On singularities of real algebraic sets and applications to kinematics

Abstract We address the question of identifying non-smooth points in V ℝ ( I ) {{\bf{V}}}_{{\mathbb{R}}}(I) the real part of an affine algebraic variety. Two simple algebraic criteria will be formulated and proven. As an application, we investigate the configuration spaces of the planar four-bar linkage and the delta robot and prove that all singularities are CS-singularities.


Introduction
For any zero set = ( ) , there is the question of identifying points where X is not locally a submanifold of n .
The standard approach to this problem is to look for points ∈ p X, where the rank of the Jacobian of ( … ) g g , , k 1 drops below the height of I, which is the codimension of ( ) I V . Unfortunately, this is in general not enough to imply that X is not locally a submanifold. Obviously problems arise, if I is not radical or equidimensional (cf. Examples 1.1(ii) and (iii)) and techniques to handle this are well known (although not computationally feasible in some cases), but there are more intricate difficulties for real algebraic sets, where the localization of the reduced coordinate ring is not regular and = ( ) X I V is still a smooth submanifold of n at this point (cf. Examples 1.1(vi)).
The following examples illustrate different kinds of behaviour of real algebraic sets at points where the Jacobian drops rank.

Examples 1.1
In all examples, we use the notation m ≔ 〈 〉 ≤ [ ] shows the expected behaviour. m A is not regular and = ( ) is not locally a manifold at the origin.
x xy x y , , 2 . Then = ( ) X I V is just the y-axis, which is locally a manifold at the origin, although m A is not regular. The problem here is clearly that I is not a radical ideal, i.e.
localized at m is regular. In theory, I can be calculated algorithmically with Gröbner base methods (e.g. radicalðIÞ calculates the radical in the computer algebra system (CAS) SINGULAR). Unfortunately, the computation is unfeasible in many cases. But we will see a useful criterion to decide if A is already reduced, which is the case for many polynomial systems, which originate from engineering problems, see Proposition 2.3. is the union of the x-axis, the y-axis and the plane given by = z 1. At the origin X is certainly not locally a submanifold, but the rank of the Jacobian at the origin equals = I ht 1. Note that I is radical, but not equidimensional and m A is not regular. In this case, we need to calculate an equidimensional decomposition before applying the Jacobian criterion. Again this is possible with Gröbner base methods (primdecGTZðIÞ calculates a primary decomposition in SINGULAR) but as hard as the computation of the radical. However, for many problems in kinematics A is (locally) a complete intersection ring, which implies that I is equidimensional, see Proposition 2.3.
(iv) Let = 〈 + 〉 ≤ [ ] I x y x y z , , 2 2 . Then = ( ) X I V is the z-axis, which is a submanifold of 3 , although the rank of the Jacobian drops at any point of X below I ht and I is radical and equidimensional. This difficulty only appears in real geometry, since = ( ) X I V is not locally a complex manifold at any point of the z-axis. The problem is that , is a regular local ring.
There are algorithms to compute the real radical = ( ) x (e.g. realradðIÞ computes the real radical over in SINGULAR) but this computation is harder than that of the usual radical.

Also, we have in general
⋅ [ ] ≠ ⋅ [ ] = ( ) I x I x X Īr r (see Example (v)), in contrast to the usual radical. If this is the case not much can be gained by computing ( ) X I . We present a very useful criterion by T. Y. Lam [1] to check for an ideal I whether ( ) = X I I , see Proposition 2.6.
(v) Let = 〈 − 〉 ≤ [ ] I x y x y 5 , 3 3 and = ( ) X I V , which is just the line given by = x y 5 3 . The Jacobian drops rank at the origin but X is an analytic submanifold of 2 . Note that ( ) = I X I , but ( ) = 〈 − 〉 I X x y 5 3 .
(vi) This example motivated this paper. Let = 〈 + − 〉 ≤ [ ] I y x y x x y 2 , 3 2 4 and = ( ) X I V . We will see that m A is not regular and even ( ) = X I I , but = ( ) X I V is the analytic submanifold of 2 shown in Figure 1. Thus, we have established that = ( ) I X I and m A not regular does not imply that X is nonsmooth at the origin.
The reason here is that some analytic branches are not visible in the real picture. We can decompose + − y x y x 2 3 2 4 in the ring of convergent power series { } x y , : x y x x y y x y y 2 1 1 1 1 .
is negative for y close to zero, hence the real zero set of ( , coincides with the real zero set of + − y x y x 2 3 2 4 on the domain of g. Since (∂ ∂ ) ≠ ( ) g g , 0 ,0 x y at the origin, X 2 is clearly a submanifold there.
Here m A is not regular and ( ) = X I I again. But in this case X is not locally an analytic submanifold at the origin although the real picture looks very "smooth," which is because X is a C 3 -submanifold (but not C 4 ).
It is well known that any real algebraic set which is (locally) a smooth ( ∞ C )-manifold is also a real analytic manifold (see Proposition 3.3), so any "nonanalytic" point is at the most "finitely differentiable." This example emphasizes the need for an algebraic criterion to algebraically discern between the singularities seen in the last two examples, because the real picture can be very deceiving. Criteria to identify points that are not locally topological submanifolds are beyond the scope of this article, although we will see that we can rule out this case in a lot of situations. ( + ).
In this paper we want to show strategies how to deal effectively with all the problems seen in the examples, arising in the study of singular points of real algebraic sets. This is of great importance in the theory of linkages [2,3], when studying local kinematic properties, since the configuration space of a linkage will usually be given as a real algebraic set. Thus far, there is no broad consensus in the kinematics community how to handle points with rank drop in the Jacobian of the constraint equations. However, it is largely accepted that a configuration space (CS)-singularity should be defined as nonmanifold point of the configuration space. But then it is difficult to conclusively identify those points as the previous examples show.
This problem gained momentum when it was observed [4] that a closed 6R-chain exists with rank drop in the Jacobian of the constraint equations but smooth configuration space nevertheless. In this example, the singular configuration turned out to be an embedded point of the configuration space and would vanish upon taking the radical of the ideal of algebraic constraint equations. The same phenomenon can be seen in Example (ii). Proposition 2.3 will show that this can only happen for overconstrained mechanism.
For further discussion we refer the reader to Sections 7 and 8, where we investigate the configuration spaces of two types of linkages with the developed techniques. This will demonstrate how to identify CSsingularities for a large class of linkages. We will be able to address all questions raised in [5].
As mentioned earlier, the problem underlying Example (vi) requires some novel theoretical machinery. One of our main results will be: Theorem 1.2. Let Y be an irreducible -variety embedded in n . Assume Y is normal at ∈ p Y and p is in the euclidean closure of the real nonsingular points of Y. Then p is a manifold point of = ∩ Y Y n if and only if Y is nonsingular at p.
Here, manifold point means any point ∈ p Y such that Y is locally a smooth submanifold of n at p, see Definition 3.2. We will see that this property and the fact that p is a limit point of real nonsingular points are intrinsic to Y, so we could also formulate Theorem 1.2 in a coordinate invariant way (without specifying an embedding).
Note that any variety which is locally a complete intersection and for which the codimension of the singular locus is greater than 1 is normal according to Serre's criterion [6, Theorem 39]. Thus, Theorem 1.2 provides a useful tool for the analysis of many algebraic sets Y with > Y dim 1. For example, an immediate corollary for dimensions greater than 1 is that any isolated hypersurface singularity is either isolated in Y or a nonmanifold point.
Note also that if p is not a limit point of the real nonsingular points of Y one can find a euclidean neighbourhood U of p such that U does not contain any real nonsingular points of Y. We can now replace Y by the Zariski closure ′ Y of ∩ U Y . If we iterate this procedure we find an algebraic set Ỹ and a euclidean neighbourhood Ũ of p such that ∩ = ∩ Y U Y Ũ˜˜and p is in the euclidean closure of the nonsingular real points of Ỹ . Since p will be a manifold point of Y if and only if it is a manifold point of Y we can now analyse Ỹ in place of Y.
If Y is not normal, one has the option to calculate a normalization → φ Z Y : . We will see in Example 5.3 how to utilize φ and Theorem 1.2 to analyse many algebraic sets with > Y dim 1. We will also see in which cases this approach does not work.
For algebraic curves normalization yields the following result:  3 Thus, lying over the origin is , , 2 , , .
Hence, there is exactly one real point ( ) 0, 0, 0 lying over the origin and this is a simple root. According to Theorem 1.3, the origin must be a manifold point of ( ) f V .
Further curve examples and also a discussion on how the conditions of Theorem 1.3 can be checked symbolically by CAS like SINGULAR can be found in Example 6.5.
Before the examples from kinematics in Sections 7 and 8, the paper is structured as follows: in Sections 2 and 3 we review some well-known facts from commutative algebra, real algebra and differential geometry, which will enable us to make precise the notion of manifold point and deal with Examples (i)-(v). We will also focus on base extensions of affine algebras, which is very handy if one needs to extend results gained by algorithmic calculations in polynomial rings over to polynomial rings over .
In Section 4, we build the theoretical foundation for local analysis of real algebraic sets. Central to the exposition is Theorem 4.3, which gives an algebraic condition for manifold points and shows together with Risler's analytic Nullstellensatz that this is an intrinsic property. Finally, we can derive Theorem 1. x will not be prime in general and symbolic calculations in x are not possible. Instead, we investigate the integral closure of the local ring to divide the associated primes of the extended ideal. The main result Proposition 5.2 goes back to Zariski and Samuel [7] and was extended by Ruiz x I x¯. Ultimately, in Section 6 we formulate and prove Theorem 6.4, which is a ring-theoretic formulation of Theorem 1.3 and decides the case completely for real algebraic curves. This extends results of [9].
This paper is mostly build on the work of Risler [10], Efroymson [11], and Ruiz [8]. For further reading regarding local properties of real algebraic sets the author also recommends O'Shea and Charles [12] for their work on geometric Nash fibres and real tangent cones.

Algebraic preliminaries
In this section, let be a field such that

Remark. ( )
A red denotes the reduction / ( ) A 0 without nilpotents. The stacking of subscripts in m (( ) ) A red p is admittedly horrible but we will see in Proposition 2.2 that there is some freedom in the choice of the coefficient field. So we can get rid of the complexification and/or the reduction in Definition 2.1 if I is radical and/or m ≤ A p .

Base change
We review some facts from commutative algebra regarding extensions of the coefficient field.
Recall that for any prime ideal p ≤ A of a commutative ring the height of p is defined as the supremum of lengths of chains of prime ideals ending at p. Thus, the height of p coincides with p A dim . For an arbitrary ideal ≤ I A, the height of I means the infimum of the heights of all prime ideals containing I. We will write I ht to denote the height of I. It is well known that A noetherian implies Proposition 2.2. Let be any field extension of . Then is regular for one and then all associated primes P of p .
Remark. Since we require ⊂ , is a perfect field in particular and therefore separable over . This means (note that ⊂ does not need to be algebraic) that every finitely generated subextension is separably generated over , see [15, A1.2]. (i) and (ii) would work for any field extension of any field , whereas (iii) and (iv) are in general wrong if ⊂ is not separable.

Proof. (i) and (ii) follow because [ ] ⊂ [ ]
x x¯is a faithfully flat ring extension. (iii) is a consequence of the fact that any reduced -algebra is geometrically reduced [16, Lemmas 10.42.6 and 10.44.6]. We will show (iv) with the general Jacobian criterion [14,Theorem 5.7.1], since there appears to be no reference in the usual literature on commutative algebra.
First choose any associated prime P of p and let p, P denote the preimages of p and P in [ ] , since the tensor product commutes with direct sums. Consequently, , as stated in the beginning of Section 2.

Now assume p
A is a regular local ring and choose an associated prime q of I with q p ⊂ˆ(note that there should be only one prime with this property, otherwise p A would not be regular). Then we conclude that q = h ht from the Jacobian criterion. Now any associated prime of q has height h as well [7, VII Theorem 36] and one of them is contained in P . But then P ( ) A is regular according to the general Jacobian criterion.
On the contrary, assume that P ( ) A is regular. Then there exists an associated prime Q of I with Q P ⊂â is a prime ideal associated with I. Now But then q r p = ⊂ˆ. Also, Q q = ≥ h ht ht . Consequently, p A is regular according to the general Jacobian criterion. □

Locally complete intersection rings
arising from engineering problems, it is often the case that x Ī is locally a complete intersection [15, p. 462] and in particular Cohen-Macaulay. This has very useful implications: . Then (i) I is equidimensional, i.e. any associated prime of I has the same dimension − n k. (ii) Let J be the ideal of the k-minors of the Jacobian of ( … ) , then I is radical.
Proof. The first statement is just the unmixedness theorem [15,Corollary 18.14]. Note also that any associated prime of I is minimal over I.

Real algebra
We review some facts from real algebra. Most of them can be found in [1] or [17].
Definition 2.4. Let B be any commutative ring and ≤ I B an ideal. B is called (formally) real, if and only if any equation which is either the smallest real ideal containing I or B if there are no real ideals between I and B, cf. [17]. Therefore, I is real if and only if = I I r .
The analogue to Hilbert's Nullstellensatz in real algebraic geometry is as follows: x be any ideal. Then x (e.g. realrad in SINGULAR), but to the author's knowledge, all implemented algorithms so far only compute over since there is ambiguity in the ordering of field extensions of (in SINGULAR we get realradðx 3 À 5y 3 Þ ¼ x 3 À 5y 3 ).

Examples 2.7
(i) is clearly not real, since + = i 1 0 2 2 , but and are. Also, any domain B is real if and only if its field x y x y z , , 2 2 from Example 1.1(iv). I is not real, since ∉ x y I , . We see easily from the definition that ∈ x y I , r and from the real Nullstellensatz follows that ∉ I 1 r . Hence, x y x y 5 , 3 3 from . This is different for the standard radical, see , since this is a polynomial of degree 3. Also, the local ring at ( ) x y , 0 0 is regular with the Jacobian criterion, hence according to Proposition 2.

Analytic preliminaries
In this section, we set = , . Any open subset ⊂ U n is meant to be euclidean open. f is called analytic to an open set ⊂ V n , with where → ρ : n d is the projection to the first d coordinates. Note that it needs to be checked that this gives a local definition. We leave this task to the reader.
Proposition 3.1. Let ⊂ X n be any set and ∈ p X. The following conditions are equivalent: (a) There is an open neighbourhood U of p such that ∩ X U is an analytic (smooth, holomorphic) submanifold of n .
(b) There exists a permutation → π : n n of coordinates such that is locally the graph of an analytic (smooth, holomorphic) mapping at ( ) π p . (c) For a generic choice of ∈ ( ) A n GL , , ( ) A X is locally the graph of an analytic (smooth, holomorphic) mapping at Ap.
Remark. Generic choice means as usual the complement of a proper algebraic subset of ( ) n GL , . Such a subset is dense in ( ) n GL , in the euclidean topology. Any smooth mapping parameterizing a real algebraic set will be a smooth semi-algebraic mapping whose component functions are known to be Nash functions [17, Definition 2.9.3, Proposition 8.1.8] and in particular analytic. We get the following proposition:

Local real algebraic geometry
We now assume = and that the origin is a singular point of a real algebraic set X. So we have an ideal As we have seen in Example 1.1(vi) we need to investigate the extension of I in the ring of convergent/formal power series. The following notations will be used in the rest of the paper: Since the ring extensions We will also need the fact that is the r-adic completion of : Now we define the following ideal of { } x , which is usually called the vanishing ideal of the set germ ( ) X,0 [10]. We can do a similar construction in [[ ]] x , but since ( ) f p is not defined in general for elements ∈ p n , x , we need to replace points in n with tuples of formal Puiseux series without constant term. See x Î¯0 euclidean neighbourhood with converging on and 0 on ,ˆ¯ˆ. Proof. First, let the origin be a manifold point of X (of dimension d). According to Proposition 3.1, we find w.l.o.g an analytic parameterization of X: where U is an euclidean neighbourhood of the origin in d and ( ) = Ψ 0 0. We set 1 1 1 and claim that = L Iˆ.
Clearly we have ⊂ L Iˆ, so let ∈ a Iˆ. Since ( ) = Ψ 0 0, we can compose a and Ψ and get a converging power series which follows because ∘ a Ψ is identically zero close to the origin. We now set  (3) and x L is a regular local ring. We will use Nagata's Jacobian criterion [8, This means that the submatrix comprising the first ( − ) n d columns of the Jacobian matrix of ( … ) − g g , , n d 1 evaluated at the origin has full rank. Now let U be a euclidean environment of the origin in n such that U is contained in the region of convergence of all g i . We set ′ ≔ { ∈ | ( ) = } X x U g x i 0, for all .
The next proposition collects some well-known facts on the relationship of the rings , and . is also an easy consequence of results in [8]. We will carry out a proof for completeness sake. , which states that = ′ I Î in the complex setting. Then we see easily from Proposition 4.5 why there is no need in complex algebraic geometry to consider the completion of to decide if ˆi s regular. Because for = we have = / ( ) = 0 if is reduced, and is regular iff is regular.
In the real case, it is not enough for I to be real to imply the realness of ′ I , see Example 1.1(vi), hence Iˆis in general bigger than ′ I and the nonregularity of does not imply the nonregularity of ˆ. On the other hand, if is regular, then is regular and real, hence also = . In Section 6, we analyse thoroughly how to check that ″ I is real for = I dim 1. But there is also a useful criterion for many higher-dimensional algebraic sets. This leads directly to the first of our main results listed in the introduction. We repeat it here for completeness sake: Let Y be an irreducible -variety embedded in n . Assume Y is normal at ∈ p Y and p is in the euclidean closure of the real nonsingular points of Y. Then p is a manifold point of = ∩ Y Y n if and only if Y is nonsingular at p.
Proof. According to Corollary 4.6 and Theorem 4.7, the only thing we need to show is that ( ) Y I is real. But this follows from the simple point criterion, Proposition 2.6. □ Corollary 4.8. Let X be a normal, irreducible real algebraic variety embedded in euclidean space. Any isolated singularity of X is either a nonmanifold point of X or isolated in X.
See Example 5.3 for a demonstration of Theorem 1.2 and Corollary 4.8. If X is not normal, we can calculate a normalization of X and utilize Efroymson's criterion again. This will be the subject of Section 5.

Normalization and analytic branches
In order to decompose the extended ideal ″ I , we examine the normalization of , which can be compared to the normalization of . We assume again = , but also require ≤ [ ] I x to be a radical ideal, with minimal decomposition x Ī , we then have the following minimal primary decomposition of the zero ideal: where p i is the prime ideal generated by p ′ i in for = … i s 1, , . From now on we will use the notation p = / i i and for any reduced ring A we will write A for the integral closure of A in its total ring of fractions. The following lemma collects some well-known facts about the integral closure of reduced local rings. Recall that r is the maximal ideal of . and n ′ ij is one of the k i maximal ideals of i . We also have the following minimal primary decomposition We now want to compare the normalization of and the completion of , so we need to investigate what form n can take for n ≤ maximal. Since n n = ( ) ′ i for some i and n ′ ≤ i maximal we assume that is a domain. The following exposition is taken from [8, Section VI.4], which can be checked for details. Because r n = / ⊂ / is an algebraic field extension it must be n / = , . We distinguish between the following three cases:  Proof. The only thing missing from the proof in [8] is to take into account nondomains , so we need to check = ×…× .
. The real part of = ( ) Y I V is plotted in Figure 2. Since the singular locus of Y is just the origin and A is a complete intersection ring, Y must be normal according to Serre's criterion [6, Theorem 39] or use [15,Theorem 18.15].
Thus, because the origin is not isolated in ( ) I V , it must be a nonmanifold point of ( ) I V according to Corollary 4.8.
x y z x y z , , 2 2 . The real part of = ( ) Y I V is called the Whitney umbrella and plotted in Figure 3. It demonstrates nicely the scope and limits of Theorem 1.2 and Corollary 4.6.
x y z x y z , , 2 2 be the coordinate ring of Y. Since Y is not normal we cannot apply Theorem 1. Since B is the coordinate ring of a nonsingular variety Z it must be integrally closed and ψ is the normalization of A. ( + ). Figure 3: The singular locus of Y is clearly given by the z-axis = x y , 0. Let = ( ) p z 0, 0, 0 be any point on the real part of this axis. We want to use Corollary 4.6 to decide if p is a manifold point of = ( ) Y I V , so we have to check, if the completion of the local ring of Y at p is real. Because is real if and only if is real and normalization commutes with completion in the sense of Proposition 5.2 it is enough to show that all points … q q , , k 1 lying over p in the normalization are real and the completion of the local ring of Z at q i is real for all i.¹ Now . It follows that is not real so we cannot use Corollary 4.6. However, since there are no real points lying over p, this means that p is not a limit point of the real nonsingular points of Y [11]. Thus, we can find an euclidean neighbourhood U of p not containing any real nonsingular points of Y. The Zariski-closure of ( ) ∩ I U V is then just the z-axis = ( ) Y x y Ṽ , and the real part of Ỹ is a submanifold of 2 .

Real algebraic curves
Now we apply the theory of Section 5 to singularities of real algebraic curves. Let . Then, the structure of = / ( ) 0 r will be especially nice, since the real radical of an associated prime q of ′′ I will be either q itself or the maximal ideal m m x : x be a prime with q = − n ht 1. Then Proof. Any ideal is real if and only if it is radical and all associated primes are real [11]. So q r is the intersection of all real primes containing q. We only have to show that m″ is real, but this is clear, since  if and only if one of the following two conditions is true (a) There is exactly one real maximal ideal n ≤ lying over r m = ⋅ and r ⋅ is n-reduced. (b) All the maximal ideals n ≤ are not real. In this case, the origin is an isolated point of ( ) I V .
Proof. First, let q q ( ) = ∩…∩ 0 i i Now we will show the following assertion: Clearly q / i is real if and only if q / i is real, since they are contained in the quotient field of q / i , so we need to show that n ( ) * i is real if and only if n i is real. If n i is not real, then n / ≅ i and one can see from the construction before Theorem 5.2 that n ( ) * i will not be real (since n ( ) On the other hand, let n i be real, then n ( ) * i will be the n i -adic completion of the local ring n i . Since is normal of dimension 1, we also know that n i is regular according to Serre's regularity criterion R 1 [6, Theorem 39]. Then n ( ) * i is regular too [13,Proposition 11.24], with residue field n / = i . Hence, n ( ) * i must be real because of [1,Proposition 2.7]. This proves assertion (5).
Next, we consider where q′ i is the preimage of q i in [[ ]] x . As one checks easily q i is real if and only if q′ i is real.
If none of the n i is real, then none of the q′ i is real and according to Lemma 6.1, we would get Now we examine the case that exactly one n i is real, w.l.o.g. we choose n 1 real. Then q ′′ = ′ I 1 r . We have the following commutative diagram: First, we assume that r n n n ⋅ = ⋅ 1 1 1 . We proceed in two steps.   So r is n 1 -reduced. This completes the proof. ] / x y T T N , , , 1 2 , where N is referenced in SINGULAR by the handle norid. The previous output is a minimal primary decomposition over of the ideal J generated by (〈 〉) ψ x y , in B. See also equation (2), where we calculated just that for the same example.
The second entry in both lists is the radical of the respective primary component of J. Since they are the same, the primary ideals in this decomposition must be prime and J is radical, i.e. any zero of J is simple. We see that there is exactly one real zero lying over the origin (from ideal [1] in the primary decomposition) and this is a simple zero. Thus, the origin is a manifold point of ( ) f V . We should note that ⊗ B is the normalization of ⊗ A, because any normal -algebra is geometrically normal [16,Section 10.160.4]. This means there is no problem with calculating the integral closure of A in SINGULAR over the rational numbers.
is the cusp from Figure 4. Let us run Listing 1 with the second line replaced by ideal I = x^2 -y^5. Then, we get the output: : for the normalization of A. Also, we denote with J the ideal generated by (〈 〉) ψ x y , in B. We see from the output above that there is exactly one real zero q lying over the origin. But this is not a simple zero, because the maximal ideal n q is not a primary component of J, i.e. J is not n q -reduced. According to Theorem 6.4 the origin is not a manifold point of ( ) g V .
6 . The real curve ( ) g V is plotted in Figure 5. Listing 1 with the second line replaced by ideal I = y^3 -y^4 + y * x^4 -x^6 gives the following output: : for the normalization of A. Also, we write J for the ideal generated by (〈 〉) ψ x y , in B. From the output we see that J is reduced and a prime ideal over . This means any root of J in is simple. It remains to check how many real roots J has. For this we could analyse the cubic, which is the first generator of J in the output above. But we can also use the real root counter of the SINGULAR library rootsmr:lib [27]: Hence, there is only one real root lying over the origin and this is a simple root. Thus, the origin is a manifold-point of ( ) h V .

CS-singularities of the four-bar linkage
In a basic description a linkage is a collection of rigid bodies connected by joints. Its configuration space is the set of all assembly configurations and can in general be represented by a real algebraic set, see Figure 6 and equation (8) for an example. A nonmanifold point of the configuration space is called a CS-singularity. In such a configuration, the linkage will exhibit degenerate kinematic behaviour. Recently, efforts have been made in the kinematics community to define and categorize kinematic singularities of linkages in a rigorous way [2,3,28].
As pointed out in the introduction and in [5, p. 227] one needs to check for every singularity of the configuration space whether it is a CS-singularity. In this section, we apply the theory developed so far to identify all CS-singularities of the four-bar linkage ( Figure 6). This linkage is one of the oldest and most widely used planar mechanisms in Kinematics and Mechanical Engineering. In its common form it consists of four bars of length l l l l , , , 1 2 3 4 connected in a circular arrangement by revolute joints (hinges) with one bar fixed to the ground.
Conditions on the design parameters l i such that there exist points with a rank drop in the constraint equations are well known, see e.g. [5,29] (Grashof condition (9)). Lesser known are methods to show that these points are CS-singularities, i.e. nonmanifold points. We will do this in a purely computational   algebraic way with our theory on algebraic curves. See also [30,Theorem 1.6] or [31] for a different method , since other cases can be treated in a similar way. Since = I dim 1 equidimensional we need to analyse the ideal J generated by I and all the 3-minors of the Jacobian of ( ) p p p , ,  . To get the equations of the strict transform on this chart, we need to remove the exceptional divisor, so we have to calculate the saturation ≔ ( 〈 〉 ) ∞ J I y : .
y This can easily be achieved with the command sat in SINGULAR. But again we have to be careful to check whether the Gröbner basis calculations stay valid for all assumed values for l l , 2 4 , so we will calculate the saturation manually. First we calculate ∩ 〈 〉 I y y , which we get by eliminating t of + 〈( − ) 〉 I t t y 1 .
y On singularities of real algebraic sets and applications to kinematics  1807 Now we divide any generator of ∩ 〈 〉 I y y by y and after checking that all coefficients of the leading monomials will not be zero after substitution of values for l l , 2 4 we normalize the generators and get the following Gröbner basis of = ( 〈 〉) J I y : In this last section, we want to apply some of our results to a linkage with a configuration space of dimension greater than 1. We will do this for the delta robot of Figure 7, but the same argument can be used for many other linkages, e.g. the five-bar linkage or the 3RRR-planar parallel manipulator of [5].
The delta robot is a parallel linkage that consists of three identical limbs which carry a platform serving as a Cartesian positioning device (Figure 7). It was developed in 1985 by a research team under the supervision of Reymond Clavel who described it first in his PhD thesis [33].
(c) A is a product of normal domains. In particular, the irreducible components of Y a b , (if there are more than one) have empty intersection.
With (a)-(c) above we can apply Corollary 4.8 and have the result that any real singularity of Y a b , is either a nonmanifold point of X a b , or isolated in X a b , . □ Listing 2: Analysis of the singular locus of the delta configuration space. On singularities of real algebraic sets and applications to kinematics  1811