0.0017
0.0015
0.0014
0.0013
0.0011
0.00104
5
2
3
2
3
0
2
4
Inner-side dimension (B) for uni-tube section in (mm)
generations
fig.(4) correlation between inner side dimension and no. of generations.
Total deflection
(mm)
generations
Fig.(5) correlation between the deflection and no. of generations.
Total weight (N)
generations
Fig.(6) correlation between robot weight and no. of generations.
Variable (VAR.)
generations
Fig.(7) correlation between (Variable) and no. of generations.
References:
[1]. P.S. shiakolas, D. koladiya and J.kebrle “optimum robot design based on task specifications using evolutionary techniques and kinematic, dynamic, and structural constraints”, international journal of inverse problems in engineering, volume 10, number 4, pp 359-375, 2002.
[2]. Marcus P., Peter K., Xiaolong F.A., Johan A. and doniel W. “industrial robot design optimization in the conceptual design phase”, IEEE –mec. Of robot, 2004.
[3]. Abdel –Malek K. and Paul B., “criteria for the design of manipulator arms for high stiffness to weight ratio”, SME journal of manufacturing systems, Vol.17, No.3, pp.209-220.
[4]. E.I. Rivin, mechanical design of robots, 1988, Mc Grow –Hill, Inc, New York.
[5]. E.j. Hearn “Mechanics of materials”, Pergamon press, 1977.
تقليل الخطاء في الذراع الألي (الروبوت) على أساس التصميم الأمثل ولنسبة جساءة الى وزن عالية.
د. أحمد عبد الحسين علي
جامعة بغداد / كلية الهندسة
قسم الميكانيك
الخلاصة:
في هذا البحث تم دراسة تاثير استخدام المقطع الثلاثي الأنابيب الدائريه لأجل تقليل الخطاء الطرفي في الذراع الألي ، تم اجراء مقارنه بين المقطع الثلاثي الأنابيب و المقطع المربع التقليدي للذراع الألي ، الدراسه بينت بانه لكلا الذراعين ذات المقطع الثلاثي و المربع والذان لهما نفس الوزن ممكن تقليل الخطاء بحدود %33 .
تم كتابة برنامج باستخدام MathCAD لحساب أقل وزن للذراع الآلي ذي المقطع المربع الذي يمكن أن يتحمل الأوزان المسلطه ويعطي أقل تشوه.
الجزء الثاني من البرنامج يقوم بعملية الأمثليه لأجل ايجاد ابعاد المقطع ذي الأنابيب الثلاثيه الدائريه والذي له نفس وزن الذراع ذي المقطع المربع وله تشوه أقل من نضيره المربع.
Defining: E,� EMBED Equation.3 ���, m1,m2,m3, � EMBED Equation.3 ���,T,S, � EMBED Equation.3 ���
� EMBED PBrush ���
� EMBED Equation.3 ���
2
� EMBED Equation.3 ���
� EMBED Equation.3 ���
1
� EQ �� EMBED Equation.3 ���
� EMBED Equation.3 ���
m3
m1
m2
3
Fig.(8) flow chart of the program built-up by the use of MathCAD soft ware
Yes
Yes
No
No
Finding:� EMBED Equation.3 ���� EMBED Equation.3 ���
Optimization for tri-tube section
h>D , Wtii-tube=Wuni-tube, D� EMBED Equation.3 ���2T
Maximize (I), find h,D
1
1
Finding minimum (VAR=Wrobot� EMBED Equation.3 ���)
Saving generations
If
Wrobot>Fac.*Wmin
If
� EMBED Equation.3 ���
Calculation of: � EMBED Equation.3 ���,� EMBED Equation.3 ���,Wrobt
For B1 =B1+0.001
For B2=B2+0.001
For B3=B3+0.001
Calculation of B3, B2, B1 and Wmin=q3*L3+q2*L2+q1*L1
PAGE
25
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