Real Time Di ec Kinema ic P oblem Compu a ion o he 3PRS
obo Using Neu al Ne wo ks
Asie Zubiza e a, Mikel La ea, Eloy I igoyen, I zia Cabanes, E a Po illo
Au oma ic Sys em and Con ol Enginee ing Dp ., Uni e si y o he Basque Coun y (UPV/EHU)
Abs ac
Ge ing a eliable calcula ion o he Di ec Kinema ic P oblem (DKP) is one o he main
challenges o he implemen a ion o Real-Time (RT) con olle s in Pa allel Robo s. In he
gene al case, he solu ion o he DKP needs, among o he calcula ed a iables, he es ima ion
o he obo pose in e ms o he senso s placed on ac ua o s. The e o e, ob aining all hese
a iables equi es he use o i e a i e p ocedu es which employ high compu a ional ime.
A i icial Neu al Ne wo ks ha e been p oposed o implemen he complex DKP equa-
ion mapping in he li e a u e due o hei uni e sal app oxima o p ope y. Howe e , he
p oposals in his a ea do no conside he Real Time implemen a ion o he ANN based
solu ion, and no app oxima ion e o s compu a ional ime analysis is ca ied ou . In his
wo k, a me hodology ha uses A i icial Neu al Ne wo ks (ANNs) o app oxima e he DKP
is p oposed. Based on he 3PRS pa allel obo , a comp ehensi e s udy is ca ied ou in
which se e al ne wo k con igu a ions a e p oposed o app oxima e he DKP. Mo eo e , o
demons a e he e ec i eness o he app oach, he p oposed ne wo ks a e e alua ed consid-
e ing no only hei app oxima ion capabili ies, bu also hei Real Time pe o mance in
compa ison wi h he adi ional i e a i e p ocedu es used in obo ics.
Keywo ds: Pa allel Robo s, Kinema ic P oblem, A i icial Neu al Ne wo k
1. In oduc ion
Cu en obo ic applica ions demand mo e p ecision, speed and load handling capabili ies
in o de o ul il p oduc i i y and economic goals. The classical mechanical s uc u e used o
hese applica ions is he single kinema ic chain app oach o se ial obo s, which is composed
by a single se ies o elemen s ha connec a ixed base wi h he end e ec o o ool. This
s uc u e p o ides wide ope a ional wo kspace and lexibili y, al hough i s load handling
capabili y and dynamic pe o mance a e usually limi ed.
IThis wo k was suppo ed in pa by he Go e nmen o Spain unde p ojec DPI2012-32882 and
UPV/EHU unde g an UFI11/28
Email add esses: [email p o ec ed] (Asie Zubiza e a), [email p o ec ed] (Mikel
La ea), [email p o ec ed] (Eloy I igoyen), [email p o ec ed] (I zia Cabanes),
[email p o ec ed] (E a Po illo)
P ep in submi ed o Neu ocompu ing May 2, 2016
This is he Manusc ip e sion o a Published Wo k ha appea ed in inal o m in Neu ocompu ing 271: 104-114 (2018) To access he inal
edi ed and published wo k see h ps://doi.o g/10.1016/j.neucom.2017.02.098
© 2017. This manusc ip e sion is made a ailable unde he CC-BY-NC-ND 4.0 license h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/
In o de o sa is y indus y’s equi emen s, in he las decade, he in e es on al e na i e
obo ic s uc u es has g own. Pa allel Robo s [1] ha e been p oposed as a sui able app oach
o high dynamic pe o mance asks, i.e., high speed and accele a ion, p ecision o high
load handling asks. Thei ou s anding pe o mance in hese a eas is due o hei mul iple
kinema ic chains, o pa allel s uc u e, in which a mo ing pla o m, whe e he ool is loca ed,
is connec ed o a ixed one by means o se e al se ial links. This s uc u e p o ides highe
s i ness han he one o se ial obo s, allowing load o be dis ibu ed among he limbs.
Mo eo e , posi ioning e o s can be compensa ed wi hin he mul iple limbs, and he s uc u e
can e en be designed o p esen lowe ine ia by loca ing he ac ua o s in he ixed base.
Howe e , he complex, mul iple loop s uc u e o pa allel obo s p esen s some disad-
an ages. Fo ins ance, hei wo kspace is much smalle han he one o se ial obo s, and
hei s uc u e is due o p esen singula i ies in his wo kspace. One o he mos impo an
handicaps is ela ed o he con ol o hese obo s, as he Di ec Kinema ic P oblem (DKP)
needs o be sol ed. The DKP calcula es he pose o he Tool Cen e Poin (TCP) o he
obo in e ms o he mo ion o he ac ua o s, and is manda o y o p ope obo con ol. In
he gene al case, howe e , his p oblem has no analy ical solu ion due o he complexi y o
he equa ions, and i e a i e nume ical p ocedu es ha e o be used o es ima e he posi ion
o he end e ec o in each con ol loop.
New on-Raphson (N-R) app oach is widely used o he calcula ion o he DKP [2]. This
p ocedu e equi es an ini ial guess o he end e ec o loca ion and an inde ini e numbe
o i e a ions, con e ging o a local solu ion ha sa is ies he complex kinema ic equa ions
o he obo . The numbe o i e a ions is highly dependan on he ini ial guess and i s
dis ance o he solu ion, and o each i e a ion, he Jacobian o he equa ion sys em has o
be calcula ed and in e ed. Hence, he compu a ional cos o his app oach is usually high,
which ep esen s a key issue when implemen ing ad anced con olle s o pa allel obo s.
Being high speed asks one o he main a eas o applica ion o pa allel obo s, se e al
wo ks ha e ocused on op imising he compu a ional cos o he DKP. Th ee main g oups
can be de ec ed among he p oposed solu ions. The i s g oup o app oaches is ocused on
educing he complexi y o he nonlinea kinema ic model o pa allel obo s by ob aining
an uni a ia e polynomial ha can be sol ed e icien ly. Examples o applica ion can be
ound o se e al pa allel obo s, such as he 4 deg ees-o - eedom (do ) Sch¨on lies pla o m
de ailed in [3], o he 3 do econ igu able MaPaMan [4], being he mos s udied one he 6
do Gough-S ewa pla o m [5, 6, 7]. Al hough e ec i e, hese app oaches a e e y sensi i e
o measu emen e o s in p ac ice, as a small se o da a is used o es ima e he es o he
a iables.
The second g oup is ocused on in oducing ex a senso da a ha allows an analy i-
cal solu ion o he DKP. Fo ha pu pose, addi ional senso s ha e o be in oduced in he
mechanical s uc u e in o de o measu e non-ac ua ed join mo ion. This app oach has
demons a ed o educe DKP compu a ional cos signi ican ly while main aining high es i-
ma ion accu acy [8, 9]. Howe e , he use o ex a senso s inc eases he cos o he obo ,
and adds addi ional complexi y o i s calib a ion.
Finally, he hi d g oup is ocused on he use o app oxima o s ha simpli y he highly
coupled and nonlinea kinema ic equa ions. The main hypo hesis behind hese app oaches
2
is ha a bounded bu no exac accu acy is needed when es ima ing he end e ec o pose.
Hence, uni e sal app oxima o s can be used i he app oxima ion e o is kep wi hin he
equi ed bounds. In his con ex , he use o Taylo se ies was p oposed by Wang, e al [10]
o calcula e he DKP o he Gough-S ewa pla o m. Among o he app oaches, he use o
A i icial Neu al Ne wo ks (ANN) ha e also been p oposed due o hei na u e as uni e sal
app oxima o s [11]. ANNs simula e he beha iou o biological neu al ne wo ks by means
o a se o ma hema ical ep esen a ions o neu ons, which a e g ouped in in e connec ed
laye s. Each neu on combines i s mul iple inpu s o compu e i s ou pu by means o a
nonlinea ac i a ion unc ion, which allows hem o lea n high complex inpu -ou pu da a
ela ions such as he kinema ic ela ions o pa allel obo s.
Few wo ks ha e ocused on he use o ANN as app oxima o s o he DKP sol ing in
pa allel obo s. In hese, a chi ec u es based on Radial Basis Func ion (RBF) [12, 13], poly-
nomial neu al ne wo ks [14] o Adap i e-ne wo k-based uzzy in e ence sys em [15] ha e
been p oposed. Howe e , he mos popula one is he Mul i Laye Pe cep on (MLP) a -
chi ec u e, wi h applica ions in he calcula ion o he DKP o se e al pa allel obo s, such
as he Hexa [16], he Hexapod [17, 18], he 3RRR plana obo [13] o he 3RRR sphe ical
obo [19].
The a o emen ioned wo ks demons a e he po en ial o ANNs as an e ec i e solu ion
o he compu a ion o he DKP in pa allel obo s. Howe e , mos o he wo ks a e ocused
on he quali y o he es ima ion, and do no conside wo impo an issues: he de ini ion o
he bounded e o o de e mine he alidi y o he ANN con igu a ion, and he Real Time
compu a ional cos o he ANN based app oach in compa ison wi h adi ional ones [20].
This wo k is ocused on he Real Time implemen a ion o ANN o sol e he DKP o
pa allel obo s, p o iding wo main con ibu ions in his a ea: a) a me hodology o de e mine
he bes ANN a chi ec u e is p o ided based on a bounded e o c i e ia; and b) a Real Time
compu a ional cos s udy is ca ied ou o de e mine he e ec i eness o he app oach and
o discuss i s alidi y o RT con ol applica ions.
Fo ha pu pose, he 3PRS pa allel obo has been selec ed as he s udy case and is
p esen ed in Sec ion 2. The DKP sol ing app oaches based on ANN a e discussed in Sec ion
3. The ANN aining p ocedu e and app oxima ion pe o mance is analysed in Sec ion 4, in
which he selec ion c i e ia based on bounded accu acy e o is discussed. The Real Time
pe o mance o he p oposed ne wo ks is e alua ed in Sec ion 5. Finally, he mos impo an
ideas a e summa ised.
2. The 3PRS Pa allel Robo
The 3PRS is a lowe mobili y pa allel obo , composed by h ee PRS (p isma ic- o a y-
sphe ical) limbs ha connec a ixed base wi h a mo ing pla o m whe e he end e ec o , o
ool, is a ached (Fig. 1). This s uc u e p o ides h ee deg ees o eedom: displacemen
along he global zaxis, and wo independen o a y mo ions θxand θyalong he global x
and yaxes. Howe e , due o he con igu a ion o he obo , mo emen in he xand yaxes
and he θzangle is no null. This e ec is known as pa asi ic mo ion [21].
3
hp
hp
a
Figu e 1: 3PRS Pa allel Robo
The e exis se e al applica ions o his obo , such as dish posi ioning, sola acking
o ene gy gene a ion, machining o es ing de ices. In o de o achie e he desi ed mobili y,
he obo is ac ua ed by h ee linea mo o s AiBi ha a e a ached o he ixed base in a
symme ical con igu a ion. Th ee limbs BiCi ans e he mo ion o he ac ua o s o he
mo ing pla o m whe e he TCP Pis loca ed. Fo ha pu pose, one end o he limb, Bi
is composed by a slide and a o a y join , while he o he , Ci, p esen s a sphe ical join .
This con igu a ion es ic s he mo ion o he limb o he πiplane (Fig. 2), de ined by Ai,
Biand Ci. The mo ing pla o m has a egula iangula geome y, and i s heigh is hp.
The kinema ic ela ions o he 3PRS pa allel obo a e based on he loop closu e equa ions
(Fig. 2), which ela e he mo ion o he ac ua o s qaand he pose o he end e ec o loca ed
in he mo ing pla o m.
p+dO
i−li−bi−ai=03×1,i= 1,2,3(1)
whe e he i s wo e ms de ine he pose o he end e ec o , being p=x y z Ti s
posi ion in he ixed ame O(x, y, z) and dO
i he o ien a ion ec o o each e ex o he
mo ing pla o m, which is calcula ed as,
dO
i=R di(2)
which ep esen s he p ojec ion o he cons an ec o di=PCi(de ined in he mo ing
ame) in he ixed ame,
di= − sin (2π(i−1)/3) −hpTi=1,2,3 (3)
dide ines he geome y o he mo ing pla o m, being = 0.3638 (m) he adius o he
mo ing pla o m and hp= 0.04 (m) i s heigh .
The o a ion ma ix R ela es he o ien a ion o he mo ing ame P(u, , w) wi h espec
o he ixed one O(x, y, z). I Roll-Pi ch-Yaw (θx, θy, θz) no a ion is selec ed,
4
R=
ux xwx
uy ywy
uz zwz
=
cθzcθycθzsθysθx−sθzcθxsθzsθx+cθzsθycθx
sθzcθycθzcθx+sθzsθysθxsθzsθycθx−cθzsθx
−sθycθysθxcθycθx
(4)
whe e cand ss and o cosine and sine igonome ical ope a ions, espec i ely.
hp
hp
a
Figu e 2: Kinema ic loop
The h ee las e ms o Eq. 1 a e de ined in e ms o join a iables. liis he ec o
ela ed o limb BiCi, whose leng h li= 0.9805 (m) is cons an ,
li=li
−cos (2π(i−1)/3) sin ϕi
−sin (2π(i−1)/3) sin ϕi
cos ϕi
i= 1,2,3 (5)
biis he posi ion ec o o he slide join Bi, which depends on he ac ua ed a iables
qai=|AiBi|and he cons an linea guide angle α=−0.7854 ( ad).
bi=
qaicos αcos(2π(i−1)/3)
qaicos αsin(2π(i−1)/3)
qaisin α
i= 1,2,3 (6)
Finally, aiis he cons an posi ion ec o o each o he h ee linea guides wi h espec
o he ixed ame, de ining he ixed pla o m geome y,
ai=ai acos (2π(i−1)/3) − asin (2π, (i−1)/3) 0 Ti=1,2,3 (7)
whe e a= 0.425 (m) is he adius o he linea guides disposi ion.
I a dis ance cons ain l2
i=||li|| is imposed o each o he ec o ial closu e loop equa ions
(Eq. 1), he ela ionship be ween he ou pu coo dina es x=x y z θxθyθzTand
5
he inpu join a iables associa ed o he mo ion o he ac ua o s qa=qa1qa2qa3T
can be de ined as ollows,
Γi+3 =q2
ai+Biqai+Ci= 0 i= 1,2,3(8)
whe e Bi= (−2CAix cos α)/cos(2π(i−1)/3)
−2CAiz sin α−2CAiy cos αsin(2π(i−1)/3)
Ci=||CAi|| +h2
b−l2
i+ (2 hbCAix sin α)/cos(2π(i−1)/3)
+2 hbCAiy sin αsin(2π(i−1)/3) −2hbCAiz cos α
being CAi=px+R di−ai=CAix CAiy CAiz T, o i= 1,2,3.
As s a ed p e iously, he 3PRS p esen s only 3 deg ees o eedom, being z,θxand θy
he independen ou pu a iables. In o de o de ine he kinema ic equa ion sys em o he
obo , he ela ionship be ween hese independen ou pu a iables and he pa asi e mo ions
x,yand θzneeds o be modelled. Hence, h ee cons ain equa ions ha e o be in oduced.
As he mo ion o limbs BiCiis cons ained o a plane πi(Fig. 2)[21],
Γ1=hpwy− uy−y= 0
Γ2=hpwx+
2(ux− y)−x= 0
Γ3= x−uy= 0
(9)
whe e wx,wy, x, y,uxand uydepend on he RPY Eule angles (θx, θy, θz) (Eq.4).
The equa ion sys em ob ained by combining Eqs. 8 and 9 de ines he kinema ic equa ions
used o sol e he Di ec Kinema ic P oblem.
3. Sol ing he Di ec Kinema ic P oblem
As s a ed p e iously, in obo ic applica ions he loca ion o he end-e ec o associa ed o
he mo ing ame P(u, , w) wi h espec o he ixed ame O(x, y, z) is essen ial o execu e
obo ic asks. As his measu emen is no possible in he gene al case, he Di ec Kinema ic
P oblem needs o be sol ed, so ha he ou pu a iables ha de ine he end-e ec o pose
x, can be es ima ed in e ms o he measu able ac ua o a iables qa. Hence, he DKP is,
in ac , a mapping p oblem,
x= (qa) (10)
whe e as de ailed in he p e ious sec ion, xis a ec o con aining he ca esian posi ion and
o ien a ion o he pla o m and qais he inpu join a iable ec o associa ed o he mo ion
ac ua o s.
Howe e , being highly nonlinea , an analy ical exp ession o he mapping canno be
ound in he gene al case. The classical app oach is o sol e he mapping locally, by using
an i e a i e sol ing p ocedu e based on an ini ial guess, i.e, New on-Raphson app oach.
Howe e , i is also possible o y o de ine an ANN o lea n he mapping be ween he
inpu s and ou pu s. These al e na i es will be discussed nex .
6
3.1. Classical app oach: New on-Raphson
New on-Raphson app oach is based on he use o he Jacobian J o i e a e om an
ini ial guess o he solu ion. In o de o apply i o he DKP sol ing case, he alues o
he ac ua ed join a iables qaa e measu ed and conside ed known, and an ini ial guess o
all he a iables in x(independen and pa asi e) is equi ed. Then, he nume ical i e a i e
p ocess is de ined as ollows,
xk+1=xk−J−1Γ(xk,qa) (11)
whe e Γ(xk,qa) = 0, o he pa icula case o he 3PRS obo , is he 6 equa ion sys em
de ined in Eqs. 8 and 9, and J=∂Γ/∂x is he 6 ×6 Jacobian o he pa allel obo .
The i e a i e p ocedu e s ops when a maximum numbe o i e a ions has been eached,
o he ela i e e o be ween i e a ions is less han a p ede ined alue. Al hough i quickly
con e ges o he solu ion, his app oach equi es in e ing he Jacobian ma ix o he obo ,
which can be ime consuming.
3.2. A i icial Neu al Ne wo k app oach
ANNs a e ma hema ical cons uc s ha emula e he biological unc ions o neu al ne -
wo ks. Thei s uc u e is based on a i icial neu ons, ha emula e biological ones by he use
o nonlinea ac i a ion unc ions. ANNs ha e demons a ed o be uni e sal app oxima o s
[11], being able o ob ain a bounded e o in unc ion app oxima ion depending on he size
o he hidden laye . This way, hey ha e been used in a wide ange o applica ions. As
de ailed in he in oduc ion, in he obo ics li e a u e he Mul i Laye Pe cep on (MLP)
s uc u e [11] is one o he mos used due o he ex ensi e li e a u e exis ing on i s aining
and pe o mance uning. Hence, in his wo k, his a chi ec u e is selec ed o pe o m he
compa a i e s udy.
I should be no ed ha ANNs need o be ained so ha hey lea n he mapping p oblem.
This p ocedu e equi es o de ine a se o examples, he T aining Se , so ha he weigh s
and biases ha de ine he pa ame e s o he ANN can be uned o i he mapping p oblem
using he Backp opaga ion Me hod. The aining o an ANN is no a i ial ask, and i
usually is he mos complex and ime consuming one. The pa icula aining p ocedu e o
he ANNs o he s udy case o he 3PRS pa allel obo is analysed in Sec ion 4.
Once ained, he ANN can ep oduce he mapping p oblem a e de ining i s inpu s.
Al hough app oxima ion e o s can a ise, depending on he quali y o he aining, he
execu ion o he ANN is ela i ely as . The p opaga ion s age in he neu al ne wo k akes
he inpu s o he ne wo k and mul iplies hem by he weigh s connec ing o he nex laye .
Once he neu ons o he hidden laye ha e hei weigh ed inpu s added, he esul passes
h ough he ac i a ion unc ion o p opaga e he signal o he nex laye . All in all, he
p opaga ion s age is a se o ma ix mul iplica ions, sums and nonlinea ac i a ion unc ions.
The compu a ional cos o his p ocedu e depends on he numbe o neu ons and signals o
p ocess, and will be analysed in Sec ion 5.
Fo he pa icula case o he 3PRS pa allel obo , and he Di ec Kinema ic P oblem
mapping, di e en ANN opologies can be chosen. An impo an issue o be conside ed is
7
he ou pu con igu a ion o he ne wo k, i.e., he numbe o ou pu s o be es ima ed. I
seems easonable ha a complex mapping p oblem will equi e less hidden laye neu ons
i i s ou pu s a e es ima ed sepa a ely by di e en ANNs. Hence, based on his easoning,
h ee di e en ANN based DKP es ima o app oaches can be de ined based on he na u e
o he h ee ou pu s o be ob ained (z,θxθy):
•Single 3-N-3 s uc u e, ha handles he en i e mapping p oblem in i s globali y (Fig.3),
i.e, om he h ee ac ua o senso posi ions qa1,qa3,qa2, he ne es ima es he indepen-
den ou pu a iables (z,θxθy).
•Two s uc u es, one 3-N-1 o es ima ing he ou pu posi ion zand o he 3-N-2 ha
p o ides he es ima ion o he ou pu o ien a ion (θx, θy) (Fig.4).
•Th ee 3-N-1 s uc u es, one o each ou pu (Fig.5).
The las a iable o be de e mined is he size o he hidden laye (N). Howe e , he
lack o an es ablished me hod o ob ain N ela ed o a conc e e p oblem, equi es he use
o an expe imen al app oach o de e mine his a iable. The p ocedu e o de e mine he
app op ia e hidden laye neu on numbe is analysed in he nex sec ion.
N
inpu ou pu
hidden
laye
qa1
2
qa
3
qa
z
θx
y
θ
Figu e 3: Single 3-N-3 ne
4. ANN aining me hodology o DKP implemen a ion
In o de o de e mine he bes al e na i e among he p oposed ne wo k con igu a ions
(all coo dina es 3 −N−3, o ien a ion coo dina es in 3 −N−2, and single coo dina es
3−N−1), an i e a i e ba ch aining p ocedu e has been designed. Fo his eason, o
each ne wo k con igu a ion a se o neu al ne wo ks has been e alua ed, whose hidden laye
neu on numbe N a ies om 5 o 90 wi h in e als o 5 neu ons. The ini ialisa ion o he ne
pa ame e s (weigh s and biases) has been ca ied ou applying Nguyen-Wid ow app oach.
Fu he mo e, o educe he e ec o andom weigh ini ialisa ion, o each alue o N, 10
ne s ha e been ained.
Acco ding o he uni e sal app oxima o p ope y as s a ed in [11], any ype o ac i a ion
unc ion can be implemen ed in he designed MLP. Howe e , due o he exis ence o bo h
8
inpu ou pu
hidden
laye
N
qa1
2
qa
3
qa
N
θx
y
θ
z
Figu e 4: Posi ion and O ien a ion: 3-N-1 z& 3-N-2 (θx,θy) ne s
N
inpu ou pu
hidden
laye
N
N
qa1
2
qa
3
qa
z
θx
y
θ
Figu e 5: Th ee independen 3-N-1 (z,θx,θy) ne s
posi i e and nega i e inpu -ou pu da a, he hype bolic angen and linea ac i a ion unc-
ions ha e been selec ed o he hidden and ou pu laye espec i ely [22] o all ne wo ks
9
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Neu ons in hidden laye
0
50
100
150
ime (µs)
3-N-3 ANN Time Pe o mance
no m
h
o
dno m
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Neu ons in hidden laye
0
50
100
150
ime (µs)
3-N-2 ANN Time Pe o mance
no m
h
o
dno m
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Neu ons in hidden laye
0
50
100
150
ime (µs)
3-N-1 ANN Time Pe o mance
no m
h
o
dno m
Figu e 12: ANN Time Pe o mance s hidden neu on numbe
App oach Single Posi ion+O ien a ion Independen N-R
Con igu a ion 3-55-3 3-20-1 z3-15-2 3-20-1 z3-15-1 θx3-10-1 θy
Time (µs) 80.19 45.24 40.65 45.24 39.77 36.18 675.71
85.89 121.19
Table 3: To al Pe o mance Times o each DKP sol ing app oach
6. Conclusions
An e icien calcula ion o he Di ec Kinema ic P oblem in Pa allel Robo s is c i ical
o designing Real Time con olle s. This e ec i eness is ela ed o ob aining p ecise obo
pose alues below a maximum e o limi , a each sample ime o execu ion.
Al hough he classical app oach o sol e he DKP is he New on-Raphson algo i hm,
pa adigms such as ANN ha e also been p oposed. ANNs p o ide ad an ages o e he NR
app oach, as hei compu a ional cos is lowe . Howe e , hey p o ide an app oxima ion o
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he solu ion, and a p ope aining p ocedu e has o be de ined o achie e an app op ia e
accu acy.
In his wo k, a comp ehensi e s udy o Mul ilaye Pe cep on A i icial Neu al Ne wo k
accu acy and ime pe o mance in DKP sol ing o a 3PRS pa allel obo has been p e-
sen ed. Taking in o accoun he mapping p oblem p esen ed by his obo , se e al ANN
con igu a ions ha e been p oposed: a single 3-N-3 ne wo k, wo ne wo ks (posi ion and o i-
en a ion), and a single ne wo k o each ou pu . Each con igu a ion has been es ed wi h a
numbe o hidden laye neu ons in he ange [5-90].
Fo he aining p ocess, a se o examples o ANN unning (192.238) and ANN es ing
(2.836.219) ha e been picked om he 3PRS e ec i e wo kspace, being di e en examples in
bo h se s. Fu he mo e, in o de o a oid he dependency o ANN pa ame e s ini ialisa ion,
each o hose s uc u es we e ained 10 imes, selec ing he bes i e a ion as ep esen a i e.
The maximum app oxima ion e o has been conside ed as he pe o mance index in o de
o de ine he bes ne wo ks.
To ex ac conclusions abou he goodness o ANN in his p oblem, he ne wo ks ha e
been e alua ed in accu acy and in Real-Time, compa ing he esul s wi h he classical N-
R me hod. This way, he bes s uc u es o each con igu a ion ha e been: 3-20-1 o z
posi ion, 3-15-1 o θxo ien a ion, and 3-10-1 o θyo ien a ion; 3-15-2 o o ien a ion and
3-20-1 z o posi ion; 3-55-3 o posi ion and o ien a ion coo dina es. I is necessa y o no e
ha he wo s pose es ima ion esul s ha e been ob ained close o he wo kspace limi s,
whe e poin s wi h singula i ies eme ge. Fo u he wo ks, a p e ious s udy could enhance
he dis ibu ion o pose coo dina es chosen as examples o he aining p ocess.
O he ele an aspec o his wo k is he analysis made in compu a ional cos , du ing
execu ion in a Real ime pla o m. In o de o e alua e he pe o mance and e iciency
o bo h me hods, N-R and ANN, a deep analysis in ime consuming has been made in a
Indus ial PC. Conside ing ha he in oduced code in such pla o m will consume di e en
amoun s o ime in each phase, ou s eps ha e been e alua ed sepa a ely. The mos ele an
alue has been loca ed in o he ime ( h) equi ed o e alua e he ou pu s o he hidden laye .
Ob iously, he en i e sum o he execu ion ime inc eases as he numbe N o hidden laye
neu ons g ows up. As i has been p esen ed in sec ion 5, he s uc u e 3-55-3 has eached he
less ime consuming a e, being in o 80.19 µs, ob aining a alue below a le el o magni ude
gi en by N-R me hod. These expe imen s alida e he usage o he ANNs o sol e he DKP
bo h conside ing compu a ional cos and e o pe o mance.
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