applied
sciences
A icle
In luence o he App oach Di ec ion on he
Repea abili y o an Indus ial Robo
Michal Voce ka 1,* , Róbe Huˇnady 2, Ma in Haga a 2, Zdenko Bobo ský1, Tomáš Ko 1
and Václa K ys 1
1Depa men o Robo ics, Facul y o Mechanical Enginee ing, VŠB—Technical Uni e si y Os a a,
708 00 Os a a, Czech Republic; [email p o ec ed] (Z.B.); [email p o ec ed] (T.K.);
acla [email p o ec ed] (V.K.)
2Depa men o Applied Mechanics and Mechanical Enginee ing, Facul y o Mechanical Enginee ing,
Technical Uni e si y o Košice, 042 00 Košice, Slo akia; [email p o ec ed] (R.H.);
[email p o ec ed] (M.H.)
*Co espondence: [email p o ec ed]
Recei ed: 13 No embe 2020; Accep ed: 1 Decembe 2020; Published: 5 Decembe 2020


Abs ac :
The a icle aims o p o e he hypo hesis, ha an app oach di ec ion in luences epea abili y
a a ge poin o a ajec o y. Unlike mos esea ches ha deal wi h absolu e accu acy, his pape is
ocused on de e mining he achie able epea abili y and he in luence o he di ec ion o app oach
on i . To p o e he hypo hesis, se e al measu emen s a e pe o med unde di e en condi ions,
on indus ial obo ABB IRB1200. To e i y and con i m he esul ob ained om he esol e s loca ed
on he indi idual axes o he obo , he measu emen s a e eplica ed using high-speed digi al image
co ela ion came as. Using an ex e nal measu ing de ice, he eal epea abili y o he obo endpoin
is de e mined. The measu emen p o ed he co ec ness o he hypo hesis, i.e., he dependence o he
app oach di ec ion on epea abili y was p o ed. Fu he mo e, eal de ia ions we e measu ed and he
ex en o his in luence on he obo epea abili y was de e mined.
Keywo ds: obo epea abili y; DIC; obo p ecision; in luence o app oach di ec ion
1. In oduc ion
Indus ial obo s and manipula o s a e gene ally conside ed o be uni e sal sys ems ha can be
ope a ed in any indus ial condi ions, in any way. Howe e , he pu pose o he manipula o is gi en by
he moun ed ool—i s ins alla ion conside ably limi s he e sa ili y o he manipula o . In indus ial
condi ions, i is he ule ha he manipula o is a pa o a single-pu pose machine, p oduc ion line,
o o he equipmen . In hese cases, i is usually possible o ope a e only in a speci ic way, usually
pe o ming one speci ic ask cyclically. The e o e, in cases whe e i is necessa y o inc ease he p ecision
o he manipula o , i may be no necessa y o inc ease he absolu e accu acy in he en i e wo king
en elope o he obo , bu i migh be su icien o inc ease epea abili y only o speci ic a ge s.
In e ms o indus ial manipula o p ecision, wo di e en alues a e gene ally add essed—accu acy
and epea abili y. Accu acy is he abili y o he obo o each a ool cen e poin s (TCPs) p og ammed
pose wi h espec o he obo
´
s base ame. Repea abili y is abili y o he obo o e u n i s TCP ( ool
cen e poin ) o he same posi ion, epea edly om he same di ec ion.
The accu acy o he manipula o is a ec ed by se e al ex e nal in luences. In addi ion o
empe a u e, manu ac u ing inaccu acies, sha de lec ion and clea ances in bea ings and gea boxes,
i can be, o example, he ac ha he esol e s (o encode s) a e loca ed a he beginning o he d i e
uni , and he e o e, do no de ec an e o ha occu s on he mo o and gea box [
1
]. Repea abili y is
caused by clea ances, esolu ion o posi ion senso s, he mal expansion, and o he in luences. I does
Appl. Sci. 2020,10, 8714; doi:10.3390/app10238714 www.mdpi.com/jou nal/applsci
Appl. Sci. 2020,10, 8714 2 o 24
no include he coo dina e sys em se ing e o s, o he han he modelled a m leng h, manu ac u ing,
and assembly de ia ions (e.g., non-pe pendicula i y o non-pa allelism o join s and guides).
While absolu e accu acy can a y by up o 15 mm [
2
] compa ed o he ideal model in he case o
la ge manipula o s, and i is necessa y o use an ex e nal de ice, such as a lase acke o measu ing
and inc easing accu acy, epea abili y is usually in he ange o en hs o hund ed hs o a mm [
1
].
Howe e , i depends on he ope a ing condi ions a lo . The alues gi en by he manu ac u e usually
apply o a s eady s a e unde ideal condi ions and o a new manipula o . Ne e heless, he e a e
applica ions in which an o dina y le el o epea abili y may no appea o be su icien , especially o
less igid obo s uc u es.
Fo example, ABB o e s he Absolu e Accu acy and CalibWa e Op ion o i s obo s [
2
].
This unc ionali y aims o elimina e he di e ence be ween an ideal manipula o in a Compu e -aided
design (CAD) en i onmen and he eal obo ins alled on a ac o y si e. The co ec ion sol es no only
he mechanical p ope ies and inaccu acies o he pa icula manipula o bu also de ia ions caused by
an inaccu a ely de ined payload. An ad an age is also he in e changeabili y o he obo “piece by
piece” i necessa y, as he accu acy o obo s, calib a ed by his me hod, is consis en . The di e ence
be ween an ideal i ual and a eal manipula o is usually in he ange o 8–15 mm, depending on he
ype o obo and he concu ence o manu ac u ing ole ances and de ia ions. This echnique uses a
lase acke o measu e accu acy a one hund ed a ge s wi hin he en i e obo wo kspace, based on
his measu emen , calib a ion wi hin he en i e obo wo king en elope is p o ided. The esul ing
accu acy is gene ally 0.5 mm, bu ypically no lowe [2].
Calib a ion me hods based on he use o a lase acke [
3
,
4
] a e commonly used in indus ial
applica ions. Fu he esea ch in his ield ocuses o example on an addi ional inc ease o accu acy o
his me hod by making a big numbe o measu emen s (in he o de o ens o housands) as desc ibed
in [
5
]. This is achie ed by a measu ing sys em ha communica es wi h he obo con olle and he lase
acke , while he obo con igu a ions a e chosen wi h he help o a deep neu al ne wo k. Lase acke s
a e o en used as a e e ence measu emen o e alua e he p ope ies o o he calib a ion me hods [
6
–
9
].
Ano he use o lase s o obo calib a ion is in he o m o lase in e e ome y-based sensing and
measu emen as desc ibed in [
10
]— his me hod p o ides eal- ime dynamic posi ion measu emen s
wi h high accu acy, high sampling a e, and la ge wo king space. A 3D measu emen sys em consis ing
o a comme cially a ailable elescoping ball ba sys em is p oposed in [
9
]. A e calib a ing he obo
in selec ed 72 poses o he end-e ec o wi h espec o he base, e i ica ion in 10,000 andom obo
con igu a ions made by compa ison wi h a lase acke con i med app oxima ely wo imes be e
accu acy o he obo han be o e he calib a ion. A geome ic me hod o calib a ion is also desc ibed
in [
11
] wi h a special ocus on inding he bes calib a ion poses o he obo o achie e he highes
possible imp o emen in posi ioning accu acy. The au ho s in [
12
] poin ou a p oblem ha applies
o all calib a ion me hods and is ela ed o he in luence o backlash o he obo join s. This e ec is
desc ibed in he a icle oge he wi h a p oposed me hod o he elimina ion o his sou ce o addi ional
e o . A commonly used g oup o me hods o obo calib a ion is based on pho og amme y. One o
he app oaches is o use a 3D pho og amme y-based measu ing de ice o ack he posi ion and
o ien a ion o he obo end-poin in eal- ime and use his in o ma ion as eedback o he obo
con olle [
13
]. Pho og amme y is also applied in o -line obo calib a ion [
6
]. Resea ch has also been
done in he a ea o use o a comme cially-a ailable po able pho og amme y sys em in indus ial
obo calib a ion, he au ho s in [
7
] compa e his app oach wi h a lase acking calib a ion wi h e y
good esul s— he accu acy o pho og amme y is only sligh ly wo se. The esea ch in [
14
] desc ibes
he use o a came a moun ed on he obo end-e ec o and a calib a ion boa d placed in he wo kplace
whe e he came a can see i mos o he ime. This p oposed calib a ion me hod also wo ks on-line.
A simila solu ion wi h a single came a moun ed nea he obo end-poin was in oduced also in [
15
]
and he achie ed imp o emen in obo accu acy was be ween h ee o six imes. As a as high-speed
came as a e conce ned, au ho s in [
16
] u ilized he Phan om 2511 came a wi h up o 25,000 ames
pe second o asce ain he accu acy o an indus ial obo no only in he absolu e alue bu also
Appl. Sci. 2020,10, 8714 3 o 24
conce ning he mo emen di ec ion. Such a sys em can also be used o measu e dynamic mo emen ,
oscilla ion, ib a ion, e c. Applica ion o s e eo- ision o obo calib a ion has also been he subjec
o esea ch, o example in [
17
] a me hod o inc easing he accu acy o an indus ial milling obo
wi h he help o a s e eo came a is desc ibed. In addi ion, ano he esea ch was done using low-cos
sys ems [18] o lase -came a- iangula ion [19].
Unlike pho og amme y, he digi al image co ela ion me hod enables ull- ield 3D displacemen
measu emen s, allowing he co ela ion sys em o be used in bo h ib a ion analysis and de o ma ion
analysis o componen s. The main ad an age o e poin measu emen is he possibili y o including
mo e da a in he p ocessing o esul s. Since he measu emen is ca ied ou on a su ace, in addi ion o
posi ion in o ma ion, i is also possible o ob ain in o ma ion abou he di ec ion o he su ace no mal.
I he high-speed came as a e used, i is possible o analyze kinema ics, dynamics, and ib a ion o he
manipula o in de ail, o o measu e i s modal p ope ies. The s udy in [
20
] desc ibes a mo ion analysis
o a wo-a m se ice obo as i passes be ween wo posi ions. I was a la ge-scale applica ion designed
o econs uc he mo emen o he manipula o in space. The measu emen esul s we e 3D ajec o ies
o h ee join nodes and ib a ion esponses in hese nodes du ing obo mo emen . The au ho s
in [
21
] p oposed an algo i hm o simple i e a i e lea ning con ol o a obo , whose aim was o educe
geome ic e o s occu ing in he echnology p ocess known as single poin inc emen al shee o ming.
They apply he 3D digi al image co ela ion me hod o measu e he geome ic e o along he ool pa h.
In addi ion o posi ion measu emen , he DIC me hod can also be used in ope a ional ib a ion analysis
o expe imen al modal analysis. Se e al pape s ha e been published on his opic [
22
–
27
]. In he
pape [
26
], he au ho s p oposed a highly e icien p ocedu e o p ocessing la ge amoun s o da a
ep esen ing he ib a ion ime esponses o he analyzed s uc u e. This p ocedu e was subsequen ly
implemen ed in he DICMAN 3D so wa e applica ion o modal es e alua ion. In ano he wo k [
27
],
he au ho s wen e en u he by p esen ing a echnique ha makes i possible o use a DIC sys em
o measu e ib a ion esponses du ing body mo emen . I is based on pos -p ocessing nume ical
elimina ion o he igid body mo ion componen s. They used his echnique o measu e he modal
shapes and ope a ing shapes o he o a ing disk. These app oaches can be ad an ageously used in
he analysis o dynamic beha io o obo s and manipula o s, e.g., o educe dynamic load when
accele a ing, decele a ing, o op imizing ajec o y.
Mos o he scien i ic publica ions a e ocused on measu emen and inc ease o he obo absolu e
accu acy. Acco ding o he aim o his a icle, i is also necessa y o men ion hose ha add ess he
measu emen o epea abili y.
A eam o esea che s om he Uni e si y o Žilina desc ibed a me hodology o pose- epea abili y
measu emen , based on he use o lase in e e ome e and digi al indica o [
28
]. This pape uses he
ecommended p ocedu e desc ibed in he ISO 9283:1998—in e na ional s anda d ha desc ibes how
pe o mance cha ac e is ics should be speci ied [
29
]. Expe imen al measu emen o obo epea abili y
is also desc ibed in a pape om Robe Mo is Uni e si y, USA [30].
We mus no o ge o men ion a icles ocused on a compa a i e e alua ion o indus ial obo s
using di e en ypes o me hods [31–33].
I is clea om he abo e o e iew ha he a ea o inc easing he accu acy o epea abili y o
manipula o s is an a ea, ha has long been paid a en ion o, and se e al possible solu ions ha e been
p esen ed o inc ease he p ecision. Howe e , none o hese me hods examines in mo e dep h he
in luence o a i al di ec ion a he desi ed a ge .
The pape aims o p o e he dependence o he a i al di ec ion o he a ge on epea abili y and
o de e mine he ange o di ec ional de ia ion o a speci ic obo by a se o measu emen s.
2. Condi ions and P ocedu e o Hypo hesis Ve i ica ion
•
Acco ding o he au ho s’ assump ion, he e is a dependence be ween he di ec ion o he obo
a i al a he a ge and he epea abili y ha can be achie ed a his a ge . Based on he summa y
Appl. Sci. 2020,10, 8714 4 o 24
s a ed abo e, he au ho s claim ha his in luence is no conside ed in he case o obo epea abili y
inc easing a emp s done by o he esea che s.
•
In he i s phase, only obo d i es esol e da a will be used o e i ica ion. These da a should
al eady p o e he dependence o he di ec ion o app oach on he epea abili y o he obo in he
measu ed a ge . Fo his eason, he join a iables will be con e ed (acco ding o he o wa d
kinema ics ask) o he endpoin de ia ion (x, y, z, and he o al de ia ion). Since he da a om he
e ol e s do no include he e o ha occu s in he gea box, a m de lec ion, he mal expansion,
and also does no include he in luence o manu ac u ing ole ances and de ia ions o indi idual
pa s o he manipula o mechanical s uc u e, he second phase o measu emen will ollow.
•
In he nex phase, he a ge posi ion and o ien a ion will be measu ed by he 3D DIC me hod,
and he de ia ion in he epea abili y spec um will be e alua ed wi h high p ecision.
•
As a esul o he measu emen , he e will be a g aphical ep esen a ion o he measu ed a ge
icini y and he deg ee o de ia ion alid o i s di ec ions o app oach.
•
The esul o his expe imen al measu emen will be alid o a speci ic indus ial obo , wi h a
de ined solu ion o ins alla ion in he wo kplace and o a speci ic a ge .
3. Desc ip ion o he Measu ed Robo
The ABB IRB1200 5/0.9 indus ial obo was chosen o he expe imen s. I is a compac angula
six-axis obo wi h IP40 p o ec ion, which is designed o wo k in clean indoo spaces, o applica ions
such as packaging, handling, assembly, e c.
Acco ding o he manu ac u e , he epea abili y o he manipula o is 0.025 mm and he obo
can be se in any o ien a ion a any angle. Table 1con ains he basic echnical pa ame e s o he obo .
The o wa d each o he obo is 901 mm wi h a load capaci y o 5 kg (wi h he cen e o g a i y a a
maximum dis ance o 100 mm om he lange).
Table 1. Basic pa ame e s o IRB1200 5/0.9 [1].
Posi ion Repea abili y 0.025 mm
Max. TCP speed 8.9 m/s
Max. TCP accele a ion 36 m/s2
Accele a ion ime—1 m 0.06 s
Robo weigh 54 kg
The expe imen s a e pe o med a he wo ks a ion wi h a pai o he ABB IRB1200 obo s— he
obo s a e moun ed on a pedes al so ha he bases o he obo s a e no in a ho izon al posi ion bu a e
inclined by an angle o 40
◦
in he y-axis o he coo dina e sys em o each o he obo s. This is no a
s anda d moun ing posi ion o he obo , in his case, he i s axis o he obo is loaded by o que e en
i i main ains a posi ion (any o he han 0
◦
) o o a es a a cons an speed. This ac di ec ly a ec s he
accu acy and epea abili y o he obo .
The obo was wa med-up be o e e e y measu emen and kep a a cons an empe a u e as
much as possible. This is a s anda d ou ine o e e y p ecise obo p og amming ask. Be o e he
obo a ge s a e p ecisely calib a ed, he obo is kep in he mo emen o some ime, o wa m up i s
s uc u e in a way i will be wa med du ing i s p oduc ion
Fo measu emen , he obo is equipped wi h payloads o a ious pa ame e s (Figu e 1, Table 2),
which simula es eal ope a ing condi ions, whe e obo s a ely ca y minimal mass. The e is a
black-and-whi e speckle pa e n o subsequen e i ica ion o epea abili y by he DIC me hod glued
on he on su ace o he payload.
Appl. Sci. 2020,10, 8714 5 o 24
Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 o 23
Figu e 1. Payload side iew.
Table 2. Payload pa ame e s.
Payload
(Kg)
L
(mm)
Ø
(mm)
Cen e o G a i y
z (mm)
0.04
10
100
5
2.538
42
100
21
4.647
54
120
27
4. The Expe imen
This chap e desc ibes he i s phase o measu emen , which aims o p o e he in luence
be ween he obo app oach di ec ion o he end a ge and he epea abili y ha can be achie ed a
his a ge .
Be o e s a ing he expe imen , i was necessa y o de e mine he minimal oscilla ion
s abiliza ion ime when he obo is s opped. The s abiliza ion ime was chosen based on he esul s
o he dynamic measu emen o he obo a m s abiliza ion du ing i s sudden s op/s a . The
necessa y da a we e ob ained om h ee-axis accele ome e s applied in wo places—di ec ly on he
e ec o (on he side o he analyzed payload—see Figu e 2) and he base o he obo . A ypical
e ec o oscilla ion eco ded in one o he accele ome e axes is shown in Figu e 3. Al hough in his
case, he a m can be conside ed s abilized a e app oxima ely 0.6 s, o a highe deg ee o ce ain y
in he measu emen he ime o s abiliza ion was ipled.
Figu e 2. Robo payload wi h applied black-and-whi e speckle pa e n.
Figu e 1. Payload side iew.
Table 2. Payload pa ame e s.
Payload (Kg) L (mm) Ø (mm) Cen e o G a i y z (mm)
0.04 10 100 5
2.538 42 100 21
4.647 54 120 27
4. The Expe imen
This chap e desc ibes he i s phase o measu emen , which aims o p o e he in luence be ween
he obo app oach di ec ion o he end a ge and he epea abili y ha can be achie ed a his a ge .
Be o e s a ing he expe imen , i was necessa y o de e mine he minimal oscilla ion s abiliza ion
ime when he obo is s opped. The s abiliza ion ime was chosen based on he esul s o he dynamic
measu emen o he obo a m s abiliza ion du ing i s sudden s op/s a . The necessa y da a we e
ob ained om h ee-axis accele ome e s applied in wo places—di ec ly on he e ec o (on he side o
he analyzed payload—see Figu e 2) and he base o he obo . A ypical e ec o oscilla ion eco ded
in one o he accele ome e axes is shown in Figu e 3. Al hough in his case, he a m can be conside ed
s abilized a e app oxima ely 0.6 s, o a highe deg ee o ce ain y in he measu emen he ime o
s abiliza ion was ipled.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 o 23
Figu e 1. Payload side iew.
Table 2. Payload pa ame e s.
Payload
(Kg)
L
(mm)
Ø
(mm)
Cen e o G a i y
z (mm)
0.04
10
100
5
2.538
42
100
21
4.647
54
120
27
4. The Expe imen
This chap e desc ibes he i s phase o measu emen , which aims o p o e he in luence
be ween he obo app oach di ec ion o he end a ge and he epea abili y ha can be achie ed a
his a ge .
Be o e s a ing he expe imen , i was necessa y o de e mine he minimal oscilla ion
s abiliza ion ime when he obo is s opped. The s abiliza ion ime was chosen based on he esul s
o he dynamic measu emen o he obo a m s abiliza ion du ing i s sudden s op/s a . The
necessa y da a we e ob ained om h ee-axis accele ome e s applied in wo places—di ec ly on he
e ec o (on he side o he analyzed payload—see Figu e 2) and he base o he obo . A ypical
e ec o oscilla ion eco ded in one o he accele ome e axes is shown in Figu e 3. Al hough in his
case, he a m can be conside ed s abilized a e app oxima ely 0.6 s, o a highe deg ee o ce ain y
in he measu emen he ime o s abiliza ion was ipled.
Figu e 2. Robo payload wi h applied black-and-whi e speckle pa e n.
Figu e 2. Robo payload wi h applied black-and-whi e speckle pa e n.

Appl. Sci. 2020,10, 8714 6 o 24
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 o 23
Figu e 3. The s abiliza ion p ocess o he obo e ec o , eco ded a e a s op a he a ge poin .
The lowcha in Figu e 4 desc ibes he expe imen p ocess. The ABB IRB 1200 obo equipped
wi h a cylind ical payload whose TCP lies a he in e sec ion o he on plane o he cylinde and i s
axis o symme y cycles con inuously o he same poin , bu om di e en di ec ions ( om di e en
s a ing poin s). The mo emen de ini ion is en e ed pa ame ically in o he con ol applica ion
(w i en in C #), which uns on a compu e and collec s da a om he obo . The pa ame e s o he
obo du ing he measu emen a e con ained in Table 3.
Figu e 4. Expe imen con ol sys em lowcha .
Figu e 3. The s abiliza ion p ocess o he obo e ec o , eco ded a e a s op a he a ge poin .
The lowcha in Figu e 4desc ibes he expe imen p ocess. The ABB IRB 1200 obo equipped
wi h a cylind ical payload whose TCP lies a he in e sec ion o he on plane o he cylinde and i s
axis o symme y cycles con inuously o he same poin , bu om di e en di ec ions ( om di e en
s a ing poin s). The mo emen de ini ion is en e ed pa ame ically in o he con ol applica ion (w i en
in C #), which uns on a compu e and collec s da a om he obo . The pa ame e s o he obo du ing
he measu emen a e con ained in Table 3.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 o 23
Figu e 3. The s abiliza ion p ocess o he obo e ec o , eco ded a e a s op a he a ge poin .
The lowcha in Figu e 4 desc ibes he expe imen p ocess. The ABB IRB 1200 obo equipped
wi h a cylind ical payload whose TCP lies a he in e sec ion o he on plane o he cylinde and i s
axis o symme y cycles con inuously o he same poin , bu om di e en di ec ions ( om di e en
s a ing poin s). The mo emen de ini ion is en e ed pa ame ically in o he con ol applica ion
(w i en in C #), which uns on a compu e and collec s da a om he obo . The pa ame e s o he
obo du ing he measu emen a e con ained in Table 3.
Figu e 4. Expe imen con ol sys em lowcha .
Figu e 4. Expe imen con ol sys em lowcha .
Appl. Sci. 2020,10, 8714 7 o 24
Table 3. Robo measu emen mo emen pa ame e s.
C ossing Speed ( o he De aul Posi ion) 400 mm/s
App oach speed ( o he measu ed a ge ) Vmax (max. 8900 mm/s)
S abiliza ion ime (bo h he de . pos. and a ge )
2 s
All s a ing posi ions wi hin one cycle a e loca ed a he same dis ance om he measu ed
a ge — he de aul posi ions o m a sphe ical su ace c ea ed acco ding o he speci ied diame e and
densi y using he p inciple o sphe ical Fibonacci la ice [
34
]. The coo dina es (x, y, z) a e upda ed by
he con ol applica ion and co espond o poin s on he sphe ical su ace wi h he de ined diame e .
The o ien a ion a he de aul posi ions is no changed. Due o he o ien a ion o he manipula o ,
only he on hemisphe e is conside ed.
The sphe ical Fibonacci la ice is used as he simples solu ion o he e en dis ibu ion o he
de ined numbe o poin s on a sphe ical su ace. Poin s a e dis ibu ed e enly ega dless o he
diame e . By he e en dis ibu ion, i is mean ha he dis ance be ween e e y wo adjacen poin s is
he same.
The p ecise desc ip ion o e e y single de aul posi ion is no necessa y, he esul should be he
sui abili y o a i al om all he di ec ions o he des ina ion a ge , shown by a g adien be ween
mul iple poin s. F om such a g aph, i will be possible o de e mine ela i ely accu a e alues o he
e o caused by he di ec ion o app oach o a poin , anywhe e in he hemisphe e, i.e., e en a ound
p ecisely speci ied poin s (Figu e 5).
Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 o 23
Table 3. Robo measu emen mo emen pa ame e s.
C ossing Speed ( o he De aul Posi ion)
400 mm/s
App oach speed ( o he measu ed a ge )
Vmax (max. 8900 mm/s)
S abiliza ion ime (bo h he de . pos. and a ge )
2 s
All s a ing posi ions wi hin one cycle a e loca ed a he same dis ance om he measu ed
a ge — he de aul posi ions o m a sphe ical su ace c ea ed acco ding o he speci ied diame e and
densi y using he p inciple o sphe ical Fibonacci la ice [34]. The coo dina es (x, y, z) a e upda ed by
he con ol applica ion and co espond o poin s on he sphe ical su ace wi h he de ined diame e .
The o ien a ion a he de aul posi ions is no changed. Due o he o ien a ion o he manipula o ,
only he on hemisphe e is conside ed.
The sphe ical Fibonacci la ice is used as he simples solu ion o he e en dis ibu ion o he
de ined numbe o poin s on a sphe ical su ace. Poin s a e dis ibu ed e enly ega dless o he
diame e . By he e en dis ibu ion, i is mean ha he dis ance be ween e e y wo adjacen poin s is
he same.
The p ecise desc ip ion o e e y single de aul posi ion is no necessa y, he esul should be
he sui abili y o a i al om all he di ec ions o he des ina ion a ge , shown by a g adien be ween
mul iple poin s. F om such a g aph, i will be possible o de e mine ela i ely accu a e alues o he
e o caused by he di ec ion o app oach o a poin , anywhe e in he hemisphe e, i.e., e en a ound
p ecisely speci ied poin s (Figu e 5).
Figu e 5. (A): Poin s, e enly dis ibu ed on he sphe ical su ace. (B): Semi-sphe e wi h o igin
o ien a ion.
A o al o 24 cycles we e pe o med wi h di e en alues o he adius o he men ioned sphe e
(50, 100, 150, 200, 250, 300 mm), he numbe o s a ing posi ions (10, 20, 40, 80, 160), and he numbe
o epe i ions (10, 20, 30). All combina ions o he pa ame e s o pe o med measu emen s a e lis ed
in Tables 4–8. Visualiza ion o he loca ion o he hemisphe e is shown in Figu e 6. The measu ed
a ge has he same o ien a ion as he TCP ( ool coo dina e sys em), and is loca ed a he dis ance o
+500 mm in he di ec ion o WCS (wo ld coo dina e sys em) axis. The WCS is shown in Figu e 6 on
he le . The posi ion and o ien a ion o he TCP ( ool coo dina e sys em) a e displayed on he obo
ool. In he middle o Figu e 6 is he hemisphe e, along i s su ace, he de aul posi ions a e e enly
dis ibu ed.
Table 4. Pa ame e s o measu emen s o he in luence o he numbe o epe i ions.
Measu emen
Poin s
Repea
Weigh (kg)
Radius (mm)
1
10
10
2.538
50
2
9
10
2.538
300
3
9
20
2.538
300
4
11
30
2.538
300
Figu e 5.
(
A
): Poin s, e enly dis ibu ed on he sphe ical su ace. (
B
): Semi-sphe e wi h o igin o ien a ion.
A o al o 24 cycles we e pe o med wi h di e en alues o he adius o he men ioned sphe e
(50, 100, 150, 200, 250, 300 mm), he numbe o s a ing posi ions (10, 20, 40, 80, 160), and he numbe o
epe i ions (10, 20, 30). All combina ions o he pa ame e s o pe o med measu emen s a e lis ed in
Tables 4–8. Visualiza ion o he loca ion o he hemisphe e is shown in Figu e 6. The measu ed a ge
has he same o ien a ion as he TCP ( ool coo dina e sys em), and is loca ed a he dis ance o +500 mm
in he di ec ion o WCS (wo ld coo dina e sys em) axis. The WCS is shown in Figu e 6on he le .
The posi ion and o ien a ion o he TCP ( ool coo dina e sys em) a e displayed on he obo ool. In he
middle o Figu e 6is he hemisphe e, along i s su ace, he de aul posi ions a e e enly dis ibu ed.
Table 4. Pa ame e s o measu emen s o he in luence o he numbe o epe i ions.
Measu emen Poin s Repea Weigh (kg) Radius (mm)
1 10 10 2.538 50
2 9 10 2.538 300
3 9 20 2.538 300
4 11 30 2.538 300
Appl. Sci. 2020,10, 8714 8 o 24
Table 5. Pa ame e s o measu emen s o he in luence o he numbe o s a ing posi ions.
Measu emen Poin s Repea Weigh (kg) Radius (mm)
5 10 10 2.538 300
6 20 10 2.538 300
7 40 10 2.538 300
8 80 10 2.538 300
9 160 10 2.538 300
Table 6. Pa ame e s o measu emen s o he in luence o he adius dimension.
Measu emen Poin s Repea Weigh (kg) Radius (mm)
10 40 10 2.538 50
11 41 10 2.538 100
12 40 10 2.538 150
13 40 10 2.538 200
14 41 10 2.538 250
15 39 10 2.538 300
Table 7. Pa ame e s o measu emen s o he in luence o he payload.
Measu emen Poin s Repea Weigh (kg) Radius (mm)
16 40 10 0.04 300
17 40 10 2.538 300
18 40 10 4.647 300
Table 8. Pa ame e s o measu emen s o he in luence o he measu ed a ge posi ion.
Measu emen Poin s Repea Weigh (kg) Radius (mm) Shi (WCS) (mm)
19 41 10 2.538 150 +150 z
20 41 10 2.538 150 −150 z
21 39 10 2.538 150 −150 x
22 39 10 2.538 150 +150 x
23 42 10 2.538 150 +150 y
24 41 10 2.538 150 −150 y
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 o 23
Table 5. Pa ame e s o measu emen s o he in luence o he numbe o s a ing posi ions.
Measu emen
Poin s
Repea
Weigh (kg)
Radius (mm)
5
10
10
2.538
300
6
20
10
2.538
300
7
40
10
2.538
300
8
80
10
2.538
300
9
160
10
2.538
300
Table 6. Pa ame e s o measu emen s o he in luence o he adius dimension.
Measu emen
Poin s
Repea
Weigh (kg)
Radius (mm)
10
40
10
2.538
50
11
41
10
2.538
100
12
40
10
2.538
150
13
40
10
2.538
200
14
41
10
2.538
250
15
39
10
2.538
300
Table 7. Pa ame e s o measu emen s o he in luence o he payload.
Measu emen
Poin s
Repea
Weigh (kg)
Radius (mm)
16
40
10
0.04
300
17
40
10
2.538
300
18
40
10
4.647
300
Table 8. Pa ame e s o measu emen s o he in luence o he measu ed a ge posi ion.
Measu emen
Poin s
Repea
Weigh (kg)
Radius (mm)
Shi (WCS) (mm)
19
41
10
2.538
150
+150 z
20
41
10
2.538
150
−150 z
21
39
10
2.538
150
−150 x
22
39
10
2.538
150
+150 x
23
42
10
2.538
150
+150 y
24
41
10
2.538
150
−150 y
Figu e 6. Robo wo kspace and measu ed a ge .
Figu e 6. Robo wo kspace and measu ed a ge .
Appl. Sci. 2020,10, 8714 9 o 24
5. Expe imen al Resul s
To con i m he hypo hesis ha he endpoin epea abili y depends, among o he hings, on he
di ec ion o app oach o his poin , i was necessa y o ealize he selec ed measu emen s in h ee
di e en ways. Fo each o hese measu emen s, a se o s a ing poin s was c ea ed, hen h ee
di e en measu emen s we e execu ed unde he same condi ions, wi h he di e ence ha in he i s
measu emen he poin s we e selec ed in o de om he lowes o he highes alue o he y-axis (y
MIN
→
y
MAX
), he second measu emen was in he opposi e di ec ion (y
MAX →
y
MIN
) and in he hi d one,
he poin s we e chosen in andom o de .
The join a iables (j
1
—j
6
(
◦
)) we e always ead and hen he alues o he same poin s we e
compa ed when measu ing in a di e en o e all o de .
F om he g aphs (Figu e 7) i is e iden ha he de ia ion depends only on he posi ion o he
s a ing poin ela i e o he endpoin . Whe he hese s a ing poin s o he app oach a e selec ed in
o de om one side o he o he one, o comple ely andomly, is no impo an . This esul , among o he
hings, excludes acciden al e o s o o he in luences ha would cause andomness in hese esul s.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 o 23
5. Expe imen al Resul s
To con i m he hypo hesis ha he endpoin epea abili y depends, among o he hings, on he
di ec ion o app oach o his poin , i was necessa y o ealize he selec ed measu emen s in h ee
di e en ways. Fo each o hese measu emen s, a se o s a ing poin s was c ea ed, hen h ee
di e en measu emen s we e execu ed unde he same condi ions, wi h he di e ence ha in he i s
measu emen he poin s we e selec ed in o de om he lowes o he highes alue o he y-axis (yMIN
→ yMAX), he second measu emen was in he opposi e di ec ion (yMAX → yMIN) and in he hi d one,
he poin s we e chosen in andom o de .
The join a iables (j1—j6 (°)) we e always ead and hen he alues o he same poin s we e
compa ed when measu ing in a di e en o e all o de .
F om he g aphs (Figu e 7) i is e iden ha he de ia ion depends only on he posi ion o he
s a ing poin ela i e o he endpoin . Whe he hese s a ing poin s o he app oach a e selec ed in
o de om one side o he o he one, o comple ely andomly, is no impo an . This esul , among
o he hings, excludes acciden al e o s o o he in luences ha would cause andomness in hese
esul s.
The wa e o ms o indi idual measu emen s almos comple ely o e lap wi h only a minimal
di e ence be ween he i s “yellow” measu emen (yMIN → yMAX), in which he poin s we e execu ed
om he lowes y-axis alue o highes . Red (yMAX → yMIN), was execu ed in he opposi e o de and
blue using a comple ely andom o de o poin s. All h ee wa e o ms ha e been so ed om YMIN o
YMAX, due o he possibili y o compa ing he same poin s in hese measu emen s.
Figu e 7. De ia ions o join s (°) in he o de ed and andom selec ion o de aul posi ions.
Acco ding o he ma hema ical model, he posi ion and de ia ions o he ool cen e poin can
be calcula ed based on he o wa d kinema ics ask, bu his alue a ies in he spec um o absolu e
inaccu acy.
To de e mine he epea abili y and he eal de ia ions om he selec ed a ge , i is necessa y o
measu e he p ecision a he endpoin (TCP). Fo his pu pose, he 3D DIC me hodology will be
applied in he second phase, o e i y he esul s.
Figu e 7. De ia ions o join s (◦) in he o de ed and andom selec ion o de aul posi ions.
The wa e o ms o indi idual measu emen s almos comple ely o e lap wi h only a minimal
di e ence be ween he i s “yellow” measu emen (y
MIN →
y
MAX
), in which he poin s we e execu ed
om he lowes y-axis alue o highes . Red (y
MAX →
y
MIN
), was execu ed in he opposi e o de and
blue using a comple ely andom o de o poin s. All h ee wa e o ms ha e been so ed om Y
MIN
o
YMAX, due o he possibili y o compa ing he same poin s in hese measu emen s.
Acco ding o he ma hema ical model, he posi ion and de ia ions o he ool cen e poin
can be calcula ed based on he o wa d kinema ics ask, bu his alue a ies in he spec um o
absolu e inaccu acy.
To de e mine he epea abili y and he eal de ia ions om he selec ed a ge , i is necessa y
o measu e he p ecision a he endpoin (TCP). Fo his pu pose, he 3D DIC me hodology will be
applied in he second phase, o e i y he esul s.
Appl. Sci. 2020,10, 8714 16 o 24
To ob ain in o ma ion abou he ans o ma ion coo dina es o he poin s o he analyzed objec ,
Dan ec Dynamics co ela ion de ices use an algo i hm based on a pseudo-a ine ans o ma ion [
38
].
I he ans o ma ion pa ame e s o possible displacemen , elonga ion, shea , and dis o ion o he
ace a
0
–a
7
(Figu e 18) a e conside ed, acco ding o he abo e-men ioned algo i hm he ans o ma ion
coo dina es (u, ) can be calcula ed as ollows:
u(a0,a1,a2,a3,e
x,e
y)=a0+a1
e
x+a2
e
y+a3
e
xe
y
u(a4,a5,a6,a7,e
x,e
y)=a4+a5
e
x+a6
e
y+a7
e
xe
y(1)
Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 o 23
I he ans o ma ion pa ame e s o possible displacemen , elonga ion, shea , and dis o ion o he
ace a0–a7 (Figu e 18) a e conside ed, acco ding o he abo e-men ioned algo i hm he ans o ma ion
coo dina es (u, ) can be calcula ed as ollows:
𝑢(𝑎0, 𝑎1, 𝑎2, 𝑎3, 𝑥, 𝑦)= 𝑎0+ 𝑎1𝑥 + 𝑎2𝑦 + 𝑎3𝑥𝑦
𝑢(𝑎4, 𝑎5, 𝑎6, 𝑎7, 𝑥, 𝑦)= 𝑎4+ 𝑎5𝑥 + 𝑎6𝑦 + 𝑎7𝑥𝑦
(1)
Figu e 18. T ans o ma ion pa ame e s used in he algo i hm based on a pseudo-a ine ans o ma ion.
The measu emen p ocedu e was he same as in he case o obo join a iables measu emen .
The lowcha (Figu e 4) was only ex ended (Figu e 19) as ollows ( h ee images a e always aken):
Figu e 19. Flowcha o he expe imen con ol sys em (DIC).
To de e mine he e ec o obo d i , Measu emen 4 (Table 1, Figu e 20) was pe o med. D i
can se iously a ec he epea abili y and accu acy i he obo s uc u e empe a u e changes
signi ican ly. The d i could be p e en ed by keeping he s uc u e a he op imal wo king
empe a u e (by cooling in wi h ans in ound y applica ions, o example). Measu emen 4 esul s
howe e , p o ed, ha o condi ions unde which all he measu emen s ook place, he in luence o
d i is no signi ican . The p e-wa m-up ou ine was used be o e e e y measu emen o ge he
sys em o i s op imal ope a ing empe a u e.
Figu e 18.
T ans o ma ion pa ame e s used in he algo i hm based on a pseudo-a ine ans o ma ion.
The measu emen p ocedu e was he same as in he case o obo join a iables measu emen .
The lowcha (Figu e 4) was only ex ended (Figu e 19) as ollows ( h ee images a e always aken):
Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 o 23
I he ans o ma ion pa ame e s o possible displacemen , elonga ion, shea , and dis o ion o he
ace a0–a7 (Figu e 18) a e conside ed, acco ding o he abo e-men ioned algo i hm he ans o ma ion
coo dina es (u, ) can be calcula ed as ollows:
𝑢(𝑎0, 𝑎1, 𝑎2, 𝑎3, 𝑥, 𝑦)= 𝑎0+ 𝑎1𝑥 + 𝑎2𝑦 + 𝑎3𝑥𝑦
𝑢(𝑎4, 𝑎5, 𝑎6, 𝑎7, 𝑥, 𝑦)= 𝑎4+ 𝑎5𝑥 + 𝑎6𝑦 + 𝑎7𝑥𝑦
(1)
Figu e 18. T ans o ma ion pa ame e s used in he algo i hm based on a pseudo-a ine ans o ma ion.
The measu emen p ocedu e was he same as in he case o obo join a iables measu emen .
The lowcha (Figu e 4) was only ex ended (Figu e 19) as ollows ( h ee images a e always aken):
Figu e 19. Flowcha o he expe imen con ol sys em (DIC).
To de e mine he e ec o obo d i , Measu emen 4 (Table 1, Figu e 20) was pe o med. D i
can se iously a ec he epea abili y and accu acy i he obo s uc u e empe a u e changes
signi ican ly. The d i could be p e en ed by keeping he s uc u e a he op imal wo king
empe a u e (by cooling in wi h ans in ound y applica ions, o example). Measu emen 4 esul s
howe e , p o ed, ha o condi ions unde which all he measu emen s ook place, he in luence o
d i is no signi ican . The p e-wa m-up ou ine was used be o e e e y measu emen o ge he
sys em o i s op imal ope a ing empe a u e.
Figu e 19. Flowcha o he expe imen con ol sys em (DIC).

Appl. Sci. 2020,10, 8714 17 o 24
To de e mine he e ec o obo d i , Measu emen 4 (Table 1, Figu e 20) was pe o med. D i can
se iously a ec he epea abili y and accu acy i he obo s uc u e empe a u e changes signi ican ly.
The d i could be p e en ed by keeping he s uc u e a he op imal wo king empe a u e (by cooling
in wi h ans in ound y applica ions, o example). Measu emen 4 esul s howe e , p o ed, ha o
condi ions unde which all he measu emen s ook place, he in luence o d i is no signi ican .
The p e-wa m-up ou ine was used be o e e e y measu emen o ge he sys em o i s op imal
ope a ing empe a u e.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 o 23
Figu e 20. Manipula o epea abili y (mm), Measu emen 4.
Acco ding o he “3D Map (Figu e 21) i is e iden , ha a a gi en posi ion o he poin ela i e
o he obo , he epea abili y can be inc eased simply by changing he di ec ion o app oach o he
a ge Figu e 21 shows Measu emen s 14 and 16 ( adius 250/300 mm, 40 di e en de aul posi ions,
di e en ca ied weigh ) displaying i s o al e o s. The poin s a e so ed om YMIN o YMAX i.e., om
he bo om o he semi-sphe e o i s op.
F om he igu es, i is appa en ha unde he de ined condi ions, he mos con enien di ec ion
o app oach is om he VII oc an o Measu emen 14 and he op (z-axis) in case o Measu emen
16.
Figu e 21. (A): To al e o (mm), Measu emen 14. (B): To al e o (mm), Measu emen 16.
Fo he selec ed measu emen s, he eal epea abili y alues (mm) measu ed by DIC a e
displayed in Figu es 22 and 23. F om he Measu emen 14 esul s (Figu e 22), i is e iden ha he
ca ied weigh in luences he epea abili y p edominan ly in he middle sec ions (mid- ange o
Figu e 20. Manipula o epea abili y (mm), Measu emen 4.
Acco ding o he “3D Map (Figu e 21) i is e iden , ha a a gi en posi ion o he poin ela i e o
he obo , he epea abili y can be inc eased simply by changing he di ec ion o app oach o he a ge
Figu e 21 shows Measu emen s 14 and 16 ( adius 250/300 mm, 40 di e en de aul posi ions, di e en
ca ied weigh ) displaying i s o al e o s. The poin s a e so ed om Y
MIN
o Y
MAX
i.e., om he
bo om o he semi-sphe e o i s op.
F om he igu es, i is appa en ha unde he de ined condi ions, he mos con enien di ec ion
o app oach is om he VII oc an o Measu emen 14 and he op (z-axis) in case o Measu emen 16.
Fo he selec ed measu emen s, he eal epea abili y alues (mm) measu ed by DIC a e displayed
in Figu es 22 and 23. F om he Measu emen 14 esul s (Figu e 22), i is e iden ha he ca ied weigh
in luences he epea abili y p edominan ly in he middle sec ions (mid- ange o poin ’s y alue) o
e o g aphs. In he case o Measu emen 16 (Figu e 23), he di e ence in epea abili y alues wi hin
he measu ed hemisphe e is no signi ican .
Appl. Sci. 2020,10, 8714 18 o 24
Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 o 23
Figu e 20. Manipula o epea abili y (mm), Measu emen 4.
Acco ding o he “3D Map (Figu e 21) i is e iden , ha a a gi en posi ion o he poin ela i e
o he obo , he epea abili y can be inc eased simply by changing he di ec ion o app oach o he
a ge Figu e 21 shows Measu emen s 14 and 16 ( adius 250/300 mm, 40 di e en de aul posi ions,
di e en ca ied weigh ) displaying i s o al e o s. The poin s a e so ed om YMIN o YMAX i.e., om
he bo om o he semi-sphe e o i s op.
F om he igu es, i is appa en ha unde he de ined condi ions, he mos con enien di ec ion
o app oach is om he VII oc an o Measu emen 14 and he op (z-axis) in case o Measu emen
16.
Figu e 21. (A): To al e o (mm), Measu emen 14. (B): To al e o (mm), Measu emen 16.
Fo he selec ed measu emen s, he eal epea abili y alues (mm) measu ed by DIC a e
displayed in Figu es 22 and 23. F om he Measu emen 14 esul s (Figu e 22), i is e iden ha he
ca ied weigh in luences he epea abili y p edominan ly in he middle sec ions (mid- ange o
Figu e 21. (A): To al e o (mm), Measu emen 14. (B): To al e o (mm), Measu emen 16.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 18 o 23
poin ’s y alue) o e o g aphs. In he case o Measu emen 16 (Figu e 23), he di e ence in
epea abili y alues wi hin he measu ed hemisphe e is no signi ican .
Figu e 22. Manipula o epea abili y (mm), Measu emen 14.
Figu e 23. Manipula o epea abili y (mm), Measu emen 16.
Figu e 22. Manipula o epea abili y (mm), Measu emen 14.
Appl. Sci. 2020,10, 8714 19 o 24
Appl. Sci. 2020, 10, x FOR PEER REVIEW 18 o 23
poin ’s y alue) o e o g aphs. In he case o Measu emen 16 (Figu e 23), he di e ence in
epea abili y alues wi hin he measu ed hemisphe e is no signi ican .
Figu e 22. Manipula o epea abili y (mm), Measu emen 14.
Figu e 23. Manipula o epea abili y (mm), Measu emen 16.
Figu e 23. Manipula o epea abili y (mm), Measu emen 16.
Measu emen s 20 and 21 (Figu es 24–26) we e pe o med unde he same condi ions, in di e en
measu ed a ge s (see Table 8). As hese wo measu ed a ge s a e no oo a apa and obo axis
o ien a ion was no oo di e en , he e is only a sligh di e ence be ween hese wo esul s. Howe e ,
he di e ence be ween he mos con enien /leas sui able di ec ion o a i al is almos 0.09 mm o
Measu emen 20 (0.07 mm in Measu emen 21) and so his could be he epea abili y inc ease i he
app oach ajec o y will be changed acco ding o he in o ma ion om he g aphs.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 19 o 23
Measu emen s 20 and 21 (Figu es 24–26) we e pe o med unde he same condi ions, in di e en
measu ed a ge s (see Table 8). As hese wo measu ed a ge s a e no oo a apa and obo axis
o ien a ion was no oo di e en , he e is only a sligh di e ence be ween hese wo esul s. Howe e ,
he di e ence be ween he mos con enien /leas sui able di ec ion o a i al is almos 0.09 mm o
Measu emen 20 (0.07 mm in Measu emen 21) and so his could be he epea abili y inc ease i he
app oach ajec o y will be changed acco ding o he in o ma ion om he g aphs.
Figu e 24. (A): To al e o (mm), Measu emen 20. (B): To al e o (mm), Measu emen 21.
Figu e 25. Manipula o epea abili y (mm), Measu emen 20.
Figu e 24. (A): To al e o (mm), Measu emen 20. (B): To al e o (mm), Measu emen 21.
Appl. Sci. 2020,10, 8714 20 o 24
Appl. Sci. 2020, 10, x FOR PEER REVIEW 19 o 23
Measu emen s 20 and 21 (Figu es 24–26) we e pe o med unde he same condi ions, in di e en
measu ed a ge s (see Table 8). As hese wo measu ed a ge s a e no oo a apa and obo axis
o ien a ion was no oo di e en , he e is only a sligh di e ence be ween hese wo esul s. Howe e ,
he di e ence be ween he mos con enien /leas sui able di ec ion o a i al is almos 0.09 mm o
Measu emen 20 (0.07 mm in Measu emen 21) and so his could be he epea abili y inc ease i he
app oach ajec o y will be changed acco ding o he in o ma ion om he g aphs.
Figu e 24. (A): To al e o (mm), Measu emen 20. (B): To al e o (mm), Measu emen 21.
Figu e 25. Manipula o epea abili y (mm), Measu emen 20.
Figu e 25. Manipula o epea abili y (mm), Measu emen 20.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 20 o 23
Figu e 26. Manipula o epea abili y (mm), Measu emen 21.
7. Conclusions
The hypo hesis ha he epea abili y o he obo a he a ge depends, among o he hings, on
he di ec ion o app oach o i , has been p o en.
The g aphs (Figu es 7 and 11) show ha he e o does no occu andomly ( he de ia ion o he
app oach o he a ge was he same ega dless o he o de in which he s a ing poin s we e
e alua ed). By ecalcula ing he esul s o he measu ed alues o he join a iables o he posi ioning
de ia ion o he ool cen e poin , he epea abili y was de e mined (inaccu acies ha belong o he
indi idual di ec ions o app oach a he a ge ).
Many in luences a ec he epea abili y alue. Especially he he mal expansion will always
a ec any high-p ecision de ice. Fo his eason, he obo was wa med-up be o e e e y
measu emen and kep a a cons an empe a u e as much as possible.
The da a gi en by he esol e s canno be used o de e mine he mos accu a e di ec ion o an
app oach o a a ge . I is caused by he ac , ha in hese da a, he a m and sha de lec ion is no
con ained. Resol e da a ha e been used only o p o e he hypo hesis. The eal accu a e da a could
be gi en only by he ex e nal measu emen de ice, DIC in his case.
To e i y he asce ainmen and especially o ob ain eal and accu a e alues o obo
epea abili y, all measu emen s we e eplica ed wi h he use o he 3D DIC me hodology. The esul s
o hese measu emen s a e displayed no only in he o m o box-plo g aphs bu also in 3D maps,
ha be e illus a e he achie ed esul s.
The dependence o he di ec ion o app oach on he epea abili y, achie able a he a ge , was
e i ied a se e al poin s. In Measu emen s 19–24, he a ge poin was shi ed by 150 mm in bo h
di ec ions o all he wo ld coo dina e sys em axes. The hypo hesis was, he e o e, e i ied no only
unde di e en pa ame e s o he obo ’s mo emen bu also a se en di e en a ge s.
The deg ee o his dependence a ies wi h espec o o he pa ame e s, such as he weigh
ca ied o he app oach speed. Fo example, in Measu emen 20 (Figu es 24 and 25), he di e ence
om he (ze o, exac ) alue is +0.019 (max) and +0.106 (min). I , o example, depending on he na u e
o he p ocess and he wo kplace, i is possible o app oach om he mos ad an ageous di ec ion,
Figu e 26. Manipula o epea abili y (mm), Measu emen 21.
Appl. Sci. 2020,10, 8714 21 o 24
7. Conclusions
The hypo hesis ha he epea abili y o he obo a he a ge depends, among o he hings, on he
di ec ion o app oach o i , has been p o en.
The g aphs (Figu es 7and 11) show ha he e o does no occu andomly ( he de ia ion o
he app oach o he a ge was he same ega dless o he o de in which he s a ing poin s we e
e alua ed). By ecalcula ing he esul s o he measu ed alues o he join a iables o he posi ioning
de ia ion o he ool cen e poin , he epea abili y was de e mined (inaccu acies ha belong o he
indi idual di ec ions o app oach a he a ge ).
Many in luences a ec he epea abili y alue. Especially he he mal expansion will always a ec
any high-p ecision de ice. Fo his eason, he obo was wa med-up be o e e e y measu emen and
kep a a cons an empe a u e as much as possible.
The da a gi en by he esol e s canno be used o de e mine he mos accu a e di ec ion o an
app oach o a a ge . I is caused by he ac , ha in hese da a, he a m and sha de lec ion is no
con ained. Resol e da a ha e been used only o p o e he hypo hesis. The eal accu a e da a could be
gi en only by he ex e nal measu emen de ice, DIC in his case.
To e i y he asce ainmen and especially o ob ain eal and accu a e alues o obo epea abili y,
all measu emen s we e eplica ed wi h he use o he 3D DIC me hodology. The esul s o hese
measu emen s a e displayed no only in he o m o box-plo g aphs bu also in 3D maps, ha be e
illus a e he achie ed esul s.
The dependence o he di ec ion o app oach on he epea abili y, achie able a he a ge ,
was e i ied a se e al poin s. In Measu emen s 19–24, he a ge poin was shi ed by 150 mm in bo h
di ec ions o all he wo ld coo dina e sys em axes. The hypo hesis was, he e o e, e i ied no only
unde di e en pa ame e s o he obo ’s mo emen bu also a se en di e en a ge s.
The deg ee o his dependence a ies wi h espec o o he pa ame e s, such as he weigh ca ied
o he app oach speed. Fo example, in Measu emen 20 (Figu es 24 and 25), he di e ence om he
(ze o, exac ) alue is +0.019 (max) and +0.106 (min). I , o example, depending on he na u e o he
p ocess and he wo kplace, i is possible o app oach om he mos ad an ageous di ec ion, he e may
be a e inemen compa ed o he leas ad an ageous di ec ion, by 0.087 mm in he o de o he o al
de ia ion o epea able accu acy.
This esea ch p esen s a new oppo uni y o inc ease he epea abili y o any obo ic a m.
Acco ding o he knowledge o he su ounding space and i s in luence on he ool cen e poin
de ia ion, he ajec o y could be sligh ly changed o imp o ed so ha he epea abili y will be
inc eased. Au ho s in his esea ch used he DIC came as o measu e he eal s a e, bu any ex e nal
de ice, which is accu a e enough would be used.
The p o ided measu emen is no pe manen , egula calib a ion will be equi ed as in any o he
accu a e de ice.
In a p ac ical applica ion, a high p ecision assembly ask, o example, he p ocedu e would be as
ollows:
1. P epa e he obo and ex e nal echnologies o an ope a ional s a e;
2. Se DIC came as, de ine a con ol applica ion inpu s (o ien a ion o measu ed a ge , numbe o
poin s on sphe e, numbe o epe i ions om e e y single ini ial poin , e c.);
3. Run he obo ’s mo emen o he measu ed a ge , measu e he eached posi ion by DIC;
4.
E alua e he sphe e, c ea e he “3D map”, place i in o a igu e o wo ks a ion, o ge an o ien a ion
o sphe e acco ding o he su ounding ins alla ion;
5.
Wi h he in o ma ion o he mos and leas sui able app oaches o di ec ion, e-p og am he obo
mo ion o ge he be e epea abili y.

Appl. Sci. 2020,10, 8714 22 o 24
8. Fu u e wo k
The au ho s plan o use he desc ibed p ecision measu emen me hodology o measu e he
p ecision o he obo ’s mo emen along he ajec o y. A e de e mining he inaccu acies, i will
ce ainly be possible o compensa e o hese, bu he ques ion is o wha ex en i will be possible o
op imize he mo emen along he ajec o y.
Au ho Con ibu ions:
M.V.: So wa e, w i ing— e iew and edi ing, in es iga ion R.H.: me hodology, alida ion
M.H.: so wa e, in es iga ion V.K.: concep ualiza ion, w i ing—o iginal d a , esou ces, unding acquisi ion,
T.K.: so wa e, w i ing— e iew and edi ing, isualiza ion Z.B.: me hodology, supe ision, da a cu a ion, o mal
analysis, alida ion, p ojec adminis a ion. All au ho s ha e ead and ag eed o he published e sion o
he manusc ip .
Funding:
This wo k was suppo ed by he Eu opean Regional De elopmen Fund in he Resea ch Cen e o
Ad anced Mecha onic Sys ems P ojec , p ojec numbe CZ.02.1.01/0.0/0.0/16_019/0000867 wi hin he Ope a ional
P og amme Resea ch, De elopmen and Educa ion and p ojec suppo ed by he Scien i ic g an agency o he
Minis y o Educa ion, Science, Resea ch and Spo o he Slo ak Republic and he Slo ak Academy o Sciences,
p ojec numbe VEGA 1/0355/18: he use o expe imen al me hods o mechanics o e inemen and e i ica ion o
nume ical models o mechanical sys ems wi h a ocus on composi e ma e ials.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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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 ilia ions.
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