International audienceThis paper investigates a situation pointed out in a recent paper, in which a non-singular change of assembly mode of a planar 2-RPR-PR parallel manipulator was realized by encircling a point of multiplicity 4. It is shown that this situation is, in fact, a non-generic one and gives rise to cusps under a small perturbation. Furthermore , we show that, for a large class of singularities of multiplicity 4, there are only two types of stable singularities occurring in a small perturbation: these two types are given by the complex square mapping and the quarto mapping. Incidentally , this paper confirms the fact that, generically, a local non-singular change of solution must be accomplished by encircling a cusp point

A Peidro

C Innocenti

H Bamberger

H Whitney

J Callahan

M Zein

PR McAree

Coste, Michel

Wenger, Philippe

Chablat, Damien

HAL-Rennes 1

Hidden cuspsMichel Coste, Philippe Wenger, Damien ChablatTo cite this version:Michel Coste, Philippe Wenger, Damien Chablat. Hidden cusps. 15th International Symposiumon Advances in Robot Kinematics, Jun 2016, Grasse, France. 2016. <hal-01309185>HAL Id: hal-01309185https://hal.archives-ouvertes.fr/hal-01309185Submitted on 29 Apr 2016HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Hidden cuspsMichel Coste, Philippe Wenger and Damien ChablatAbstract This paper investigates a situation pointed out in a recent paper, in which anon-singular change of assembly mode of a planar 2-RPR-PR parallel manipulatorwas realized by encircling a point of multiplicity 4. It is shown that this situation is,in fact, a non-generic one and gives rise to cusps under a small perturbation. Fur-thermore, we show that, for a large class of singularities of multiplicity 4, there areonly two types of stable singularities occurring in a small perturbation: these twotypes are given by the complex square mapping and the quarto mapping. Inciden-tally, this paper confirms the fact that, generically, a local non-singular change ofsolution must be accomplished by encircling a cusp point.Key words: parallel robot, cusp, non-generic singularity, perturbation1 IntroductionThe non-singular change of assembly mode in parallel manipulators, first observedby C. Innocenti and V. Parenti-Castelli [1], is often associated with the presence ofcusps and the non-singular change of assembly mode is realized by turning around acusp point, or a cuspidal edge of the singularity surface (see for instance [2, 3, 4, 5]).It has also been reported that non-singular change of assembly modes can be real-ized by following an “alpha curve” (i.e. a fold curve intersecting itself transversally)[6, 7], and that the presence of cusps is not necessary for the existence of non-singular assembly mode changes [8].Michel CosteIRMAR, Universite´ de Rennes 1 – CNRS, e-mail: michel.coste@univ-rennes1.frPhilippe WengerIRCCyN, CNRS – E´cole Centrale de Nantes, e-mail: philippe.wenger@irccyn.ec-nantes.frDamien ChablatIRCCyN, CNRS – E´cole Centrale de Nantes, e-mail: damien.chablat@irccyn.ec-nantes.frAuthors partially supported by ANR-14-CE34-0008-01 Kapamat12 Michel Coste, Philippe Wenger and Damien ChablatA recent paper [9] exhibits an example of a 2-dof parallel manipulator with anisolated singularity of multiplicity 4 of the inverse kinematics mapping, such thatcircling around the image of this singularity in the joint space results in a non-singular assembly mode change; moreover, after a second loop around the singular-ity, one is back in the same assembly mode. There is no cusp in the picture, but weintend to explain in the present paper that actually the cusps are hidden. Precisely,the singularity of multiplicity 4 is not a stable singularity, which means that it dis-appears under a small perturbation of the geometry of the manipulator, giving riseto three cusp points; in the joint space, the isolated singularity is perturbed into adeltoid curve with three cusps. Hence, circling around the singularity of multiplicity4 was actually circling around 3 degenerate cusps.H. Whitney [10] has shown that the only stable singularities of mappings be-tween spaces of dimension 2 are folds and cusps. Any other higher order singularitybecomes a combination of folds and cusps after perturbation, which amounts to saythat these higher order singularities are degenerations of folds and cusps. We shallshow that the case study of the perturbation of the 2-dof manipulator actually de-scribes two main cases of singularities of multiplicity 4 (complex square and quartomappings) leading to two different perturbations (the former with three cusps, thelatter with one cusp).2 A case study2.1 2RPR-PR with higher order singularitiesWe begin with the example given in [9]. It is a 2RPR-PR planar manipulator witharchitecture described in Figure 1.Fig. 1 Architecture of 2RPR-PRThe output coordinates are the angle ϕ and the y-coordinate of the revolute jointB which is constrained to move on the vertical axis. The input coordinates are theHidden cusps 3square `21 and `22 of the lengths of the legs A1B1 and A2B2. The equations for theinverse kinematic mapping are`2i = y2−2bi y sin(ϕ)+a2i −2ai bi cos(ϕ)+b2i for i= 1,2 (1)The Jacobian matrix Jac of the inverse kinematic mapping and its Jacobian determi-nant J (up to a factor 4) are:Jac = 2(−b1 y cos(ϕ)+a1 b1 sin(ϕ) y−b1 sin(ϕ)−b2 y cos(ϕ)+a2 b2 sin(ϕ) y−b2 sin(ϕ))J = (b1+b2) cos(ϕ)y2+(a1 b1−a2 b2) sin(ϕ)y− (a1+a2)b1 b2 sin(ϕ)2 .(2)The possible cusp points and higher order singularities may be detected by addingto J = 0 the equationsJac(−JyJϕ)=(00), (3)where Jϕ and Jy denote the partial derivatives of J with respect to ϕ and y. Theseequations express that the curve of singular points in the workspace either has asingularity or has a tangent vector in the kernel of the Jacobian matrix. We observethat (ϕ,y) = (0 mod pi,0) satisfy Jac = the zero matrix (and hence also J = 0 andequations (3) hold); these singularities are not cusps, but higher order singularities.2.2 An exampleWe compute the singularities in the workspace and in the joint space for an examplewith a1 = 3, a2 = 7, b1 = 6, b2 = 5, which is the same as the one considered in [9].In this case we can check that the only real solutions of J = 0 and the equations (3)are (ϕ,y) = (0 mod pi,0).For (ϕ,y) = (0,0), we have `1 = 3, `2 = 2. Developing the equations for theinverse kinematic mapping and for J in a neighbourhood of (0,0) we get`21−9 = y2+12yϕ+18ϕ2+h.o.t. , `22−4 = y2−10yϕ+35ϕ2+h.o.t.J = 11y2−17yϕ−300ϕ2+h.o.t. , (4)where h.o.t. stands for “higher order terms”. This shows that the singularity is ofmultiplicity 4, and that the point (ϕ = 0,y = 0) is a node of the curve of singular-ities (the discriminant ∆ = (−17)2− 4× 11× (−300) of the quadratic part of thedevelopment of J at (0,0) is positive).For (ϕ,y) = (pi,0), we have `1 = 9, `2 = 12. Developing `21−81, `22−144 and Jin a neighbourhood of (pi,0), with ψ = φ −pi , we get`21−81 = y2−12yψ−18ψ2+h.o.t. , `22−144 = y2+10yψ−35ψ2+h.o.t.J = −11y2+17yψ−300ψ2+h.o.t. . (5)4 Michel Coste, Philippe Wenger and Damien ChablatThis shows that the singularity is also of multiplicity 4, and that the point (ϕ =pi,y = 0) is an isolated double point of the curve of singularities (the discriminant∆ = 172− 4× (−11)× (−300) of the quadratic part of the development of J at(pi,0) is negative).Figure 2 (a) represents the workspace of the manipulator; it must be understoodthat the right side (ϕ = 3pi/2) has to be identified with the left side (ϕ =−pi/2). Thesingularity curve is represented in thick blue; one can see the node at ϕ = 0,y = 0and the isolated double point at ϕ = pi,y= 0. The dash-dot black curve is the levelcurve `1 = a1 + b1 = 9 and the dashed red curve is the level curve `2 = a2 + b2 =12. The numbers in the zones delimited by these curves indicate the correspondingimages by the inverse kinematic mapping in the joint space.ϕy123412341234123412(a) Workspace `1`21 234(b) Joint spaceFig. 2 Workspace and joint space of the manipulatorThe joint space is represented in Figure 2 (b) (the same figure appears in [9]).One can see the image of the singularity curve in blue. Inside the domain delimitedby this curve, the direct kinematic problem has four solutions, except at the image(`1 = 9, `2 = 12) of the isolated singularity point where there is one solution ofmultiplicity 4 and two other solutions. Above each point of the image singularitycurve, there are two double solutions, except at the point `1 = 3, `2 = 2 (image ofthe node) where there is one solution of multiplicity 4. The zones numbered 1,2,3,4are the images of the zones with the corresponding numbers in the workspace.It can be seen that circling around the isolated singularity point in the joint space,following the numbering 1-2-3-4-1, yields a non-singular assembly mode changeleading from a zone numbered 1 in the workspace touching the isolated singularitypoint to the other one. A second loop makes one return to the initial assembly mode.This phenomenon cannot be faithfully represented in a 3-dimensional reduced con-figuration space: one cannot have a ramp turning around the singular configurationand returning to the start level after two turns without an artificial self-intersection.Hidden cusps 5One can see in this example a non-singular assembly mode change by circlingaround a singularity which is not a cusp. The hidden cusps are revealed by slightlyperturbing the geometry of the manipulator.2.3 Revealing the hidden cuspsThe manipulator is modified so that the revolute joint B on the platform is no longeron the line B1B2, but at a distance d from this line: see Figure 3.Fig. 3 The modified manipulatorWe compute the example with a1 = 3, a2 = 7, b1 = 6, b2 = 5 and d = 3. Theequations for the inverse kinematic mapping are now`21 = y2−6y(cos(ϕ)−2sin(ϕ))−36cos(ϕ)−18sin(ϕ)+54`22 = y2−2y(3cos(ϕ)+5sin(ϕ))−70cos(ϕ)+42sin(ϕ)+83 , (6)and the Jacobian determinant is, up to a constant factor,J = 11y2 cos(ϕ)− y(30 cos(ϕ)+17 sin(ϕ)+33)+390 cos(ϕ)2−30 cos(ϕ) sin(ϕ)−300 (7)We can detect the cusps or higher order singularities by solving the system formedby J = 0 and equations (3). We get four real solutions, which are(ϕ '−0.0023,y' 2.9069), (ϕ ' 2.6492,y'−2.2190),(ϕ '−2.7368,y'−1.2968), (ϕ ' 3.0855,y' 2.6935) (8)It can be checked that all four points are actually cusp points.The curve of singularities in the workspace is represented in thick blue in Figure4 (a). Note that it retains the overall features of the original singularity curve inFigure 2 (a), except for the node at (0,0) which is simplified in two non-intersecting6 Michel Coste, Philippe Wenger and Damien Chablatbranches and the isolated double point at (pi,0) which has evolved into an oval. Thecharacteristic curves in the workspace (defined in [11]) are represented in green.One can recognize the cusp points as the points of tangency of the characteristiccurves with the curve of singularities: there is one cusp point on one branch of thesimplification of the node, and three cusp points on the oval obtained by perturbingthe isolated double point of the singularity curve.ϕyP1P2(a) Workspace `1`2Q(b) Joint spaceFig. 4 Workspace and joint space of the modified manipulatorFigure 4 (b) represents the joint space. One can see the four cusps, three on thecentral deltoid (image of the oval) and one on a branch of the outer curve (the twobranches have also a crossing point). There are six solutions to the direct kinematicproblem inside the deltoid, and 4, 2 or 0 solutions as one proceeds towards the outerregions.Note that the connected zone in the workspace encircling the large green deltoidis not a uniqueness domain [11]: it is a 2-sheeted covering of the zone around thedeltoid in the joint space. This latter zone is not simply connected, so we cannotdeduce that a connected component of its preimage is a uniqueness domain.The picture of the joint space shows that circling around the isolated singularityin the joint space was actually circling around three degenerate cusps. The dashedcircle from Q to Q around the deltoid in the joint space lifts to the dashed trajectoryfrom P1 to P2 in the workspace; a second turn on the circle completes the circuitfrom P2 back to P1.Hidden cusps 73 The general mathematical picture: unfolding of a singularity ofmultiplicity 4 of a mapping of surfacesWe explain here how the observations made for the modified 2RPR-PR fit into a gen-eral mathematical framework. We begin with by recalling two examples describedin [12].3.1 Complex square mapping and its unfoldingThe first example is given by f : (x,y) 7−→ (u = x2 − y2, v = 2xy) which is thecomplex square function z 7→ z2, written in real and imaginary parts; this shows thatevery point in R2 is the image by f of two points in R2, except the origin whichis the image of the origin only. The Jacobian determinant of f is, up to a constantfactor, x2 + y2. The only singularity of f is at the origin, and this singularity hasmultiplicity 4 (the dimension of the quotient algebra R[x,y]/(x2− y2,2xy)).Now we perturb the mapping f tof˜ : (x,y) 7−→ (u= x2− y2+4ax, v= 2xy+4by) . (9)The Jacobian determinant of f˜ becomes, up to a constant factor, (x+a+b)2+y2−(a−b)2. If b 6= a, the set of singular points of f˜ is the circle with centre (−a−b,0)and radius |a− b|. There are three cusp points on this circle, and the image curvein the (u,v) plane is a deltoid with three cusps. A point inside the deltoid has fourpreimages, outside two. Circling around the deltoid permutes the two preimages (ascircling around the origin does for the complex square root).Figure 5 (a) shows the situation at the source (coordinates (x,y)) and at the target(coordinates (u,v)), in the case a = 1, b = −1. The blue curves are the curves ofsingularities. The green curve at the source is the characteristic curve; the cuspspoints are the points where the blue and green curves are tangent.y vx u(a) Perturbation of complex squarey vx u(b) Perturbation of quartoFig. 5 Perturbations of the complex square mapping and of the quarto mapping8 Michel Coste, Philippe Wenger and Damien Chablat3.2 Quarto mapping and its unfoldingThe second example is the mapping g : (x,y) 7−→ (u= x2, v= y2); this mapping isnamed “quarto” because it folds the plane x,y) onto the first quadrant of the plane(u,v), which is covered by four sheets. The Jacobian determinant of g is, up to aconstant factor, xy. The set of singular points of g is the union of the two axes.Let us perturb the mapping g tog˜ : (x,y) 7−→ (u= x2+2ay, v= y2+2bx) . (10)The Jacobian determinant becomes, up to a constant factor, xy− ab. This is theequation of an equilateral hyperbola, if ab 6= 0. Its image by g˜ is a curve in the (u,v)plane with two branches, one of which has a cusp; inside the cusp, each point hasfour preimages by g˜, between the branches two, and zero elsewhere.Figure 5 (b) represent the situation at the source and at the target in the casea= b= 1 in the same way as for the preceding example. One can see the cusp pointat the source.3.3 General caseThe two examples above are actually the complete list of the stable singularitiesthat can be obtained by perturbing a singularity of multiplicity 4 where the 2× 2Jacobian matrix is the zero matrix. These are the elliptic (complex square case) andhyperbolic (quarto case) Σ 2 singularities which are studied in [13], Part I §3. Thenotation Σ 2 means that the Jacobian matrix has corank 2, i.e. is the zero matrix indimension 2, and in this case multiplicity 4 is equivalent to the fact that the discrim-inant ∆ of the quadratic part of the Taylor expansion of the Jacobian determinant atthe singularity is nonzero. The elliptic case corresponds to ∆ < 0 and the hyperboliccase to ∆ > 0.We can now return to the example of the 2RPR-PR. We have ∆ > 0 at the sin-gularity (ϕ,y) = (0,0) (see (4)): we are here in the case ”quarto mapping” and wecan clearly see the relevant parts of Figure 4 corresponding to Figure 5(b). We have∆ < 0 at the singularity (ϕ,y) = (pi,0) (see (5)): we are now in the case ”complexsquare mapping” and we can compare the relevant parts of Figure 4 with Figure5(a).4 ConclusionWe have shown that the singularities of multiplicity 4 that appear in the study ofthe kinematics of the 2RPR-PR are not generic and give rise to cusps under a smallperturbation. We have also shown that these singularities belong to a family of sin-Hidden cusps 9gularities which splits in two cases according to the sign of the discriminant of thequadratic part of the Jacobian determinant : the ”complex square mapping” caseand the ”quarto mapping” case which are well known in the theory of singularitiesof differentiable mappings. In the first case, the singularity is isolated and circlingaround it in the joint space results in an exchange of two solutions to the direct kine-matic problem. A small perturbation to stable singularities gives three cusp pointswhich were in some sense ”hidden” in the singularity of multiplicity 4, and so onecan argue that this example does not invalidate the rule that, generically, local non-singular assembly mode changes arise by circling around cusps.We have limited our study to the 2-dof case. In a future work, we shall exam-ine the perturbation of the second example in [9], which is interesting because itgives a fully parallel generic 3-RPR manipulator with properties similar to the oneswe have seen for the constrained 2RPR-PR. We shall also discuss in more detailscharacteristic surfaces and uniqueness domains [11].References1. Innocenti, C., Parenti-Castelli, V.: Singularity-free evolution from one configuration to anotherin serial and fully-parallel manipulators. J. Mech. Des. 120, 73-79 (1998)2. McAree, P.R., Daniel, R.W.: An explanation of never-special assembly changing motions for3-3 parallel manipulators. The International Journal of Robotics Research 18(6): 556-574(1999)3. Zein, M., Wenger, P., Chablat, D.: Singular curves in the joint space and cusp points of 3-RPRparallel manipulators. Robotica 25(6), 717-724 (2007)4. Caro, S., Wenger, P., Chablat, D.: Non-singular assembly mode changing trajectories of a6-DOF parallel robot. In: Proceedings of the ASME 2012 International Design EngineeringTechnical Conferences & Computers and Information in Engineering Conference IDETC/CIE2012, pp. 1245-1254 (2012)5. Husty, M., Schadlbauer, J., Caro, S., Wenger,P.: The 3-RPS manipulator can have nonsingularassembly-mode changes. In: Computational Kinematics, Mechanisms and Machine ScienceVol. 15, pp. 339-348. Springer (2014)6. Bamberger, H., Wolf, A., Shoham, M.: Assembly mode changing in parallel mechanisms.IEEE Transactions on Robotics 24(4), 765-772 (2008)7. Macho, E., Altuzarra, O., Pinto, C., Hernandez, A.: Transitions between multiple solutionsof the direct kinematic problem. In: Advances in Robot Kinematics: Analysis and Design,pp. 301-310. Springer (2008)8. Coste, M., Chablat, D., Wenger, P.: Nonsingular change of assembly mode without any cusp.In: Advances in Robot Kinematics, pp. 105-112. Springer (2014)9. Peidro, A., Marin, J.M., Gil, A., Reinoso, O.: Performing nonsingular transitions betweenassembly modes in analytic parallel manipulators by enclosing quadruple solutions. Journalof Mechanical Design 137(12) ,122302-122302-11 (2015)10. Whitney, H.: On Singularities of Mappings of Euclidean Spaces. I. Mappings of the plane intothe plane. Annals of Mathematics 62, 374-410 (1955)11. Wenger, P., Chablat, D.: Uniqueness domains in the workspace of parallel manipulators. In:I.F.A.C-Symposium on Robot Control (SYROCO97), pp. 431-436 (1997)12. Callahan, J.: Singularities and plane maps. American Mathematical Monthly 81, 211-240(1974)13. Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of differentiable maps, Vol. 1.Birkha¨user (1985)

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