A comprehensive performance evaluation of different mobile manipulators used as displaceable 3D printers of building elements for the construction industry

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 19. Fu e , B.; Poullain, P.; Ga nie , S. 3D p in ing o cons uc ion based on a complex wall o polyme - oam and conc e e. Addi . Manu . 2019,28, 58–64. [C ossRe ] 20. Li, V.C.; Bos, F.P.; Yu, K.; McGee, W.; Ng, T.Y.; Figuei edo, S.C.; Ne s, K.; Mech che ine, V.; Ne ella, V.N.; Pan, J.; e al. On he eme gence o 3D p in able Enginee ed, S ain Ha dening Cemen i ious Composi es (ECC/SHCC). Cem. Conc . Res. 2020,132, 106038. [C ossRe ] 21. Dha mawan, A.G.; Sedo e, B.W.C.; Foong, S.; Soh, G.S. An agile obo ic sys em moun ed on sca old s uc u es o on-si e cons uc ion wo k. Cons . Robo . 2017,1, 15–27. [C ossRe ] 22. Ba d, J.; Cupko a, D.; Washbu n, N.; Zeglin, G. Robo ic conc e e su ace inishing: A moldless app oach o c ea ing he mally uned su ace geome y o a chi ec u al building componen s using P o ile-3D-P in ing. Cons . Robo . 2018,2, 53–65. [C ossRe ] 23. Dö le , K.; Hack, N.; Sandy, T.; Gi hale , M.; Lussi, M.; Walze , A.N.; Buchli, J.; G amazio, F.; Kohle , M. Mobile obo ic ab ica ion beyond ac o y condi ions: Case s udy Mesh Mould wall o he DFAB HOUSE. Cons . Robo . 2019,3, 53–67. [C ossRe ] 24. Reinha d , D.; Ti chkosky, N.; Bicke on, C.; Wa , R.; Wozniak-O’Conno , D.; Candido, C.; Cab e a, D.; Page, M.; Bohnenbe ge , S. Towa ds onsi e, modula obo ic ca bon- ib e winding o an in eg a ed ceiling s uc u e. Cons . Robo . 2019,3, 23–40. [C ossRe ] 25. Bos, F.; Wol s, R.; Ahmed, Z.; Sale , T. Addi i e manu ac u ing o conc e e in cons uc ion: Po en ials and challenges o 3D conc e e p in ing. Vi ual Phys. P o o yp. 2016,11, 209–225. [C ossRe ] 26. Da alab, O.; Kazemian, A.; Khoshne is, B. Pe spec i es on a BIM-in eg a ed so wa e pla o m o obo ic cons uc ion h ough Con ou C a ing. Au om. Cons . 2018,89, 13–23. [C ossRe ] 27. Mech che ine, V.; Ne ella, V.N.; Will, F.; Nä he , M.; O o, J.; K ause, M. La ge-scale digi al conc e e cons uc ion–CONP in 3D concep o on-si e, monoli hic 3D-p in ing. Au om. Cons . 2019 ,107, 102933. [C ossRe ] 28. Pe o , A.; Amziane, S. 3D P in ing in Conc e e: Gene al Conside a ions and Technologies. 3D P in . Conc S a e A Chall. Digi . Cons . Re olu . 2019, 1–40. [C ossRe ] 29. Kaspe zyk, C.; Kim, M.K.; B ilakis, I. Au oma ed e-p e ab ica ion sys em o buildings using obo ics. Au om. Cons . 2017,83, 184–195. [C ossRe ] 30. Pe o , A.; Rangea d, D.; Pie e, A. S uc u al buil -up o cemen -based ma e ials used o 3D-p in ing ex usion echniques. Ma e . S uc . 2016,49, 1213–1220. [C ossRe ] 31. de So o, B.G.; Agus í-Juan, I.; Hunhe icz, J.; Joss, S.; G ase , K.; Habe , G.; Adey, B.T. P oduc i i y o digi al ab ica ion in cons uc ion: Cos and ime analysis o a obo ically buil wall. Au om. Cons . 2018 , 92, 297–311. [C ossRe ] 32. Ngo, T.D.; Kashani, A.; Imbalzano, G.; Nguyen, K.T.; Hui, D. Addi i e manu ac u ing (3D p in ing): A e iew o ma e ials, me hods, applica ions and challenges. Compos. Pa B Eng. 2018,143, 172–196. [C ossRe ] 33. Buchanan, C.; Ga dne , L. Me al 3D p in ing in cons uc ion: A e iew o me hods, esea ch, applica ions, oppo uni ies and challenges. Eng. S uc . 2019,180, 332–348. [C ossRe ] 34. Wang, C.; Liu, X.; Yang, X.; Hu, F.; Jiang, A.; Yang, C. T ajec o y acking o an omni-di ec ional wheeled mobile obo using a model p edic i e con ol s a egy. Appl. Sci. 2018,8, 231. [C ossRe ] 35. Pappala do, C.M.; Guida, D. Fo wa d and In e se Dynamics o a Unicycle-Like Mobile Robo . Machines 2019,7, 5. [C ossRe ] 36. Ma ínez-Rocamo a, A.; Ga cía-Al a ado, R.; Casano a-Medina, E.; González-Böhme, L.F.; Aua -Cheein, F. Pa ame ic P og amming o 3D P in ed Cu ed Walls o Cos -E icien Building Design. J. Cons . Eng. Manag. 2020,146, 04020039. [C ossRe ] 37. Gi hale , M.; Sandy, T.; Dö le , K.; B ooks, I.; Buckingham, M.; Rey, G.; Kohle , M.; G amazio, F.; Buchli, J. Mobile obo ic ab ica ion a 1: 1 scale: The in si u ab ica o . Cons . Robo . 2017,1, 3–14. [C ossRe ] 38. Keb ia, P.M.; Al-Wais, S.; Abdi, H.; Naha andi, S. Kinema ic and dynamic modelling o UR5 manipula o . In P oceedings o he 2016 IEEE In e na ional Con e ence on Sys ems, Man, and Cybe ne ics (SMC), Budapes , Hunga y, 9–12 Oc obe 2016; pp. 4229–4234. 39. Vallejo-Ala cón, M.; Cas o-Lina es, R.; Velasco-Villa, M. Unicycle- ype obo & quad o o leade - ollowe o ma ion backs epping con ol. IFAC-Pape sOnLine 2015,48, 51–56. 40. Mo eno, J.; Clo e , E.; Lupiañez, R.; T esanchez, M.; Ma ínez, D.; Pallejà, T.; Casano as, J.; Palacín, J. Design, implemen a ion and alida ion o he h ee-wheel holonomic mo ion sys em o he assis an pe sonal obo (APR). Senso s 2016,16, 1658. [C ossRe ] [PubMed] Sus ainabili y 2020,12, 4378 17 o 17 41. Li, X.; Zell, A. Mo ion con ol o an omnidi ec ional mobile obo . In In o ma ics in Con ol, Au oma ion and Robo ics; Sp inge : Be lin/Heidelbe g, Ge many, 2009; pp. 181–193. 42. Kamel, M.A.; Zhang, Y. Decen alized leade - ollowe o ma ion con ol wi h obs acle a oidance o mul iple unicycle mobile obo s. In P oceedings o he 2015 IEEE 28 h Canadian Con e ence on Elec ical and Compu e Enginee ing (CCECE), Hali ax, NS, Canada, 3–6 May 2015; pp. 406–411. 43. Song, Z.; Ren, H.; Zhang, J.; Ge, S.S. Kinema ic analysis and mo ion con ol o wheeled mobile obo s in cylind ical wo kspaces. IEEE T ans. Au om. Sci. Eng. 2015,13, 1207–1214. [C ossRe ] 44. Raj, J.; Raghuwaiya, K.; Vanualailai, J.; Sha ma, B. Na iga ion o Ca -Like Robo s in Th ee-Dimensional Space. In P oceedings o he 2018 5 h Asia-Paci ic Wo ld Cong ess on Compu e Science and Enginee ing (APWC on CSE), Nadi, Fiji, 10–12 Decembe 2018; pp. 271–275. 45. Pa le, B.; Pandey, A.; Pa hi, D.; Jagadeesh, A. A e iew: On pa h planning s a egies o na iga ion o mobile obo . De . Technol. 2019,15, 582–606. [C ossRe ] 46. Ra hinam, S.; Manyam, S.G.; Zhang, Y. Nea -Op imal Pa h Planning o a Ca -Like Robo Visi ing a Se o Waypoin s Wi h Field o View Cons ain s. IEEE Robo . Au om. Le . 2019,4, 391–398. [C ossRe ] 47. Sun, Z.; Dai, L.; Liu, K.; Xia, Y.; Johansson, K.H. Robus MPC o acking cons ained unicycle obo s wi h addi i e dis u bances. Au oma ica 2018,90, 172–184. [C ossRe ] 48. Tahmasebinia, F.; Niemelä, M.; Eb ahimzadeh Sepasgoza , S.; Lai, T.; Su, W.; Reddy, K.; Shi owzhan, S.; Sepasgoza , S.; Ma oquin, F. Th ee-Dimensional P in ing Using Recycled High-Densi y Polye hylene: Technological Challenges and Fu u e Di ec ions o Cons uc ion. Buildings 2018,8, 165. [C ossRe ] 49. Tomé, A.; Vizo o, I.; Valença, J.; Júlio, E. Inno a i e Me hod o Au oma ic Shape Gene a ion and 3D P in ing o Reduced-Scale Models o Ul a-Thin Conc e e Shells. In as uc u es 2018,3, 5. [C ossRe ] 50. Bogue, R. 3D p in ing: The dawn o a new e a in manu ac u ing? Assem. Au om. 2013 ,33, 307–311. [C ossRe ] 51. Kie zmann, J.; Pi , L.; Be hon, P. Dis up ions, decisions, and des ina ions: En e he age o 3-D p in ing and addi i e manu ac u ing. Bus. Ho iz. 2015,58, 209–215. [C ossRe ] 52. Iza d, J.B.; Dubo , A.; He é, P.E.; Cabay, E.; Culla, D.; Rod iguez, M.; Ba ado, M. La ge-scale 3D p in ing wi h cable-d i en pa allel obo s. Cons . Robo . 2017,1, 69–76. [C ossRe ] 53. Uni e sal Robo s. 2020. A ailable online: h ps://www.uni e sal- obo s.com/es/ (accessed on 20 Ap il 2020). 54. Scaglia, G.; Mon oya, L.Q.; Mu , V.; di Sciascio, F. Nume ical me hods based con olle design o mobile obo s. Robo ica 2009,27, 269–279. [C ossRe ] 55. Scaglia, G.; Se ano, E.; Rosales, A.; Albe os, P. Linea in e pola ion based con olle design o ajec o y acking unde unce ain ies: Applica ion o mobile obo s. Con ol Eng. P ac . 2015 ,45, 123–132. [C ossRe ] 56. Deepyaman, M.; Ayan, A.; Mi hun, C.; Ami , K.; Ramdoss, J. Tuning PID and PI λ D µ con olle s using he in eg al ime absolu e e o c i e ia. In P oceedings o he 4 h In e na ional Con e ence on In o ma ion and Au oma ion o Sus ainabili y ICIAFS, Colombo, S i Lanka, 12–14 Decembe 2008; pp. 457–462. 57. Bos, F.P.; Ahmed, Z.Y.; Wol s, R.J.; Sale , T.A. 3D p in ing conc e e wi h ein o cemen . In High Tech Conc e e: Whe e Technology and Enginee ing Mee ; Sp inge : Cham, Swi ze land, 2018; pp. 2484–2493. c 2020 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h p://c ea i ecommons.o g/licenses/by/4.0/).