aa r X i v : . [ m a t h . A T ] D ec GENERIC SINGULAR CONFIGURATIONS OF LINKAGES
DAVID BLANC AND NIR SHVALB
Abstract.
We study the topological and differentiable singularities of the con-figuration space C (Γ) of a mechanical linkage Γ in R d , defining an inductivesufficient condition to determine when a configuration is singular. We show thatthis condition holds for generic singularities, provide a mechanical interpretation,and give an example of a type of mechanism for which this criterion identifies allsingularities. Introduction
The mathematical theory of robotics is based on the notion of a mechanism con-sisting of links, joints, and rigid platforms. The mechanism type is a simplicial (orpolyhedral) complex T Γ , where the parts of dimension ≥ linkage (or mechanism) Γ itself is determinedby assigning fixed lengths to each of the links of T Γ . See [Me, Se, T] and [F] forsurveys of the mechanical and topological aspects, respectively.0.1 . Configuration spaces. Here we concentrate on the most prevalent type ofmechanism T Γ : namely, a finite 1-dimensional simplicial complex (undirected graph),with N vertices and k edges. Note that a rigid platform is completely specified bylisting the lengths of all its diagonals (i.e., the distance between any two vertices), sowe need not list the platforms explicitly. Our results actually hold also for the casewhen some links of Γ are prismatic (or telescopic) – i.e., have variable length – butfor simplicity we deal here with the fixed-length case only.A length-preserving embedding of the vertices of the linkage Γ in a fixed ambientEuclidean space R d is called a configuration of Γ. In applications, d is mostcommonly 2 or 3. The set of all such embeddings, with the natural topology (anddifferentiable structure), is called the configuration space of Γ, denoted by C (Γ).Such configuration spaces have been studied intensively, with the hope of extractinguseful mechanical information from their topological or geometric properties. Muchof the mathematical literature has been devoted to the special case when Γ is a closedchain (polygon): see, e.g., [FTY, Hau, HK, KM1, KM2, MT]. However, the generalcase has also been treated (cf. [Ho, Ka, KTs, KM3, OH, SSB1, SSB2]).0.2 . Singularities. There are two main types of singularities which arise in robotics.The kinematic singularities of a mechanism, which appear as singularities of work andactuation maps defined on C (Γ) ( § Date : June 17, 2018.1991
Mathematics Subject Classification.
Primary 70G40; Secondary 57R45, 70B15.
Key words and phrases. configuration space, workspace, robotics, mechanism, linkage, kinematicsingularity, topological singularity. and have been studied intensively (see, e.g., [GA], [Me, § topological or differentiable singularities of the configuration space C (Γ) itself have not received much attention in the literature since [Hu], aside fromsome special examples (see, e.g., [F, KM2] and [ZBG]).For any linkage Γ, the configuration space C (Γ) is the zero set of a smoothfunction λ : R Nd → R k (see § C (Γ) is typically a smoothmanifold (when ~ ∈ R d is a regular value of λ ), and even if not, “most” points of C (Γ) are smooth, since a simple necessary condition for a point V in C (Γ) to besingular is that Rank(d λ V ) < k . Thus we are in the common situation where it isrelatively straightforward to identify configurations which are possibly singular, butnot so easy to pinpoint when this is in fact so.Our goal in this paper is threefold:(a) To provide a straightforward inductive description of a sufficient condition fora configuration V to be differentiably singular (in fact, this will imply that V is even a topological singularity) – see Proposition 3.8 and Theorem 4.9.(b) To show that this condition applies generically (that is, to all but a positive-codimension subset of the singular locus Σ) – see Remarks 3.7 and 4.8.(c) To obtain a mechanical interpretation for all singularities in the configurationspace of a linkage Γ as a tangential conjunction of two kinematic singularitiesof type I (cf. [GA]) for complementary sub-mechanisms of Γ – see Remark4.10.The third goal is completely achieved only in the plane (for d = 2), since themodel we use for configuration spaces is not completely realistic for rigid rods in R .See Remark 1.7 below for an explanation of the difficulties involved.0.3. Remark.
Since the function f : R Nd → R k defining the configuration space isa quadratic polynomial (cf. § C (Γ) is actually a real algebraic variety. Thusany topological or differentiable singularity V is in particular an algebraic singularity(cf. [Sh, Ch. II, § d = 2) –see [KM3, Ki, JS]. Thus our results here appear to be statements about any realalgebraic variety.However, the point we wish to make here is not that the cone singularities are themost common ones in algebraic varieties; it is rather the mechanical interpretation ofthe generic singularities, and the mechanical underpinnings of the inductive processdescribed in Section 4.In fact, while the topological, differentiable, and geometric structures on configu-ration spaces of linkages can be used to study their mechanics (cf. [KM2, KTe]), thealgebraic structure usually plays no role (but see [C]).0.4 . Organization. In Section 1 we briefly review some of the basic notions usedin this paper. In Section 2, various concepts of local equivalences of configurationspaces are defined; these help to simplify the study of singular points. In Section 3we explain the role played by pullbacks of configuration spaces. This is applied inSection 4 to provide an inductive construction, which is used both to describe thesufficient condition mentioned in § ENERIC SINGULAR CONFIGURATIONS OF LINKAGES 3
Acknowledgements.
We wish to thank the referee for his or her comments.1.
Background on configuration spaces
We first recall some general background material on the construction and basicproperties of configuration spaces. This also serves to fix notation, which is notalways consistent in the literature.1.1.
Definition.
Consider an abstract graph T Γ with vertices V and edges E ⊆ V .A linkage (or mechanism ) Γ of type T Γ is determined by a function ℓ : E → R + specifying the length ℓ i of each edge e i in E = { e i = ( u i , v i ) } ki =1 (subject tothe triangle inequality as needed). We write ~ℓ := ( ℓ , . . . , ℓ k ) ∈ R E for the vectorof squared lengths.The set of all embeddings of V in an ambient Euclidean space R d is an openmetric subspace of ( R d ) V , denoted by Emb d ( T Γ ). We have a squared length map λ : Emb d ( T Γ ) → R E with λ ( u i , v i ) := k ϕ ( u i ) − ϕ ( v i ) k , and the configuration space of the linkage Γ = ( T Γ , ℓ ) is the metric subspace C (Γ) := λ − ( ~ℓ ) of Emb d ( T Γ ).A point V ∈ C (Γ) is called a configuration of Γ. Note that λ is an algebraic functionof V ∈ R dN (which is why the lengths were squared), so C (Γ) is a real algebraicvariety.1.2. Remark.
By [Hi, I, Theorem 3.2], we know that C (Γ) is a smooth manifold if ~ℓ is a regular value of λ : that is, if its differential d λ V is of maximal rank forevery V ∈
Emb d ( T Γ ) with λ ( V ) = ~ℓ .However, for some mechanism types T Γ , this condition may not be generic: thereexist mechanism types T Γ and an open set U in R dN consisting of non-regular valuesof F Γ . This means that for each ~ℓ ∈ U , the configuration space C (Γ ~ℓ ) := λ − ( ~ℓ )has at least one configuration V ∈ C (Γ ~ℓ ) such that λ not a submersion at V . See[SSB2] for an example.1.3 . Isometries of configuration spaces. The group Euc d of isometries of theEuclidean space R d acts on the space C (Γ). When Γ has a rigid “base platform” P of dimension ≥ d −
1, this action is free. In this case we can work with the“restricted configuration space” C (Γ) / Euc d , and the quotient map has a continuoussection (equivalent to choosing a fixed location in R d for P ). See § W of R d ) may be fixed by certain transformations (those fixing W ), so the actionof Euc d is not free.1.4. Definition.
Choose a fixed vertex x ⋆ of Γ as its base-point : the action of thetranslation subgroup T ∼ = R d of Euc d on x ⋆ is free, so its action on C (Γ) isfree, too, and we call the quotient space C ∗ (Γ) := C (Γ) /T the pointed configurationspace for Γ. Thus C (Γ) ∼ = C ∗ (Γ) × R d , and a pointed configuration (i.e., an elementof C ∗ (Γ)) is simply an ordinary configuration expressed in terms of a coordinateframe for R d with the origin at x ⋆ .If we also choose a fixed link ~ v in Γ starting at x ⋆ , we obtain a smooth map p : C ∗ (Γ) → S d − which assigns to a configuration V the direction of ~ v . The fiber DAVID BLANC AND NIR SHVALB b C ∗ (Γ) of p at ~ e ∈ S d − will be called the reduced configuration space of Γ. Notethat the bundle C ∗ (Γ) → S d − is locally trivial.1.5. Definition.
A mechanism Γ may be equipped with a special point x e – inengineering terms this is the “end-effector” of Γ, whose manipulation is the goal ofthe mechanism. We think of ∆ := { x e } as a sub-mechanism of Γ (more generally,we could choose any rigid sub-mechanism). Assuming that the base-point x ⋆ ofΓ is not x e , the inclusion j : ∆ ֒ → Γ induces a map of configuration spaces j ∗ : C ∗ (Γ) → C (∆), whose image W is called the work space of the mechanism. The work map ψ : C ∗ (Γ) → W of Γ is the factorization of j ∗ through W (which is notalways a smooth manifold).1.6. Example.
Now consider a closed 5-chain Γ , as in Figure 1, with end-effector x e = x (2) . Here the direction of ~ v := x (4) − x (0) is fixed. x (0) x (1) x (3) x (2) x (4) Figure 1.
Closed 5-chain Γ
The work space of each of the two open sub-chains of Γ starting at x (0) andending at x (2) is a closed annulus. Therefore, W is the intersection of these twoannuli (see Figure 2), i.e. a curvilinear polygon in R , whose combinatorial typedepends on the lengths of the links. Figure 2.
The lens-shaped work space W for Γ Remark.
The configuration spaces studied in this paper are mathematical models,which take into account only the locations of the vertices of Γ, disregarding possibleintersections of the edges. In the plane, there is some justification for this, since wecan allow one link to slide over another. This is why this model is commonly used(cf. [F, KM1]; but see [CDR]). However, in R the model is not very realistic, sinceit disregards the fact that rigid rods cannot pass through each other. ENERIC SINGULAR CONFIGURATIONS OF LINKAGES 5
Thus a proper treatment of configurations in R must cut our “naive” version ofEmb d ( T Γ ) (and thus C (Γ) and C ∗ (Γ)) along the subspace of configurations whichare not embeddings of the full graph T Γ . The precise description of such a “realistic”configuration space Conf( T Γ ) is quite complicated, even at the combinatorial level,which is why we work here with Emb d ( T Γ ), C (Γ), and C ∗ (Γ) as defined in § C (Γ) has a dense open subspace U (Γ) consistingof embeddings of the full graph (including its edges), which may be identified with adense open subset of Conf( T Γ ). We observe that even such a model Conf( T Γ ) isnot completely realistic, in that it disregards the thickness of the rigid rods.Unfortunately, the generic singularities we identify here are not in U (Γ). Never-theless, in some cases at least, our method of replacing one singular configuration byanother (see Section 2 below) allows us to replace the generic singularity in C (Γ) \ U (Γ)with a configuration in U (Γ ′ ), for a suitable linkage Γ ′ . See Section 5 for an exam-ple of this phenomenon (which also occurs in the 3-dimensional version of the linkagedescribed there).2. Local equivalences of configuration spaces
Let Γ and Γ ′ be two linkages. We would like to think of points in the respec-tive configuration spaces as being equivalent if they are both smooth, or both have“similar” singularities. Since these concepts are local, we make the following:2.1. Definition.
Two configurations V in C (Γ) and V ′ in C (Γ ′ ) are:(a) locally equivalent if there are neighborhoods U of V in C ∗ (Γ) and U ′ of V ′ in C ∗ (Γ ′ ), and a homeomorphism f : U → U ′ with f ( V ) = V ′ .(b) locally product-equivalent if there are neighborhoods W of V in C ∗ (Γ) and W ′ of V ′ in C ∗ (Γ ′ ) equipped with homeomorphisms W ∼ = U × R k (taking V to ( V , x )) and W ′ ∼ = U ′ × R m (taking V ′ to ( V ′ , y )), aswell as a homeomorphism f : U → U ′ with f ( V ) = V ′ .See [KM3] for other formulations of this and similar notions.Evidently, any two smooth configurations in any two configuration spaces are locallyproduct-equivalent.In the next section we decompose our configuration spaces into simpler factors(locally), gluing them along appropriate work maps. The singularities of the configu-ration spaces translate into work singularities on the factors, so we need an analogousnotion of work maps being locally equivalent (at smooth configurations), or locallyequivalent up to a Euclidean factor:2.2. Definition. If i : ∆ ֒ → Γ and i ′ : ∆ ֒ → Γ ′ are inclusions of a common rigidsub-mechanism ∆ (usually a single point) in two distinct linkages, and V ∈ C ∗ (Γ), V ′ ∈ C ∗ (Γ ′ ) are two smooth configurations, we say that i ∗ and ( i ′ ) ∗ are(a) work-equivalent at ( V , V ′ ) if there are neighborhoods U of V , U ′ of V ′ ,and W of i ∗ ( V ) = ( i ′ ) ∗ ( V ′ ), and a diffeomorphism f making the following DAVID BLANC AND NIR SHVALB diagram commute:(2.3) U Mm | ①①①①①①①①① i ∗ | U (cid:21) (cid:21) ✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰✰ f ∼ = / / U ′ (cid:17) q ●●●●●●●●● ( i ′ ) ∗ | U ′ (cid:9) (cid:9) ✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓ C ∗ (Γ) i ∗ (cid:15) (cid:15) C ∗ (Γ ′ ) ( i ′ ) ∗ (cid:15) (cid:15) C ∗ (∆) C ∗ (∆ ′ ) W h ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ ( (cid:8) ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ (b) S -equivalent at ( V , V ′ ) if there are neighborhoods W ∼ = U × R k of V and W ′ ∼ = U ′ × R m of V ′ and a homeomorphism f : U → U ′ as in § i ∗ factors through the projection π : W → U and ( i ′ ) ∗ factors through π ′ : W ′ → U ′ in such a way that the diagram analogous to(2.3) commutes.An important example of these notions is provided by the following simple mech-anism:2.4. Definition. An open k -chain is a linkage Γ k op , where T Γ is a connected lineargraph with k + 1 vertices (where all but the endpoints x (0) and x ( k ) are ofvalency 2), with lengths ( ℓ , . . . , ℓ k ). See Figure 3 below. It is natural to choose thebase-point x ⋆ := x (0) (fixed at the origin, say) to define the pointed configurationspace C ∗ (Γ k op ), and x e := x ( k ) as end-effector.The resulting workspace W is S d − × [ m, M ], for fixed 0 < m < M , where m = min {| P ki =1 ± ℓ i |} and M = P ki =1 ℓ i are respectively the minimal andmaximal possible distances of x e from x ⋆ . The spherical (or polar) coordinate θ ∈ S d − is the direction of the vector ~ v = x e − x ⋆ .A closed ( k + 1 )-chain is a linkage Γ k +1cl , where T Γ is a cycle with k + 1 vertices(of valency 2), having lengths ℓ = | x (1) − x (0) | , ℓ = | x (2) − x (1) | , . . . , ℓ k +1 = | x (0) − x ( k ) | (see Figure 1).A prismatic closed ( k +1 )-chain Γ k +1pcl has the same T Γ , with lengths ( ℓ , . . . , ℓ k )as for Γ k +1cl , but with the last link prismatic – that is, the length ℓ = | x (0) − x ( k ) | varies in the range m ≤ ℓ ≤ M .2.5. Lemma ([G]) . The work map ψ of an open chain is a submersion, unless V isaligned (that is, all links have a common direction vector ~ w in R d at V ). In thiscase the ( d − -dimensional subspace Im(d ψ ) V is orthogonal to ~ w . Clearly the configuration spaces of an open k -chain and the corresponding prismaticclosed ( k + 1)-chain are isomorphic. However, the following result will be useful inunderstanding the work map singularities of an open chain, by allowing us to disregardits ( d − Proposition. If Γ k op is an open k -chain with links ( ℓ , . . . , ℓ k ) , then the pointed configuration space C ∗ (Γ k op ) is S -equivalent at any configuration V to the reduced configuration space b C ∗ (Γ k +1pcl ) of a closed prismatic ( k + 1 )-chain. ENERIC SINGULAR CONFIGURATIONS OF LINKAGES 7
Figure 3.
Coordinates for the open chain
Proof.
We may choose ( θ, φ , . . . , φ k − ) ∈ ( S d − ) k as local coordinates for the smoothconfiguration space C ∗ (Γ k op ) near V , where φ i is the spherical angle between thevectors x ( i − x ( i ) and x ( i ) x ( i +1) (see Figure 3), and θ is as in § ~ v = ~ ).Thus in a coordinate neighborhood U ∼ = R k ( d − of V the work map i ∗ : U → R d − × [ m, M ] factors as ( π θ , ρ ), where π θ ( θ, φ , . . . , φ k − ) = θ is the projection,and ρ ( θ, φ , . . . , φ k − ) = k x ( k ) − x (0) k ∈ [ m, M ].Now for each ℓ ∈ [ m, M ], the fiber ρ − ( ℓ ) is diffeomorphic to the configurationspace C ∗ (Γ k +1cl ) of a closed chain having k + 1 links of lengths ( ℓ , . . . , ℓ k , ℓ ).As in § C ∗ (Γ k +1cl ) ∼ = S d − × b C ∗ (Γ k +1cl ), so C ∗ (Γ k +1cl ) is locally product-equivalent to b C ∗ (Γ k +1cl ), and in fact C ∗ (Γ k +1cl ) is S -equivalent to b C ∗ (Γ k +1cl ) withrespect to ∆ = { x ( k ) } . As ℓ varies, we obtain the mechanism Γ k +1pcl .If ~ v := x e − x ⋆ vanishes at V , but V is not aligned, then the work map ψ is a submersion at V , and the same holds for C ∗ (Γ k +1pcl ), so they are S -equivalent.If ~ v = z at V and V is aligned, choose the coordinate θ be the direction of thealignment vector ~ w . (cid:3) . Decomposing the work map. Consider an arbitrary mechanism Γ with base point x ⋆ and work map ψ : C ∗ (Γ) → R d for the end-effector x e . Note that C ∗ (Γ) is locally diffeomorphic tothe product S d − × b C ∗ (Γ) ( § b C ∗ (Γ) ֒ → C ∗ (Γ) → S d − (for ~ v := x e − x ⋆ ∈ S d − ) is locally trivial (assuming ~ v does not vanish). If we chooselocal spherical coordinates S d − × R + for the work space W ⊆ C (∆) ⊆ R d , thework map ψ : C ∗ (Γ) → W ⊆ S d − × R + may be written locally in the form(2.8) ψ = Id S d − × ˜ ψ : S d − × b C ∗ → S d − × R + for some smooth function ˜ ψ : b C ∗ → R + (which is the work function for the associatedreduced configuration space). Note that the derivative of the work function ψ maythus be written in the form:(2.9) (d ψ ) ( ~ v , ˆ V ) = (cid:18) I d − ( ∇ ˜ ψ ) ˆ V (cid:19) . which shows that d ψ has rank d or d − Proposition. If V = ( ˆ V , V ′ ) ∈ C ∗ (Γ) is a smooth configuration for a mecha-nism Γ with work function ψ = Id S d − × ˜ ψ as in (2.8) , with x e = x ⋆ , and ˆ V is a DAVID BLANC AND NIR SHVALB non-degenerate singular point of ˜ ψ , then C ∗ (Γ) is S -equivalent at V to an alignedconfiguration of an open n -chain for some n ≥ .Proof. By the Morse Lemma (cf. [Ma, Theorem 2.16]) we may choose local coordinates ~ t = ( t , . . . , t k − d +1 ) for b C ∗ (Γ) near ˆ V (where k = dim C ∗ (Γ)), so that ˜ ψ has theform(2.11) ˜ ψ ( ~ t ) = a + j X i =1 t i − k − d +1 X i = j +1 t i . On the other hand, by Proposition 2.6 the configuration space C ∗ (Γ n op ) for anopen n -chain at any configuration V ( n ) is S -equivalent to the reduced configurationspace b C ∗ (Γ n +1pcl ) at some configuration ˆ V ( n +1) , where Γ n +1pcl is a prismatic closed( n + 1)-chain. The reduced work mapˆ φ : b C ∗ (Γ n +1pcl ) → γ ⊆ W ⊆ R d assigns to each ˆ V ∈ b C ∗ (Γ n +1pcl ) the length of the variable link (with γ ∼ = [ m, M ], thesegment of possible lengths).As shown in [MT, Theorem 5.4], ˆ φ is a Morse function, having (non-degenerate)singular points precisely at the aligned configurations ˆ V ( n +1) of the closed chainΓ n +1pcl . Although Milgram and Trinkle do not calculate the index of ˆ φ at ˆ V ( n +1) ,their computation of the Hessian of ˆ φ in [MT, Key Example, p. 255], combined withFarber’s proof of [F, Lemma 1.4] for the planar case, show that this index is equal to n − k , where k is the number of forward-pointing links in the configuration ˆ V ( n +1) .Thus by the Morse Lemma again we may choose an aligned configuration ˆ V ( n +1) and local coordinates in b C ∗ (Γ n +1pcl ) around it so that ˆ φ too has the form (2.11),and thus C ∗ (Γ) is S -equivalent at V to b C ∗ (Γ n +1pcl ) at ˆ V ( n +1) . By Proposition 2.6it is then readily seen to be S -equivalent to C ∗ (Γ n op ) at the corresponding alignedopen-chain configuration V ( n ) . (cid:3) Pullbacks of configuration spaces
We now describe a procedure for viewing the configuration space of an arbitrarylinkage Γ as a pullback, obtained by decomposing Γ into two simpler sub-mechanisms.The basic idea is a familiar one – see, e.g., [MT].3.1 . Pullbacks.
Let Γ k op denote an open chain which is a sub-mechanism of Γ (cf. § ′ denote the mechanism obtained from Γ by omitting the k linksof Γ k op (and all vertices but x (0) and x ( k ) ). For simplicity we choose x ⋆ := x (0) as the common base-point of Γ, Γ k op , and Γ ′ , and x e := x ( k ) as the commonend-effector of Γ k op and Γ ′ . See Figure 4.The work space of both mechanisms Γ ′ and Γ k op (i.e., the set of possiblelocations for x e ) is contained in R d , and we have work maps ψ : C ∗ (Γ ′ ) → R d and φ : C ∗ (Γ k op ) → R d which associate to each configuration the location of x e .Note that the pointed configuration space C ∗ (Γ k op ) is a manifold (diffeomorphic to( S d − ) k ) with a natural embedding i : C ∗ (Γ k op ) ֒ → R kd , and similarly j : C ∗ (Γ ′ ) → ENERIC SINGULAR CONFIGURATIONS OF LINKAGES 9 x (k) x ( x ( k-1) Figure 4.
Decomposing Γ into two sub-mechanisms R M for a suitable Euclidean space R M . This can be done, for example, by usingthe position coordinates in R d for every vertex in Γ.Let X := C ∗ (Γ k op ) × R M and Y := R d × R M , and define h : X → Y to bethe product map φ × Id R M and g : C ∗ (Γ ′ ) → Y to be ( ψ, j ), so that g is anembedding of C ∗ (Γ ′ ) as a submanifold in Y . Since we have a pullback square:(3.2) C ∗ (Γ) (cid:15) (cid:15) / / C ∗ (Γ k op ) φ (cid:15) (cid:15) C ∗ (Γ ′ ) ψ / / W ⊆ R d , C ∗ (Γ) may be identified with the preimage of the subspace C ∗ (Γ ′ ) ⊆ Y under h .Let V ′ ∈ C ∗ (Γ ′ ) and V ( k ) ∈ C ∗ (Γ k op ) be matching configurations with ψ ( V ′ ) = φ ( V ( k ) ), and let x ∈ X be the configuration ( V ( k ) , j ( V ′ )), so that h ( x ) = g ( V ′ ):(3.3) x ∈ X h (cid:15) (cid:15) = C ∗ (Γ k op ) ∋ V ( k ) φ (cid:15) (cid:15) × R M ∋ j ( V ′ ) Id (cid:15) (cid:15) h ( x ) ∈ Y = R d ∋ φ ( V ( k ) ) × R M ∋ j ( V ′ ) V ′ ∈ C ∗ (Γ ′ ) ?(cid:31) g O ψ ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ (cid:3) j ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ We want to know if the point
V ∈ C ∗ (Γ) defined by ( V ′ , V ( k ) ) is singular.By [Hi, I, Theorem 3.3], V is smooth if h ⋔ C ∗ (Γ ′ ) – i.e., h is locally trans-verse to C ∗ (Γ ′ ) at the points x ∈ X and V ′ ∈ C ∗ (Γ ′ ), which means thatIm dh x + T V ′ ( C ∗ (Γ ′ )) = T V ′ ( Y ) = R d × R M .Since Id R M is onto, this is equivalent to:(3.4) Im(d φ ) V ( k ) + Im(d ψ ) V ′ = R d . Generic singularities in pullbacks. Clearly, the failure of (3.4) is a necessary condition for V = ( V ′ , V ( k ) ) to be singular in C ∗ (Γ). Note that if (3.4)does not hold, then neither (d φ ) V ( k ) n nor (d ψ ) V ′ n is onto R d . By Lemma 2.5, thefirst implies that the configuration V ( k ) n for the open chain Γ k op must be aligned,while the second implies that (d ψ ) V ′ n is of rank < d . Definition.
Given a pullback diagram as in (3.2), a configuration ( V ′ , V ( k ) ) ∈ C ∗ (Γ) ⊆ C ∗ (Γ ′ ) × C ∗ (Γ k op ) will be called generically non-transverse if ˆ V ′ is anon-degenerate singular point of ˜ ψ , and x (0) = x ( k ) .3.7. Remark.
Note that since ˜ ψ : b C ′∗ → R + is an algebraic function, generically itwill be a Morse function, so any singular point ˆ V ′ is non-degenerate. Likewise, inthe moduli space Λ = R k + for open k -chains, the subspace of moduli λ for whichΓ k op has no aligned configurations with x (0) = x ( k ) is Zariski open in Λ. Thusamong the potentially singular configurations of C ∗ (Γ) (i.e., those for which (3.4)fails), the generically non-transverse ones are indeed generic.3.8. Proposition.
Given a pullback diagram (3.2) , any generically non-transverseconfiguration ( V ′ , V ( k ) ) is the product of a Euclidean space with a cone on a ho-mogeneous quadratic hypersurface, so in particular it is a topological singularity of C ∗ (Γ) .Proof. Since ˆ V ′ is a non-degenerate singular point of ˜ ψ , by Proposition 2.10 thework map ψ : C ∗ (Γ ′ ) → W ⊆ R d is work-equivalent to the work map η of an openchain Γ n op at some aligned configuration V ( n ) . Thus the pullback diagram (3.2)may be replaced by one of the form(3.9) C ∗ (Γ) (cid:15) (cid:15) / / C ∗ (Γ k op ) φ (cid:15) (cid:15) C ∗ (Γ n op ) η / / W ⊆ R d , so that C itself is S -equivalent at ( V ( k ) , V ( n ) ) to the configuration space of a closedchain with ( n + k ) links at an aligned configuration (since φ and η were non-transverse). This is known to be the cone on a homogeneous quadratic hypersurface,by [F, Theorem 1.6] and [KM2, Theorem 2.6], so it is topologically singular. (cid:3) Inductive construction of configuration spaces
We now define an inductive process for studying the local behavior of a configura-tion V of a linkage Γ. This consists of successively discarding open chains of Γ whilepreserving the local structure.4.1 . The inductive procedure. We saw in § C ∗ (Γ) asa pullback of two configuration spaces C ∗ (Γ k op ) and C ∗ (Γ ′ ), where the first iscompletely understood, and the second is simpler than the original C ∗ (Γ).This idea may now be applied again to C ∗ (Γ ′ ): by repeatedly discarding (oradding) open chain sub-mechanisms, we construct a sequence of pullbacks(4.2) C (Γ n +1 ) (cid:15) (cid:15) / / C (Λ n ) φ n (cid:15) (cid:15) C (Γ n ) ψ n / / R d , ENERIC SINGULAR CONFIGURATIONS OF LINKAGES 11 for 1 ≤ n < M , where each Γ n − is a sub-mechanism of Γ n , with Γ = Γ M , andΛ n is an open chain in R d (so C (Λ n ) is a product of ( d −
1) -spheres). Themaps ψ n and φ n are work maps for the common endpoint of Γ n and Λ n .Each configuration V for Γ determines a sequence of pairs V ′ n +1 = ( V ′ n , V ( k ) n ) in C (Γ n +1 ), as in (4.2), where V ( k ) n is necessarily a smooth point of C (Λ n ). Evidently,if V ′ n is a smooth point of C (Γ n ), V ′ n +1 will be, too, if (3.4) holds.4.3. Remark.
Note that there is usually more than one way to decompose a givenlinkage Γ as in § n , Λ n ) M − n = k (Γ M = Γ) of this tree will be called a decomposition of Γ.This flexibility can be very useful in applying the inductive procedure (see § . Generic singularities in C ∗ (Γ) . Our goal is to use this procedure to studysingular configurations of C (Γ). Here we start with the simplest case, which is alsothe generic form of singularities in configuration spaces, as we shall see below.Thus we assume by induction that V ′ n is a smooth configuration, but (3.4) fails .Our goal is to analyze this failure in the generic case, and then show that in this case V ′ n +1 is a singular point. Eventually, we would like to use this to deduce that theoriginal configuration V is singular, too.In § § V ′ n +1 ∈ C (Γ n +1 ) is defined by a pair of smooth configurations ( V ′ n , V ( k ) n ), but(3.4) fails, then generically at least, V ′ n +1 is a topological singularity. However,this does not yet guarantee that the corresponding configuration V in C ∗ (Γ) itselfis singular (unless Γ = Γ n +1 , of course).4.5. Example.
Let Γ be a planar closed 4-chain with links of lengths ℓ (1) , ℓ (2) , ℓ (3) ,and ℓ (4) . See Figure 5. l ( l ( l ( l ( x ( x (2) Figure 5.
Workspace for the point x (2) of a closed 4-chainGenerically, b C ∗ (Γ ) is a smooth 1-dimensional manifold, with local parametergiven by θ (the angle between v (1) and v (3) , say). However, if ℓ (1) + ℓ (3) = ℓ (2) + ℓ (4) ,then b C ∗ (Γ ) has a topological singularity – a node – at the aligned configurationˆ V where the links v (1) and v (3) face right, say, and v (2) and v (4) face left (see [F, Theorem 1.6]). In fact, if there are no further relations among ℓ (1) , . . . , ℓ (4) ,this is the only singularity, and b C ∗ (Γ ) is a figure eight (the one point union of twocircles). We can think of Γ as being decomposed into two sub-mechanisms Γ ′ and Γ ′′ , each an open 2-chain: Γ ′ consisting of v (1) and v (2) , and Γ ′′ of v (3) and v (4) . Note that ˆ V := ( V ′ , V ′′ ), where V ′ and V ′′ are both aligned.In this case we can describe C ∗ (Γ ) explicitly in terms of the work map φ : C ∗ (Γ ) → R (for the vertex x e := x (2) ), which is a four-fold covering map atall points but V : in a punctured neighborhood of V , neither V ′ nor V ′′ can bealigned, and each independently can have an “elbow up” (+) or “elbow down” ( − )position, which together provide the four discrete configurations corresponding to asingle value of φ . In b C ∗ (Γ ), taken together, these give four different branches ofthe curve (parameterized by θ ) – which coincide at V . See Figure 6. (+,+) ( +,-) (-,-) ( -,+) Figure 6.
The four branches of b C ∗ (Γ )Now assume given a linkage Γ in which Γ = Γ as above (with ℓ (1) + ℓ (3) = ℓ (2) + ℓ (4) ). Assume that to obtain Γ we add an open 2-chain Λ Γ , havingvertices x (0) , x (3) , and x (4) , with k x (0) x (4) k = ℓ (5) and k x (3) x (4) k = ℓ (6) . Wetherefore now have a rigid triangle △ x (0) x (3) x (4) (with x (4) in “elbow up” or“elbow down” position relative to the edge x (0) x (3) ). Thus C ∗ (Γ ) = C ∗ (Γ ) × {± } ,and the singularity at V ′ := ˆ V is unaffected.In the last stage Γ = Γ is obtained by adding another open 2-chain Λ := Γ with one new vertex x (5) , k x (4) x (5) k = ℓ (7) and k x (5) x (1) k = ℓ (8) . We require theconfiguration V (2)2 of Λ in which x (1) , x (4) , and x (5) are aligned to coincidewith the aligned configuration V ′ = ˆ V of Γ (and thus V ′ = ( V ′ , +) of Γ ).The effect of adding Λ is to prevent the open chain Γ ′ = x (0) x (1) x (2) fromever being in an “elbow down” position, thus eliminating two of the four branches of b C ∗ (Γ ) (see Figure 6), so V := ( V ′ , V (2)2 ) (which reduces to ˆ V in C ∗ (Γ )) is not singular in C ∗ (Γ).To show that this is indeed so, consider an alternative decomposition of Γ (seeRemark 4.3 above), in which we start with the closed 5-chain Γ = x (4) x (5) x (1) x (2) x (3) ,with base point x (3) . See Figure 7. Note that V ′ corresponding to ˆ V is non-singular in C ∗ (Γ ). When we add the open 2-chain Λ = x (3) x (0) x (1) , we seethat the configuration V ( k )1 corresponding to V is aligned, but since the work map φ : C ∗ (Γ ) → R determined by the work point x (1) is a submersion at V ′ ,condition (3.4) holds at V = ( V ′ , V ( k )1 ), so V is smooth.4.6 . Singularities in the inductive process. In Example 4.5 we saw that asingularity appearing at one stage in the inductive process described in § ENERIC SINGULAR CONFIGURATIONS OF LINKAGES 13 x ( x ( x ( x ( x ( x (2) Figure 7.
An alternative decomposition of Γdisappear at a later stage. However, in that case the reason was that the alignedconfiguration V (2)2 of Λ = Γ matched up in (4.2) with the aligned configuration V ′ of Γ .4.7. Definition.
For any linkage Γ, a configuration
V ∈ C ∗ (Γ) will be called generically non-transversive if for some decomposition (Γ n , Λ n ) M − n = m of Γ = Γ M (see § V ′ m , V ( k ) m ) ∈ C ∗ (Γ m ) × C ∗ (Λ m ) is generically non-transverse inthe sense of Definition 3.6, and the open chain configurations V ( k ) n ∈ C ∗ (Λ n ) are notaligned for M > n ≥ m .4.8. Remark.
As noted in Remark 3.7, the condition that the original pair ( V ′ m , V ( k ) m )is generically non-transverse is indeed generic, in the sense that it occurs in a sub-variety of C ∗ (Γ m ) × C ∗ (Λ m ) of positive codimension. Since the work maps eachopen chain φ n : C ∗ (Λ n ) → R d are algebraic for each n > m , the subvariety of C ∗ (Γ n ) × C ∗ (Λ n ) consisting of pairs ( V ′ n , V ( k ) n ) for which V ′ n corresponds to V ′ n − (and eventually to V ′ m ) and V ( k ) n is aligned form a subvariety of positivecodimension, so the condition that V is generically non-transversive in the sense ofDefinition 4.7 is indeed generic among the singular points of C ∗ (Γ).4.9. Theorem.
For any linkage Γ , a generically non-transversive configuration V is a topological singular point of C ∗ (Γ) – in fact, the product of a cone on ahomogeneous quadratic hypersurface by a Euclidean space.Proof. Let ( V ′ m , V ( k ) m ) be a generically non-transverse configuration of C ∗ (Γ m +1 ) ⊆ C ∗ (Γ m ) × C ∗ (Λ m ), so by Proposition 3.8 it is the cone on a homogeneous quadratichypersurface. By induction on the decomposition (Γ n , Λ n ) M − n = m , we may assume thatat the n -th stage the configuration V ′ n ∈ C ∗ (Γ n ) has a neighborhood U of thestated form. By Definition 4.7 we know that the work map φ n : C ∗ (Λ n ) → R d is asubmersion at V ( k ) n , so it is work-equivalent at V ( k ) n (Definition 2.2) to a projection π : R N n → R d (see [L, Theorem 7.8]). Therefore, in the pullback C ∗ (Γ N +1 ) theconfiguration V ′ n +1 = ( V ′ n , V ( k ) n ) has a neighborhood U × R N n − d – which isagain of the required form. (cid:3) Remark.
Note that if Γ = Γ M has a decomposition (Γ n , Λ n ) M − n = m as in § V ′ n +1 = ( V ′ n , V ( k ) n ) for each n , then the configuration V = V ′ M ∈ C ∗ (Γ n ) = C ∗ (Γ) is smooth, of course. Thus we obtain a mechanicalinterpretation of all differentiable singularities in any configuration space: namely,they must occur at a kinematic singularity of type I for some sub-mechanism Γ n of Γ – that is, a (smooth) configuration V ′ n ∈ C ∗ (Γ n ) at which the work map ψ n : C ∗ (Γ n ) → R d is not a submersion (see [GA]).In fact, more than this is required, since at the same point V another sub-mechanism– namely, the open chain Λ n – must be aligned, and it must be “co-aligned”with V ′ n in the sense that together they are S -equivalent to an aligned closed chain(see proof of Proposition 3.8). We call this situation a conjunction of two kinematicsingularities. 5. Example: a triangular planar linkage
We now consider an explicit example, which shows how all singular configurationsof a certain type of planar linkage can be identified, by making use of a non-trivial S -equivalence.5.1 . Parallel polygonal linkages. In [SSB2], a certain class of mechanisms werestudied, called parallel polygonal linkages . These consist of two polygonal platforms .The first is the fixed platform, which is equivalent to fixing in R d the initial point x ( i )0 of each of k open chains (called branches ) (1 ≤ i ≤ k ), of lengths n (1) , . . . , n ( k ) ,respectively. The terminal point x ( i ) n ( i ) of the i -th branch is attached to the i -th vertexof a rigid planar k -polygon P , called the moving platform. See Figure 8. Figure 8.
A pentagonal planar mechanismIn the planar case, it was shown in [SSB2, Proposition 2.4] that a necessary condi-tion for a configuration V of such a linkage Γ to be singular is that one of the followingholds:(a) Two of its branch configurations V ( i ) and V ( i ) are aligned, with coincidingdirection lines Line( x ( i )0 , x ( i ) n ( i ) = Line( x ( i )0 x ( i ) n ( i ).(b) Three of its branch configurations are aligned, with direction lines in the sameplane meeting in a single point P (see Figure 9).For simplicity we assume that k = 3, so the two platforms are triangular.5.2. Remark.
In the type (a) singularity there is obviously a sub-mechanism Γ ′ whichis isomorphic to an aligned closed chain, so the corresponding configuration V ′ issingular. Evidently, the caveat exemplified in § V is singular in C ∗ (Γ). ENERIC SINGULAR CONFIGURATIONS OF LINKAGES 15 P ω ω ω Figure 9.
Singular configuration of type (b)5.3 . A sub-mechanism and its equivalent open chain.
We shall now showthat the same holds (generically) for type (b), using the approach of Section 3.Consider the sub-mechanism Γ ′ of Γ obtained by omitting the third branch (butretaining the fixed platform), with base point at x := x (3)0 (the fixed endpoint ofthe omitted branch), and work point at y := x (3) n (3) (the moving endpoint of thisbranch). Let V ′ be the configuration of Γ ′ corresponding to V of case (b) above(so in particular the remaining two branches are aligned).Assume that the first branch has n := n (1) links, and the second has n ′ := n (2) links. We may then choose “internal” parameters ( φ , . . . , φ n ) for the first branch,and ( ρ , . . . , ρ n ′ ) for the second branch (as in the proof of Proposition 2.6). Wecan then express the lengths ℓ = k x (1)0 x (1) n k and m = k x (2)0 x (2) n ′ k as functions of( φ , . . . , φ n ) and ( ρ , . . . , ρ n ′ ), respectively. Note that Γ ′ has n + n ′ + 1 degreesof freedom, so one additional parameter is needed. Two obvious choices are one ofthe “base angles” φ = ∠ ( x (1) n x (1)0 x (2)0 ) or ρ = ∠ ( x (2) n ′ x (2)0 x (1)0 ) for the two branches(see Figure 10). x (3) x (1) x n' (2) x n'' (3) x= y= x (2) x n (1) =(0,0) =(c,0) Figure 10.
The sub-mechanism Γ ′ However, for our purposes we shall need a different parameter, defined as follows:Let z be the meeting point of the direction lines Line( x (1)0 x (1) n ) and Line( x (2)0 x (2) n ′ )for the two branches (this is the point P of Figure 9). As our additional parameterwe take the angle θ between the direction line Line( x , y ) for the (missing) third branch and the line Line( y , z ) (see Figure 11). Note that θ = 0 or π in our specialconfiguration V ′ . Letting N := n + n ′ + 1, the standard parametrization for theopen N -chain Γ N op defines a (local) diffeomorphism F : C ∗ (Γ ′ ) → C ∗ (Γ N op ). x y m Figure 11.
The parameters θ , m , and ℓ for the sub-mechanism Γ ′ In order to show that F is a work-equivalent at V ′ to an aligned configuration V ( N ) of Γ N op (Definition 2.2), we must show that V ′ is a generic singularity for Γ ′ – that is, that the reduced work map ˜ ψ : b C ∗ (Γ ′ ) → R has an (isolated) singularityat V ′ , where ˜ ψ assigns to any configuration V of Γ ′ the length ˜ ψ ( V ) = k x , y k .It is difficult to write ˜ ψ explicitly as a function of θ : for this purpose it is simplerto use φ or ρ as above. However, if we fix the lengths ℓ = ℓ ( φ , . . . , φ n ) and m = m ( ρ , . . . , ρ n ′ ) of the direction vectors for the two branches, the resultinglinkage ˜Γ ′ is a planar closed 4-chain with one degree of freedom (parameterizedby φ , say), and the third vertex y of the moving triangle traces out a curve in R ,called the coupler curve for ˜Γ ′ (cf. [Hal, Ch. 4]). Therefore, the infinitesimal effectof a change in φ is the rotation of y about the point z described above (called theinstantaneous point of rotation for ˜Γ ′ ). In particular, the angle θ also changes, sowe deduce that d θ/ d φ = 0 at the aligned configuration V ′ . This allows us toinvestigate the vanishing of d ˜ ψ/ d φ instead of d ˜ ψ/ d θ .This is the point where we are assuming genericity of V ′ : it might happen thatthe coupler curve is singular precisely at this point, in which case d θ/ d φ mayvanish, so we are no longer guaranteed that θ is a suitable local parameter. But suchinstances of case (b) are not generic.Since in the reduced configuration space b C ∗ (Γ ′ ) we do not allow rotation ofΓ ′ about the base-point x ⋆ = ( x , y ), we may assume that x (1)0 = (0 ,
0) and x (2)0 = ( c, a := k x (1) n x (2) n ′ k and b := k x (1) n y k for the (fixed) sides of themoving triangle (with fixed angle γ between them), as in Figure 12.We find that the following identities hold:( ℓ + s ) cos φ = c − ( m + t ) cos ρ ( ℓ + s ) sin φ = ( m + t ) sin ρa cos( ψ − γ ) = c − m cos ρ − ℓ cos φa sin( ψ − γ ) = m sin ρ − ℓ sin φ ENERIC SINGULAR CONFIGURATIONS OF LINKAGES 17 x y z b s a t c m Figure 12.
Angles and lengths in the sub-mechanism Γ ′ (where ψ is the angle between side b and the x -axis), and: a = ( c − m cos ρ − ℓ cos φ ) + ( m sin ρ − ℓ sin φ ) ˜ ψ = ( ℓ cos φ + b cos ψ − x ) + ( ℓ sin φ + b sin ψ − y ) . After differentiating we find:d(cos ρ )d φ = ℓtms sin ρ and d(cos ψ )d φ = − ℓs sin ψ , and we deduce that d ˜ ψ/ d φ vanishes if and only if: bℓ sin( φ − ψ )+ bs sin( ψ − φ )+( s cos φ, s sin φ ) · ( − y , x )+( b cos ψ, b sin ψ ) · ( − y , x ) = 0 . This formula expresses the fact that the area of the triangle △ x (1)0 zx is the sum of theareas of the quadrangle x (1)0 x (1) n yx and △ x (1) n zy , which holds if and only if x (1) n yx are aligned. From the formulas for ℓ = ℓ ( φ , . . . , φ n ) and m = m ( ρ , . . . , ρ n ′ ) wesee that d ˜ ψ/ d φ i and d ˜ ψ/ d ρ j all vanish at V ′ (as for any aligned open chain),so in case (b) ∇ ˜ ψ = 0, taken with respect to ( θ, φ , . . . , φ n , ρ , . . . , ρ n ′ ). Since allbut one of the parameters are the standard internal angles for open chains, we cancheck that the Morse indices for the reduced work maps of Γ ′ and Γ N op matchup at V ′ and F ( V ′ ), showing that F is indeed a work-equivalence (see proof ofProposition 2.10). Thus we may apply Proposition 3.8 to deduce that V ′ is a conesingularity.5.4. Summary.
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ENERIC SINGULAR CONFIGURATIONS OF LINKAGES 19 [ZFB] D.S. Zlatanov, R.G. Felton, & B. Benhabib, “Singularity analysis of Mechanisms andRobots via a Motion–Space Model of the Instantaneous Kinematics”,
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Dept. of Mathematics, U. Haifa, 31905 Haifa, Israel
E-mail address : [email protected] Dept. of Industrial Engineering, Ariel Univ. Center, 47000, Ariel, Israel
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