sus ainabili y
A icle
A Comp ehensi e Pe o mance E alua ion o
Di e en Mobile Manipula o s Used as
Displaceable 3D P in e s o Building
Elemen s o he Cons uc ion Indus y
Robe Guamán Ri e a 1, Rod igo Ga cía Al a ado 2and Alejand o Ma ínez-Rocamo a 3
and Fe nando Aua Cheein 1,*
1Depa men o Elec onics Enginee ing, Uni e sidad Técnica Fede ico San a Ma ía, Valpa aíso 1680, Chile;
[email p o ec ed]
2Depa men o Design and heo y o A chi ec u e, Uni e sidad de Bio-Bio, Concepción 1202, Chile;
ga [email p o ec ed]
3A DiTec, Depa men o A chi ec u al Cons uc ions II, IUACC, Highe Technical School o Building
Enginee ing, Uni e sidad de Se illa, A . Reina Me cedes 4-a, 41012 Se illa, Spain; [email p o ec ed]
*Co espondence: [email p o ec ed]
Recei ed: 23 Ma ch 2020; Accep ed: 2 May 2020; Published: 27 May 2020


Abs ac :
The cons uc ion indus y is cu en ly echnologically challenged o inco po a e new
de elopmen s o enhancing he p ocess, such as he use o 3D p in ing o complex building s uc u es,
which is he aim o his b ie . To do so, we show a sys ema ic s udy ega ding he usabili y and
pe o mance o mobile manipula o s as displaceable 3D p in ing machine y in cons uc ion si es,
wi h emphasis on he h ee main di e en exis ing mobile pla o ms: he ca -like, he unicycle
and he omnidi ec ional (mecanum wheeled), wi h an UR5 manipula o on hem. To e alua e i s
pe o mance, we p opose he p in ing o he ollowing building elemen s: helical, squa e, ci cula and
mesh, wi h di e en sizes. As me ics, we conside he o al con ol e o obse ed in he obo s and
he o al acking e o associa ed wi h he ene gy consumed in he ac i i y o ge a mo e sus ainable
p ocess. In addi ion, o u he es ou wo k, we cons ained he obo wo kspace hus esembling
eal li e cons uc ion si es. In gene al, he s a is ical esul s show ha he omnidi ec ional pla o m
p esen s he bes esul s –lowes acking e o and lowes con ol e o – o ci cula , helicoidal and mesh
building elemen s; and ca -like pla o m shows he bes esul s o squa e-like building elemen . Then,
an inno a i e pe o mance analysis is achie ed o he p in ing o building elemen s, wi h a con ibu ion
o he educ ion o ene gy consump ion.
Keywo ds: mobile manipula o ; 3D p in ing; acking ajec o y
1. In oduc ion
Cons uc ion au oma ion (CA) is de ined as he in eg a ion o in elligen machines (e.g., building
obo s and/o embedded and dedica ed sys ems) [
1
], p in ing me hods (such as 3D p in ing,
conc e e p in ing, con ou c a ing, D-Shape [
2
]), adi ional cons uc ion me hods [
3
] and ad anced
cons uc ion echnologies (such as i) ex usion-based AM and ii)binde je ing) [4]. CA p omises se e al
bene i s o he cons uc ion indus y o enhance he p oduc i i y, quali y, and sus ainabili y o a chi ec u al
p ac ices and building cons uc ion [5].
Addi i e Manu ac u ing (AM) echnologies allow he ho izon o he 3D p in ing o be expanded
in he cons uc ion sec o . In his con ex , binde je ing AM is a echnology ha pe o ms he selec i e
deposi ion o a binde solu ion h ough a p in nozzle on o a p e iously deposi ed powde laye [
6
].
Sus ainabili y 2020,12, 4378; doi:10.3390/su12114378 www.mdpi.com/jou nal/sus ainabili y
Sus ainabili y 2020,12, 4378 2 o 17
Fu he mo e, he addi i e manu ac u ing o conc e e ex usion-based has ecen ly employed in he
cons uc ion indus y. This echnology in ol es ex uding he cemen -based ma e ial wi h nozzles o
di e en sizes o build a laye ed s uc u e model [7].
The echnology o 3D p in ing in cons uc ion applica ions can be conside ed as a p ocess whe e
a ious ma e ials a e successi ely solidi ied laye upon laye by ex usion, o o m solid models [
8
].
The majo applica ions o 3D p in ing in la ge scale in as uc u e a e di ided in o h ee scena ios:
D-Shape [
9
], con ou c a ing (CC) [
10
] and conc e e p in ing [
11
]. The o me , D-Shape, is based on
binde injec ion and, despi e i s high e ec i eness, i is only used o cus omized cons uc ions [
12
].
Fo he second case, CC, i is a compu e -con olled me hod wi h po abili y and cos imp o emen s:
i enables o build smoo h su aces in a sho ime [
13
,
14
]. Finally, conc e e p in ing is e y simila o
CC, bu i is associa ed wi h he de elopmen o componen s wi hou o mwo k [15].
The success o 3D p in ing in CA depends on he quali y o he ma e ials. The p in able mix u e
is simila and a ies in composi ion om he adi ional cemen pas e. Fo his eason, he p in ing
ma e ials mus ha e app op ia e heological and composi ional p ope ies ha allow: easy ex usion,
s ong adhesion be ween he p in ing laye s, a oid he collapse o he s uc u e and main ain he
p in ing pa e n du ing and a e he ma e ial deposi ion p ocess [
4
]. In his con ex , [
16
] p esen s an
ul a-high pe o mance conc e e (UHPC) de eloped o accele a e he p in ing p ocess and o imp o e
he mechanical s eng h o laye -by-laye cons uc ion. One o he main mo i a ions o conside UHPC
is he high mechanical pe o mance o build conc e e-based s uc u es, which is an al e na i e o mo e
adi ional cons uc ion me hods (see [
17
–
19
] o u he de ails). Fu he mo e, o achie e s uc u al
in eg i y, du abili y, eliabili y, and obus ness wi hou any suppo s uc u es. Thus, e . [
20
] p esen s
a 3D p in ing based on Enginee ed Cemen i ious Composi es (ECC), whose ad an ages a e sus ainable
mix design, heology con ol, and long- e m du abili y o he 3D p in ing.
Addi i e Manu ac u ing (AM) echnology using he ex usion-based me hod has been imp o ing
wi h he in eg a ion o obo ic sys ems; speci ically, a obo manipula o handling he ex uso and
con olling he ma e ial deposi ion [
21
–
23
]. Fo example, in [
24
,
25
] i is shown he cus omisa ion o
a obo ic a m in he cons uc ion indus y, aimed a p in ing conc e e walls o di e en geome ies.
The main ad an age o using obo manipula o s is hei wo kload capaci y, hei accu acy in
epe i i e asks and he lexible p og amming [
26
,
27
]. Ne e heless, obo manipula o s in cons uc ion
applica ions a e mainly used as ixed machine y, wi hou in e ac ing wi h he en i onmen o mo ing
wi hin he cons uc ion si e [16].
The cons uc ion indus y uses obo s o assembly and disassembly o a ious componen s.
In pa icula , e . [
28
] iden i ies ou s ages o assembling building componen s: (i) no assembly
componen s, (ii) assembly o a la ge elemen o med wi h se e al small componen s, (iii) placemen o
componen s in he inal posi ion, and (i ) assembly o non-p in ed ex e nal componen s. In his con ex ,
e . [
29
] p esen s a Robo ic P e ab ica ion Sys em (RPS) ha allows he au oma ic disassembly o a
p e ab ica ed s uc u e and de e mina es he needs and gaps in knowledge in he cu en p e ab ica ion.
Mo eo e , as a mechanical ools, obo ic a ms ha e ad an ages ha a ac he a en ion o he
cons uc ion indus y, such as hei dex e i y and hei eachabili y. Howe e , such ad an ages also
es ic he scale size o he cons uc ion: when ixed in he g ound, a obo manipula o wi h an
ex uso (used o p in ing) can only p in elemen s ha all wi hin i s wo kspace. O he wise, he obo
has o be manually displaced [30–33].
To o e come he la e p oblem, in his wo k we p opose o s udy he usabili y o a obo ic a m
moun ed on a mobile pla o m (also obo ic), o co e bigge a eas o he cons uc ion si e and o
p in con ex geome ies ( o example, he walls o a oom). By in eg a ing a mobile pla o m o he
obo manipula o used as 3D p in e , we inc ease i s dex e i y, i s wo kspace and i s capabili ies.
Howe e , since we use he same obo manipula o , we change he mobile pla o m o ind which
one is he mos app op ia e o be used in he cons uc ion si e. In pa icula , we es he ollowing
con igu a ions: unicycle, ca -like and ou wheeled omnidi ec ional wi h mecanum wheels [
34
,
35
].
We es he pe o mance o he mobile manipula o moun ed on he h ee di e en pla o ms in e ms
Sus ainabili y 2020,12, 4378 3 o 17
o e o and e o , when p in ing se e al ypes o building elemen s. Fo he emainde o his wo k,
we will e e o he obo ic pla o m as he mobile manipula o .
The selec ion o p ope machine and p in ing s a egy acco ding o he building pieces p in ed
is c ucial o a success ul e iciency and e sa ili y o his eme ging echnology o 3D-p in ed
cons uc ion wi h obo ic sys ems. In his con ex , he e icien use o obo ic pla o ms is ela ed
o hei ope a ional powe consump ion [
36
]. When he obo pe o ms a p in ing ask, he powe
consump ion a ies signi ican ly du ing he ope a ion o he obo , and comp ise he ac ua ion sys ems
ha will in e ac wi h he en i onmen and he cons uc ion p ocess.
Fo example, he use o ene gy esou ces o he mobile manipula o depends on he in e ac ion
o he pla o m wi h he en i onmen and he mo ion i execu es. Fo his eason, hei ene gy use is
go e ned by he i egula i y o he su ace o he cons uc ion en i onmen , he complexi y o he p in
model, and he payload i suppo s. In his wo k, we ela e he ene gy consump ion o he cos unc ion
in e ms o minimum ajec o y acking e o wi h he mo ion speeds o he mobile manipula o .
As p e iously s a ed, his wo k is ocused on s udying he usabili y o mobile manipula o
pla o ms as 3D p in ing machine y in cons uc ion si es, wi h emphasis in he di e en exis ing
mobile pla o ms, in o de o p o ide a no el e iew and sugges adequa e me hodologies o
buildings cons uc ion. P e ious wo k has been done on ma e ials, machines and/o speci ic p in ing
expe imen s, bu lack o igo ous s udies and gene al app oaches abou p in ing p ocedu es wi h
mobile obo pla o ms. Mo emen and ope a ion o obo s a e complex asks o be combined in a
p in ing p ocess o la ge pieces, and building execu ion equi es a di e si y o elemen s o be p in ed
in di e en loca ions, hen o es and de ine p ope s a egies is essen ial o de elop he cons uc ion
wi h obo s. In his con ex , [
8
,
37
] exposes s a egies o mobile sys ems o p in ing building elemen s
bu do no s udy he oolpa h in ela ion o he machine ype and piece design. Such p ocedu e is
ele an o de elop he p in ing me hodology and de ine e ec i e app oaches and equipmen s.
This wo k is o ganized as ollows: Sec ion 2shows he ajec o y p o iles in he cons uc ion
en i onmen , he ma hema ical de i a ion o he mobile manipula o model, he s a egy ollowed
by he mobile manipula o ajec o y acking con olle , and he me ics ollowed o ind mobile
pla o m pe o mance. Sec ion 3p esen s he esul s using he mobile manipula o wi h he h ee
di e en mobile pla o ms p e iously in oduced o di e en building scena ios. Sec ion 4shows a
discussion abou he asks pe o med by he mobile manipula o in he cons uc ion en i onmen s.
Finally, he conclusions a e shown in Sec ion 5.
2. Ma e ials and Me hods
Figu e 1shows he gene al scheme o he a chi ec u e p oposed in his wo k o e alua ing he
pe o mance o mobile manipula o s in he 3D p in ing case o building elemen s. To he le , we ha e
wo e e ence p o iles: ci cula and squa e, in wo di e en iews: 2D ( wo dimensional) and 3D.
Such p o iles ep esen he building elemen s o be p in ed. When sen o he manipula o , he building
elemen s a e con e ed in o pa h e e ences o be acked by he end e ec o (whe e he ex uso is
loca ed) o he manipula o . We conside h ee mobile pla o ms whe e he manipula o is going o
be moun ed: unicycle ype, ca -like ype (Acke man) and omnidi ec ional ype ( ou wheeled wi h
mecanum wheels). The ajec o y e e ences a e hen ans o med in o mo ion con ol commands
h ough a linea algeb a con olle o ensu e ha he sys em mobile manipula o beha es as expec ed:
i s end e ec o — o which he nozzle is a ached—mo es desc ibing he building elemen o be p in ed.
In he ollowing subsec ions, each pa o Figu e 1is explained in de ail.
Sus ainabili y 2020,12, 4378 4 o 17
Omnidi ec ional UnicycleCa like
Linea Algeb a
Con olle
2D p o ile
3D p o ile
2D p o ile
3D p o ile
0
0
0.1
z [m]
2
0.2
y [m]
0
0.3
4
x [m]
2
4
66
Desi ed
Ac ual
3
0.8
2
y [m]
3
x [m]
12
0.9
z [m]
1
1
Desi ed
Ac ual
4
8
y [m]
6
x [m]
6
4
0.5
28
z [m]
1
Desi ed
Ac ual
0.2
0
0.4
4
1
0.6
z [m]
x [m] y [m]
2
2
0.8
30
Desi ed
Ac ual
Re e ence p o ile Resul Mobile manipula o
Figu e 1.
Cons uc ion p ocess o a ajec o y p o ile de eloped by a mobile obo ic sys em.
Mobile manipula o wi h h ee con igu a ions o obo mobile. The obo ic s uc u e is o med
o base mobile (unicycle, ca -like and omnidi ec ional) and manipula o .
2.1. Robo Manipula o
The obo ic a m used o pe o m 3D p in ing o he building elemen s is he UR5, manu ac u ed
by Uni e sal Robo s. This manipula o has six deg ees o eedom and a con ol uni ha p o ides basic
join s con ol as well as a companion compu e -compu a ional sys em in cha ge o da a p ocessing
and b oadcas ing. I suppo s a payload o 5 kg; i s maximum speed is
π
ad/s and i s epea abili y is
±0.1 mm. Table 1shows he main ea u es o his manipula o .
Table 1. Robo manipula o : UR5 main ea u es.
Fea u e Value
Weigh 18.4 kg
Payload 5 kg
Reach 850 mm
Join Ranges ±2π
Speed π ad/s
I/O powe supply 24 V 2 A
Communica ion TCP/IP 100 Mbi :IEEE 802.3u, 100BASE-TX
P og aming Polyscope g aphical use in e ace
Tempe a u e The obo can wo k in a empe a u e ange o 0–50 deg ees
One o he main ad an ages o his obo is ha i o e s low-le el p og aming wi h high cycle
ime. The obo can be modelled using Dena i -Ha enbe g pa ame e s o geome ical model and i
has been used in no el applica ions in di e en indus ies [
38
]. Al hough we used he UR5 obo ic
a m in his wo k, he p ocedu e ollowed and p esen ed he ein can be ex ended o o he manipula o s.
2.2. Mobile Robo
Table 2shows he kinema ic models in con inuous and disc e e- ime o he non-holonomic
(i.e., unicycle and ca like) and holonomic (i.e., omnidi ec ional obo ) pla o ms s udied in his
wo k. We chose such h ee mobile pla o ms since hey a e he mos used ones as epo ed in he
li e a u e [39–41]
. Howe e , as also s a ed o he UR5 case, i needed, he p ocedu e p esen ed in his
wo k can be adap ed o o he ypes o pla o ms.
As i is shown in Table 2, he h ee obo s ha e h ee deg ees o eedom, named as
x
,
y
and
ψ
,
which apply o o a ion and ansla ion o he obo in he plane, conside ing only plana e ains.
In he case o he unicycle obo , i can u n a ound i s con ol poin [
42
]; howe e , he ca -like model
has o ollow a ci cle based pa h [
43
], making i impossible o u n wi hou displacing [
44
]. On he
o he hand, he omnidi ec ional ehicle can displace a any di ec ion in he plane. Mo e in o ma ion
ega ding he kinema ic (and dynamic) cons ain s o each obo ic ype can be ound in [45–47].
Sus ainabili y 2020,12, 4378 5 o 17
Table 2. Kinema ic model o each mobile pla o m.
Model Con inuos Time Disc e e Time
Unicycle [39]
˙
x=µcos ψ−aωsin ψx(n+1)=x(n)+µ(n)cos ψ(n)−aω(n)sin ψ(n)
˙
y=µsin ψ+aωcos ψy(n+1)=y(n)+µ(n)sin ψ(n)+aω(n)cos ψ(n)
˙
ψ=ω ψ(n+1)=ψ(n)+ω(n)
Ca -like [40]
˙
x=µcos ψ−aωsin ψx(n+1)=x(n)+µ(n)cos ψ(n)−aω(n)sin ψ(n)
˙
y=µsin ψ+aωcos ψy(n+1)=y(n)+µ(n)sin ψ(n)+aω(n)cos ψ(n)
˙
ψ=µ
L an (δ)ψ(n+1)=ψ(n)+µ(n)
L an δ(n)
Omnidi ec ional [41]
˙
x=1
4γ 1−1
4β 2+1
4γ 3+1
4β 4x(n+1)=x(n)+1
4γ 1(n)−1
4β 2(n)+1
4γ 3(n)+1
4β 4(n)
˙
y=1
4β 1−1
4γ 2+1
4β 3+1
4γ 4y(n+1)=y(n)+1
4β 1(n)−1
4γ 2(n)+1
4β 3(n)+1
4γ 4(n)
˙
ψ=− 1
4(l+d)− 2
4(l+d)+ 3
4(l+d)+ 4
4(l+d)ψ(n+1)=ψ(n)− 1(n)
4(l+d)− 2(n)
4(l+d)+ 3(n)
4(l+d)+ 4(n)
4(l+d)

Sus ainabili y 2020,12, 4378 6 o 17
In Table 2,
µ
and
ω
a e he linea and angula eloci ies o he unicycle and ca -like obo ;
a
is
he dis ance be ween he cen e o he pla o m and he cen e o mass in global coo dina es;
δ
is he
heading o he ca -like;
L
is he leng h be ween axles o he ca -like obo and
(l+d)
is he dis ance
om he cen e o he wheel o he cen e o mass o he omnidi ec ional obo ;
1
,
2
,
3
,
4
a e
wheel eloci ies o he omnidi ec ional obo . Fu he mo e,
γ
is de ined by
(cos ψ−sin ψ)
and
β
is
de ined by (sin ψ+cos ψ). Su ix n ep esen s sampling ime.
2.3. Mobile Manipula o
As s a ed in Sec ion 1, when he obo manipula o is a ached o one o he mobile pla o ms
men ioned in Sec ion 2.2, he combined sys em becomes a mobile manipula o . I s kinema ic is he
esul o also combining he manipula o kinema ics wi h he mobile obo kinema ics. Hence, we use
he Dena i -Ha en e g con en ion o de i e he mobile manipula o kinema ic model, ollowing he
guidelines p e iously published by [38].
Table 3shows he kinema ic model o he obo ic a m moun ed on he h ee di e en mobile
pla o ms. Fo unicycle and ca -like con igu a ions, linea and angula eloci y de ines he mo ion o
he mobile pla o m. The o ien a ion o he unicycle obo is de ined by angula o a ion. Howe e ,
in he ca -like con igu a ion, he o a ion is a unc ion o he linea eloci y and he leng h o he mobile
pla o m. Fo he omnidi ec ional obo , o a ion and linea eloci y a e a unc ion o he angula
eloci y desc ibed by he wheels.
Table 3. Kinema ic model o he mobile manipula o .
Mobile Manipula o Kinema ic Model
Unicycle [˙
xee ˙
yee ˙
zee]T=J[µ ω ˙
θ1˙
θ2˙
θ3˙
θ4˙
θ5]T
Ca -like [˙
xee ˙
yee ˙
zee]T=J[µµ
L˙
θ1˙
θ2˙
θ3˙
θ4˙
θ5]T
Omnide ec ional [˙
xee ˙
yee ˙
zee]T=J[ 1 2 3 4˙
θ1˙
θ2˙
θ3˙
θ4˙
θ5]T
whe e,
J=∂hee
∂ (x,y,ψ,θ1, ..., θ5)(1)
and
hee = [xee yee zee]T
is he posi ion o he end-e ec o –wi h espec o some global e e ence
ame–, and
[θ1θ2θ3θ4θ5]T
a e i s join angles. Fu he mo e,
[˙
xee ˙
yee ˙
zee]T
is he eloci y o
he end-e ec o ;
˙
θ1
,
˙
θ2
,
˙
θ3
,
˙
θ4
and
˙
θ5
a e angula eloci ies o he join s. Figu e 2shows he
manipula o moun ed on h ee mobile pla o ms (unicycle, ca -like and omnidi ec ional wi h ou
mecanum wheels). Angles and eloci ies a e ep esen ed acco ding o a global e e ence ame
G
,
R
is
he local coo dina e o he mobile obo ,
M
ep esen s he local coo dina e o he manipula o UR5 and
he coo dina e o he end-e ec o is
Oe
. The manipula o links a e ep esen ed by
d1
,
a2
,
a3
,
d4
, and
d5, and he heigh o he mobile pla o m is h1.
Sus ainabili y 2020,12, 4378 7 o 17
x1
y1
z1
x
y
z
x2
y2z2x4
y4
z4
M
x3
y3z3
x5
y5
z5
xee
yee zee
xG
yG
zG
G
d
1
a2
a3
d
5
d
4
d
6
Oe
R
h
1
(a)
x2
y2z2
x4
y4
z4
x3
y3z3
x5
y5
z5
xee
yee zee
a2
a3
d
5
d
4
d
6
Oe
xG
yG
zG
G
x1
y1
z1
x
y
z
M
R
d
1
h
1
(b)
x
y
z
R
xG
yG
zG
G
x2
y2z2
x4
y4
z4
x3
y3z3
x5
y5
z5
xee
yee zee
a2
a3
d
5
d
4
d
6
Oe
y1
z1
Mx1
d
1
h
1
(c)
Figu e 2.
Mobile manipula o model: obo ic a m wi h six deg ee eedom moun ed in mobile pla o m;
(
a
) shows he unicycle case, (
b
) shows he ca -like case and (
c
) he omnidi ec ional case wi h ou
mecanum wheels.
2.3.1. Wo kspace Res ic ion
Since his wo k is ocused on analysing he usabili y o mobile manipula o s in cons uc ion
en i onmen s, i becomes necessa y o also analyse he wo kspace su ounding he pla o m due o
he ac ha i s mo ion is cons ained by he en i onmen layou , he e ain and he ask es ic ions,
he la e wi h he aim o gua an eeing he e ec i eness and sa e y o he ope a ion.
In ou wo k, he i s cons ain impossed o he sys em is he es ic ion o he wo kspace, by wo
imes he maximum ex ension o he obo manipula o . Thus, i limi s he displacemen o he pla o m
by allowing i no mo e han one ime he obo manipula o ex ension, o ensu e he sa e y o he
ope a ion. O he c i e ia migh apply since i is o be no ed ha he pla o m does no mo e eely
in a cons uc ion si e. In he end, he idea is o conside he design equi emen s and shapes o he
building elemen s o e alua e he mos app op ia e wo kspace, o hus a oid excu ing asks a he
maximum poin o ope a ion, o o in ade an a ea beyond he p in ed building elemen [
19
]. Figu e 3
summa izes he es ic ions conside ed in his wo k o all he building elemen s s udied, whe e he
solid ed line ep esen s he ou e limi imposed on he mobile pla o m, his limi may no exceed
wice he maximum leng h o he obo ic a m; he solid blue line is he inne limi and canno exceed
he maximum a m leng h.
Sus ainabili y 2020,12, 4378 8 o 17
Figu e 3.
Cons ain s o he wo kspace. The solid ed line ep esen s he maximum limi o ope a ion
o he mobile pla o m and he minimum limi is in a solid blue line. The e o e, he mobile obo should
no espass bo h squa es.
2.4. Building Elemen s
The cons uc ion indus y uses complex building elemen p o iles o build houses and buildings.
These su aces esul in high p o ile p ojec s o iconic a chi ec u e designs. The main challenges o
building elemen p o iles a e hei complex geome ies, size and epea abili y o he ask, which can
a ec he pe o mance o he obo ic sys ems [
48
,
49
]. The e o e he need o using mobile manipula o s.
We gene a e ou ajec o y p o iles ollowing he guidelines p esen ed in [
26
]. Such p o iles a e
hen con e ed in o e e ence ajec o ies ed o he mobile manipula o , pa ame ized in ime and
disc e ized wi h sampling ime o 0.1 s (howe e , o he sampling c i e ion migh apply). The building
elemen s s udied in his wo k a e: ci cle, helical, mesh and squa e; which a e also depic ed in Figu e 4.
M
e
s
h
Ci cle Squa e
Hel
i
c
al
Figu e 4. Building elemen s p oposed in his wo k.
We es ed h ee di e en sizes o he abo e building elemen : small (leng h = 1 m, wid h =
1 m, heigh = 1.2 m); medium (leng h = 2 m, wid h = 2 m, heigh = 1.2 m); and la ge (leng h = 5 m,
wid h = 5 m
, heigh = 1.2 m), ollowing he guidelines p esen ed in [
50
,
51
]. Such building elemen s a e
uni o m, excep o he co ne s in squa e building elemen s, whe e he change is ab up and migh
a ec he o ien a ion o he mobile pla o m. Fo he ci cle and helical case, he diame e p oposed was
o 1 m (small size), 2 m (medium size) and 4 m (la ge size).
The gene a ion o he building elemen s is di ided in o 2D laye s o gene a e 3D model geome ies.
Each building elemen is ep esen ed by ci cula and squa e geome ies. Addi ionally, o p in ing
pu poses, we added he ollowing cons ain s: p in ing speed be ween 0.01 and 1 ms
−1
[
52
]; maximum
Sus ainabili y 2020,12, 4378 9 o 17
mobile pla o m speed o 0.7 ms
−1
; and we limi ed he maximum angles o he join s o
−
2
π
o 2
π
ad [53].
2.5. Mo ion Con ol
In his wo k, we p opose a ajec o y acking algo i hm o he mobile manipula o o ensu e
3D p in ing o building elemen s a cons an speed. To his end, a ajec o y acking algo i hm is
implemen ed using he con olle p oposed by Scaglia e al. [
54
]. This con olle gua an ees eloci y
egula ion be ween he end e ec o and he p in ing ask.
The kinema ic model p esen ed in Table 3can be exp essed as ollows:
˙
h( )= Γ,˙
θ1,˙
θ2,˙
θ3,˙
θ4,˙
θ5(2)
The con ol a iables o he mobile pla o m a e ep esen ed by
Γ
. We hen de ined
Γ
o each
mobile pla o m and eplaced i in Equa ion (2), whe e
Γ=µ
,
ω
ep esen s he con ol a iables o he
unicycle obo ,
Γ=µ
,
µ
L
is de ined o a ca -like obo and
Γ= 1
,
2
,
3
,
4
ep esen s he con ol
signals o he omnidi ec ional obo .
The con inuous sys em shown in Equa ion (2) can be ew i en disc e ized using an Eule app oach
as shown below, o he h ee models:
h(n+1)=h(n)+Z(n+1)To
nTo
Γ,˙
θ1,˙
θ2,˙
θ3,˙
θ4,˙
θ5d (3)
h(n+1)∼
=h(n)+T0 Γ(n),θ1(n),θ2(n),θ3(n),θ4(n),θ5(n)(4)
Γ(n),θ1(n),θ2(n),θ3(n),θ4(n),θ5(n)=J(n)










Γ(n)
θ1(n)
θ2(n)
θ3(n)
θ4(n)
θ5(n)










(5)
whe e
J(n)
is he disc e e Jacobian ma ix o
J
om Table 3a each sampling ime. The alues o
h( )
a disc e e ime
=nTo
, whe e
To
is he sampling pe iod, and
n∈{0, 1, 2, . . .}
, a e deno ed as
h(n)=hxee(n),yee(n),zee(n)iT
and
h(n+1)=hxee(n+1),yee(n+1),zee(n+1)iT
. The kinema ic model o
he mobile manipula o , is de ined by:




xee(n+1)−xee(n)
To
yee(n+1)−yee(n)
To
zee(n+1)−zee(n)
To




=J(n)U(n)(6)
whe e U(n)=hΓ(n)θ1(n)θ2(n)θ3(n)θ4(n)θ5(n)iT. The p oposed con ol law is de ined by:
Uc(n)=J+
(n)




xeed(n+1)−kxxeed(n)−xee(n)−xee(n)
yeed(n+1)−kyyeed(n)−yee(n)−yee(n)
zeed(n+1)−kzzeed(n)−zee(n)−zee(n)





, (7)
whe e
J+
(n)
is he Jacobian pseudo-in e se ma ix o he mobile manipula o , he posi ion o
he end-e ec o is de ined by
[xee(n)yee(n)zee(n)]T
amed in a global e e ence sys em,
and K= [kxkykz)]T
is he se o uning pa ame e s. The desi ed pa h o he end-e ec o is gi en
Sus ainabili y 2020,12, 4378 16 o 17
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c
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