Design and Experiment of Triangular Prism Mast with Tape-Spring Hyperelastic Hinges
Yang et al. Chin. J. Mech. Eng.
Design and Experiment of Triangular Prism Mast with Tape-Spring Hyperelastic Hinges
Hui Yang 2
Hong-Wei Guo 0
Yan Wang 1
Rong-Qiang Liu 0
Meng Li 3
0 China State Key Laboratory of Robotics and System, Harbin Institute of Technology , Harbin 150001 , China
1 China Electronics Technology Group Corporation No.38 Research , Hefei 230088 , China
2 College of Electrical Engineering and Automation, Anhui University , Hefei 230601 , China
3 Qian Xuesen Laboratory of Space Technology, China Academy of Space Technology , Beijing 100094 , China
Because of the limited space of the launch rockets, deployable mechanisms are always used to solve the phenomenon. One dimensional deployable mast can deploy and support antenna, solar sail and space optical camera. Tape-spring hyperelastic hinges can be folded and extended into a rod like configuration. It utilizes the strain energy to realize self-deploying and drive the other structures. One kind of triangular prism mast with tape-spring hyperelastic hinges is proposed and developed. Stretching and compression stiffness theoretical model are established with considering the tape-spring hyperelastic hinges based on static theory. The finite element model of ten-module triangular prism mast is set up by ABAQUS with the tape-spring hyperelastic hinge and parameter study is performed to investigate the influence of thickness, section angle and radius. Two-module TPM is processed and tested the compression stiffness by the laser displacement sensor, deploying repeat accuracy by the high speed camera, modal shape and fundamental frequency at cantilever position by LMS multi-channel vibration test and analysis system, which are used to verify precision of the theoretical and finite element models of ten-module triangular prism mast with the tape-spring hyperelastic hinges. This research proposes an innovative one dimensional triangular prism with tape-spring hyperelastic hinge which has great application value to the space deployable mechanisms.
Deployable mechanism; Triangular prism mast; Tape-spring hyper-elastic hinges; Static analysis; Modal experiment
Conventional articulated truss structures are composed
of mechanical hinges which can meet accuracy and
stiffness requires of space mission. But those structures have
some disadvantages, such as large weight, high friction
and energy-wasting features. Tape-spring hyperelastic
(TSH) hinges, which are folded elastically can self-deploy
by releasing stored strain energy, which consist of a
fewer component parts, can be manufactured
]. Flexible hinges have several advantages for
space applications, including a low
mass-to-deployedstiffness ratio, cost, and self-latch . With the increasing
demand, flexible hinges have been widely used as folding
and deployment mechanisms in deployable structures,
such as synthetic aperture radars (SARs) [
arrays and antenna booms. Tape-spring hinges have been
used in the Japanese Mars orbiter PLANT-B for bar-like
deployment structures of the thermal plasma analyzer
The U.S. Air Force Research Laboratory (AFRL) [
used TSH hinge in the main truss’s folding longeron
elements, which provided considerable snap-through
force to drive and lock the main truss. Imperial
College London Santer [
] proposed a concertina-folded
magnetometer boom with TSH hinges for CubeSat use.
Watt et al. [
] proposed a TSH hinge with two sets of
wheels held together by wires wrapped around them, and
deploying impact is reduced for the added damp. The
Mars Advanced Radar Express spacecraft [
consisted of two 20 m dipoles and a 7 m monopole which
were slotted at certain intervals to stow them in a much
a Folded status b Deployed status
Figure 1 One modulus TPM with Hyperelastic hinges
Figure 2 Stretched diagram of TPM
hyperelastic hinge rope
small size. Silver et al. [
] proposed an integral folding
hinge to deploy camera and investigated axial loading,
bending induced buckling response. Schioler et al. and
Seffen et al. [
] analyzed buckling properties of
single layer TSH hinge based on Timoshenko theory. Seffen
et al.  got sample points by finite element method and
obtained fitting nonlinear mechanical models by single
value decomposition method of TSH hinge. Guan et al.
] designed a TSH hinge for solar sail and
investigated its buckling properties by finite element method.
Bai et al. , Yan et al. [
], Wang et al. [
investigated geometrical and mechanical properties of
ultrathin-walled lenticular collapsible composite tube in fold
deformation. Yang et al. [
] optimized the
geometric parameters of TSH hinge to improve driving moment
and reduce deploying impact, and established two
different theoretical models to analyze the stability of
deployment status for the TSH hinges. However, there are still
some engineering problems for the TSH hinge applying
to a deployable mechanism.
This paper proposed a new ten-module
triangular prism mast (TPM) with TSH hinges. Static
bending stiffness and compression stiffness theoretical
models are established. The compression stiffness tests
are performed to verify the accuracy of the static
theoretical models of the TPM. Finite element model of
tenmodule TPM are establish by ABAQUS and geometrical
parameter study are analyzed for bending and twisted
modal frequencies. Two-module TPM is developed to
test the fundamental frequencies and related modal
shapes which are used to verify the accuracy of the
tenmodule TPM (Additional file 1).
2 Design and Static Analysis
2.1 Structures Design
One module TPM with tape-spring hinges is shown in
Figure 1. Adjacent two-module is connected by
triangular frame. The length of triangular frame is 469 mm,
stowed and deployed longitudinal lengths of ten-module
prism mast are 475.2 mm and 5278 mm respectively. All
components except standard parts are manufactured by
duralumin 2A12. Each modulus TPM with TSH hinge
contains three longitudinal links and two triangular
frames. The TSH hinges are used to drive and lock TPM.
Lateral area of each modulus has a series of flexible
Kevlar rope with tensile force 30 N which ensures high
stiffness of the deployed prism mast and reduces deploying
impact. There is one resin-based carbon fiber drive pipe
one each end of the tensile lock which prevent
involvement during deploying process.
The TSH hinge folded with 180°, the two triangular
frame close to each other, and three longitudinal links
folded between the two triangular frames when the
mast stowed. The tape-spring drive the mast to deploy.
After deploying, the TSH hinge restore original shape,
the Kevlar ropes tensile and the TPM was rigidified to a
2.2 Static Analysis
Bending stiffness analysis has been analyzed in Ref. [
Thus, stretching stiffness and compression stiffness will
be derived in this paper.
2.2.1 Stretching Stiffness
Axial force F0 is applied on each point A, B and C.
Circumcircle radius of cross section is R1 = lb √3. Stretch
force diagram is shown as Figure 2.
By resolving the forces in the x-direction at point A, the
equation is obtained as follows.
2Fr0 sin β0 + Fl = F0,
where Fl is axial force of longitudinal link, Fr0 is initial
pretension of the rope, lb is the cross link length, β0 is the
angle between line GD and line ED.
When the TPM is under stretching, deformation of
longitudinal link and diagonal rope has following related
equation based on geometric deformation condition.
where δir is rope deformation in each modulus, δli is
longitudinal stretch deformation of each modulus, lr is rope
length, Fr is the force in rope, Er is the material Young
modulus for rope, Ar is cross section area of the rope, l1
is the length of longitudinal rigid link, l2 is the length of
the TSH hinge, Fl is the force in the longitudinal link, E1
is the material Young modulus for longitudinal link, A1 is
the cross-section area of the longitudinal link, n1 is the
number of the tape spring, a11 is the unit stretching
stiffness of the TSH hinge.
Stretch deformation δl of the TPM is
n2 · 3F0(2l1 + l2) ,
δl = n2δli,
δr = δli sin β0,
i Fr lr
δr = Er Ar
2.2.2 Compression Stiffness
One end of the TPM is fixed and a compressive force 3F0
is applied on the other end. Due to initial tension Fr0 of
rope initial deformation is δr0 = Fr0lr/(ErAr), which leads
to a critical compress value 3F0′ on the end. Based on
the geometric deformation conditions longitudinal link
deformation of each module is written as follows:
When compressive force is applied at point A,
equilibrium equation is gotten as follows:
Simultaneous Eq. (2c), Eq. (5) and Eq. (6), critical
unloaded compressive force is expressed as follows:
F ′ = Er Ar sin β0 E21lA11 + n1la211
If axial compressive force is no less than the critical
unloaded compressive force, unloaded compressive
stiffness equals to sum total compression stiffness of
longitudinal rigid links and hyperelastic tape-spring hinges, that
where n2 is the module number of the TPM, EA is the
stretching stiffness of the TPM.
Simultaneous Eqs. (1), (2) and (3), stretching stiffness
of the TPM is obtained as follows:
EA = 6Er Ar sin3 β0 + 2l31(2l1 + l2)
E1A1 + l2(n11−Eνt )
If the TPM has no rape, the stretching stiffness is
rearranged as follows:
EA = 6Er Ar sin3 β0.
The compression stiffness is only related to the rope
stiffness ErAr and initial angle β0.
If axial compressive force is more than the critical
unloaded compressive force, the total deformation of the
where δ′r is deformation of the rope with only axial
compress force, δr0 is deformation of the rope under initial
Single module longitudinal link and rope deformation
meets the geometric deformation conditions. When 3F0
is lower than F′, simultaneous Eq. (2), Eq. (3), Eq. (6) and
Eq. (9) equivalent compression stiffness of the TPM is
EA = 6Er Ar sin3 β0 + 3 1 −
2Fr0 sin β0
(2l1 + l2)
E21lA11 + l2(n11−Eνt )
It is found that when 3F0 is more than F′ the
compression stiffness is related to the compression stiffness of the
longitudinal link and the TSH hinges; when 3F0 is lower
than F′, the compression stiffness changed with axial
load. If the end compress load 3F0 is much small, that is
3F0 = 12F0 sin3 β0 · E21lA11 + l2(1n1−Etν2) · ErlArr .
The compression stiffness can be written as follows
3 Deploying State Modal Analysis
3.1 Modal Analysis
Due to nonlinear characteristics of the TSH hinge modal
analysis of the TPM is performed by ABAQUS. In finite
element model x-axis is along direction of transverse
link, y-axis is along longitudinal link and z-axis points
from section center of transverse links to point of the
other two transverse links. Materials of the TSH hinge,
rigid link, transverse link and rope are Ni36CrTiAl,
stainless steel, aluminum alloy and Kevlar respectively.
Longitudinal link, transverse link and the TSH hinge are set up
with four nodes that are fully integrated to reduce shell
elements (S4R). Rope is modeled by two nodes and three
dimensional elements (T3D2). Weld is defined to model
connection between ropes and transverse links.
Reference point (RP) is established at each joint which are
given mass and inertial properties. Multi-point coupling
is applied to model the connection of transverse links.
The joint hinges are modeled by defining Hinge
connection. Contact of tape springs are modeled by defining Tie
Six reference points at the root of the TPM restrain
three translational Degree of Freedoms (DOFs) and
the TPM is in the state of cantilever. Firstly, the TPM is
performed on static analysis with 30 N pretension force
and deformation is got. Then, modal analysis is done by
subspace method with rope prestress deformation. Total
length for ten-module TPM is 5.26 m. Figure 3 shows the
FEM of the ten-module TPM with TSH hinges.
The five order modal shapes of ten modules TPM with
TSH hinge are shown in Figure 4. The five order mode
frequencies and mode shape description are listed in
3.2 Parameter Study
Compared to conventional rigid deployable mechanism
the TSH hinge introduces flexible influence. Thus, it is
necessary to analyze the effect of geometric parameters,
such as thickness t, cross-section radius R, center angle
φ and separation distance s, to mode frequency and
propose method of increasing stiffness. Table 2 lists
frequencies of ten-module TPM under different thickness. It is
shown that bend frequency increases 3.695%–11.84%
and twist frequency increases 0.186%–0.221% when the
thickness changes from 0.12 mm to 0.14 mm.
Table 3 lists ten-module mast frequencies under
different tape central angles. It can be seen that the bend
frequency increases by 3.654%–6.156% and twist frequency
increases by 0.393%–0.434% when the central angle
changes from 80° to 100°.
Frequencies of the ten-module TPM under
different radiuses are listed in Table 4. It can be seen that
bend frequencies increase by 2.674%–5.343% and twist
frequencies increase by 0.372%–0.41% when radius
change from 18 mm to 22 mm.
Frequencies of ten-module TPM under different
separations are listed in Table 5. It can be seen that bend
frequencies decrease by 0.0428%–0.0919% and twist
frequencies increase by 0.134%–0.141% when the separation
changes from 16 mm to 20 mm.
It can be concluded that geometric parameters have
greater influence on bend stiffness than twist stiffness.
Sensitivity of the geometric parameter is from large to
small as follows: thickness, central angle, radius and
separation. What’s more, front three parameters have
enhanced effect on bend stiffness and the last one has
4 Experiment Investigation
4.1 Two‑module TPM
To validate the precision of static stiffness theoretical
model, two units triangular prism mast is developed.
Figure 5 is folding and deployment configurations of
triangular prism mast with hyperelastic hinges.
Two adjacent units triangular prism mast are closed to
each other by locating pins and fastened to a work holder
by a rope when it is folded. At this time, the longitudinal
links are stowed into the prism frames and tension ropes
are located to grooves. After releasing the tension ropes,
the triangular prism mast is deploying by the driving of
the TSH hinges. Spherical wheels support the mast to
reduce the influence of gravity.
4.2 Compression Stiffness Test
Figure 6 is compression stiffness test diagram of the
TPM. Weights are applied at the end of the TPM for
10 kg at each time. Compress load and displacement for
the two-modulus TPM are listed in Table 6.
It can be calculated that experimental compression
stiffness is 2.324 × 106 N/m and theoretical value is
2.167 × 106 N/m. Relative error between the
experimental and theoretical value is 7.08%. The main reason for
the phenomenon is that equivalent stiffness of the TSH
hinges is much smaller, contact stiffness between the tape
springs should be considered.
4.3 Deploying Repeat Accuracy Test
Figure 7 shows deploying configuration for the two
modulus TPM. High-speed camera is employed to
capture deploying process of the TPM. Spherical hinges and
elastic rope are used to eliminate the effect of weight and
friction. Figure 8 shows longitudinal displacement-time
relationship for five times deploying processes. Table 7
lists compress load and displacement for the
40 f /Hz 60 80
a 1st –X response
20 40 f /Hz 60
b 1st –Y response
40 f /Hz 60
c 1st –Z response
40 f /Hz 60 80
d 2nd –X response
40 f /Hz 60
e 2nd –Y response
40 f /Hz 60
f 2nd –Z response
20 40 f /Hz 60 80 100 20 40 f /Hz 60
g 3rd –X response h 3rd –Y response
Figure 10 Vibration test curves of six nodes for two-module TPM at cantilever position
40 f /Hz 60
i 3rd –Z response
a 1st, bend
b 2nd, torsion
c 3rd, bend
d 4th, torsion
e 5th, bend
Based on longitudinal displacement for five times
deploying test standard deviation of the experimental
value is 1.688 mm. Thus, deploying repeat accuracy is
4.4 Modal Analysis of Cantilever Position
LMS multi-channel vibration test and analysis system is
employed to measure fundamental frequencies and mode
shapes which consists of exciting hammer, acceleration
sensor which can test three directional accelerations,
dynamic signal acquisition system and data processor.
The longitudinal direction of the TPM is set as x-axis,
outer normal direction of cross section is set as y-axis
and vertical downward direction is set as z-axis which
is selected as force hammer stimulating direction. The
measurements of three acceleration sensors, which are
located at three nodes on the crossbeam, are divided into
three groups. Then, integral modal superposition is
carried out. The exciting point is set at one end of the
crossbeam. Test apparatus and geometry diagram of cantilever
position are shown in Figure 9. Three vertexes on the
bottom are constrained points. Alphabets a and b stand for
the order of two times measurement. Location of point
b3 is closest to the exciting point. Vibration test curves of
six nodes for two-module TPM at cantilever position are
shown in Figure 10. Modal shapes for the TPM at
cantilever position are shown in Figure 11. Modal test results for
the two-modulus TPM at cantilever position are listed in
Table 8. It can be seen that acceleration sensor at point
b3 has a larger response at initial phase; response curves
of two group acceleration sensors are concentrated on
middle- and low frequency. The first order frequency of
the TPM at cantilever position is 13.02 Hz and
corresponding mode shape is bending.
Then, modal analysis of the two-module TPM at
cantilever position and the modeling method is identical to
the ten-module TPM by ABAQUS. Modal shape
simulation results for the two-module TPM at cantilever
position are shown in Figure 12.
The model with black lines is undeformed
configuration and the model with colored lines is deformed
configuration. Comparison between modal test and simulation
results for the two-modulus TPM at cantilever position
is listed in Table 9. It can be seen that the mode shapes
between simulated and experimental results are
consistent compared to Figures 11, 12 and Table 9. The relative
errors between simulated and experimental results for
frequencies are no more than 5.501%. It indicates that the
finite element model for the ten-module TPM is accurate.
Structure design and deploying modal analysis of the
tenmodulus TPM are performed. The two-modulus TPM is
processed and its mechanical properties are investigated
(1) The experimental and theoretical compression
stiffness static models are 2.324 × 106 N/m and
theoretical value is 2.167 × 106 N/m. It verifies the accuracy
of the static theoretical models.
(2) Geometric parameters have greater influence on
bend stiffness than twist stiffness. Sensitivity of the
geometric parameter is from large to small as
follows: thickness, central angle, radius and
(3) The deploying repeat accuracy of the two-module
TPM is 1.688 mm which is tested by the high-speed
(4) The veracity of the finite element model of the
tenmodule TPM at cantilever position is validated
by modal test of the two-module TPM. The first
fundamental frequency of the ten-module TPM
is 2.561 Hz and the corresponding mode shape is
Additional file 1. Brief introduction of the manuscript.
H-WG and R-QL was in charge of the whole trial; HY wrote the manuscript; YW
and ML assisted with sampling and laboratory analysis. All authors have read
and approved the final manuscript.
Hui Yang, born in 1986, is currently a lecturer at College of Electrical Engineering
and Automation, Anhui University, Hefei, China. She got the doctor degree from
Harbin Institute of Technology, China, in 2015. Her research interest includes
deployable mechanism, membrane antenna, triangular rollable and
collapsible boom, tape-spring hyperelastic hinge, multiobjective optimization design,
deployment dynamics and finite element analysis. E-mail: huiyang_0431@163.
Hong-Wei Guo, born in 1980, is currently an associate professor at Harbin
Institute of Technology, China. His main research interest includes space
manipulator system vibration control, large deployable structure and energy
absorber optimization. E-mail: .
Yan Wang, born in 1986, is currently a senior engineer at China Electronics
Technology Group Corporation No.38 Research, Hefei, China. His research interest
includes configuration synthesis and design of deployable truss structures.
Rong-Qiang Liu, born in 1965, is currently a professor at Harbin Institute of
Technology, China. His main research interest includes wearing robot, military
civil power robot and large deployable structure research. E-mail: liurq@hit.
Meng Li, born in 1985, is currently a senior engineer at Qian Xuesen
Laboratory of Space Technology, China Academy of Space Technology. He got the
doctor degree from Harbin Institute of Technology in 2013. His main research
interest includes the optimization design of energy-absorber structures,
impact dynamics and finite element method. E-mail: .
The authors declare that they have no competing interests.
Ethics Approval and Consent to Participate
Supported by National Natural Science Foundation of China (Grant No.
51605001), Joint Funds of the National Natural Science Foundation of China
(Grant No. U1637207), and Anhui University Research Foundation for Doctor
(Grant No. J01003222).
Springer Nature remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.
 K A Seffen. On the behavior of folded tape-springs . Journal of Applied Mechanics-Transactions of the ASME , 2001 , 68 ( 3 ): 369 - 375 .
 J Block , M Straubel , M Wiedemann . Ultralight deployable booms for solar sails and other large gossamer structures in space . Acta Astronautica , 2011 , 68 ( 7-8 ): 984 - 992 .
 M Mobrem , D S Adams . Deployment analysis of the lenticular jointed antennas onboard the Mars express spacecraft . Journal of Spacecraft and Rockets , 2009 , 46 ( 2 ): 394 - 402 .
 K Oya , J Onoda . Characteristics of carpenter tape hinge made of TiNi alloy . Proceedings of the 43rd AIAA /ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Denver, Colorado, April 22-25 , 2002 .
[5 ] H M Y C Mallikarachchi , S Pellegrino . Quasi-static folding and deployment of ultrathin composite tape-spring hinges . Journal of Spacecraft and Rockets , 2011 , 48 ( 1 ): 187 - 198 .
 Y Wang , R Q Liu , H Yang , et al. Design and deployment analysis of modular deployable structure for large antennas . Journal of Spacecraft and Rockets , 2015 , 52 ( 4 ): 1101 - 1110 .
 Y Wang , H W Guo , H Yang , et al. Deployment analysis and optimization of a flexible deployable structure for large synthetic aperture radar antennas . Proceedings of the Institute of Mechanical Engineers Part G Journal of Aerospace Engineering , 2016 , 230 ( 4 ): 615 - 627 .
[8 ] H M Y C Mallikarachchi , S Pellegrino . Deployment dynamics of ultrathin composite booms with tape-spring hinges . Journal of Spacecraft and Rockets , 2014 , 52 ( 2 ): 604 - 613 .
 B Zi , Y Li . Conclusions in theory and practice for advancing the applications of cable-driven mechanisms . Chinese Journal of Mechanical Engineering , 2017 , 30 ( 4 ): 763 - 765 .
 H G Yang , L Ding , H B Gao , et al. Experimental study and modeling of wheel's steering singkage for planetary exploration rovers . Journal of Mechanical Engineering , 2017 , 53 ( 8 ): 99 - 108 . (in Chinese)
 M Santer , A Sim , J Stafford . Testing of a segmented compliant deployable boom for CubeSat Magnetometer Missions . Proceedings of the 52nd AIAA / ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Denver, Colorado, April 4- 7 , 2011 .
 A M Watt , S Pellegrino . Tape-spring rolling hinges . Proceedings of the 36th Aerospace Mechanisms Symposium , Glenn Research Center, May 15 -17, 2002 : 1 - 14 .
 D S Adams , M Mobrem . Lenticular jointed antenna deployment anomaly and resolution onboard the mars express spacecraft . Journal of Spacecraft and Rockets , 2009 , 46 ( 2 ): 403 - 410 .
 M Mobrem , D S Adams . Deployment analysis of lenticular jointed antennas onboard the mars express spacecraft . Journal of Spacecraft and Rockets , 2009 , 46 ( 2 ): 394 - 402 .
 M J Silver , P Warren . Dynamic modeling of the folding of multi-shell flexible composites . Proceedings of the 51st AIAA /ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Orlando, Florida, April 12-15 , 2010 .
 T Schioler , S Pellegrino . A bistable structural element . Proceedings of the Institute of Mechanical Engineers Part C Journal of Engineering Sciences , 2008 , 222 : 2045 - 2051
 K A Seffen , S Pellegrino . Deployment dynamics of tape springs . Proceedings of the Royal Society of London Series A , 1999 , 455 ( 1983 ): 1003 - 1048 .
 K A Seffen , Z You , S Pellegrino . Folding and deployment of curved tape spring . International Journal of Mechanical Sciences , 2000 , 42 : 2055 - 2073 .
 F L Guan , X Y Wu , Y W Wang. The mechanical behavior of the double piece of tape spring . Proceedings of the 6th International Conference on Intelligent Computing, Changsha, China, August 18-21 , 2010 : 102 - 110 .
 J Wang , F L Guan , Z Z Zhou . Design and analysis of tape hinge for deployable structures . Journal of Astronautics . 2007 , 28 ( 3 ): 720 - 726 . (in Chinese)
 J B Bai , J J Xiong , J P Gao , et al. Analytical solutions for predicting in-plane strain and interlaminar shear stress of ultra-thin-walled lenticular collapsible composite tube in fold deformation . Composite Structures , 2013 , 97 ( 2 ): 64 - 75 .
 Y Yan , H B Wang , Q Li , et al. Finite element simulation of flexible roll forming with supplemented material data and the experimental verification . Chinese Journal of Mechanical Engineering , 2016 , 29 ( 2 ): 342 - 350 .
 W Wang , P Y Lian , S X Zhang , et al. Effect of facet displacement on radiation field and its application for panel adjustment of large reflector antenna . Chinese Journal of Mechanical Engineering , 2017 , 30 ( 3 ): 578 - 586 .
 H Yang , H W Guo , Y Wang , et al. Mechanical modeling and analysis for hyperelastic hinge in a triangular prism deployable mast . Journal of Astronautics , 2016 , 37 ( 3 ): 275 - 281 . (in Chinese)
 H Yang , R Q Liu , Y Wang , et al. Experiment and multiobjective optimization design of tape-spring hinges . Structural and Multidisciplinary Optimization , 2015 , 51 ( 6 ): 1373 - 1384 .