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A unified design for lightweight robotic arms base

publicité
Research Article
A unified design for lightweight robotic
arms based on unified description
of structure and drive trains
International Journal of Advanced
Robotic Systems
July-August 2017: 1–14
ª The Author(s) 2017
DOI: 10.1177/1729881417716383
journals.sagepub.com/home/arx
Haibin Yin, Shansheng Huang, Mingchang He and Junfeng Li
Abstract
This article presents a unified design for lightweight robotic arms based on a unified description of structure and drive trains. In
the unified design, the drive trains and structural dimensions are parameterized as design variables, and a major objective
minimizes the total mass of robotic arms satisfying the constraint conditions and design criteria. To implement the optimization
problem, a mapping relationship between mass and torque of drive trains is introduced as their power–density curves, which
enable a unified description of structure and drive trains combining with the dynamics of robotic arms. In this implementation of
unified design, there are two modules: structure optimization and drive trains design. The finite element method with nonlinear
programming by quadratic Lagrange algorithm is adopted to implement the structure optimization. Moreover, the dynamic
analysis in MSC ADAMS is achieved to design the drive trains of robotic arms. This method could uniformly evaluate all
components of robotic arms in mass and continuously search the global optimal results. Finally, a design example on this unified
design is compared with a referenced design to illustrate the validity and advantage of the proposed scheme.
Keywords
Lightweight robotic arms, unified description, unified design, drive trains
Date received: 25 July 2016; accepted: 29 May 2017
Topic: Service Robotics
Topic Editor: Marco Ceccarelli
Associate Editor: Yukio Takeda
Introduction
Robotic arms have been widely used in industrial production, agriculture, service and space explorations, and so on.
However, most of these existing robotic arms have obvious
disadvantages: low payload–weight ratio, bulky structure,
high power consumption, and low safety in human–robot
coexistence environment.1 As a result, lightweight design
on robotic arms is required to meet the requirements in high
performance and special tasks such as space manipulation.2
To address the above problems, many researchers have
launched on the design of lightweight robotic arms, which
is a complex system including drive trains design, structure
design, dynamic control, and so on.3
Drive trains account for large proportion of whole
weight of robotic arms, so the lightweight design on drive
trains integrating motors and gears is very promising. To
improve the power–density of motors for robotic joints,
Seo and Rhyu proposed a design of axial flux permanent
magnet brushless DC (BLDC) motors.4 Jing et al. studied
the structure optimization of BLDC motors by finite element method (FEM) to increase its power–density.5 However, these investigations on the power–density of motors
Key Laboratory of Hubei Province for Digital Manufacture, School of
Mechanical and Electric Engineering, Wuhan University of Technology,
Hubei, China
Corresponding author:
Haibin Yin, Wuhan University of Technology, Luoshi Road No. 122,
Hongshan, Wuhan, Hubei 430070, China.
Email: [email protected]
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License
(http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/
open-access-at-sage).
2
are not enough. Chedmail and Gautier proposed a method
on the optimum selection of drive trains to minimize the
total mass of robotic arm with given trajectories.6 Pettersson and Ölvander presented a multiobjective optimization
on the drive trains of an industrial robotic arm.7 Zhou et al.8
presented an optimization problem on drive train of lightweight robotic arm to select its motors and gearboxes from
a commercial catalog. Moreover, Park et al. reported a new
modular drive train for a dual arm robot manipulator to
make the robotic arm lighter and safer in human–robot
coexistence environment.9 However, these researches
mainly focused on the drive trains design, disregarding the
influence on the structure from the renewal of drive trains.
There are some reported optimization designs on structure to obtain the optimal dimensions of robotic arms.
Albers and Ottnad used a topology optimization to optimize the structural dimensions of links for lightweight
robotic arms based on FEM.10 Zhang et al. introduced an
integrated modular design approach by FEM and dynamic
simulation to design a lightweight, high stiffness, and compact robotic arm.11 Shiakolas et al. proposed a structural
optimization of robotic arms with task specifications,12 in
which the length and cross-sectional parameters were optimized to minimize the required joint torque by dynamic
analysis. In addition, Schrock et al. designed a wheelchairmounted robotic arm in carbon fiber and polycarbonate to
achieve lightweight design and enhance the structural stiffness.13 However, these researches mainly focused on the
structural optimization to improve the dynamic performance without considering its influence on drive trains.
Structure and drive trains are interactional and expected
to consider together. An integrated approach, which simultaneously implemented the structural and drive trains optimization, was recently proposed to design a lightweight
anthropomorphic arm.14 However, this structural optimization only considered the dimensions of links while disregarded the parameters of joint shell; in addition, the index
number of motors and gears were defined as the design
variables of drive trains optimization, so these nonassociated and discrete design variables resulted in a large
amount of computations and locally optimal results. Therefore, a complete description on structure and effective optimization design is required to design globally optimal
lightweight robotic arms.
In this article, the whole structure consisting of links and
joint shell is optimized, and drive trains including motors
and gears are designed. These design variables of structure
and drive trains are generous and interactional, so their
characteristics and laws are studied and a unified design
based on the inherent laws of structure and drive trains is
expected to design lightweight robotic arms.
This unified design consists of two modules: structure
optimization and drive trains design. The FEM with
nonlinear programming by quadratic Lagrange (NLPQL)
algorithm15 is utilized to optimize structural dimensions of
links by means of minimizing its mass and satisfying the
International Journal of Advanced Robotic Systems
constraints. Besides the structural dimensions of links, the
dimensions of joint shell are also considered. These dimensions of joint shell are dependent on the type of selected
drive trains, which are in turn dependent on the whole
structure of robotic arms. Therefore, the unified design is
a complex optimization problem, and an inherent law of
structure and drive trains is required to clear up their relationship and solve this problem. Above all, the objective
minimizes the total mass of robotic arms in this study, so
the mass of drive trains is one of the principal variables in
the inherent laws. In addition, the torque of drive trains is
also key variables as it is a bridge linking the parameters
and mass of structure. As a result, a mathematic formulation between the mass and torque of drive trains is proposed
as the basic theory of the inherent law, which enables all
components of robotic arms to evaluate uniformly in mass.
The mathematic formulation is defined as power–density,
which is developing with technique because it is related to
the materials, topology structure, and speed of drive trains.
This study achieves a fitting curve as joint’s power–density
based on candidate motors and gears reflecting current
technique. The fitting power–density and dynamics of
robotic arms formulate the unified description of drive
trains and structure. The unified design performs static
structural analysis in ANSYS Workbench and dynamic
analysis in MSC ADAMS (MSC. Software, 2013 Version).
Finally, an example on the unified design is compared with
a referenced design14 to demonstrate its effectiveness and
advantages of the proposed scheme.
A design scheme of lightweight
robotic arms
In this section, a design scheme and some considerations
are introduced as an optimization problem of lightweight
robotic arms. In the optimization problem, design variables
on links and joints are defined, and a major objective minimizes the total mass of the robotic arm satisfying the corresponding constraint conditions and design criteria.
Scheme model and considerations
Figure 1 shows a scheme model of lightweight robotic arms
with five degree of freedom (DOF), which includes 2-DOF
shoulder, 2-DOF elbow, 1-DOF wrist, and a hand gripper.
This study focuses on the lightweight design of robotic
arms, so that mass distribution of robotic arms is supposed
to be as close as possible to the driving joint. For example,
the joint 4 is arranged at location closing joint 3 to reduce
the excited vibration.
In addition, the aluminum alloy is chosen as the material
of structure to meet the requirements of lightweight robotic
arms. Meanwhile, in order to achieve better dynamic performance and more compact structure, BLDC motors with
higher power density and harmonic drive gearboxes (HGs)
with higher reduction ratios are utilized to form drive
Yin et al.
3
Joint 3
θ3
Link 1
Joint 4
θ4
Link 2
Joint 2
θ2
Joint 5
Harmonic
Drive
Gearbox
Joint 1
θ1
Planetary
Gearhead
Motor
End-effector
Ecoder
Components of drive train for
joint 1, 2, 3 and 5
Components of drive
train for joint 4
Figure 1. Scheme model of lightweight robotic arms.
trains. In this work, the HGs are used as transmission components for all other joints except for joint 4 due to only a
slender space in link 2, so a slender planetary gearhead
(PG) is utilized in joint 4 to increase the torque. Moreover,
an electric gripper with servo drive acts as the end effector
to pick up target.
This lightweight design is a study on general method and
not for specific tasks, so kinematics analysis is not the necessary work in this research. However, the motions of robotic
arms are one of conditions in dynamic analysis; therefore, a
pick-and-place operation (PPO) in joint space is specified as
the trajectories to calculate the required torques of robotic
arms, and the dynamic equation is generally described as
_ þ GðqÞ ¼ t g
MðqÞ€
qþVðq; qÞ
(1)
where M denotes the inertia matrix of the robotic arm, V is
the vector of centrifugal and Corilolis forces, G is the vector related to gravitational forces, and the vector t g consists
of the output torques from the gears in all joints. q ¼ [y1, y2,
y3, y4, y5]T is the vector of joint trajectories.
For complex robotic arms, it is difficult to formulate an
accurate and parameterized mathematic model to optimize
them. Hence, the dynamic model of the robotic arm is established by the virtual prototype technology in MSC ADAMS
to implement dynamic simulation as accurate as possible.
Design variables
To optimize robotic arms, the design variables including
structural dimensions of links and parameters of joints are
defined as follows.
L2
Assembly
dimensions
L1
L3
R1-1
l1
b1
a1
R1
b2
a2
H1
r1
R2-1
l2
r2
R2
Structural
dimensions
Figure 2. Parameterized structural dimensions of links.
Parameterized structural dimensions of links. The structure is
one of major components and is optimized as the basis of
joint optimization for lightweight robotic arms. As shown
in Figure 2, the robotic arm is parameterized in assembly
(marked in red) and structural dimensions (marked in
black) of links. The assembly dimensions determine the
kinematic performance of robotic arms. To obtain a higher
kinematics performance of the robotic arm in the overall
workspace, the assembly dimension L2 is set to 0.7 times L1
referring the maximum values of global conditioning index
(GCI).16 Moreover, to limit the computational work, some
assembly dimensions H 1 and L 3 ; some structural
4
International Journal of Advanced Robotic Systems
Whdi
Harmonic drive gearbox
Rhdi
i=2,3,5
i=1
Rs1
Δsi
Wsi
Ws1
δs1
δsi
Δs1
Rsi
Modular structure for joint 2,3, and 5
Modular structure for joint 1
Figure 3. Modular joints.
dimensions of links, such as lengths l1 and l2; and outer
radius R1 and R2 are assumed as constants. While the other
structural dimensions of links, such as inner radius r1 and
r2, the major axes a1 and a2 of ellipse groove and the minor
axes b1 and b2 are defined as the design variables of links.
For the sake of brevity, here the design variables of links
are described as a vector ul ¼ [r1, r2, a1, a2, b1, b2].
Design variables of joints. Similarly, joints are also major
components and account for large proportions of whole
mass of a lightweight robotic arm, so their optimization
potential is huge. The joint design adopts a modular
approach to improve the efficiency of optimization design,
and the detailed configurations of modular joint are shown
in Figure 3.
The modular joints consist of drive trains and shells,
and the design variables uJ of joints are accordingly
classified into two groups: the first group are the rated
torques of the drive trains, which are denoted by a vector ud ¼ [ud1, ud2, ud3, ud4, ud5]. As depicted in Figure 3,
the second group are the shell dimensions of joints,
which are defined as a vector us ¼ [us1, us2, us3, us4,
us5] and described as
us1 ¼ ½Rs1 ; Ws1 ; us2 ¼ ½Rs2 ; Ws2 ; us3 ¼ ½Rs3 ; Ws3 ;
us4 ¼ ½Rs4 ; Ws4 ; us5 ¼ ½Rs5 ; Ws5 (2)
where the shell dimensions us4 of joint 4 are kept as constants due to a slender drive train being fixed in the hollow
link 2. In addition, the independent dimensions such as ds1,
Ds1, ds2, Ds2, ds3, Ds3, ds5, and Ds5 have been optimized in
previous investigation17 minimizing the mass of robotic
arm with the constraints of strength and stiffness, and are
directly used in this study.
As shown in Figure 3, the gearboxes as the major components of drive trains directly determine the shell dimensions of joints, considering the fit in assembly process and
lightweight objective. The previous study17 summarized
the relationships between shell dimensions and selected
drive trains as
"
8
>
<½R ; W 1
hdi
hdi
½Rsi ; Wsi ¼
0
>
:
0
#
4 sgnðiÞ 2 sgnði 1Þ
½r2 ; l2 ;
;
i 6¼ 4;
i ¼ 4;
(3)
where Rhdi and Whdi are the maximum radius and width of HGs
from the product catalog. For a candidate drive train, the major
dimensions are correlative with its rated torque. However, the
rated torque of the drive train is continuous variables and
difficult to continuously map shell dimensions during optimization. Therefore, a segmented function about the shell dimensions mapping the rated torque of joints is represented by
cand
cand
½Rhdi ; Whdi ¼ ½Rhdk
; Whdk
;
½Rs4 ; Ws4 ¼ ½r 2 ; l2 ;
Tdðk1Þ < udi Tdk ; i 6¼ 4
Tdðk1Þ < ud4 Tdk ; i ¼ 4
(4)
where Tdk is the rated torque of the kth candidate drive
cand
cand
train, and [Rhdk
; Whdk
] is the corresponding dimensions.
k is the number of the candidate drive trains. According to
equations (3) and (4), the mapping relationship between ud
and us is described in a general formulation as
us ¼ Fc ðud Þ
(5)
As a result, the design variables of the robotic arm can be
briefly denoted by a vector X ¼ [ul, uJ], where ul and uJ ¼ [ud,
us] are the design variables of links and joints, respectively.
Yin et al.
5
max rmc
; tg;i
maxfjtg ðt; XÞjgi Tg;i
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð Dt
:
3 1
udi
3
max
¼
tg;i
ðt; XÞ dt ¼
; maxfj yðtÞjgi Ng;i
Dt 0
S2
(8)
.
Figure 4. Model of drive train and load.
Moreover, these design variables related to the structural
strength and dynamic performance of the robotic arm are to
be updated in the whole design process and have to meet some
corresponding constraints and design criteria.
Constraints and design criteria
To implement the optimization design, the constraints on
structural strength and design criteria of drive trains are
introduced in this section.
Constraints on structural strength. Since the renewal of structural dimensions and design variables of joints will inevitably lead to the changes of structural strength, some
constraints on structural strength of the robotic arm making
it safety should be formulated as
S1 sm ðXÞ sy ; S1 dm ðXÞ dmp
(6)
where sm denotes the maximum Von-Mises equivalent
stress of the robotic arm and sy is the yield strength of the
structural material. dm is the maximum total deformation of
the robotic arm, and dmp is the maximum permissible deformation. S1 is a structural safety coefficient, and these constraints are evaluated through the static structural analysis
in ANSYS Workbench.
Design criteria of drive trains. The drive trains consist of
motors and gears, and a schematic model of drive train with
load is depicted in Figure 4. The required toque tgi of the ith
joint of robotic arm can be obtained through the dynamic
calculation, and the corresponding required torque of motor
tmi can be transformed as
tm;i
Jg €
tg ðtÞ
¼
Jm þ 2 yðtÞ þ
i
(7)
where Jm represents the inertia of motor rotor, and Jg
denotes the inertia of gear rotor. represents the gear ratio,
and is its transmission efficiency.
To ensure the drive trains with enough ability to drive
payload under given trajectories, the following design
criteria of drive trains are expressed as
max
rmc
max
where Tg;i
, tg;I
, and Ng;i
are the maximum permissible
output torque, the root mean cubic (rmc) of the required
output torque, and maximum permissible input speed of the
ith gear, respectively. S2 is the choice safety coefficient,
which is generally chosen a value larger than 1 to ensure
the reliability of joint design.
The driving ability of drive trains derives from the
motors, so their driving ability is also checked as
max rms
maxfjtm ðt; XÞjgi Tm;i
; tm;i
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
1 Dt
rated
max
_
¼
tm;i ðt; XÞ dt Tm;i
; maxfjyðtÞjg
i Nm;i
Dt 0
(9)
max
rms
rated
max
, tm;i
, Tm;I
, and Nm;i
are the maximum perwhere Tm;i
missible output torque, the root mean square (rms) of the
required torque, the rated torque, and the maximum permissible output speed of the ith motor, respectively.
These above design criteria ensure the driving capability
of motors and gears, but the lightweight demands limit their
capacity. To obtain suitable candidate drive trains satisfying driving capability and lightweight, the combining
design criteria of motors and gears are described as
rated
max
max
Tm;i
udi ; Nm;i
Ng;i
(10)
Objective function
In this study, the objective of optimization problem minimizes the total mass of the robotic arm satisfying constraint
conditions and design criteria, and the objective function is
formulated as
min f ðXÞ ¼ f 1 ðul Þ þ f 2 ðuJ Þ
(11)
where f1(ul) and f2(uJ) ¼ md(ud)þms(us) are the mass of
links and joints, respectively. In addition, all design variables X are summarized as follows
X ¼ ½ul ; uJ ; ul ¼ ½r 1 ; r 2 ; a1 ; a 2 ; b1 ; b2 ; uJ ¼ ½ud ; us (12)
These design variables are expected to meet aforementioned constraint conditions and design criteria, which are
equations (5), (6), (8), and (9). In addition, the design variables of links need meet the specific range as
ul 2 ½ullower ; ulupper (13)
where ullower and ulupper are the lower and upper limits of all
design variables of links. To simplify the description in
6
International Journal of Advanced Robotic Systems
sections, all the constraints and design criteria are rewritten
in a general formulation as
gðul ; uJ Þ 0
(14)
Figure 5. Unified description of structure and drive trains.
Implementation of the unified design
The design variables are generous and the optimization
problem is complex, so the optimization design is divided
into two modules: structure optimization and drive trains
design. The former is performed by means of computer
software with NLPQL algorithm, while the latter is implemented by dynamic analysis in MSC ADAMS. Meanwhile,
the design variables of the structure and drive trains are
interactional; therefore, a unified description makes two
modules interact and achieve the optimal result in the
whole design.
Structure optimization
Nowadays, finite element analysis, such as ANSYS, is
widely used in structural analysis and optimization
design of products. In this study, a digital model is built
to optimize the structural dimensions of links in ANSYS
Workbench, where an NLPQL algorithm is integrated to
obtain the optimal results. The NLPQL algorithm15 is an
effective method to solve nonlinear optimization problems. In structure optimization, considering the joint
parameters as constants, the objective function is
expressed as
min f ðXÞ ¼ f1 ðul Þ þ f2 ðCJ Þ
subject to : gðul ; uJ ¼ CJ Þ 0
(15)
where CJ represents the joint parameters, which is fixed in
structure optimization, but is updated when considering the
feedback of drive train design.
Basic theory of the unified description
After achieving the structure optimization, the dynamic
analysis is able to get the required torques of joints, which
is obtained by assuming the joint parameters as constants in
dynamic model. However, the calculated torques will result
in new joint parameters, such as joint mass, which are
expected to renew their values in dynamic model. To realize the renewal, a mapping relationship between the rated
torque and mass of drive trains is required and defined as
power–density curve. As shown in Figure 5, the torque is a
bridge of structural dynamics and power–density curve of
drive trains, which enable the structure and drive trains to
evaluate uniformly in mass. Therefore, the mapping laws of
power–density curve of drive trains are the basic theory of
unified description and introduced as follows.
Roos has been presented the mathematic formulations of
rated torque Trated m and mass mm of the homogenous
motors,18 which are described as
Tmrated ¼ Cm lm rm2:5
(16)
mm ¼ pm lm rm2
(17)
where Cm is constant for a specific motor type with same
coiling condition. lm and rm are the radius and length of the
rotor, respectively. m is the average mass density of motor.
These parameters are associated with the topological structure and coiling method of the motor.
By substituting equation (16) into equation (17), a mapping relationship between the rated torque Trated m and
mass mm of the ith motor can be described as
!
0:5
pm rm;i
rated
mm;i ¼
(18)
Tm;i
Cm
This mapping law implies that the rated torque of a
motor is positively correlated with its mass when the parameters m and Cm are constant.
In addition, gears are also one of major components of
drive trains and applied to increase torque. If a gear is
working at rated speed, its transmission efficiency and gear
ratio are assumed as constant. Based on the above hypothesis, when the motor is operating at rated torque, the output
torque of the ith drive train disregarding the inertia of
output axis can be expressed as
rated
T0;i ¼ Tm;i
(19)
where T0 is output torque of the drive train, and and are
transmission efficiency and gear ratio, respectively. For
homogenous gears working at approximate rated speed, the
transmission efficiency might be same. Considering the
influence of the inertia from the motor and gear, the rated
output torque of a drive train reflecting its driving capacity
can be modified as
Jg €
rated
2
udi ¼ Tm Jm þ 2 jyj max
(20)
i
Meanwhile, the mass md of the drive train consisting of the
motor’s mass mm and the gear mass mg can be expressed as
md;i ¼ mm;i þ mg;i
(21)
Then, a mapping relationship between the mass and
rated torque of drive trains can be formulated by means
of equations (18) and (19). Moreover, the mapping relationship of drive trains represents their power–density
curve, which is related to the type of the drive trains. For
Yin et al.
7
Structure optimization
Drive trains design
Initialization of the robotic arm
X0 = [ul0, uJ0], f(X0)=f1(ul0)+f2(uJ0)
Parameterized dynamic model
in MSC.ADAMS
Parameterized FE model with constant
boundary conditions and given load
Inverse dynamic simulation with
given trajectories and load to obtain τg
udb = S 2 τ grmc
Define the optimization problem and
convergent condition εN in ANSYS
Workbench and then implement structure
optimization using NLPQL algorithm
min f(X)=f1(ul)+f2(CJ)
S.T.
g(ul, uJ=CJ)≤0
ul b
f(X )=f1(ul )+f2(uJ0),
X* = [ulb, uJ0]
2
b
d4
), i ≠ 4
uJ b
f(Xb)=f1(ulb)+f2(uJb),
Xb = [ulb, uJb]
|f (Xb)-f (X*)|<e
YES
Update design point X : X = X
0
b
di
f2(uJb)=md(udb)+ms(usb), uJb=[udb, usb]
udb=[ud1b,ud2b,ud3b,ud4b,ud5b], usb=Fc(udb)
NO
b
1
i=1
Obtain the optimal parameters of links
satisfying the minimum condition εN
*
Σ F (u ) +F (u
5
md (udb ) =
0
b
Stop
Figure 6. Routine of the unified design.
a certain type of drive trains, the mapping relationship can
be generally marked as
md;i ¼ Fj ðudi Þ
(22)
where Fj is a mapping function related to the jth type drive
trains, which is classified according to their topological
structure, rated speed, and materials. Based on the
power–density curve, the unified description is built and
the unified design is performed.
The routine of the unified design
As a result, the unified design is executed to minimize the
total mass of robotic arms based on the unified description
of the structure and drive trains. Figure 6 depicts the routine
of the unified design, which consists of the structure optimization of links and drive trains design.
Firstly, the initial parameters X0 ¼ [ul0, uJ0] of the robotic
arm are given to start the structure optimization, where the
design variables ul0 of links are updated and the design
variables uJ0 of joints are fixed. In this module, a parameterized finite element model is built to evaluate the structural
constraints using ANSYS Parametric Design Language in
batch mode. Meanwhile, the parameterized model integrating NLPQL algorithm is achieved to optimize the structural
dimensions of links by means of minimizing the mass of
links, and the minimum condition is "N.
When finishing structure optimization, the optimal design
variables ulb of links are obtained to update the parameters of
links in dynamic model built by means of the virtual prototype technology in MSC ADAMS. Then, an inverse
dynamic calculation using the parameterized dynamic model
of robotic arm is executed to determine the required torques
t g of the drive trains under the conditions of given trajectories and load. According to the required torques, the rated
output torque of drive train are determined, and then the
mass md(udb) of drive trains is determined by equation
(22) and the corresponding shell dimensions of joints can
be determined through equation (5). These calculated mass
of drive trains and shell dimensions of joints are feedback to
structural model in ANSYS workbench to calculate again,
until the difference between f(X*) ¼ f1(ulb) þ f2(uJ0) and
f(Xb) ¼ f1(ulb) þ f2(uJb) is less than a defined condition e.
Design example
In this section, the unified design and a referenced design
approach14 are implemented to compare the optimization
on a 5-DOF lightweight robotic arm, and their results are
compared to verify the validity of the unified design.
Referenced design approach
The referenced design integrated the structure optimization
and the optimum selection of drive trains from catalog for a
lightweight robotic arm, so it is complex approach and
needs a large number of computations to choose the optimal drive trains. To understand the referenced design easily, an optimization problem14 is presented as follows
8
International Journal of Advanced Robotic Systems
In the complex algorithm, the design point minimizing
objective function value is defined as the best point Xref,b,
on the contrary, that is the worst point Xref,w. In order to
obtain the optimal point, a new design point Xref,new is
formulated by executing the algorithm and will replace the
worst point Xref,w if it is a better point. Then, the definitions
about the best and worst points are repeated and the optimization algorithm is executed. Finally, the optimization
design will be terminated until the difference between the
objective value f(Xref,b) at the best design point and the
value f(Xref,w) at the worst design point is less than given
tolerance " by means of the repeated iterative calculation.
The detailed design approach and algorithm process can be
referred in literature.14
Input trajectory and parameters of the robotic arm
Figure 7. The flow chart of the referenced design.
min :
f ðX ref Þ ¼ mm ðumref Þ þ mg ðugref Þ þ ml ðulref Þ
design variables : X ref ¼ ½umref ; ugref ; ulref subjective to :
GðX ref Þ 0
(23)
ref
where the design variables X include the index numbers of
ref
ref
ref
ref
ref
, um2
, um3
, um4
, um5
], the index numbers
motors umref ¼ [um1
ref
ref
ref
ref
ref
ref
], and the design
of gearboxes ug ¼[ug1 , ug2 , ug3 , ug4 , ug5
ref
variables of links ul ¼[r1, r2, a1, a2, b1, b2], respectively.
mm, mg, and ml are mass function of motors, gears, and links,
respectively. G is simplified mark of all constraint conditions
and design criteria. The referenced design disregarded the
optimization on the dimensions of joint shell, which is considered in the unified design.
In the referenced design approach, the calculation process of the optimization problem is presented as follows.
As shown in Figure 7, before implementing the optimization problem, we need to specify several feasible design
points (X ref,1 , X ref,2 , . . . , X ref,m ), where the number
of design points is m ¼ 25 in this referenced design, and
these design points satisfy above constraints. The values
in these design points are not always integral number during the calculation process. However, the design variables
umref and ugref are defined as index numbers from the product
categories of motors and gearboxes, which are integral
number. Therefore, a round function14 is introduced to deal
with the design variables x ¼ [umref , ugref ] and expressed as
xint ;
xint x < xint þ 0:5
xDV ¼ roundðxÞ ¼
xint þ 1; xint þ 0:5 x < xint þ 1
(24)
where xDV represents the results of rounding up the design
variables x ¼ [umref , ugref ], and xDV is used to update the
motors and gearboxes in the referenced design.
In the example, the target payload is defined as 2 kg, which
acts on the end effector as a mass point. The assembly dimensions determining the kinematic properties of the robotic arm
are accordingly given as H1 ¼ 230 mm, L1 ¼ 380 mm, L2 ¼
270 mm, and L3 ¼ 108 mm, which can meet the requirement
of specific workspace of the lightweight robotic arm.16
The trajectories as the input conditions of dynamic
simulation are required to specify. This example will maneuver a PPO,17 and the PPO in joint space starts from an
initial position θ0 at time t ¼ 0 and stops to final position θe
at time t ¼ DT, which can be expressed as
θðtÞ ¼
8
>
>
>
>
>
>
>
>
>
<
0 1
t
t 1 [email protected] A;
t1
t 2 ½0; t1 t 2 ½t 1 ; DT t 1 vðt 0:5t1 Þ;
>
2
0
1 3
>
>
>
>
t DT þ t1A 5
>
>
t 1 ; t 2 ½DT t 1 ; DT v4t 0:5t1 [email protected]
>
>
:
t1
(25)
where s(t) ¼ t 3t þ 2.5t , v ¼ (θe θ0)/(DT t1). The
initial and final positions are set to θ0 ¼ [0, p/2, 0, 0, p]T
and θe ¼ [8p/9, p/6, p/6, p, 2p/3]T. Time DT and t1 are set
to 4 and 1 s, respectively.
In this example, the structure material is chosen as aluminum alloy (6061-T6) and its yield strength is sy ¼ 270
MPa. Meanwhile, to evaluate the constraints on structural
strength and rigidness, the structural safety coefficient is set
to S1 ¼ 2, the maximum permissible deformation is defined
as dmp ¼ 1 mm, and the minimum condition "N is 0.000001.
In addition, both the error conditions e and "C are set to
0.0001, and the choice safety coefficient is set to S2 ¼ 1.3.
Besides the structural parameters, the candidate motors
and gears are required to optimize the joints of lightweight
robotic arms. In this example, the EC series of BLDC
motors19 from the MAXON Motor Corporation, Switzerland, are applied to design joint motors. In addition, the
fourth joint adopts a GP series PG from the MAXON Motor
Corporation due to its location in a slender space, whereas
6
5
4
Yin et al.
9
Figure 8. Technical data of the candidate harmonic gearboxes, PGs, and motors. (a) Candidate harmonic gearbox.20 (b) Candidate
PGs.19 (c) Candidate BLDC motors.19 BLDC: brushless DC; PG: planetary gearhead.
other joints take the HGs with HFUC-2UH Series Units20 as
transmission elements. The related technical data of the candidate harmonic gearboxes, PGs, and motors are described
as column chart in Figure 8(a) to (c), respectively. Each
candidate motor or gear provides four parameters including
mass (mm or mg), rated torque (Tmrated or Tgrated ), maximum
speed (Nmmax or Ngmax ), and torque (Tmmax or Tgmax ).
Based on the descriptions of the product catalog, the
gear efficiency is dependent on its operation speed. In this
work, all candidate motors have near-identical rated speeds
ranging about 3000–5000 rmin1, under the action of gears
with reduce ratios ig ¼ 113 for PG and ig ¼ 100 for HG, so
the combination of motors and gears makes the robotic
joints work at an expected speed of 30–50 rmin1. Therefore, the gear efficiency of each candidate gear is set to ¼
70%, which is an average value from the product catalog
based on the approximate rated speed.
Initialization of design variables and
power–density curve
In this design example, the initial values and ranges of
design variables of links are required and listed in Table 1.
In addition, the power–density curve, which is a mathematic relationship between mass and torque of drive trains, is
required to optimize the drive trains. In theory, this relationship exists and depends on rated speed, structure, and
Table 1. Range and initial values of the structural dimensions of
links (unit: millimeter).
Design
method
Unified design and referenced design approach
Design
r1
r2
a1
a2
b1
b2
variables
Range
(24,28) (23,26) (100,145) (50,75) (30,45) (25,40)
Initial
24
23
100
50
30
25
materials of drive trains. Furthermore, the power-density
formula is developing and variable with technical progress.
Therefore, this study adopts a fitting curve based on a series
of candidate drive trains reflecting the current technology. In
this case, we need to combine motors and gears as candidate
drive trains according to the combination criteria mentioned
in equation (10). These drive trains used to fit a curve should
possess same structure, materials, and rated speed (actually
approximate due to the limitation of the number of samples).
In this study, two kinds of gears are adapted, so there are two
types of drive trains. The first-type drive trains (A1–A7)
composed of the motors and HG are listed in Table 2. Similarly, the second-type drive trains (B1–B7) composed of the
motors and PG are listed in Table 3.
As shown in Tables 2 and 3, the mass md and rated torque
ud of drive trains are computed by means of equations (20)
10
International Journal of Advanced Robotic Systems
Table 2. The first type candidate drive trains.
1
2
3
First type drive
train number
md (kg)
ud (Nm)
1
2
3
4
5
6
7
A1
A2
A3
A4
A5
A6
A7
0.536
0.565
0.750
0.781
0.890
1.030
1.450
2.106
5.195
7.574
9.984
12.480
18.252
24.882
First Type Fiting Curve
First Type Candidate Drive Trains
Second Type Fitting Curve
Second Type Candidate Drive Trains
1.8
1.6
Mass of drive train md (kg)
HG number
2
Motor
number
HG: harmonic drive gearbox.
B7
1.4
A7
1.2
A6
1
md=F1(ud)
B6
A5
A3
0.8
A4
B5
0.6
A1
B4
md=F2(ud)
A2
0.4
B3
B1
0.2
Table 3. The second type candidate drive trains.
PG number
1
2
3
0
Motor
number
Second type drive
train number
md (kg)
ud (Nm)
1
2
3
4
5
6
7
B1
B2
B3
B4
B5
B6
B7
0.240
0.269
0.304
0.601
0.710
1.140
1.580
2.170
5.644
7.950
10.848
13.560
20.120
25.420
and (21), respectively. As a result, the mapping relationship
between the mass and torque of drive trains is obtained by
curve fitting. As shown in Figure 9, the first type power–
density curve md ¼ F1(ud) is obtained based on the first type
candidate drive trains listed in Table 2, while the second type
power–density curve md ¼ F2(ud) is achieved using the second type candidate drive trains, as shown in Table 3. Both
the power–density curves will be utilized to optimize the
drive trains of lightweight arms in the unified design.
In the optimization, the initial components of drive
trains are roughly estimated using statics when the robotic
arm with initial structural parameters spreads to limit position. In the unified design, the design variables of drive
trains are rated output torque ud, whereas their design variables in referenced design are the index number of motors
(umref ) and gears (ugref ). In both design approaches, the initial
components (red symbol in Table 4) of drive trains are
same, but their expression in optimization is completely
different as shown in Table 4.
Results and discussions
Based on the same input conditions, constraints, and initial
parameters, both design approaches on a lightweight
robotic arm are achieved to compare their results. The
referenced design approach takes 346 iterative calculations
to obtain the optimal lightweight robotic arm, while the
unified design only spends 46 iterative calculations to get
a lighter structure.
B2
5
10
15
Torque of drive train ud (Nm)
20
25
Figure 9. Mapping relationship between the mass and torque of
drive trains.
Table 4. The initial components and values of design variables of
drive trains.
Unified design
Referenced design approach
Joint index
Drive train
(udi)
Motor
ref
(umi
)
Gearbox
(ugiref )
Drive
train
1
2
3
4
5
24.882 (A7)
24.882 (A7)
18.252 (A6)
10.848 (B4)
9.984 (A4)
7
7
6
4
4
3
3
2
2
2
A7
A7
A6
B4
A4
Figure 10 illustrates the convergences of partial design
variables of links based on the referenced design approach
and unified design, respectively. Table 5 lists the optimal
results of all design variables of links based on both designs
and a group of trade-off results, respectively. Compared to
the results of referenced design, the optimal values of unified design are larger; this result implies the optimal slots
and inner space of links is shortly larger in the unified
design. The trade-off result is one in-between case and
introduced later. In addition, the convergences of design
variables of links based on the unified design are uniformly
forward to the optimal results, while their convergences
based on the referenced design are fluctuating toward the
objective values due to the discrete design variables and
rounding treatment in computation of drive trains, which
affect the parameters of links.
Figure 11 depicts the convergence of partial design variables of motors and gearboxes based on the referenced
design. Similar to the case in Figure 10(a), the convergences in Figure 11 are fluctuating and finally stable in the
optimal values. This fluctuating convergences result from
limiting the design variables of motors and gears as discrete
Yin et al.
11
(b)
27
26
r1 (mm)
r1 (mm)
(a)
25
24
0
50
100
150
200
250
Iterative calculation number
300
26
25
50
Iterative calculation number
26
25
r2 (mm)
r2 (mm)
27
24
0
350
26
24
23
0
28
50
100
150
200
250
Iterative calculation number
300
350
25
24
23
0
10
20
30
40
Iterative calculation number
50
Figure 10. Convergences of the partial design variables of links. (a) r1 and r2 based on the referenced design approach. (b) r1 and r2
based on the unified design.
Table 5. The optimal results of design variables of links (unit:
millimeter).
Design
variables
Unified
design
Referenced
design
Trade-off
results
a1
a2
b1
b2
r1
r2
144.33
71.13
43.52
39.78
27.31
25.88
141.85
68.56
42.74
38.63
26.77
25.49
143.22
70.51
43.15
38.82
26.90
25.61
integrals and rounding these decimals in calculation. These
limitations of discrete variables and round function (24)
leads to the disorder and rough computations, so that the
referenced design needs a large number of computations to
obtain the final results, which are not global optimal
solutions.
Based on the unified design, Figure 12(a) depicts two
fitting power–density curves and the optimal results of
drive trains, and Figure 12(b) shows the convergences of
design variables of drive trains along its power–density
curves of the joints 1–4 at the 21th, 31th, 39th, and 46th
iterations, respectively. The power–density curves express
the inherent law of candidate drive trains integrating the
motors and gears, so that the optimal results can be continuously searched and quickly convergent along the law
curves. Only four iterations are used to obtain the optimal
results of drive trains, and other iterations (464 ¼ 42) are
executed in structural optimization. Compared to the root
joints, the end joints are more quickly convergent, for
example, the four iterations of joint 4 are almost overlapping together due to small effect from structure and other
joints. Especially, joint 5 is completely unaffected from
structure and other joints, so it is convergent at one point
around A1 as shown in Figure 12(a), which does not need to
be locally enlarged in Figure 12(b).
To compare the optimal results of all joints based on
two designs, the optimal components (in red) and values
of design variables of drive trains based on the unified
design and referenced design are listed in Table 6. In the
referenced design, the design variables of joints are the
index number of motors and gears, and convergent to a
series of integrals, which are basis to check the components according to Tables 2 and 3. In the unified design,
the design variables of joints are output rated torque and
convergent to some exact values, such as 12.83 (A5, A6),
which represents the optimal value locating between A5
and A6 on curve.
The exact results can be referred to design the drive
trains of a lightweight robotic arm, but the integrating
design on motors and gears is a challenging work. In the
practice, actuators of joints adopt the available motors and
gears selected from product catalog to integrate drive
trains, which discretely distribute on the fitting curve.
These candidate drive trains approaching and exceeding
these exact values are chosen as the optimal components
to ensure them with certain redundant capacity adapting
much wider range tasks. For example, the optimal result
is 12.83 Nm locating between A5 and A6, so that the candidate A6 is the optimal component of joint 1. In this case,
the optimal components used in both optimized lightweight
robotic arms are completely same, although the design
variables based on the unified design are continuous and
fine values while counterparts based on the referenced
design are discrete values.
Figure 13(a) depicts the convergences of both the best
and worst design points based on the complex algorithm in
the referenced design. Figure 13(b) shows the convergence
of the whole mass of robotic arm by means of the unified
design and trade-off design. Comparing the convergence
rates of the objective function for the two design methods,
we can summarize that the unified design can significantly
reduce the number of iterative calculation. Moreover, the
12
International Journal of Advanced Robotic Systems
(b)
7
motor 1
6.5
6
5.5
0
50
100
150
200
250
Iterative calculation number
300
Index number
Index number
(a)
6
3.5
2
0
350
3
50
100
150
200
250
Iterative calculation number
300
350
2
gearbox 1
2.5
2
1.5
0
50
100
150
200
250
Iterative calculation number
300
Index number
Index number
motor 3
5
gearbox 3
1.5
1
0.5
0
350
50
100
150
200
250
Iterative calculation number
300
350
(b)
2
Mass of drive train md (kg)
1.6
1.4
1.2
1
md=F1(ud)
B7
A7
A6
0.8
A4
B5
0.6
A1
B4
md=F2(ud)
A2
0.4
B3
B1
0.2
B2
0
5
10
15
Torque of drive train ud (Nm)
20
at 21th
at 31th
at 39th
at 46th
1.1
1
0.9
A5
0.8
12 12.5 13 13.5 14 14.5 15 15.5
Torque of drive train for Joint 1 (Nm)
B6
A5
A3
1.2
25
1
0.9
0.8
0.7
at 21th
at 31th
at 39th
at 46th
A3
Mass of drive train md (kg)
1.8
Mass of drive train md (kg)
First Type Fitting Curve
First Type Candidate Drive Trains
Optimal Drive Trains for Joint 1
Optimal Drive Trains for Joint 2
Optimal Drive Trains for Joint 3
Optimal Drive Trains for Joint 5
Second Type Fitting Curve
Second Type Candidate Drive Trains
Optimal Drive Trains for Joint 4
Mass of drive train md (kg)
(a)
Mass of drive train md (kg)
Figure 11. The convergence of partial design variables of motors and gears based on the referenced design. (a) Motor 1 and gearbox 1.
(b) Motor 3 and gearbox 3.
0.6
6.5 6.75 7 7.25 7.5 7.75 8
Torque of drive train for Joint 3 (Nm)
1.3
1.2
1.1
at 21th
at 31th
at 39th
at 46th
A6
1
0.9
15 15.5 16 16.5 17 17.5 18 18.5
Torque of drive train for Joint 2 (Nm)
0.25
0.225
0.2
at 21th
at 31th
at 39th
at 46th
B1
0.175
0.15
1 1.25 1.5 1.75 2 2.2
Torque of drive train for Joint 4 (Nm)
Figure 12. Convergence of drive trains along power–density curves based on the unified design. (a) Power–density curves and optimal
results of drive trains. (b) Convergence of design variables of drive trains for joint 1–4.
Table 6. The optimal components and values of design variables
of drive trains.
Unified design
Joint
index
Drive train (udi)
1
2
3
4
5
12.83 15.64 7.00 1.18 2.42 (A5, A6)
(A5, A6)
(A2, A3)
(0, B1)
(A1, A2)
Referenced design
Motor
ref
(umi
)
Gearbox
(ugiref )
Drive
train
6
6
3
1
2
2
2
1
1
1
A6
A6
A3
B1
A2
optimized mass of the robotic arm based on the unified
design is 7.62 kg, while the optimized mass based on the
referenced design approach is 8.15 kg. The results illustrate
that the unified design has better feasibility and efficiency
to achieve a more lightweight design for robotic arms. The
trade-off design can obtain one in-between mass, which is
7.982 kg.
The red points in Figure 13(b) represent the mass trend
of the robotic arm only considering structure optimization, while the blue points imply the mass changing trend
in the drive train design. These data reflect the interaction
between the structure and drive trains, so the unified
design based on the unified description of structure and
drive train is important. Considering the feedback of optimized drive trains, the structure is optimized again until
the difference between f(X*) and f(Xb) is less than the
defined condition e.
The more lightweight robotic arm is obtained in unified
design when the optimal values of drive trains are considered. If we weigh to select the optimal components (red
symbol in Table 6) as drive trains of robotic arm after quick
Yin et al.
13
(a)
(b)
10.5
10
9.5
9
8.5
11
Structure optimization
Drive train design
Tradeoff design
10.5
Mass of the robotic arm (kg)
Mass of the robotic arm (kg)
The worst design point
The best design point
10
9.5
9
8.5
8
8
0
50
100
150
200
250
Iterative calculation number
300
350
7.5
0
10
20
30
40
Iterative calculation number
50
Figure 13. Convergence of mass of the robotic arm. (a) f(Xref,w) and f(Xref,b) based on the referenced design. (b) f(X) based on the
unified design and trade-off design.
unified design, the optimal dimensions of links related to
these optimal components will yield the trade-off results
listed in Table 5. Moreover, the convergence (green points)
of the total mass of robotic arm results from trade-off
design (ud ¼ [18.252, 18.252, 7.574, 2.170, 5.195]), as
shown in Figure 13(b).
Conclusions
A lightweight robotic arm was designed by utilizing the
proposed optimization method, in which the design variables describing links and drive trains, constraints on structural strength and design criteria of drive trains, as well as
the objective function were formulated and developed. The
joint shell dimensions are determined by drive trains due to
depending on them, and a unified quantitative description
on both the structural dimensions of links and drive trains
was considered to reduce the computation complexity. The
results show that the proposal method can achieve an optimal design with minimum mass and high efficiency while
satisfying the constraints and work conditions and design
criteria of drive trains.
A unified design based on unified description of structure and drive trains was introduced to optimize the lightweight robotic arm. In this unified design, the motors and
gears were combined as a whole component based on the
design criteria of drive trains, and then a mathematic relationship between the mass and torque of drive trains was
presented as its power–density curve, which is bonds of the
unified description on structure and drive trains.
Finally, the unified design method was compared with
a referenced design approach to deal with the same
optimization problem on lightweight robotic arms, and
the results show that the unified design is validity. Considering the mathematic law between the mass and torque
of drive trains in the design process, the unified design
provided a high efficiency and fast computation optimization for the lightweight robotic arm. In addition, the
design variables in the unified design were continuous
renewal and convergent to fine values and result in a
lighter objective mass of robotic arm.
In this work, the mathematic law between the mass and
torque of drive trains was obtained by fitting curve method
based on the available motors and gears. The further application and theoretical study on the mathematic law remains
an open problem for future research.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) disclosed receipt of the following financial support
for the research, authorship and/or publication of this article: This
study was supported by National Natural Science Foundation of
China (Grant No. 51575409) and partially by the Fundamental
Research Funds for the Central Universities (20411009).
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