Self Contained

8/13/2019 Self Contained

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Abstract

Automation in Construction 13 (2004) 393 --407www.elsevier.com/locate/autcon

Self -contained automated construction deposition system

Robert L. Williams II a 9 , ames S. Albus b , Roger V. Bostelman b

a Department o Mechanical Engineering, Ohio University, 25 7 Stocker Centeel; Athens, OH 45701 -2979, USA'National Institute o Standards and Technologv, Gaithersburg. MO, USA

T h i s a rt ic le presents a novel autonomous sys tem concept for automated construction of houses and other buildings viadeposition o f concrete and similar materials. The overall system consists o f a novel cable-suspended mobility subsystem a self -contained extension o f the RoboCrane , a deposition nozzle subsystem, a metrology subsystem, and a material supplysubsystem. This article focuses mainly on the kinematics and statics analysis for control of the self-contained cable -suspendedmobility subsystem. We also present alternate design concepts for the mobility system. T h e purpose o f the Cartesian metrologysystem i s to provide an outer -loop controller to provide the required Cartesian pose motions despite uncertainties andunmodeled effects such as cable stretch, wear, and flexibility, plus wind loads.0 2004 Elsevier B.V. All rights reserved.

Keywonis: Automated construction; Deposition; Mobility; Metrology; RoboCrane; Cable -suspended; Self -contained; Forward pose kinematics

1. Introduction

Conventional construction cranes that can beseen at any construction site have the followingcharacteristics: nonrigid support; low payload - to -weight ratio (including counterweight); low resis -tance to wind; inaccurate control of loads; only usedto lift and coarsely position loads; limited remote,autonomous capabilities; workers are in a hazardousarea; and at any given location, only 1 degree offreedom (d i s controlled by the crane (Le., thelength o f the lift cable between the boom andobject); human workers are required with tag l inesto maintain th e lo ad 's remaining 5 degrees o f

- - _ _ _* Corresponding author. Tel.: +I-740 -593 -1096; fax: +I-74 0 -

E -mail address: [email protected] (R.L. Williams).URL: http://www.ent.ohiou.edu/ - bobw.

593 -0476.

freedom. This i s inefficient, humans have limitedstrength, and it i s dangerous.

To improve upon these undesirable character -istics, the RoboCrane was developed at NationalInstitute of Standards and Technology (NIST)[1,2,6]. The RoboCrane i s an inverted StewartPlatform wherein a moving platform i s controlledin 6 degrees o f freedom via six active cables andwinches. Not only can RoboCrane provide lift, butalso the remaining 5 degrees o f freedom are ac -tively controlled to be s t i f f and stable (over alimited range o f motion and orientations . Thisconcept was extended for a stiff, stable underwaterwork platform, wherein the platform may be co n -trolled to be stationary even if surrounding seas arcnot [4].

Inspired by the NIST RoboCrane, many research -ers have been involved with cable -suspended robots.A few o f these have focused on cable -suspended

0926 -5805/ - see front matter 0 2004 Elsevier B.V. All rights reserved.doi: 10.1016/j.autcon.2004.0I .001


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394 R.L. Ctilliams II et al. /Automation in Construction 13 2004 39 3 - 407

crane devices. Aria et al. [3] developed a 7 degreesof freedom, three -cable suspended crane - type robot

(the remaining freedoms are an moverhead gantry,plus top and bottom turntables) for an automobileassembly line, intended fo r he avy products assem -bly. Mikulas and Yang [8] present a three -cablecrane design for a lunar construction application,off -loading massive modules from a landing site,moving them, and constructing them into an oper -ational base. Viscomi et al. [ll] developed con -struction automation technology wherein Stewartplatform cranes (i.e., RoboCranes) are central. Shan -mugasundram and Moon [9] present a dynamicmodel o f a parallel link crane with positioningand orientation capabilities, with unilateral cable

constraints. Yamamoto et a]. [14] propose a crane -type parallel mechanism with three active cables forhandling heavy objects. Shiang et al. [IO] present aparallel four -cable positioning crane for offshoreloading and unloading o f cargo vessels under highsea states. A novel process for deposition o f mate-rial in construction applications i s under develop -ment by Khoshnevis [7]. Williams e t al. [13]present dynamics modeling and control for cable -based robots, ensuring only positive tensions duringal l motion.

The RoboCrane has great potential as an automatedconstruction robot system; however, its major draw-back i s that it requires rigid overhead cable supportpoints which may not exist at most construction sites.Therefore, this article introduces an economical, self -contained, movable -base construction crane for tele -operated andlor autonomous construction appli-cations. Compared to existing commercial an dproposed construction cranes, the system i s novelbecause it combines conventional rigid crane mem -bers with RoboCrane -type cable suspension and ac -tuation concepts to provide a rigid, lightweight, long -reach, overhead platform. This new concept has thepotential to address the shortcomings l is ted above forconventional c rane sys tems . Furthermore, th e self-contained design provides the required rigid overheadcable connection points. T h i s article first presents theoverall system concepts, and then mainly focuses onthe kinematics equations fo r control o f the self -contained mobility subsystem. We also consider qua -si-statics analysis to avoid configurations requiringnegative actuating cable tensions. Alternate design

concepts are also presented. Lastly, we discuss theproposed controller to ensure sufficient accuracy de -

spite real -world issues such as cable flexibility andwind forces.

2. Overall system description and application

An Automated Construction System concept i sunder development at NIST, based on self-containedextension of the RoboCrane. T h e overall systemconcept, shown constructing a building in Fig. 1,includes 4 major components (see Fig. 2): mobilitysystem (self-contained cable -suspended crane), ma -terial deposition sys tem (sl ip -form tool , metrology

system, and material supply system. The overallconcept o f automated construction o f buildingsusing free -form fabrication i s novel and integratesthese 4 systems into an advanced constructionsystem capable of manually or autonomously fabri-cating walls and structures while under manual orcomputer control, respectively. The metrology sys -tem ca n be a noncontact (e.g., laser -based) orcontact system (e.g., string - pot-based [ 121) thatmeasures the relative location o f the material depo -sition tool to a known location fo r accurate place -ment o f material from the deposition tool. Thematerial supply system feeds the deposition tool

with concrete or other material. I t ca n be a ce men ttruck or hopper, as shown in Fig. 2, with a pumpto move material to the tool. The material supplyand the metrology subsystems wi employ com -mercial products where possible. Fig. 3 shows anexperiment at NIST in (manual) deposition o fconcrete, building a portion o f a wall via a proto -type deposition nozzle with slip -form tool. Suchdeposition nozzles are also available commercially(e.g., www.curbequipment.com; www.basin -gallant.codconcretecurbsidewalk. html).

One good choice for the Cartesian metrologysystem is a serie s o f three noncontact lasers aimedat the crane deposition system (Fig. 2). This solutionprovides an accurate 6-dof(degrees of freedom) pose

T h e identification of any commercial product or trade namedoes not imply endorsement or recommendation by Ohio Universityor NET.


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R.L. Williams I/ t al. 1 Automation in Construction 13 (2004) 393407 395

mobility system (crane)

nd material deposition system

Fig. 1. Automated construction system.

measurement that i s independent o f the drive trainencoders for cable length feedback.

Now, we present an estimate o f the time requiredto build a nominal house [15.2 X 9.1 X 6.1 m 50 X 30 x20 fit>], via manual methods in this par -agraph, followed by the automated system in thefollowing paragraph. Typical o f todays constructionmethods are labor -intensive block and brick place -

ment with mortar joints. Assuming standard blockdimensions 0.2 X 0.2 X 0.4 m (8 X 8 X 16 in), therewill be 120 blocks per layer and 30 layers in th ehouse. Assuming 20 s for laying and mortaringeach block, 20 h i s required to lay the blocks forthe entire house. Then, assuming a postprocessstucco application time o f 32.3 s/m 2 (3 SIP), anadditional 2.7 h i s required for surface finishing.

Material S U D D ~ VI \ ,. System

MetrologySystem

Fig. 2. Automated construction subsystems.


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396 R.L. Wliams II et al. /Automation in Construcfion 13 2004 3 9 3 4 0 7

Fig. 3. NIST concrete deposition experiment.

Thus, we estimate almost 23 h (22 h, 40 min) i srequired fo r blocking the house via conventionalmanual methods.

Now, we estimate the time required for buildingan equivalent structure via concrete deposition us -ing the proposed automated construction systeminstead of manual block-laying. Current concreteslip -form technology e.g., www.curbmate.com)'enables large block-sized [cross section o f 0.13 X0.2 m (5 x 8 in)] deposition o f c on cre te at a rate o f4.9 m/min (16 fvmin). For the same nominal15.2 X 9.1 X 6.1 m house, using this depositioncross section, 48.8 m (160 ft) travel i s requiredper row, with 48 rows required. Therefore, the totaldeposition travel must be 2340.9 m (7680 ft), andthe total deposition time i s thus 8 h. Because theautomated syste m ca n be designed to apply thedesired finish as the walls are being constructed, noadditional time i s required for finishing. Thus,according to our est imates, the proposed automatedapproach will require just over one -third the time o fconventional methods. The automated system hasth e additional benefit o f littl e to no human super -vision required after setup.

N o t shown in our simple estimates are otherrequired processes such as reinforcement betweenlayers (one reinforcement process i s explained inRef. [7]). Also not shown in th e estimate are em -bedded processes that could be installed during thewall -build process, such as water piping and heatducts, plus electrical, phone, Internet, and otherutilities. With single block-sized layers, these utilitiescould be installed within the concrete layers; we ca n

also develop an autonomous dual -wall approach forthis.

3. Mobility system

This section presents the description, kinematics,and statics for the self -contained cable -suspendedrobot o f Fig. 1. This i s the mobility subsystem o four overall automated construction system concept.

3.I. Mobility system description

Fig. 4 shows the NIST self-contained mobilitysubsystem concept. This robot i s intended to be a

versatile, economic, accurate tool for the const ruc -tion industry. The system i s supported by a standardconstruction site dumpster, fitted with momentresisting support rods to resist tipping; the dumpsteri s rotated by a small angle 61, to move the tippingpoint forward from the front edge o f th e dumpster.

In Fig. 4, the fixed base f rame has fixed cableconnection points B1, B2, B3, and B 4. The movingmast (boom ClBo plus hinged equilateral triangleC 2C 3C 4) i s connected to the base dumpster via auniversal joint (allowing pitching and yawing) at Bo.The moving mast i s articulated via cables o f leng thsLB 1 and LB 2, whose active actuating winches aremounted at points B 1 and Bz. Therefore, vertical cablesupport point C 1 can move to increase the systemworkspace in a self-contained manner. Points B1 and

G

c2

M

Fig. 4. NIST automated construction system diagram.


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R.L. CtTlliams II et al. /Automation in Construction 13 2004 393 -407 39 1

B2 are assumed to l i e on th e right and lef t dumpstersides, but ca n be mounted anywhere along these sides,and above the top o f the dumpster as shown. Twopassive, fixed -length cables support th e equilateraltriangle, attached from fixed point B 4 over pulleys atpoint C 1 to moving points C2 and C3. The vertices o fthe moving platform ar e P1, P2, and P3, and point P i sthe centroid o f the moving platform. The depositionnozzle tip N is located at the origin o f moving frame{N>. Because RoboCrane -type devices have limitedrotations, the entire nozzle i s rotated via a turntableattached to the moving platform, with rotary variable0 T h e world coordinate frame (0) i s aligned withthe floor at the back edge o f the dumpster as shown;this fra me i s really hidden in the view of Fig. 4.

The lengths o f the nine additional active cables areL,, i = 1,2,. . .,9. A s shown in Fig. 4, cable 1 connects C 1to P,, cable 2 connects C I to P 2, cable 3 connects C 1 to4 , cable 4 connects C2 to P 1, cable 5 connects C to P3,cable 6 connects C to P 3, cable 7 connects C 3 to P 2,cable 8 connects C 4 to P I , and cable 9 connects C 4 toP 2. Cable lengths 1, 2, and 3 are controlled by activewinches mounted at B3; therefore, th e line B3C1 i sactually three (articulating because C I ca n move)cables, passing from each winch over point C 1 tomoving platform point P i i = 1,2,3; these three are theheavy -lift cables. Cables 4 and 5 are controlled byactive winches mounted at C 2, cables 6 and 7 are

controlled by active winches mounted at C and cables8 and 9 are controlled by active winches mounted at CCables 4 - 9 provide stable, rigid control o f a l l 6 degreeso f freedom in conjunction with the heavy - lift cables.

The concept o f Fig. 4 can be seen as an irregularRoboCrane with moving, self -contained, vertical sup -port points CI , C2, C 3, and C 4, controlled by cables 1through 9. T h i s RoboCrane, however, i s overactuated(three more cables than the minimum number o f sixcables for a ceiling -mounted RoboCrane and 6-dfoperation). Points C2 and C allow the moving plat -form workspace to extend beyond vertical point C 1.To maintain control in al l motions, all cable tensionsmust remain positive at a l l times.

In order to provide a more stable standard construc -tion site dumpster base, we tilt the base by a smallangle 4 B about the X axis on the back bottom comero f the dumpster (see Fig. 4). This moves the tippingpoint o f the system from the front bottom comer o f thedumpster to the end o f the moment resisting rods o f

length d As shown in Fig. 5, the mechanism analogyfor this dumpster tipping i s a slider - crank mechanism,where the crank d

B3 pivots about X th e coupler i s dM

(hinged at the back top corner o f the dumpster), and thefoot pad (spreading out the weight over the ground) i sthe slider, connected to dM with a pin joint. From thisslider - crank analogy, given the desired dumpstertilting angle $B, we can calculate the variables I Mand yM (same for both dumpster sides at different Xlocations .

The dumpster tilting in Fig. 5 could be actuated bycables and winches, one set on either side o f thedumpster, where th e motor and winch i s mounted tothe dumpster and the other en d o f the cable i smounted to the coupler dM (or vice versa).

3.2. Mast subsystem kinematics

This section presents the kinematics analysis forthe mast portion o f the NIST Automated ConstructionSystem. We wish the equilateral triangle to be ashorizontal as possible for a ll motion.

T h e role o f the mast i s to provide self -containedmobility for the RoboCrane -like portion o f the auto -mated construction system (see Fig. 4). As shown inFig. 6, the mast consists of equilateral triangle C2C 3C 4hinged via a revolute joint to boom C I Bo at point C 4 .

The ke y aspect o f this mast concept is that theconfiguration o f the equilateral triangle portion i smaintained by tw o fixed -length, passive cables. Bothcables are fixed to the dumpster at point B 4, pass overpulleys at point C,, and are fixed to moving equilateraltriangle points C 2 and C3. As cables L B ~and LB.2 move

tY M

Fig. 5. Slider - crank analogy for dumpster tipping.


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398 R.L. Williams et al . /Automation in Construction 13 (2004) 393 -407

Cl

Fig. 6. Mobility system mast.

point C1, these passive cables move and support theequilateral triangle portion passively, which in turnsupports six o f the RoboCrane -like cables for movingthe automated construction platform. At any instantduring motion, the two passive cables can be seen astwo lengths on either side o f the pulley, L C betweenpoint C1 and points C and C respectively, and 1,between moving point C, and fixed point B4. With thisdesign, having a revolute joint at C (whose axis i saligned with X 0 and X in the nominal configurationwhen the boom i s in th e YOZ, plane , both cableportions LC are guaranteed to be theoretically thesame length for al l motion, which ensures that th e 2components o f C and C ar e always the same (al-though different from the Z component o f C4 ingeneral). T h i s passive motion control for the equilat -eral triangle portion o f the mast enables a pantograph -like motion. L C and 1 both change during motion, buttheir sum i s constant, se t by design to keep theequilateral triangle as horizontal as possible duringall motion.

Fig. 7 shows a kinematic diagram o f the mast,connected to the dumpster frame at point Bo via auniversal joint allowing yaw (6,) and pitch e 2).This reference position defines both angles to bezero. The mast ca n be considered to be a 3R serial

robot connected to the dumpster with joint angles 61and d2, plus angle 63 moving the triangle withrespect to the boom. 8,, 02, and O3 are not controlleddirectly but via active cables L B ~ and LB2, andpassive cables Lcfl , . Th e origin o f frame {Bo} i smounted to point Bo; the orientation of {BO} i sidentical to that o f {B} (which i s the same as {0}but rotated by 4s).

As seen in Fig. 7, dl i s the length o f boom C IBO, d2i s the length from boom base point Bo to th e equilat-eral triangle connection point C4, and d3 i s the

Fig. 7. Mast kinematic diagram.


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Table IMast Denavit - Hartenbm parameters

R.L. WiNiams et al. /Automation in Construction 13 (2004) 3 9 3 4 0 7

1 0 0 0 9, + 90 2 90 0 0 923 0 dz 0 93

equilateral triangle side (with height h3). Movingpoint C 5 i s the midpoint o f C2C3.

Th e pitch angle 82 should be kept well away fromthe ground position because this approaches a singu -larty where cables LBl and L B ~ re collinear with themast; in this singularity, infinite force would berequired to move the mast (due to the system design,the singularity actually occurs underground, but highforces are required as the mast approaches theground). In setup, the boom will most likely be liftedby these cables from th e ground; thus, this i s anotherreason to mount points B I and B 2 above the dumpstertop by h B (this effectively moves the singularityfurther from the ground).

The Denavit - Hartenberg (DH) parameters [5] forthis serial robot are given in Table 1. Note that a jointangle offset o f 90 i s required for i= 1 (i.e., dl + 90 )because the X and X, axes are not aligned in thezero position. Note also that, in the convention o f Ref.[5], the only mast length parameter to appear in the

DH table i s d 2 because the last active frame i s {3},centered at C 4 . The other pertinent lengths must beincluded at th e next stage o f mast kinematics. In Table1, Oi are the only variables, while the remaining DHparameters are constant.

399

The homogeneous transformation matrices relatingframes {C,}, i=2,3,4,5 whose origins are points C,and whose orientation i s identical to that o f {3}), tothe world frame {Bo} can be found using symboliccomputer kinematics from the DH parameters andhomogeneous transformation relationships.

Note that in Eq. (2), we have taken advantage o fthe consecutive parallel and .23 axes; in such cases,we expect functions o f (82+d3) to simplify thekinematics equations via sum -of-angle formulas. Th elast transforms T for use in Eq. (2) are obtained byusing identity for the orientation and the constant

relative position vectors {3C j }. Substituting the DH

parameters and the constant transforms into Eq. (2)yields:

0 0 1

where we have used the abbreviations ci = cos0,and si = sine,; furthermore, ~ 2 3 cos(8, + 8 and ~ 2 3sin(0 2+e 3). The formula for i s the fourthcolumn, first three rows of Eq. (3). Other positionvectors are:


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400 R.L. Williams II e t al. 1 Automation in Construction I 3 (2004) 3 9 3 4 0 7

The orientation associated with i s not depen -dent on 03:

P I

C 2

c'1

Now, given values for 01, 02, and 03, it i s easy toevaluate the abso lu te position o f moving points Ciwith respect to {Bo} using the above formulas.Ultimately, al l vectors wil l be represented in th e (0)frame using 2T =gT$T. However, these serialangular values 01, 02, and O3 will not be knownbecause i t would increase cost and complexity unnec -essarily to add angle sensing to the passive universaljoint at Bo and passive revolute joint at C Instead,we have two choices.

(1) For inverse pose kinematics, the upper mastpoint C 1 i s specified at each instant (i t ca n bemoving). It i s convenient to specify C1 via angles8' and 0 (because CI i s constrained by the lengthd l ), using the first expression o f (4). If we wish tospecify th e pitch angle as an absolute (horizontallyreferenced) angle, we need to calculate f i rs t therelative pitch angle using th e dumpster tilting angularoffset: O2 = OzAs s - 4n. Then, we can easily calculatethe two required cable lengths L B I and L Bz using theEuclidean norm o f the appropriate vector differencesas given below:

(2) For forward pose kinematics, th e two cablelengths, L B ~and L B ~ ,are known from their winchangular feedback measurements. Upper mast pointCI i s calculated given these tw o cable lengths. FromFigs. 4 and 6, point C, i s the intersection o f threespheres: fixed mast radius d l centered at Bo, radiusLE 1 centered at B,, and radius Lm centered at B2.T h e intersection o f three spheres i s aIso th e basis forthe forward pose kinematics solution o f the nine -cable RoboCrane -like device. T h i s solution i s pre -sented in Ref. [12]. OC 1 is found from the intersec -tion o f three spheres with these center and radii: oB1,L s l ), eB2,L B 2), and (OB 0,dl). Note that this

ordering o f the three spheres i s very important toavoid singularities [12]. Given 'C I, we next calculatethis vector with respect to {BO}: 'OC =[(&I - 'PC I ;then, we can ca lcu la te passive universal joint anglesOl and e2 from an inverse position kinematics solutiono f the expression for {E oC l }={P P y P,} in Eq. (4):

Note that the resulting O2 in Eq. (7) i s a relative anglewith respect to {Bo}; the absolute (horizontallyreferenced) pitch angle must take dumpster tiltingangle 4n into account: = 82 + +B .

Now, we can determine 03; i t i s done in the samemanner for both of the above cases. Note that 0 3 i sdefined to be a relative angle (with respect to boomC I Bo) and hence requires no offset like 02. Fig. 8 showsa side view o f the mast arrangement. T h i s side views ho ws a planar representation o f the passive equilateraltriangle pantograph cables; 11 = C 1B4 11are the realportions o f th e tw o passive pantograph cables, and12= 1C I C I i s a virtual variable cable representing theplanar projections o f th e cables' portions L C (see Fig. 6and visualize th e plane ClC2C,; 1 bisects this triangle):12 = d m . Then, using th e la w o cosines:

O3 i s negative in Eq. (8) due to it s definition in Figs. 7and 8. This pantograph mechanism i s designed toattempt to maintain the equilateral triangle as near

Fig. 8. Mast s ide view.


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R.L. Williams II et al. /Automation in Construction I 3 2004 393 -407 401

25

20

15

N

10

5

Y

Fig. 9. Passive horizontal mechanism demonstration.

horizontal as possible for al l motion. I t can be exactonly at one 02 angle, but we wish i t to be close at al lother configurations. I f the triangle i s perfectly hori-zontal, th e following condition i s met: O3 = - &ABS.

Now, we s t i l l need to calcu la te the (actual) cablelengths L C for 03 determination. The two passivepantograph cables (running from B4, over pulleys atC 1, connecting to points C2 and C 3) are o f fixed

length, L = I I +L C . Therefore, L C =L - ZI . L etus fix L by design, requiring the equilateral trian -gle to be exactly horizontal (0, = - &ABS) at a nom-inal value o f th e absolute pitch angle B~ABS,~,,,) andfor the central value o f the yaw angle, 8=0&ABS n o m should be in the middle o f the allowable&ABS range, or some other nominal, often -used confi -guration. At the nominal configuration, we have L =llnorn +&nom, where 11nom = IICl n o d 4 II. We also haveLC, = J(lz ,,, 4 /4; the nominal virtual panto -graph cable length i s 1 = )ICIno m C5 n o m 11The nom -inal locations o f C I and C5 are found by substituting8, = 0 and relative angle tJZn om =02 AB Sn om- bB into thefirst and last expressions o f Eq. (4):

{ Clnom} = dlC2nam IB a

C ~ m m } = d2~2nom +h d ~1IIs?norn 1 I l 2 s z n o r n - h B } (9)

Finally, given 01, e and 0 from Eqs. (7) and a), we can calculate the position vec tors for pointsCi from Eq. (4), fo r general configurations; we canthen transform these to (0).

Our current mobility system parameters are (in m)

off, = 0.6096, and 4s = 3 O. We assume a standard6.096 X 2.438 X 2.438 base dumpster; points B 1 andB 2 are at the front of the dumpster, mounted hs = 2.438from the dumpster top. Fig. 9 shows a series o f mastmotions in th e Y o plane for our mast design.

Fig. 10 shows the horizontality results for our mastdesign, over al l motion. Mast pitch angle &ABS i s theindependent variable, while families o f curves aregiven fo r different 0, values O D, 15 , 30 , 45 ); in

this manner, one plot covers all motion. Note that in allresults (including statics later), th e motion is symmet -r i c with respect to f el .

Fig. 10 shows the - 0 results (negative for easycomparison to the mast pitch angle e2) for all motion.For horizontality, we desire O3 = - OZAB S , which i s th edashed (Ideal) line in Fig. 10. We can see that this i ssatisfied (theoretically) only at = - &ABSnom =48 , for el = 0. Away from this condition, the - 0 re -sults deviate significantly from th e desired dashed line.

Fig. I 1a shows th e Yo workspace and Fig. 1Ibshows it s associated C2C3 Zo heights, for our mast

dMz7.620, dl =24.384, d 18.999, d3=6.096, off y =

70 I I

Fig. 10. Mast angle 03.


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402 R.L. Williams II et a /Automation in Construction 13 (2004) 393407

design for a ll motion. In analyses with different mastdesigns, we discovered that good horizontality i sassociated with poor Yo workspace and vice versa,demonstrating tradeoffs between performance meas -ures in mast design.

3.3. Mobility system kinematics

The inverse pose kinematics problem i s stated:given th e required nozzle tip pose and the desiredposition o f upper mast point OC 1, calculate the 11 cable

a20

15

10

5

P 0k

-5

-1 0

-15

-2030

b20

E

z.g 10N

5

~ ......... ...............

20 0 -10

030 40 50 60 70

ea E S fdes)

Fig. 11. (a) Mast Y o workspace projectlon. @) Associated Zoheights.

lengths Li , i= 1,2,. . .,9 and L B l and Lm and 6 Thesolution to this problem may be used as the basis for apose control scheme. For the automated constructionsystem, inverse pose kinematics i s easier to solve thanforward pose kinematics: given the pose o f the nozzletip (N) tool, we first specify 0 according to deposi -tion task requirements. Then, we ca n find movingplatform cable connection points P I , P2, and P3; then,the inverse pose solution consists simply of calculatingthe cable lengths using the Euclidean norm o f theappropriate vector differences between the variousmoving and fixed cable connection points. The inversepose kinematics solution yields a unique closed -formsolution, and the computation requirements are notdemanding.

As presented in Section 3.2, the first steps in th einverse pose kinematics solution are to specify C 1 viaangles O1 and 0 using Eq. (4), calculate LB 1 and L B ~using Eq. (6), calculate O3 from Eq. (8), and thencalculate the remaining moving cable -connectionpoints Ci using Eqs. (3) and (4). With an alternatetelescoping boom design, i t i s possible to easilyspecify CI directly and then calculate angles 81 and 02.

Given [T and ON we then calculate the movingplatform pose [$TI and then moving cable connectionpoints P I, P z, and P 3: $TI = gT]gT] - I, where LTi s a function o f 0~ and the nozzle position with respectto the moving platform. Th e vector positions o f points

P i with respect to (0) are:

,2,3

Note that we must augment each position vector in(1 .O) with a '1 ' in the fourth row. T h e fixed relativevectors {Pi} are from platform geometry. Given themoving cable connection points PI, P2, and P3 in {0}from Eq. (lo), we can find thc nine unknown cablelengths. The inverse pose kinematics solution i s theEuclidean norm o f the appropriate vector differencesas shown below:

L, = IIOPI - 0c1 II L 2 = I1OP 2 - O c1 II LJ IIOP3 - O c, II

L4 = I1OPI - 0cz II L.5 = IIOP - 0 cz II L g = IIOPJ O c, II

L, = l1P2 - 0c3 11 L8 = I1OP - 0 c, II L y = IIO P Z - O c, I/(11)

The forward pose kinematics solution i s required forsimulation and sensor -based control o f the NIST Au -


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tomated Construction Mobility System. The forwardpose kinematics problem i s stated: given the 11 cablelengths Li ,2,. . .,9, andL B I , LE, and ON, calculatethe nozzle tip pose i.$Tl. For this system, forward posekinematics i s not as straightforward as inverse posekinematics. However, unlike most parallel robot for -ward pose kinematics problems, there exists a closed -form solution, and the computation requirements arenot demanding. There are multiple solutions, but,generally, the correct solution for the automated con -struction system can be easily determined.

As presented in Section 3.2, th e f ir st steps in theforward pose kinematics solution are to calculate C 1given LB1 and L B Z using the intersection o f threespheres, calculate the passive universal joint angles

el and O2 from Eq. (7), calculate O3 b m Eq. (S), andthen calculate the remaining moving cable -connectionpoints Ci using Eqs. (3) and (4). The remainingforward pose kinematics solution consists of findingthe intersection point o f three given spheres; this mustbe done three additional times in the following se -quence, one for each moving platform cable connec -tion point P i Let us refer to a sphere as a vector centerpoint c and scalar radius r: (c,r).

1. P I i s found from the intersection of: (OC2,L4),

2. P2 is found from the intersection of: eP l ,d p ),

3. P3 i s found from th e intersection of: (P 1,d p ),

ec4cfJ 9 (oclcl>.

(OC4&9), e W 2 ) .

(OP2,4), ec1 Cd.

The detailed solution for the intersection o f threespheres i s presented in Ref. [12]; that reference alsopresents discussions on imaginary solutions, singular-ities, and multiple solutions. Now, le t us finish theforward pose kinematics solution. Given Pi we ca ncalculate the orthonormal rotation matrix ipw directly,using the definition that each column o f th is matrixexpresses one o f the XY Z unit vectors o f {P) withrespect to (0) [SI. These columns are calculated as

follows, from moving platform geometry.

where O P 4 i s the midpoint of P I P 2. Given P i andIjRl, we then have [:TI, and ~?l]=$,TI$,Tl - .Finally, use \.$TI= $TJ1,$TJ to calculate the nozzletip pose.

There are two solutions to the intersection point o fthree given spheres [121; therefore, the forward posekinematics problem yields a total o f 2 4 = 16 mathe -matical solutions because we must repeat the algorithmfour times fo r the NIST Automated ConstructionMobility System. I t i s generally straightforward todetermine the correct solution using logic in theforward pose kinematics software. Fig. 12 shows anominal pose for our mobility system design Matlabsimulation.

3.4. Mast subsystem statics

This section discusses the model for quasi -statictension -based control o f the cable -suspended con -struction system. We first consider mast -moving ten -sions, then the overall mast statics model, followed bysimulation results.

3.4.1. Mast -moving tensionsNow, we consider a c ru cia l issue for moving the

mast. If the yaw angle O1 i s commanded to a value thati s very large, one o f the mast -moving cables willrequire an impossible pushing force. Fig. 13 shows

th e top view o f these mast-moving cables plus boom.When the XoYo projection o f mast BoC l becomes

0

Fig. 12. Mobility system Matlab model.


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404 R.L. Williams II et al. /Automation in Construction 13 (2004) 393407

collinear with th e XoYo projection o f cable L B I , wehave reached the positive limit on 8, (by symmetry,th e equal, negative lmit on 8, occurs when BoC l i scollinear with LB 2). We can calculate OILIM IT asfollows:

f2 i s the fraction along the Y direction where pointsB, and B 2 are mounted to the dumpster. Fig. 14 showsOILIMIT for fractions from 0 to 1 and differentdumpster tipping angle values dB = 1 O, 3 , 5 O, 7 O .

we see that f 6lLrMrT increases (which i s good)with increasingh in all cases; hrthermore, f 81LIMITdecreases (which i s bad) with increasing in al lcases. Any specific design i s a single point on Fig.14; the plots verify that, for large limits o n el, wemust move the points B, and B 2 as far forward aspossible cf 2 = 1). However, this ca use s a loss o f halfthe moment arm for lifting the mast (compared to

=O); this i s why we also raised B 1 and B2 off thedumpster in the Z direction an additional hB amount,to recover the original moment arms fo r cables LB I andLB2. All motion should be kept well away from thespecific f O I LIMIT fo r any given design, to safelyavoid the slack cable problem and the resulting cata -strophic loss o f control. For our design with @B= 3 ,the theoretical O1 limit i s 5 74.7 .

Fig. 13. &LI M IT determination.

90

80

50

400.2 0.4 0.6 0.8 1

f2 (fraction of full d s2)

3.4.2. Mast statics modelThe mast statics problem i s stated: given external

loads at points Ci ,2,3,4, plus the system config -uration, calculate the tensions in a ll cables. Internaljoint forces between rigid members are also unknown.We assume all crane members are weightless and al lcables are in tension. Now, we outline our general 3Dmast statics solution. For each o f the free bodydiagrams (not shown due to lack o f space) o f theequilateral triangle C2C 3C 4, pantograph pulley, and

boom B oC l , we ca n write two 3D vector equations o fstatic equilibrium: CF = 0 and CM=O. Therefore, wehave a total o f six scalar equations times three movingbodies, for 18 equations. However, we only have 12unknowns, scalar cable tensions tc (the same for bothpassive pantograph cables), ts l , and tB2 (the activemast -moving cable tensions), plus three 3D vectorinternal force unknowns, between the equilateral tri-angle and boom, between the pulley and boom, andbetween the boom and base dumpster. Thus, fo rsolution, we ignore six o f the statics equations; if wefollow the method now described, th e unknowns maybe found member by member. Firstly, for the equilat -

eral triangle, use only the CM, = 0 scalar equation, in13) coordinates; this equation yields th e unknown tc ,the tension in both pantograph cables C1 C and C 1 CThen, we use al l three force components in the C F = 0vector equation to find the internal force o f the boomacting on the equilateral triangle at the pin joint locatedat C Next, using the pantograph pulley free body


13/15


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406 R.L. Wi[liams II et al. /Automation in Conshuction 13 2004 393 -407

, .

Fig. 16. Alternate NIST self -contained mobility system.

-. r

C 4. I t is theoretically possible in this case to ensurethat plane C 2C 3C 4 i s always horizontal in the worldframe. Th e ji b i s a rigid link C,C4 that i s hinged viarevolute joint at point Cl. The single heavy lift cableconnects to the centroid P o f the moving platform,runs over a pulley (not shown) at point C,, and i sactuated by a heavy lift motor a t point B 3. Theprimary mast C tBo telescopes; the end member

C,C&3 i s a rigid cross -shaped member.For al l designs, we can use the same base dumpster

with tilting, the same mast -moving cables, and thesame moving platform with turntable, plus nozzle ortools.

Th e role o f the Fig. 16 mast i s again to provideself-contained mobility for the RoboCrane -like por-tion o f th e automated construction mobility robot. Jibtip point C i s actively controlled by a variable cablelength L ji b = I I I 2 onnecting C 4 to B 4 over a pulley,controlled to ensure that virtual isosceles triangleC&C, i s horizontal for all motion.

5. Proposed con t ro l l e r concept

To satisfy commanded Cartesian trajectories, wepropose the following controller. Given a series o fcommanded Cartesian poses, we u se t he inverse posekinematics solution o f Section 3.3 to calculate the

required cable lengths (all mast and moving platformcables) at each control step. Each o f these commandedcable lengths will be achieved at a high control updaterate (say 1000 Hz) via the motors and cable reels, withjoint encoders and a rotary - to - linear mapping foractual cable length feedback.

According to NIST RoboCrane hardware experi -ence, cable length sensing using encoders in the loadpath does not yield sufficient accuracy fo r the construc -tion task. Therefore, we also propose a Cartesianmetrology system (a noncontact laser -based 6 - 4 sys -tem independent o f the robot) to provide an outer-loopcontroller that can run slower (say 100 Hz) in order toprovide a servo to reduce errors in th e Cartesian pose,due to real -world issues such as modeling uncertainties,cable stretch, wear, and flexibility, plus wind loads.

In future work, we will implement this controller,along with the equations o f th is article, in a Matlab/Simulink simulation, to determine a baseline control -le r design for real -world applications. We will alsomodel cable flexibility and simulate real -world dis-turbances, such as wind loads, to test th e robustnesso f the proposed controller. However, due to NISThardware implementation and testing experience, webelieve that our proposed controller with independentCartesian metrology -based servo to task accuracy willbe s uf fic ie nt , e ve n in nonlaboratory constructionenvironments.

6. Conclusion

This article has presented tw o alternate designconcepts for a novel automated construction systembased on material deposition. We focus mainly on theself-contained, cable -suspended mobility system. TheNIST RoboCrane has been developed as a stiff, stablecrane device that controls a ll 6 degrees o f freedom ofthe load. However, the standard RoboCrane requiresrigid overhead support points for the six cables. Thiswork is an attempt to extend the RoboCrane to a

mobile, self -contained cable-suspended crane thatprovides i t s own rigid overhead cable support points.

We presented the overall system concept, and thenderived kinematics equations for control o f th e cable -suspended mobility subsystem. We also consideredstatics analysis for the mobility system mast, andidentified and calculated motion limits for avoiding


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R.L. Williams N e t al. /Automation in Construction 13 (2004) 393407 407

negative cable tensions during operation. We pre -sented alternate mobility system concepts for the

self-contained RoboCrane.The Cartesian metrology system provides a meansto achieve Cartesian trajectories in the face o f uncer -tainties and unmodeled effects such as cable stretch,wear, and flexibility, plus wind loads. Industrialrobots solve this problem by being bulky, stiff, andheavy, with large motors and low payload to weightratios. Existing construction crane systems have s t i f fbooms but also employ swinging cables that do notconstrain al l 6 degrees o f freedom. The concept ofthis article provides a lightweight system with cable -suspended actuation that ca n provide s t i fke s s in a ll 6degrees o f freedom. The metrology system will

enable accurate control despite real -world uncertain -ties and disturbances.

[41

[71

r93

Acknowledgements

The first author gratefully acknowledges supportfor this work from the NIST Intelligent SystemsDivision, via Grant #70NANl32H0130.

References

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