ma hema ics
A icle
Hyb id Op imiza ion Based Ma hema ical P ocedu e o
Dimensional Syn hesis o Slide -C ank Linkage
Al onso He nández 1, Ai o Muñoye o 2, Mónica U íza 1,* and En ique Amezua 1
Ci a ion: He nández, A.; Muñoye o,
A.; U íza , M.; Amezua, E. Hyb id
Op imiza ion Based Ma hema ical
P ocedu e o Dimensional Syn hesis
o Slide -C ank Linkage. Ma hema ics
2021,9, 1581. h ps://doi.o g/
10.3390/ma h9131581
Academic Edi o : Raimondas Ciegis
Recei ed: 31 May 2021
Accep ed: 2 July 2021
Published: 5 July 2021
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2021 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
1Facul y o Enginee ing in Bilbao, Uni e si y o he Basque Coun y (UPV/EHU), Plaza Ingenie o To es
Que edo, 48013 Bilbao, Spain; [email p o ec ed] (A.H.); [email p o ec ed] (E.A.)
2
SENER Ae oespacial, A da. de Zugaza e 56, 48992 Ge xo, Spain; ai o .munoye [email p o ec ed] (A.M.)
*Co espondence: [email p o ec ed] (M.U.)
Abs ac :
In his pape , an op imiza ion p ocedu e o pa h gene a ion syn hesis o he slide -c ank
mechanism will be p esen ed. The p oposed app oach is based on a hyb id s a egy, mixing local and
global op imiza ion echniques. Rega ding he local op imiza ion scheme, based on he null g adien
condi ion, a no el me hodology o sol e he esul ing non-linea equa ions is de eloped. The sol ing
p ocedu e consis s o decoupling wo subsys ems o equa ions which can be sol ed sepa a ely and
ollowing an i e a i e p ocess. In ela ion o he global echnique, a mul i-s a me hod based on a
gene ic algo i hm is implemen ed. The i ness unc ion inco po a ed in he gene ic algo i hm will ake
as a gumen s he se o dimensional pa ame e s o he slide -c ank mechanism. Se e al illus a i e
examples will p o e he alidi y o he p oposed op imiza ion me hodology, in some cases achie ing
an e en be e esul compa ed o mechanisms wi h a highe numbe o dimensional pa ame e s,
such as he ou -ba mechanism o he Wa ’s mechanism.
Keywo ds: pa h gene a ion; dimensional syn hesis; hyb id op imiza ion; slide -c ank mechanism
1. In oduc ion
Dimensional syn hesis consis s o inding a geome y ha enables a mechanism
o gene a e ce ain mo ion cha ac e is ics, such as ajec o ies o posi ions o elemen s.
I is ough o sol e his p oblem in ui i ely and o en equi es he implemen a ion o
speci ic me hods. Depending on he ype and amoun o p esc ibed mo ion cha ac e is ics,
i is no always possible o ob ain an exac solu ion o his p oblem, o cing us o use
op imiza ion me hods o ind an app oxima ion wi h minimal e o . The a ge mos
commonly add essed in bibliog aphies is he syn hesis ype, known as pa h gene a ion,
whe e a poin o a single deg ee o eedom mechanism is sough o un h ough a sequence
o p esc ibed posi ions. This mo ion may o may no be synch onized wi h he loca ion o
he inpu elemen , esul ing in p esc ibed o unp esc ibed iming p oblems, espec i ely.
I should be no ed ha he e also exis wo o he goals ha a e equen ly s udied, hese
being beyond he scope o his pape . These a e unc ion gene a ion, whe e he mo ion o
wo elemen s o he mechanism is synch onized, and mo ion gene a ion, whe e a sequence
o loca ions o a ce ain elemen o he mechanism is p esc ibed.
Mos o he li e a u e on dimensional syn hesis ocuses on indi idual cases, he ou -
ba hinged mechanism being he mos widely s udied. In he exis ing li e a u e, bo h he
exac syn hesis, by means o g aphic [
1
] o analy ical me hods [
2
], and he app oxima e syn-
hesis, by means o dimensional op imiza ion [
3
,
4
], ha e been s udied. On he o he hand,
he slide -c ank mechanism has been used o sol e unc ion gene a ion p oblems
[5–8]
, o
dynamic syn hesis [
9
,
10
], and o se e as an adjus able mechanism [
11
–
14
]. Howe e , no
publica ions on dimensional op imiza ion o pa h gene a ion ha e been ound. I should
be no ed ha , in addi ion o he usual kinema ic objec i es, some pape s include mo e
speci ic cha ac e is ics wi hin he e o unc ion. Fo example, in [
15
] a o mula ion is
Ma hema ics 2021,9, 1581. h ps://doi.o g/10.3390/ma h9131581 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2021,9, 1581 2 o 17
de eloped by using exac di e en ia ion ha allows o es ablishing he posi ion o he
ins an cen e o o a ion and he cen ode.
An o en-employed dimensional op imiza ion p ocedu e consis s o minimizing he
e o unc ion, o mula ed as he sum o squa ed di e ences be ween he poin s o he
disc e ized p esc ibed pa h, and hose belonging o he eal gene a ed pa h. The min-
imiza ion p ocess can be sol ed wi h di e en me hods ha can be classi ied in o wo
main g oups, hese being local and global me hods, which a e men ioned below. On he
o he hand, he e a e also publica ions ocused no so much on s udying he ma hema ical
op imiza ion echniques, bu on p oposing new ways o desc ibing he ou pu gene a ed
by he mechanism. This may esul in a mo e ad an ageous de ini ion o he op imiza ion
e o unc ion o o he elabo a ion o a lases and da abases. In ela ion o he pa h gen-
e a ion p oblems discussed in his pape , he e exis di e en app oaches o desc ibe he
ajec o ies, and p obably he mos ypical ones a e based on he Fou ie se ies [
16
] o Haa
Wa ele ans o m [
17
]. Simila ly, e e ence [
18
] desc ibes a uni ied heo y o he ha monic
cha ac e is ic pa ame e me hod o mechanism syn hesis. Apa om he dimensional
syn hesis, o he publica ions ocus on he phase ha p ecedes i , i.e., s uc u al syn hesis,
he i s s ep in he concep ual design o mechanisms. In his sense, he e can be ound
some p oposals o au oma ic algo i hms in ended o he s uc u al syn hesis o obo s and
closed-loop mechanisms [19].
In ela ion o he ma hema ical op imiza ion echniques o dimensional syn hesis,
he mos e ec i e and widely used local me hods consis o applying he null g adien
condi ion, which leads o a non-linea sys em o equa ions. This sys em o en includes
some passi e a iables ha canno be elimina ed. To sol e i , he unc ion is linea ized and
an i e a i e me hod, such as Gauss–New on, is used, s a ing om an app oxima e ini ial
solu ion p o ided by he designe . The o iginal e e ence o his ype o me hod is a pape
published in 1966 by Chi-Yeh [
20
], which was dedica ed o he ou -ba linkage. F om hen
on, se e al pape s ela ed o dimensional syn hesis o his mechanism by means o g adien
me hods ha e been published, explo ing di e en ways o imp o e he e ec i eness
o op imiza ion. As pa o hese al e na i e app oaches, in [
21
] he au ho s p oposed
modi ying he se o a iables o be op imized, conside ing he nodal coo dina es ins ead
o he usual dimensional pa ame e s, he eby allowing he elimina ion o some cons ain s
ha we e p esen in he o iginal p oblem. I is also no ewo hy ha o he publica ions
ocus on e o mula ing he e o unc ion, such as [
22
,
23
]. The au ho s o hose wo ks
p oposed o minimize he s ain ene gy o igina ed when he mechanism is o ced o un
exac ly h ough he p esc ibed ajec o y. On he o he hand, some au ho s ha e chosen o
es ima e he e o by a oiding i s sensi i i y o ansla ion and o a ion e ec s, such as [
24
],
whe e a sys em o ela i e coo dina es be ween p ecision poin s is used. Following he
same idea, he au ho s p oposed o pe o m a p io and independen phase o op imize he
ansla ion, o a ion, and scaling pa ame e s [
25
]. In addi ion o he di e en al e na i es
o cha ac e ize he design pa ame e s o he mechanism and o es ima e he esul ing e o ,
a ele an aspec o achie e good pe o mance in g adien me hods is o ca y ou an exac
calcula ion o he pa ial de i a i es, a oiding he nume ical de i a ion, since i inc eases
he compu a ional cos and esul s in a lowe e iciency. Gi en he in e es in sol ing
his p oblem, e e ence [
26
] p esen s a gene al me hod o calcula ing he exac pa ial
de i a i es om he loop equa ions p e iously iden i ied by he designe .
Despi e he la ge numbe o exis ing publica ions de o ed o op imal dimensional
syn hesis by means o g adien me hods, and o he imp o emen o hei pe o mance,
none o hem a e capable o sol ing he main limi a ion hey ha e. Un o una ely, hese
me hods a e highly sensi i e o he chosen s a ing app oxima ion, since hey a e local
in na u e and hence con e ge o he nea es minimum, which will no necessa ily be he
op imal o e all solu ion. To o e come his d awback, global me hods make i possible o
explo e he en i e space whe e solu ions can be ound. Me aheu is ic me hods a e he mos
common ones, and hey ha e been co e ed in se e al e e ences. These include gene ic
algo i hms [
27
–
29
], di e en ial e olu ion [
30
–
32
], an sea ch [
33
], k ill he d algo i hm [
34
],
Ma hema ics 2021,9, 1581 3 o 17
impe ialis compe i i e algo i hm [
35
], o neu al ne wo ks [
36
]. Ne e heless, he weakness
o heu is ic me hods in compa ison wi h g adien me hods is hei highe compu a ional
cos and a lowe con e gence a e. Fu he mo e, he e is no gua an ee ha hey will
con e ge o a minimum, nei he locally no globally.
Hyb id op imiza ion algo i hms, such as [
37
], gain g ea e s eng h when a global
me hod gene a ing seeds o s a ing mechanisms is combined wi h a good local me hod.
No mally, hyb id me hods s a by unning a global me hod o ob ain one o se e al designs
ha will la e be used as ini ial app oxima ions in a local me hod o quickly con e ge o
he nea es ela i e minimum. In his pape , as desc ibed below, a hyb id op imiza ion
app oach is p oposed.
In he case o he slide c ank mechanism analyzed in his pape , hanks o he
simplici y o i s kinema ics, i is possible o exp ess he syn hesis a iables di ec ly as
a unc ion o he dimensional and inpu pa ame e s, and hus comple ely elimina ing
he passi e a iables. A no el aspec o he p oposed app oach is he way in which he
esul ing sys em o equa ions is sol ed. Conside ing he mo e gene al case o unp esc ibed
iming syn hesis, he inal sys em o equa ions associa ed wi h he null g adien condi ion
can be di ided in o wo subsys ems wi h di e en cha ac e is ics. As will be explained,
he p ocedu e desc ibed in his a icle allows each subsys em o be sol ed sepa a ely
wi hin an i e a i e p ocess ha connec s hem oge he . This makes i possible, in some
pa icula ly simple cases, such as he wo-pa ame e slide -c ank, o sol e all he equa ions
analy ically, while in mo e complex cases, nume ical me hods mus necessa ily be adop ed.
The op imum solu ion eached will be a ela i e minimum o he e o unc ion and will be
in luenced by he ini ial app oxima ion used o sol e he sys em o equa ions nume ically.
In his pape , educing his dependency will be a emp ed by unning he local op imiza ion
algo i hm om di e en s a ing poin s p e iously selec ed by a gene ic algo i hm. The
gene a ion o he s a ing poin s could also be done by ano he ype o heu is ic me hod,
o h ough a sweeping p ocess ha gene a es andom poin s wi hin he en i e sea ch
space. E en so, he la e echnique would no achie e he mos p omising egions as he
gene ic algo i hm does, bu i would only gene a e a wide g id o di e en s a ing poin s.
The e o e, he gene ic algo i hm is he p e e ed choice.
I is impo an o highligh ha he p ocedu e desc ibed in his pape seeks o ob ain
eliable solu ions, no only om a ma hema ical ision, bu also om a p ac ical poin o
iew. Fo his eason, i will be s a ed how o a oid he ci cui de ec in he slide -c ank
mechanism, elying on he concep o b anch index. This concep was in oduced in
e e ence [
21
] o analyze he kinema ics o he ou -ba hinged linkage. The inco po a ion
o design cons ain s will be add essed by means o penal y unc ions included in he e o
unc ion o be minimized. This is essen ial o impose he G asho c i e ion and hus ensu e
ha he inpu elemen is able o ully o a e (c ank inpu ).
Finally, he e ec i eness o he p oposed me hodology will be illus a ed h ough
di e en examples. The inal solu ions ob ained in his pape a e as accu a e as he ones
eached in o he pape s when sol ing he same p oblem by using mo e complex designs,
such as ou -ba o Wa linkages.
The main no el ies and highligh s o his pape a e:
•
Deduc ion o he equa ions equi ed o he op imal dimensional syn hesis o he slide -
c ank mechanism, which cons i u es an al e na i e o he hinged ou -ba linkage
usually used in he li e a u e o sol e his ype o p oblem.
•
P oposal o an o iginal me hodology o sol e a non-linea sys em o equa ions e-
sul ing om he null g adien condi ion, based on he decoupling o wo subsys ems
o equa ions. I acili a es he esolu ion o he sys em and, in some cases, allows o
ob ain all he solu ions in an analy ical way.
•
In eg a ion o he local op imiza ion me hodology wi hin a hyb id op imiza ion
me hod, which uses a gene ic algo i hm o sea ch o he bes s a ing app oxima ions.
The i ness unc ion has been adap ed o sol e no only he p esc ibed iming p oblem,
bu also unp esc ibed iming.
Ma hema ics 2021,9, 1581 4 o 17
•
Sol ing and compa ison o examples p oposed by o he au ho s in he li e a u e
dealing wi h he ou -ba linkage. Thanks o he e ec i eness o he me hod p oposed
in his wo k, he slide -c ank mechanism, hough being simple and mo e limi ed, is
able o p o ide simila pe o mances (o e en be e in some cases) in pa h gene a ion
p oblems.
2. Ma e ials and Me hods
In his sec ion, he basis o he op imum syn hesis p ocedu e as well as he unc ioning
o he hyb id op imiza ion s a egy is p esen ed.
2.1. Bases o he Op imum Syn hesis P ocedu e
Be o e ackling he op imiza ion p ocess, he loop-closu e equa ions o he mechanism
and he posi ion equa ions o he couple poin ha aces he ajec o y will be ob ained.
In addi ion, he a iables ha ake pa in he syn hesis p ocess and hei unc ional
dependence law will be desc ibed. This enables ob aining he pa ial de i a i es ha
in e ene in he minimiza ion p ocess.
2.1.1. Syn hesis Equa ions o a Gene al Design
Fi s , he kinema ic p oblem o he slide -c ank mechanism wi h i e dimensional
pa ame e s, ep esen ed in Figu e 1, is ob ained.
Ma hema ics 2021, 9, x FOR PEER REVIEW 4 o 18
• In eg a ion o he local op imiza ion me hodology wi hin a hyb id op imiza ion
me hod, which uses a gene ic algo i hm o sea ch o he bes s a ing app oxima-
ions. The i ness unc ion has been adap ed o sol e no only he p esc ibed iming
p oblem, bu also unp esc ibed iming.
• Sol ing and compa ison o examples p oposed by o he au ho s in he li e a u e deal-
ing wi h he ou -ba linkage. Thanks o he e ec i eness o he me hod p oposed in
his wo k, he slide -c ank mechanism, hough being simple and mo e limi ed, is
able o p o ide simila pe o mances (o e en be e in some cases) in pa h gene a ion
p oblems.
2. Ma e ials and Me hods
In his sec ion, he basis o he op imum syn hesis p ocedu e as well as he unc ion-
ing o he hyb id op imiza ion s a egy is p esen ed.
2.1. Bases o he Op imum Syn hesis P ocedu e
Be o e ackling he op imiza ion p ocess, he loop-closu e equa ions o he mecha-
nism and he posi ion equa ions o he couple poin ha aces he ajec o y will be ob-
ained. In addi ion, he a iables ha ake pa in he syn hesis p ocess and hei unc-
ional dependence law will be desc ibed. This enables ob aining he pa ial de i a i es
ha in e ene in he minimiza ion p ocess.
2.1.1. Syn hesis Equa ions o a Gene al Design
Fi s , he kinema ic p oblem o he slide -c ank mechanism wi h i e dimensional
pa ame e s, ep esen ed in Figu e 1, is ob ained.
Figu e 1. Slide -c ank mechanism wi h 5 dimensional pa ame e s.
The loop-closu e equa ions a e he ollowing:
𝑎1·𝑠𝑖𝑛𝜑=𝑎5−𝑎2·𝑠𝑖𝑛𝜃
(1)
𝑠=𝑎1·𝑐𝑜𝑠𝜑+𝑎2·𝑐𝑜𝑠𝜃
(2)
F om Equa ion (1) yields:
𝑠𝑖𝑛𝜃=𝑎5 − 𝑎1·𝑠𝑖𝑛𝜑
𝑎2
(3)
Then, cos𝜃 is gi en by Equa ion (4), whe e 𝐾=±1.
𝑐𝑜𝑠𝜃=𝐾√1−(𝑎5 – 𝑎1·𝑠𝑖𝑛𝜑
𝑎2)2
(4)
The syn hesis equa ions a e as ollows:
Figu e 1. Slide -c ank mechanism wi h 5 dimensional pa ame e s.
The loop-closu e equa ions a e he ollowing:
a1·sinϕ=a5−a2·sinθ(1)
s=a1·cosϕ+a2·cosθ(2)
F om Equa ion (1) yields:
sinθ=a5−a1·sinϕ
a2
(3)
Then, cosθis gi en by Equa ion (4), whe e K=±1.
cosθ=Ks1−a5−a1·sinϕ
a22
(4)
The syn hesis equa ions a e as ollows:
x0=s−(a2−a3)·cosθ+a4·cosθ−3π
2(5)
y0=−(a2−a3)·sinθ+a5+a4·cosθ(6)
Ma hema ics 2021,9, 1581 5 o 17
Using he loop equa ions o sol e he passi e a iables
s
(Equa ion (2)) and
θ
(Equa ions (3) and (4)), and subs i u ing hem in Equa ions (5) and (6), he ollowing
exp essions a e ob ained o he syn hesis a iables, e e ing o he local sys em O’X’Y’:
x0=a1cosϕ+a3·Ks1−a5−a1sinϕ
a22
−a4·(a5−a1sinϕ)
a2
(7)
y0=(a3−a2)·(a5−a1sinϕ)
a2
cosϕ+a5+a4·Ks1−a5−a1sinϕ
a22
(8)
Rema k Rega ding B anches and Ci cui s
The sign
±
in Equa ions (7) and (8), which has been subs i u ed o
K=±
1 o
simplici y, is ela ed o he wo possible posi ions o he couple poin
P
o he same
inpu
ϕ
. This means ha wo b anches exis , each one associa ed wi h he posi i e o
nega i e alue o
K
. This ci cums ance is illus a ed in Figu e 2, in which he wo possible
con igu a ions o he mechanism, o a gi en alue o he inpu ϕ, a e ep esen ed.
Ma hema ics 2021, 9, x FOR PEER REVIEW 5 o 18
𝑥′=𝑠−(𝑎2−𝑎3)·𝑐𝑜𝑠𝜃+𝑎4·𝑐𝑜𝑠(𝜃−3𝜋
2)
(5)
𝑦′=−(𝑎2−𝑎3)·𝑠𝑖𝑛𝜃+𝑎5+𝑎4·𝑐𝑜𝑠𝜃
(6)
Using he loop equa ions o sol e he passi e a iables 𝑠 (Equa ion (2)) and 𝜃
(Equa ions (3) and (4)), and subs i u ing hem in Equa ions (5) and (6), he ollowing ex-
p essions a e ob ained o he syn hesis a iables, e e ing o he local sys em O’X’Y’:
𝑥′=𝑎1𝑐𝑜𝑠𝜑+𝑎3·𝐾√1−(𝑎5−𝑎1𝑠𝑖𝑛𝜑
𝑎2)2−𝑎4·(𝑎5−𝑎1𝑠𝑖𝑛𝜑)
𝑎2
(7)
𝑦′= (𝑎3−𝑎2)·(𝑎5−𝑎1𝑠𝑖𝑛𝜑)
𝑎2 𝑐𝑜𝑠𝜑+𝑎5+ 𝑎4·𝐾√1−(𝑎5−𝑎1𝑠𝑖𝑛𝜑
𝑎2)2
(8)
Rema k Rega ding B anches and Ci cui s
The sign ± in Equa ions (7) and (8), which has been subs i u ed o 𝐾=±1 o sim-
plici y, is ela ed o he wo possible posi ions o he couple poin 𝑃 o he same inpu
𝜑. This means ha wo b anches exis , each one associa ed wi h he posi i e o nega i e
alue o 𝐾. This ci cums ance is illus a ed in Figu e 2, in which he wo possible con ig-
u a ions o he mechanism, o a gi en alue o he inpu 𝜑, a e ep esen ed.
Figu e 2. Two possible con igu a ions o he same inpu .
The wo possible ajec o ies o poin 𝑃 associa ed wi h he di e en con igu a ions
o he couple elemen , commonly known as b anches, can be connec ed o unconnec ed,
esul ing in a unique ci cui (a unicu sal cu e), o wo ci cui s (a bicu sal cu e). In his
wo k, designs whe e he c ank inpu is able o pe o m a 360° ull o a ion a e conside ed,
meaning ha he G asho c i e ion mus be ul illed. The e o e, he wo possible b anches
will be wo unconnec ed ci cui s. To a oid b anch de ec s, all he selec ed poin s mus
ha e he same alue o 𝐾. This alue will be he one co esponding o he b anch ha
yields a minimum e o wi h espec o he desi ed pa h.
In he mos gene al case, ep esen ed in Figu e 3, he local e e ence sys em O’X’Y’
has a o a ion ela i e o he global sys em OXY, de ined by he pa ame e 𝑎6, and a ans-
la ion in he plane de ined by he pa ame e s 𝑎7 and 𝑎8.
Figu e 2. Two possible con igu a ions o he same inpu .
The wo possible ajec o ies o poin
P
associa ed wi h he di e en con igu a ions
o he couple elemen , commonly known as b anches, can be connec ed o unconnec ed,
esul ing in a unique ci cui (a unicu sal cu e), o wo ci cui s (a bicu sal cu e). In his
wo k, designs whe e he c ank inpu is able o pe o m a 360
◦
ull o a ion a e conside ed,
meaning ha he G asho c i e ion mus be ul illed. The e o e, he wo possible b anches
will be wo unconnec ed ci cui s. To a oid b anch de ec s, all he selec ed poin s mus ha e
he same alue o
K
. This alue will be he one co esponding o he b anch ha yields a
minimum e o wi h espec o he desi ed pa h.
In he mos gene al case, ep esen ed in Figu e 3, he local e e ence sys em O’X’Y’ has
a o a ion ela i e o he global sys em OXY, de ined by he pa ame e
a6
, and a ansla ion
in he plane de ined by he pa ame e s a7and a8.
The equa ions ha exp ess he syn hesis a iables in he global e e ence sys em a e
he ollowing:
xi=xi0·cos(a6)−yi0·sin(a6)+a8(9)
yi=xi0·sin(a6)+yi0·cos(a6)+a7(10)
Ma hema ics 2021,9, 1581 6 o 17
Ma hema ics 2021, 9, x FOR PEER REVIEW 6 o 18
Figu e 3. Slide -c ank mechanism wi h 8 dimensional pa ame e s.
The equa ions ha exp ess he syn hesis a iables in he global e e ence sys em a e
he ollowing:
𝑥𝑖=𝑥𝑖′·𝑐𝑜𝑠(𝑎6)−𝑦𝑖′·𝑠𝑖𝑛(𝑎6)+𝑎8
(9)
𝑦𝑖=𝑥𝑖′·𝑠𝑖𝑛(𝑎6)+𝑦𝑖′·𝑐𝑜𝑠(𝑎6)+𝑎7
(10)
Now ha all he a iables in ol ed in he syn hesis p oblem ha e been de ined, he
ollowing classi ica ion can be es ablished:
• Dimensional a iables: 𝑎1,𝑎2,…,𝑎8. These a e a iables ha de ine he leng hs o
he ba s and he ansla ion o o a ion pa ame e s o he s udied mechanism.
• Inpu a iable: 𝜑. This is an independen a iable co esponding o he deg ee o
eedom o he mechanism unde s udy.
• Passi e a iables: 𝜃,𝑠. These a e no independen a iables, bu a he depend on
he inpu and he dimensional pa ame e s.
• Ou pu a iables o syn hesis a iables: 𝑥,𝑦. These co espond o he coo dina es
o he couple poin P. In he case o pa h gene a ion syn hesis, hese a e indeed he
syn hesis a iables.
2.1.2. Op imal Design Based on he E o Func ion
The e o unc ion 𝐸, commonly used in syn hesis p oblems, is de ined as he sum
o he squa ed Ca esian dis ances be ween he p esc ibed poin s and hose ac ually gen-
e a ed:
𝐸=∑ [(𝑥𝑖−𝑥𝑖𝑑)2+(𝑦𝑖−𝑦𝑖𝑑)2]
𝑁
𝑖=1
(11)
The e o be ween p esc ibed and gene a ed ajec o y mus be minimized o ob ain
he op imal mechanism. The e a e wo op ions o ca ying ou his minimiza ion. In he
modali y known as p esc ibed iming, only he dimensional pa ame e s a e op imized,
equi ing sol ing he sys em shown in Equa ion (12). In his case, he inpu pa ame e s 𝜑𝑖
a e no a iables o be op imized, bu cons an alues (p esc ibed alues). Howe e , in an
op imiza ion known as unp esc ibed iming, bo h he dimensional pa ame e s 𝑎𝑗 and he
se o inpu pa ame e s 𝜑𝑖 a e op imized, equi ing sol ing he sys ems gi en by Equa-
ions (12) and (13). This las op ion is mo e complex bu i s po en ial o ob ain p ecise
solu ions is g ea e , since he alue o he inpu pa ame e s is no being es ic ed.
Figu e 3. Slide -c ank mechanism wi h 8 dimensional pa ame e s.
Now ha all he a iables in ol ed in he syn hesis p oblem ha e been de ined, he
ollowing classi ica ion can be es ablished:
•Dimensional a iables: a1
,
a2
,
. . .
,
a8
. These a e a iables ha de ine he leng hs o
he ba s and he ansla ion o o a ion pa ame e s o he s udied mechanism.
•Inpu a iable: ϕ
. This is an independen a iable co esponding o he deg ee o
eedom o he mechanism unde s udy.
•Passi e a iables: θ
,
s
. These a e no independen a iables, bu a he depend on he
inpu and he dimensional pa ame e s.
•Ou pu a iables o syn hesis a iables: x
,
y
. These co espond o he coo dina es
o he couple poin P. In he case o pa h gene a ion syn hesis, hese a e indeed he
syn hesis a iables.
2.1.2. Op imal Design Based on he E o Func ion
The e o unc ion
E
, commonly used in syn hesis p oblems, is de ined as he sum o
he squa ed Ca esian dis ances be ween he p esc ibed poin s and hose ac ually gene a ed:
E=
N
∑
i=1xi−xd
i2+yi−yd
i2(11)
The e o be ween p esc ibed and gene a ed ajec o y mus be minimized o ob ain
he op imal mechanism. The e a e wo op ions o ca ying ou his minimiza ion. In he
modali y known as p esc ibed iming, only he dimensional pa ame e s a e op imized,
equi ing sol ing he sys em shown in Equa ion (12). In his case, he inpu pa ame e s
ϕi
a e no a iables o be op imized, bu cons an alues (p esc ibed alues). Howe e ,
in an op imiza ion known as unp esc ibed iming, bo h he dimensional pa ame e s
aj
and he se o inpu pa ame e s
ϕi
a e op imized, equi ing sol ing he sys ems gi en by
Equa ions (12) and (13)
. This las op ion is mo e complex bu i s po en ial o ob ain p ecise
solu ions is g ea e , since he alue o he inpu pa ame e s is no being es ic ed.
∂E
∂aj
=0→
N
∑
i=1"xi−xd
i∂xi
∂aj
+yi−yd
i∂yi
∂aj#=0∀j=1, 2, . . . , n(12)
∂E
∂ϕi
=0→
N
∑
i=1xi−xd
i∂xi
∂ϕi
+yi−yd
i∂yi
∂ϕi=0∀i=1, 2, . . . , N(13)
Ma hema ics 2021,9, 1581 7 o 17
2.2. Hyb id Op imiza ion P ocedu e
Thanks o he simplici y o i s kinema ics, in he slide -c ank mechanism i is possible
o elimina e he passi e a iables. In his way, he syn hesis a iables (x,y)a e exp essed
as an explici unc ion o pa ame e s
aj
and
ϕ
, acili a ing he ob aining o he pa ial
de i a i es ha appea in Equa ions (12) and (13). A e pe o ming he subs i u ion, i
can be e i ied ha he sys em o Equa ions (12) and (13) is non-linea . This sys em can be
decomposed in o wo subsys ems o equa ions. On he one hand, Equa ion (12) is composed
o 8 equa ions, as many as he dimensional pa ame e s o he mechanism. Each o hese
equa ions has he o m
g(a1,a2. . . a8,ϕ1,ϕ2, . . . ϕN)=
0. On he o he hand, Equa ion (13)
is made up o Nequa ions, as many as poin s o p ecision has he ajec o y. Each o hese
equa ions has he o m
h(a1,a2. . . a8,ϕi)=
0. This ci cums ance is undamen al when
es ablishing he p ocedu e o sol ing he sys em o equa ions.
In addi ion, in o de o ind he op imal ini ial app oxima ion, he one ha will be
he s a ing poin o he local op imiza ion p ocess, and ha plays a ele an ole in he
p ocess, a mul i-s a app oach will be implemen ed. In he ollowing sec ions, he hyb id
p ocedu e p oposed by he au ho s o his wo k will be explained.
2.2.1. Sol ing he Equa ion Sys em o Local Op imiza ion
In his sec ion, he i e a i e algo i hm o sol e he non-linea sys em o equa ions
esul ing om he null g adien condi ion is add essed. The p ocedu e is based on he
decoupling o he wo subsys ems (12) and (13), as desc ibed below.
•Fi s phase:
This s a s om a ce ain mechanism, coming om he mul i-s a p ocedu e ha
will be explained in Sec ion 3.2. The dimensions (
a1
,
a2
,
. . .
,
a8
) o his mechanism a e
assumed cons an in his phase. The alues o hese dimensions a e subs i u ed in each o
he equa ions o subsys em (13), esul ing in a o al o Nequa ions, each wi h a unique
ϕi
.
In he simples case o he slide -c ank mechanism, wi h 2 dimensional pa ame e s
(
a1=a2=a3
and
a5=a6=a7=a8=
0), and using he ans o ma ion o hal -angle
angen , i= an ϕi
2, Equa ion (13) becomes a ou h deg ee polynomial:
2a1a4+xd
ia4 4
i+8a12+xd
ia1+2yd
ia4 3
i−12a1a4 2
i+−8a12+4xd
ia1+2yd
ia4 i+2a1a4−xd
ia4=0 (14)
F om he 4 possible alues o
i(ϕi)
, he one ha gi es he minimum e o is he
selec ed one.
Fo a design wi h 3 dimensional pa ame e s, Equa ion (13) becomes a polynomial o
deg ee 10. E en so, i is easy o ob ain he 10 oo s and de ec he co ec one p oceeding as
in he p e ious case. Howe e , wi h 4 o mo e dimensional pa ame e s, i is no longe as
easy o de e mine i s co esponding uni a ia e polynomial, no is i eally wo h i . I is
mo e p ac ical o ope a e as explained nex .
The Equa ion (13) is sol ed nume ically, s a ing om an ini ial app oxima ion, and
a i ing a a unique solu ion o he pa ame e
ϕi
. To gua an ee ha he global op imal
alue o
ϕi
is ob ained, i will be necessa y o s a om an ini ial app oxima ion ob ained
as ollows: each sum o Equa ion (11), which ep esen s he e o made in each syn hesis
posi ion
i
, is e alua ed as a unc ion o he inpu pa ame e
ϕi
along he disc e ized domain
[0,2
π
). In his way, a g aph simila o he blue cu e ep esen ed in Figu e 4will be ob ained,
whe e wo minima o he e o unc ion appea , he one indica ed on he igh being he
one wi h he lowes alue. The la e alue o
ϕi
is aken as a s a ing app oxima ion o
sol e Equa ion (13) (in ou case, using he MATLAB sol e command).
Ma hema ics 2021,9, 1581 8 o 17
Ma hema ics 2021, 9, x FOR PEER REVIEW 8 o 18
being he one wi h he lowes alue. The la e alue o 𝜑𝑖is aken as a s a ing app oxi-
ma ion o sol e Equa ion (13) (in ou case, using he MATLAB sol e command).
Figu e 4. E o unc ion o a poin 𝑖 e alua ed a [0,2π)
To illus a e his concep , in Figu e 5 he p esc ibed poin o a syn hesis posi ion 𝑖
and he ajec o y gene a ed by he mechanism o dimensions {𝑎𝑗} in he cu en i e a ion
a e indica ed. The e ec o e alua ing he sum o he e o unc ion (11) and choosing he
absolu e minimum is equi alen o a e sing he gene a ed ajec o y and selec ing he
poin o i ( he black poin ) closes o he p esc ibed one ( ed poin ). In his example, he
black poin indica ed in Figu e 5 co esponds o he absolu e minimum, 𝜑=5.88 ad,
which is he one p e iously indica ed in Figu e 4.
Figu e 5. Gene a ed ajec o y and p esc ibed poin ( ed).
• Second phase:
In his phase, he alues 𝜑𝑖ob ained in he p e ious phase will be assumed as con-
s an s, and he unknowns (𝑎1,𝑎2,…,𝑎8) will be calcula ed om he subsys em om Equa-
ion (12).
As an example, in he simpli ied pa icula case o he slide -c ank mechanism wi h
2 pa ame e s, Equa ion (12) becomes he ollowing linea sys em:
Figu e 4. E o unc ion o a poin ie alua ed a [0,2π).
To illus a e his concep , in Figu e 5 he p esc ibed poin o a syn hesis posi ion
i
and he ajec o y gene a ed by he mechanism o dimensions
aj
in he cu en i e a ion
a e indica ed. The e ec o e alua ing he sum o he e o unc ion (11) and choosing
he absolu e minimum is equi alen o a e sing he gene a ed ajec o y and selec ing
he poin o i ( he black poin ) closes o he p esc ibed one ( ed poin ). In his example,
he black poin indica ed in Figu e 5co esponds o he absolu e minimum,
ϕ=
5.88 ad,
which is he one p e iously indica ed in Figu e 4.
Ma hema ics 2021, 9, x FOR PEER REVIEW 8 o 18
being he one wi h he lowes alue. The la e alue o 𝜑𝑖 is aken as a s a ing app oxi-
ma ion o sol e Equa ion (13) (in ou case, using he MATLAB sol e command).
Figu e 4. E o unc ion o a poin 𝑖 e alua ed a [0,2π)
To illus a e his concep , in Figu e 5 he p esc ibed poin o a syn hesis posi ion 𝑖
and he ajec o y gene a ed by he mechanism o dimensions {𝑎𝑗} in he cu en i e a ion
a e indica ed. The e ec o e alua ing he sum o he e o unc ion (11) and choosing he
absolu e minimum is equi alen o a e sing he gene a ed ajec o y and selec ing he
poin o i ( he black poin ) closes o he p esc ibed one ( ed poin ). In his example, he
black poin indica ed in Figu e 5 co esponds o he absolu e minimum, 𝜑=5.88 ad,
which is he one p e iously indica ed in Figu e 4.
Figu e 5. Gene a ed ajec o y and p esc ibed poin ( ed).
• Second phase:
In his phase, he alues 𝜑𝑖 ob ained in he p e ious phase will be assumed as con-
s an s, and he unknowns (𝑎1,𝑎2,…,𝑎8) will be calcula ed om he subsys em om Equa-
ion (12).
As an example, in he simpli ied pa icula case o he slide -c ank mechanism wi h
2 pa ame e s, Equa ion (12) becomes he ollowing linea sys em:
Figu e 5. Gene a ed ajec o y and p esc ibed poin ( ed).
•Second phase:
In his phase, he alues
ϕi
ob ained in he p e ious phase will be assumed as
cons an s, and he unknowns (
a1
,
a2
,
. . .
,
a8
) will be calcula ed om he subsys em om
Equa ion (12).
As an example, in he simpli ied pa icula case o he slide -c ank mechanism wi h
2 pa ame e s, Equa ion (12) becomes he ollowing linea sys em:
4N
∑
i=1
cos2ϕi
N
∑
i=1
sin2ϕi
N
∑
i=1
sin2ϕiN
a1
a4
=
2N
∑
i=1
cosϕi·xd
i
N
∑
i=1sinϕi·xd
i+cosϕi·yd
i
(15)
Howe e , in he design cases wi h 3 o mo e dimensional pa ame e s, he subsys em
o equa ions u ns ou o be non-linea , making i necessa y o apply nume ical sol ing
Ma hema ics 2021,9, 1581 9 o 17
me hods. As a s a ing app oxima ion, he alues (
a1
,
a2
,
. . .
,
a8
) ha we e assumed as
cons an s in he p e ious phase will be aken.
•Nex s eps:
Wi h he new alues o
aj
ob ained in he second phase, we go back o he i s phase,
and hus con inue i e a i ely un il con e gence. The chosen s opping c i e ion consis s
o compa ing he cu en alues o he op imiza ion a iables
(a1,a2. . . a8,ϕ1,ϕ2, . . . ϕN)
wi h hose o he p e ious i e a ion, so ha when he di e ence is less han a speci ied
ole ance, he i e a i e p ocess will s op.
2.2.2. Implemen ing a Mul i-S a S a egy
The me hod p oposed in Sec ion 3.1 has he disad an age o being e y sensi i e o
he s a ing app oxima ion. To a oid his d awback, a mul i-s a app oach based on a
gene ic algo i hm will be used o guide he local op imiza ion me hod owa ds a sea ch o
he global op imal solu ion.
The p oposed mul i-s a app oach ob ains a su icien ly la ge se o s a ing app oxi-
ma ions, ep esen a i e o a global sweep o he design space
aj
. In his a icle, a gene ic
algo i hm is used o loca e 100 candida e solu ions ha se e as ini ial app oxima ions. I is
decided o use he gene ic algo i hm inco po a ed in MATLAB, assigning as a gumen s o
he i ness unc ion he alues o he 8 dimensional pa ame e s. The la e unc ion assigns
he op imal co espondence o inpu pa ame e s, acco ding o he p ocedu e ou lined in
he i s phase o Sec ion 3.1, and e u ns a scala alue ha quan i ies he indi idual’s
i ness as he quad a ic sum o he dis ances be ween gene a ed and p esc ibed poin s.
I is impo an o bea in mind ha , conside ing he way he i ness unc ion is posed,
each s a ing solu ion o he unp esc ibed iming p oblem will no depend on 8
+N
a iables (8 dimensional +
N
inpu s), bu only on he 8 dimensional pa ame e s, which
acili a es he explo a ion o he sea ch space by he gene ic algo i hm wi hou he need o
es ic he inpu pa ame e s.
The comple e scheme o his hyb id p ocedu e is shown in Figu e 6. As can be seen,
i consis s o execu ing a local op imiza ion om he di e en s a ing poin s ob ained
in he scan o he space o he dimensional pa ame e s
aj
. In his way, di e en local
minimums will be ob ained and he bes o hem will be selec ed.
To a oid excessi e compu a ional cos , he execu ion ime o he gene ic algo i hm
is limi ed o an accep able alue (i.e., 1 min). Bea in mind ha i will no be necessa y o
ob ain he op imal solu ions, bu ha i will be enough o be close o hem. On he o he
hand, he execu ion ime o he local me hod o be applied la e will depend on he numbe
o i e a ions pe o med, bu i gene ally consumes a ew seconds o each s a ing poin
used.
2.2.3. Inco po a ion o Design Cons ain s
Sea ching o an op imal design usually implies adap ing he gene a ed ajec o y o
a p esc ibed one. In addi ion, some design equi emen s mus be ul illed. These design
cons ain s a e ela ed o se e al aspec s, such as he maximum size o he mechanism, he
maximum and minimum leng hs o ce ain ba s, he G asho c i e ion, he ansmission
angle limi s, and so on.
Ma hema ics 2021,9, 1581 16 o 17
use o penal y unc ions. In his way, i can be ensu ed ha he c ank inpu mechanism does
no exceed a maximum a io be ween ba leng hs o o he addi ional design equi emen s
a he disc e ion o he designe .
Finally, he e ec i eness o he me hod has been p o ed by e i ying, h ough a ious
examples, ha he slide -c ank mechanism allows us o achie e solu ions wi h a p ecision
compa able o o he one deg ee o eedom mechanisms ha ha e a g ea e numbe o
dimensional pa ame e s, such as he ou -ba o he Wa ’s mechanism, wi h he addi ional
ad an age o ha ing simple kinema ics.
Supplemen a y Ma e ials:
The ollowing a e a ailable online a h ps://www.mdpi.com/a icle/10
.3390/ma h9131581/s1. The h ee ideos co esponding o he mo ion o he op imum mechanisms
a e a ailable.
Au ho Con ibu ions:
Concep ualiza ion, A.H., A.M., M.U. and E.A.; me hodology, A.H., A.M. and
M.U.; so wa e, A.M.; alida ion, A.H., A.M., M.U. and E.A.; o mal analysis, A.H.; in es iga ion,
A.H., A.M., M.U. and E.A.; esou ces, A.H. and A.M.; da a cu a ion, A.M.; w i ing—o iginal d a
p epa a ion, A.H. and A.M.; w i ing— e iew and edi ing, M.U. and E.A.; isualiza ion, A.M.;
supe ision, A.H.; p ojec adminis a ion, A.H. and M.U.; unding acquisi ion, A.H., M.U. and E.A.
All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
The au ho s wish o acknowledge inancial suppo ecei ed om he Spanish go e nmen
h ough he Minis e io de Economía y Compe i i idad (P ojec DPI2015
−
67626-P (MINECO/FEDER,
UE)), he suppo o he esea ch g oup h ough P ojec Re . IT949
−
16, p o ided by he Depa -
amen o de Educación, Polí ica Lingüís ica y Cul u a om he egional Basque Go e nmen , and
he P og am BIKAINTEK 2020 (Re . 012-B2/2020) p o ided by he Depa amen o de Desa ollo
Económico, Sos enibilidad y Medio Ambien e om he egional Basque Go e nmen .
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : No applicable.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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