Parasitic Inclinations in Cable-Driven Parallel Robots using Cable Loops

Parasitic Inclinations in Cable-Driven Parallel Robots using Cable Loops Saman Lessanibahri, Philippe Cardou, Stéphane Caro

To cite this version: Saman Lessanibahri, Philippe Cardou, Stéphane Caro. Parasitic Inclinations in Cable-Driven Parallel Robots using Cable Loops. Procedia CIRP, ELSEVIER, 2018, 70, pp.296 - 301. .

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Procedia CIRP 00 (2018) 000–000 Procedia CIRP 00 (2017) 000–000 Procedia CIRP 70 (2018) 296–301 Procedia CIRP 00 (2018) 000–000

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28th CIRP Design Conference, May 2018, Nantes, France 28th CIRP Design Conference, May 2018, Nantes, France

Parasitic Inclinations in Cable-Driven Parallel Robots using Cable Loops Parasitic Inclinations in Cable-Driven Parallel Robots using Cable Loops 28th CIRP Design Conference, May 2018, Nantes, France Saman Lessanibahriaa , Philippe Cardoubb, St´ephane Carocc Lessanibahri , Philippe Cardou , St´ephane Caro a Centrale Nantes, Saman Laboratoire des Sciences du Num´erique de Nantes, UMR CNRS 6004, 1, rue de la No¨e, 44321 Nantes, France

A new methodology functional and physical architecture of Laboratoire deto robotique, D´epartement de g´ede nie m´ecanique, e Laval, Qu´ ebec, QC, Canada Centrale Nantes, Laboratoire des analyze Sciences du Num´ethe rique Nantes, UMRUniversit´ CNRS 6004, 1, rue de la No¨ e, 44321 Nantes, France CNRS, Laboratoire Num´ erique de CNRS 6004, 1,e rue de la No¨ e, 44321 Nantes, France Laboratoiredes de Sciences robotique,duD´ epartement deNantes, g´enie m´UMR ecanique, Universit´ Laval, Qu´ ebec, QC, Canada existing products for an assembly oriented product family identification Laboratoire des Sciences du Num´erique deE-mail Nantes, UMR CNRS 6004, 1, rue de la No¨e, 44321 Nantes, France Corresponding author.CNRS, Tel.: +33-240-376-925; fax: +33-240-376-925. address: [email protected] b

a

c

∗ ∗

b

c

Corresponding author. Tel.: +33-240-376-925; fax: +33-240-376-925. E-mail address: [email protected]

Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat

Abstract École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France Abstract Cable-Driven Parallel Robots (CDPRs) also noted as wire-driven robots are parallel manipulators with flexible cables instead of rigid links. *Cable-Driven Corresponding author. +33frame, 3(CDPRs) 87 37a 54 30; E-mail address: [email protected] Parallel also noted as wire-driven parallel manipulators withthe flexible instead of CDPRs rigid links. A CDPR consists in Tel.: a Robots base Moving-Platform (MP) and a robots set of are cables connecting in parallel MP tocables the base frame. are A CDPR consists in aadvantages base frame,over a Moving-Platform (MP)robots and a in setterms of cables connecting in parallel the MP to the base frame.capacity CDPRs and are well-known for their the classical parallel of large workspace, reconfigurability, large payload well-known their advantages overofthe paralleladvantages, robots in terms large reconfigurability, payload and high dynamicforperformance. In spite all classical the mentioned one ofofthe mainworkspace, shortcomings of the CDPRslarge is their limitedcapacity orientation high dynamic In spite of alldue thetomentioned advantages, of the between main shortcomings of the CDPRs is their limited orientation workspace. Theperformance. latter drawback is mainly cable interferences andone collisions cables and surrounding environment. Hence, a planar Abstract workspace. The latter drawback mainly due to cable interferences and collisions cablesand and studied surrounding environment. a planar four-Degree-of-Freedom (DoF) is under-constrained CDPR with an articulated MP between is introduced in this paper. TheHence, end-effector is four-Degree-of-Freedom CDPRtowith anaarticulated MPdetermination, is introduced namely and studied in thisand paper. The end-effector is articulated through a cable(DoF) loop, under-constrained which enables the robot obtain modular pose orientation positioning. As a result, Inarticulated today’s business environment, theunlimited trend towards more to product andpose customization is unbroken. Due translational to thisand development, the of through a cable which enables thesingularity-free robot obtain variety a modular determination, namely orientation positioning. AsItneed a should result, the mechanism under studyloop, has an and orientation workspace in addition to a large workspace. agile and reconfigurable production systems emerged cope with various products andinproduct To design production the mechanism under study has an unlimited andofsingularity-free orientation workspace additionfamilies. to aarise large translational workspace. It should be noted that some unwanted rotational motions theto moving platform, namely, parasitic inclinations, due to the and cableoptimize loop. Finally, those systems as well as to choose the optimal product matches, product analysis methods are needed. Indeed, most of the known methods aim to be noted inclinations that some unwanted rotational motionsfor of the the mechanism moving platform, namely, parasitic inclinations, arise due to the cable loop. Finally, those parasitic are modeled and assessed at hand. analyze a product or one product family on the physical level. Different product families, however, may differ largely in terms of the number and parasitic inclinations are modeled and assessed for the mechanism at hand. nature of The components. This fact by impedes an B.V. efficient comparison and choice of appropriate product family combinations for the production c 2018 � Authors. Published Elsevier © 2018 A The Authors. Publishedisbyproposed Elsevier B.V. system. new methodology to analyze existing of products inCIRP view of their Conference functional and physical architecture. The aim is to cluster c � 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee theCIRP 28th Design 2018. Peer-review under responsibility of the scientific committee of the 28th Design Conference 2018. these productsunder in new assembly oriented product families for of thethe optimization existing assembly2018. lines and the creation of future reconfigurable Peer-review responsibility of the scientific committee 28th CIRPof Design Conference Keywords: Cable-Driven Robots, Cable Loop, Large Workspace; assembly systems. Based Parallel on Datum Flow Chain, theKinematics, physical structure of the products is analyzed. Functional subassemblies are identified, and Cable-Driven Parallel Robots, Cable Loop, Kinematics, Large and Workspace; aKeywords: functional analysis is performed. Moreover, a hybrid functional physical architecture graph (HyFPAG) is the output which depicts the similarity between product families by providing design support to both, production system planners and product designers. An illustrative example of a nail-clipper is used to explain the proposed methodology. An tensions industrialgenerated case study by on two families steering two product actuators. The of cable loopcolumns allows of us 1. Introduction thyssenkrupp Presta France is then carried out to give a first industrial evaluation of the proposed approach. tensions by twoworkspace actuators.ofThe loop allows us 1. Introduction to enlargegenerated the orientation the cable manipulator at hand, © 2017 The Authors. Published by Elsevier B.V. to enlarge the orientation workspace ofassociated the manipulator at hand, but there exists an undesired rotation to the rotation One advantage of cable-driven parallel robots that has not Peer-review under responsibility of the scientific committee of the 28th CIRP Design Conference 2018.

of cable-driven parallel of robots not yetOne beenadvantage fully exploited is the possibility usingthat thehas cables yet been fully exploited is the possibility of using the cables to transmit power directly from motor fixed to the frame to the Keywords: Assembly; Design method; Family identification to transmit power This directly from motor to the frame to the moving platform. power can then fixed be used to actuate a tool moving platform. This power canof then be usedsuch to actuate a tool or to control additional degrees freedom as rotations or towide control additional of In freedom such we as study rotations over ranges [14], fordegrees example. this article, the 1.over Introduction wideexpression ranges [14], example. In thiscircuit, article,namely, we studya cathe simplest of for a bi-actuated cable simplest of arotations bi-actuated cable circuit, cable loop, expression for controlling of the CDPR endnamely, effectoraover Due to the fast making development inloopthe of ble loop, for controlling rotations ofcable the CDPR end domain effector wide ranges. The cable the connects twoover accommunication and anmaking ongoing trend of CDPR digitization and wide ranges. Thetwo cable thelocated cable loop connects actuators through fixed pulleys at exit two points, tuators two pulleys located at attached CDPR points, digitalization, manufacturing enterprises are facing exit important Ai and Athrough it isfixed coiled around a drum to the MP. i+1 while A while it aisCDPR coiled around a drum attached to thepaper MP. challenges in today’s market environments: a incontinuing Figure 1i+1 illustrates prototype introduced this i and A Figure 1 towards illustrates a CDPR introduced in times thisan paper tendency product development and and employing a reduction cable loop.ofprototype This architecture provides unand employing a cable loop. InThis architecture an unshortened product lifecycles. isprovides an increasing limited orientation workspace ofaddition, the MP there and consequently, the limited orientation workspace of at thethe MP and consequently, the demand of two customization, same time for in aidentical global following motions arebeing generated. Translation following motions areand generated. Translation for competition with competitors all over the world. Thisidentical trend, motions oftwo the actuators, pure rotation that corresponds to motions the actuators, and rotate pure rotation that corresponds to which is of inducing development macro to micro the case while two the actuators in from reverse directions. The the case of while actuators rotate in reverse directions. Thea markets, results in diminished lot sizes to augmenting concept cabletwo loop is detailed in [14]. Indue general, by using concept of in cable loop isthe detailed [14].motions In general, byMP using product varieties (high-volume to in low-volume production) [1].a cable loop a CDPR, translational of the lead cable loop in this atensions CDPR, the translational motions the To cope with augmenting variety well asofto beMP ablelead to to equal cable on both sides ofasthe drum. Furthermore, to tensions both sides of drum. Furthermore, identify possible potentials in ofthe theequal pure cable rotation ofoptimization theon end-effector is the a result the existing different the pure rotation end-effector is a result of the different production system,ofit the is important to have a precise knowledge

but there undesired rotationapplied associated to the rotation of the MPexists whileandifferent tensions on the cable loops of the different tensionscable applied on in thethe cable loops by the MP two while actuators. Employing loops design of by two actuators. Employing in theworks, designe.g. of the the CDPRs has been the subject ofcable someloops previous the CDPRs has been thetosubject ofof some previous knowledge, works, e.g. [8,9,12]. Nevertheless, the best the authors’ [8,9,12]. Nevertheless, the bestbyofcable the authors’ knowledge, the parasitic inclinationtoinduced loops has not been ofthe the product characteristics manufactured parasitic induced by loops has notand/or addressed up inclination torange now.and Therefore, the cable main contribution ofbeen this assembled system. In and this evaluation context, theof main challenge in addressed up to now. Therefore, the main contribution of this paper lies in in this the modeling parasitic inclinamodelling and analysis is now not only withinclinasingle paper in the modeling and cable evaluation parasitic tions inlies CDPRs containing one loop.toofcope tions in CDPRs containing one loop. to determine products, a limited range or paper existing product families, Two approaches areproduct described incable the the oriTwo approaches are in to the paper toproducts determine the oribut also toofbe to described analyze define entation theable articulated MPand and ascompare a consequence itstoparasitic entation of the articulated MP as aatconsequence its parasitic new product families. can beand observed that classical existing inclinations. The first It approach aims solving the geometricoinclinations. The first approach aims at geometricoproduct families regrouped function of clients or features. static model of are the CDPR at in hand, to solving find thethe orientation of static theoriented CDPR at to find are the orientation of However, assembly product families hardly to find. the MPmodel for a of given position ofhand, its geometric center. Carricato the MP a given position of its geometric center. Carricato thefor product family level, products differ mainly in two [2]On studied an analogous problem namely, inverse geometrico[2] studied an analogous inverse main characteristics: (i) theproblem numbernamely, of components and (ii) the static problem of under-constrained CDPRs which geometricoposes major static CDPRs which poses major type ofproblem components (e.g.coupling mechanical, electrical, electronical). challenges dueoftounder-constrained the between geometry and staticchallenges to the coupling between geometry staticClassical due methodologies considering mainly singleand products equilibrium of under-constrained CDPRs. The second approach of under-constrained CDPRs. TheMP second approach orequilibrium solitary, already existing product analyze the aims at approximating the orientation offamilies the without conaims at structure approximating the orientation MP level) without conproduct on a physical level (components which sidering the geometrico-static model ofofthethe manipulator, but by sidering thegeometric geometrico-static model ofcable the manipulator, by causes difficulties regarding an efficient definition and using some properties of the loop and thebut articusing some geometric properties the cable loop and the articcomparison of different productof families. Addressing this

c 2018 The Authors. Published by Elsevier B.V. 2212-8271 � Peer-review responsibility of the scientific committee cunder 2212-8271 � 2018 The Authors. Published by by Elsevier B.V.of the 28th CIRP Design Conference 2018. 2212-8271 © 2018The The Authors. Published Elsevier 2212-8271 © 2017 Authors. Published by Elsevier B.V. B.V. Peer-review under responsibility ofofthe scientific committee of of thethe 28th CIRP Design Conference 2018. Peer-review under responsibility the scientific committee CIRP Design Conference Peer-review under responsibility of the scientific committee of the 28th28th CIRP Design Conference 2018.2018. 10.1016/j.procir.2018.02.013

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A1

297

A2 C2

C2

C1

A3 B2 u2 y p B3 u 3 u1 x

C3

B2 u2 y p B3 u 3 u1 x

p

B1

p

B1

C3 C1

Cable Loop Drum

θ

V2

Support

V3

V1

4m

End-effector

V4

yb Fb

Ob

V5

xb

V6

3m

θ

V

Path

V

V7

Fig. 2: A four-DoF planar cable-driven parallel robot with a cable loop Fig. 1: Articulated MP of a planar CDPR containing a cable loop

ulated MP. Finally, the results obtained by the two approaches for a planar CDPR with one cable loop are compared. The paper is organized into eight sections. Section 2 describes the articulated MP of the planar CDPR under study. Section 3 details the geometrico-static model of the CDPR with a cable loop. Section 4 presents two approaches to find the orientation of the MP. Section 5 introduces a method to assess the parasitic inclination of the planar CDPR with one cable loop. The parasitic inclination for a case study is presented in section 6 and the discussion is detailed in section 7. Finally, the last section concludes the paper. 2. Description of the Manipulator under Study This section describes the manipulator under study. This manipulator has a planar workspace with an articulated MP which can host different types of end-effectors with one DoF. The overall manipulator consists in a base frame and two actuated cables connecting in parallel the articulated MP to the base frame as shown in figure 2. The planar manipulator possesses four-DoF while, it is actuated by three motors through two cables. Therefore, the manipulator is considered under-actuated. The MP has two translation DoF in the xOy plane, one rotational DoF perpendicular to its translation plane. The actuation of the additional degree of freedom on the moving platform is done through a cable loop and a drum, so that no motor needs to be mounted on the moving platform. The objective of this manipulator is to provide the underlying foundation for investigating the parasitic inclinations induced by cable loops in CDPRs. Figure 1 illustrates the articulated MP of the CDPR under study. This MP is composed of a support, a drum and an end-effector. The support forms the overall body of the moving-platform and accommodates cable anchor points (B1 , B2 , B3 ) and other components. The drum operates the end-effector through the cable loop. Both the drum and the end-effector are gears such that the rotational motion of the end-effector is provided by the rotational motion of the drum. A cable (cable loop) connected to two actuators, which are not shown in Fig. 1, is wounded about the drum to make the latter rotate about its own axis. The left side of this cable is denoted C1 whereas its right side is denoted C2 . Another cable, named C3 , is connected to both the

support of the articulated MP and a third actuator. The cable loop consists in two segments each with independent cable tension (t1 and t2 ). First segment, C1 , is composed of the part of the cable loop which connects the first motor to the drum through points A1 and B1 . The second segment is denoted as C2 and connects the second motor to the drum through points A2 and B2 . 3. Geometrico-Static Model of a Planar CDPR with a Cable Loop This section presents the mathematical model of the planar four-DoF under-constrained CDPR shown in Fig. 2. The considered robot is actuated by three motors. This model defines robot geometric model along with its static equilibrium equation. Since the geometry and the statics of under-constrained CDPRs are coupled, they should be solved simultaneously. Accordingly, the loop-closure and static equilibrium equations of the CDPR are written in order to obtain its geometrico-static model of the planar CDPR under study. The three following equations express the loop-closure equations of the manipulator at hand. b

li = b ai − b p −b R p p bi ,

i = 1, 2, 3

(1)

where b li is the i-th cable vector, i.e., the coordinate vectors pointing from point Bi to point Ai .b ai = [aix , aiy ]T , p bi = [bix , biy ]T and b p = [p x , py ]T are the Cartesian vector of points Ai , Bi and P, respectively, expressed in frame Fb . b R p is the rotation matrix associated to the rotation of F p with respect to Fb and is expressed as follows:   cos θ − sin θ b (2) Rp = sin θ cos θ θ = � (xb , x p ) defines the orientation of the MP. ti , i = 1, 2, 3 is the i-th cable tension vector and it is directed from Bi toward the exit point Ai . ti = ti b ui and its magnitude is expressed as ti = �ti �2 , i = 1, 2, 3 and b ui is denoted as the i-th cable unit vetor. In order to compute the unit cable vector, b ui , we normalize, b li as follows: ui =

li , li

i = 1, 2, 3

(3)

li being the i th cable length. The following set Σ is introduced, Σ = Σ1 +Σ2 +Σ3 +C1 +C2 +C3 , which gathers the isolated parts of the robots in order to analyze its static equilibrium. Σ1 , Σ2 and

Saman Lessanibahri et al. / Procedia CIRP 70 (2018) 296–301 S. Lessanibahri et al. / Procedia CIRP 00 (2018) 000–000

298

Σ3 stand for end-effector, drum and support, respectively. From Fig. 3, the external wrenches exerted on Σ are cable tensions, ti , i = 1, 2, 3, the weight of the MP, mg, and the frictional moment or the resistance to relative motions between Σ1 and Σ2 that is denoted as m f r . The equilibrium of the external forces applied onto Σ, is expressed as follows: A1

rp rp

A12

A2

�

b

+ p hT RTp ET mg = 0

Finally, we write the equilibrium of the moments generated by cable loop about point P, the latter is the moment which drives Σ2 and consequently actuates Σ1 and is formulated as follows: r p δt + m f r = 0

(9)

L1

L2

yp r pF p

xp

P b

Σ3

we = [0

Σ1

−mg

p Tb

h RTp ET mg

m f r ]T

(13)

04 is a four dimensional zero vector and the three-dimensional cable tension vector, t is expressed as follows:

H xb

(11)

where W is the wrench matrix of the CDPR under study   b b b u1 u2 u3    p T b T T b  b p Tb T Tb W =  b1 R p E u1 b2 R p E u2 p bT3 RTp ET b u3    −r p rp 0 (12) we is the external wrench applied onto the MP

θ Σ2

p

yb

(10)

Wt + we = 04

B3

B12 B2 B1 r p r p

(8)

From Eqs. (4), (8) and (9), the static equilibrium equation of the MP is expressed as:

C3

I

Ob

i=1

p Tb T T b i R p E ti

δt = t2 − t1 C2

Fb

3 � �

r p is the radius of the drum and cable loop tension difference is expressed in the following:

t3

t2

C1

By considering (b bi − b p) = b R p p bi , i = 1, 2, 3, and (b h−b p) = b R p p h we can rewrite Eq. (5) as follows:

A3

t1

3

t = [t1

Σ

t2

t3 ] T

(14)

4. Orientation of the Moving-Platform mg Fig. 3: Moving-platform of the four-DoF planar under-constrained cable-driven parallel robot

3 �

ti b ui + mg = 0

(4)

i=1

T

In Eq. (4), m is the mass of the MP and g = [0, −g] is the gravity acceleration with g = 9.81 m.s−2 . The equilibrium of moments about point P in frame Fb is expressed as follows: 3 �� �

b

i=1

� � �T �T bi − b p ET ti + b h − b p ET mg = 0

(5)

with E=

� 0 1

� −1 0

(6)

and b h is the Cartesian coordinate vector of the MP Center of Mass (CoM) in Fb which is expressed as follows: b

h = b p +b R p p h

(7) �T � p h = h x , hy is the Cartesian coordinate vector of the CoM expressed in F p .

Here, two methodologies for finding the orientation of MP are detailed. Two approaches calculate θ for a given position of point P. In the first approach, the orientation angle θ is computed based on loop closure equations (1) and staticequilibrium equation (11). The second approach aims at finding the orientation angle θ knowing the cable tension difference δt, and the Cartesian position coordinates of point P expressed in Fb . 4.1. Orientation of the moving-platform obtained by Approach 1 In this section, the rotation angle θ of the MP is obtained while considering the three loop-closure equations defined by Eq. (1) and the static-equilibrium equations of the MP defined by Eq. (11). Accordingly, the following system of seven nonlinear equations with nine unknowns, i.e., θ, t1 , t2 , t3 , l1 , l2 ,l3 , px and py is expressed: �  �  f1 θ, t1 , t2 , t3 , l1 , l2 , l3 , p x , py = 0   � �     f2 θ, t1 , t2 , t3 , l1 , l2 , l3 , p x , py = 0    � �     f θ, t =0 , t , t , l , l , l , p , p 1 2 3 1 2 3  3 x y  �   � f θ, t =0 (15) , t , t , l , l , l , p , p  1 2 3 1 2 3 4 x y  � �     f5 θ, p x , py =0    � �     f6 θ, p x , py =0   � �    f7 θ, p x , py =0

4

Saman Lessanibahri et al. / Procedia CIRP 70 (2018) 296–301 S. Lessanibahri et al. / Procedia CIRP 00 (2018) 000–000

f1 , f2 , f3 and f4 are obtained from Eq. (11) and they are functions of variables θ, t1 , t2 , t3 , l1 , l2 ,l3 , px and py . The latter equations are expressed analytically as follows:   f1 = t1 l2 l3 a1x − px − cθ b1x + sθ b1y   (16a) + t2 l1 l3 a2x − px − cθ b2x + sθ b2y   + t3 l1 l2 a3x − px − cθ b3x + sθ b3y f2 = t1 l2 l3 a1y − py − sθ b1x − cθ b1y   + t2 l1 l3 a2y − py − sθ b2x − cθ b2y   + t3 l1 l2 a3y − py − sθ b3x − cθ b3y − l1 l2 l3 mg 



(16b)

(16c)

f4 = (t2 − t1 )r p + m f r

(16d)





where sθ = sin(θ) and cθ = cos(θ). f5 , f6 and f7 are obtained from Eq. (1) and they are functions of variables θ, p x and py . The latter equations are expressed analytically as follows: 2 2   f5 = l21 − a1x − px − cθ b1x + sθ b1y − a1y − py − sθ b1x − cθ b1y (17a) 2  2  f6 = l22 − a2x − px − cθ b2x + sθ b2y − a2y − py − sθ b2x − cθ b2y (17b) f7 =

l23

2  2  − a3x − px − cθ b3x + sθ b3y − a3y − py − sθ b3x − cθ b3y (17c)

The under-determined system of non-linear equations (15) is studied for a given b p. As a result, a system consisting in seven equations and seven unknown is obtained. Hereafter, lsqnonlin TM Matlab function is used to solve this system of seven non-linear equations. It should be noted that the following constraints are taken into account for solving the system of equations in order to make sure that cable tensions are positive: ti > 0 i = 1, 2, 3

The following equation expresses the equilibrium of the moments applied/sustained about ICR, point I, expressed in Fb . m12 + mw = 0

(19)

m12 is the moment applied onto the MP at point I due to cable tension difference δt. Then, moment m12 can be expressed as follows: m12 = r p δt

(20)

mw is the moment applied onto the MP expressed at point I due to the MP weight, which is passing through point H. mw = (b h − b i)T ET mg

(21)

under the assumption that segments A1 B1 and A2 B2 are parallel, which is valid as long as the MP is far from the points A1 and A2 . In this approach the Cartesian coordinate vector point I, b i, is computed to formulate the pure rotation of the MP about this point. ICR is the intersection point between the line L12 passing through points A12 and B12 and the line L3 passing through points A3 and B3 . The equations of lines L12 and L3 are expressed as:

f3 = t1 l2 l3 −sθ b1x − cθ b1y a1x − px − cθ b1x + sθ b1y    + t1 l2 l3 cθ b1x − sθ b1y a1y − py − sθ b1x − cθ b1y    + t2 l1 l3 −sθ b2x − cθ b2y a2x − px − cθ b2x + sθ b2y    + t2 l1 l3 cθ b2x − sθ b2y a2y − py − sθ b2x − cθ b2y    + t3 l1 l2 −sθ b3x − cθ b3y a3x − px − cθ b3x + sθ b3y    + t3 l1 l2 cθ b3x − sθ b3y a3y − py − sθ b3x − cθ b3y   − l1 l2 l3 mg hx cθ − hy sθ 

299

(18)

4.2. Orientation of the moving-platform obtained by Approach 2 This section presents a straightforward approach that enables us to obtain a sound approximation of the orientation of the MP without considering the geometrico-static model expressed in Eq. (11). This approach takes into account only the equilibrium of the moments applied/sustained about Instantaneous Center of Rotation (ICR) point regardless of the cables tension (t1 , t2 , t3 ), but the difference of cable loop tensions, namely, δt.

L12 : x(b12y − a12y ) + y(a12x − b12x ) − a12x b12y + a12y b12x = 0 (22) L3 : x(b3y − a3y ) + y(a3x − b3x ) − a3x b3y + a3y b3x = 0 (23) The Cartesian coordinate vector of points A12 and B12 , namely, b a12 = [a12x , a12y ]T and b b12 = [b12x , b12y ]T are the followings: 1 b ( a1 + b a2 ) 2

(24)

1b 1 b ( b1 + b b2 ) = b p + R p ( p b1 + p b2 ) 2 2

(25)

b

b

b12 =

a12 =

b

i being the Cartesian coordinate vector of lines L12 and L3 , i.e.,b i ≡ L12 ∩ L3 , the components of its Cartesian coordinate vector take the form: b

i = [i x , iy ]T

(26)

with i x and iy being expressed as: ix =

µ1 ν2 − µ2 ν1 , λ1 µ2 − λ2 µ1

iy =

−ν1 − λ1 i x µ1

(27)

with 1 1 1 λ1 = cθ (b1y + b2y ) + sθ (b1x + b2x ) − (a1y − a2y ) + py 2 2 2 λ2 = b3x sθ + b3y − a3y cθ − a3y + py 1 1 1 µ1 = − cθ (b1x + b2x ) + sθ (b1y + b2y ) + (a1x + a2x ) − p x 2 2 2 µ2 = b3y sθ − b3x cθ + a3x − p x 1 ν1 = cθ [(−a1x − a2x )(b1y + b2y ) + (b1x + b2x )(a1y + a2y )] 4 1 + sθ [(−a1x − a2x )(b1x + b2x ) − (b1y + b2y )(a1y + a2y )] 4 1 1 − py (a1x − a2x ) + p x (a1y − a2y ) 2 2 ν2 = (a3y b3x − a3x b3y )cθ − (a3x b3x + a3y b3y )sθ − a3x py + a3y px (28)

Saman Lessanibahri et al. / Procedia CIRP 70 (2018) 296–301 S. Lessanibahri et al. / Procedia CIRP 00 (2018) 000–000

300

Table 1: CDPR Parameters associated to the case.

In Fig. 3, all the relevant notations are illustrated. ICR point, I, is a function of θ, b p, b ai and p bi , i = 1, 2, 3. By using the following half tangent substitution in Eq. (29), Eq. (19) becomes the 6th order univariate polynomial equation (31). sin θ =

1 − t2θ 2tθ , cos θ = 1 + t2θ 1 + t2θ

(30)

C6 tθ 6 + C5 tθ 5 + C4 tθ 4 + C3 tθ 3 + C2 tθ 2 + C1 tθ + C0 = 0

(31)

Eq. (31) is a function of tθ . The obtained polynomial is solved numerically to find tθ . Then, θ can be substituted with tθ based on Eq. (30). The coefficients of the latter polynomial, C0 , C1 , ..., C6 , are detailed in 1 . Equation (31) is solved in order to find the possible inclination(s) θ of the MP for a given position of its geometric center P.

Other parameters

b

m = 12 [kg] r p = 0.03 [m] p h = [0, −0.3]T [m]

a1 = [−2.03, 3]T a2 = [−1.97, 3]T b a3 = [2, 3]T p b1 = [−0.28, 0.25]T p b2 = [−0.22, 0.25]T p b3 = [0.25, 0.25]T

(29)

θ 2 From Eqs. (26)-(30) we can rewrite Eq. (19) as follows:

Anchor point coordinates [m] b

and, tθ = tan

5

robot with a cable loop shown in Fig. 2. Eq. (33) expresses the Cartesian coordinates vector of seven via-points, namely, V1 ,...,V7 , on the prescribed path (blue path in Fig. 2). b

b

  −1 v1 = , 0   0.65 , v5 = +1

b



   −1 b −0.65 v2 = , v3 = , 0.65 +1

b

   +1 b +1 , v7 = v6 = 0.65 0 

b

0 v4 = +1 



(33)

5. Parasitic Inclinations

7. Discussion

In this paper, parasitic inclination is defined as undesired orientation of the MP that leads to inaccuracy in manipulation and positioning. This kinematic situation is an outcome of utilizing cable loop in the CDPR. Since parasitic inclination decreases the accuracy of the robot, its investigation is crucial and can be employed to minimize the parasitic inclination by optimizing the design parameters in the design stage. This section deals with the determination of the parasitic inclination, θ p , of the MP due to cable tension differences, δt, into the cable-loop. Accordingly, the following methodology is defined:

Figure 4 shows the natural inclinations θn1 and θn2 of the previous case study obtained with approaches 1 and 2 along the prescribed path, respectively. The difference between θn2 and θn1 along the prescribed path is also depicted in Fig. 4. It appears that both approaches 1 and 2 give similar results, which confirms the soundness of the assumption made in approach 2. Figure 5 shows the rotation angle θm1 and θm2 of the movingplatform obtained with approaches 1 and 2, respectively, along the prescribed path, for δt = 20 N. Finally, Fig. 6 illustrates the parasitic inclination θ p of the MP for different values of the cable tension difference δt into the cable-loop. It should be noted that θ p increases with δt. This confirms the link between δt in the cable loop and the parasitic inclination of the MP. Overall, the approach proposed in section 4.2 yields consistent results and can be applied to determine the parasitic inclination. Furthermore, this approach can be used to design the robot with respect to its parasitic inclination. This contributes to better control and more accurate positioning.

1. To determine the natural inclination θn of the MP. θn amounts to the rotation angle θ of the MP obtained with both Approaches 1 and 2 described in Secs. 4.1 and 4.2, resp., for the same tensions in both strands C1 and C2 of the cable-loop, i.e., δt = 0. 2. To determine the inclination θm of the moving-platform when tensions in both strands of the cable-loop are not the same, i.e., δt �= 0. θm can be also computed with Approaches 1 and 2 described in Secs. 4.1 and 4.2, respectively. 3. To determine the parasitic inclination, θ p of the movingplatform. θ p is the difference between θm and θn , i.e.,

8. Conclusion

This paper introduced a four-DoF planar under-constrained cable-driven parallel robot. The robot utilizes a doubleactuated cable loop system which grants an unlimited orientation workspace for the end-effector. Nevertheless, the movingplatform undergoes some parasitic inclinations that are due to (32) θ p = θm − θn the presence of a cable loop. Moreover, an analytical method to find the orientation of the moving-platform is presented. This method is validated by comparing its results with the solution 6. Case study of geometrico-static equations of the robot. Then, an approach was established to isolate the parasitic inclination induced by The rotation angle θ and the parasitic inclination θ p are comcable loop only from the natural inclination of the movingputed in this section along a given path for the design parameplatform. ters given in Tab. 1 of the four-DoF planar cable-driven parallel Some experimental validations will be conducted in the future to validate the theoretical results presented in this paper. The 1 https://drive.google.com/file/d/0B80GqJ5822jObDlXNkdOUWd6UUE/view?usp=sharing optimim design paramerers of the robot will be also searched

6

Saman Lessanibahri et al. / Procedia CIRP 70 (2018) 296–301 S. Lessanibahri et al. / Procedia CIRP 00 (2018) 000–000 × 10 -3 1

15

10

0.5

θn [o]

0 0 -0.5 -5

θn1 θn2 δθn

-10

-15 V1

V2

V3

V4

V5

-1

-1.5 V7

V6

Via-points

Fig. 4: Natural inclinations θn1 and θn2 of the moving-platform obtained with Approaches 1 and 2 along a prescribed path

0.11

15

0.1

10

θm [o]

0.08 0 0.07 -5

δθm = θm2 − θm1 [o]

0.09 5

0.06 -10

θm1 θm2 δθ m

-15 V1

V2

V3

V4

V5

V6

0.05

0.04 V7

Via-points

Fig. 5: Rotation angle θm1 and θm2 of the moving-platform obtained with Approaches 1 and 2 for δt = 20 N.

0.4

δt = 5 N δt = 1 0 N δt = 1 5 N δt = 2 0 N

0.35

0.3

θp [ ]

0.25

0.2

0.15

0.1

0.05 V1

V2

V3

V4

V5

V6

for the parasitic inclinations of its articulated moving-platform to be a minimum. References

δθn = θn2 − θn1 [o]

5

301

V7

Via-points

Fig. 6: Parasitic inclination θ p of the moving-platform for different values of the cable tension difference δt into the cable-loop

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