Panagiotis Alexander D Rahm
email: alexander.rahm@uni.lu

Panagiotis Tsaknias
email: panagiotis.tsaknias@gmail.com

GENUINE BIANCHI MODULAR FORMS OF HIGHER LEVEL, AT VARYING WEIGHT AND DISCRIMINANT

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Spaces of genuine Bianchi modular forms have been used Berger, Dembélé, Pacetti and S ¸engün to construct evidence for the Brumer-Kramer paramodularity conjecture and the Eichler-Shimura conjecture [BDPS ¸15]. This evidence however consists so far in just one Abelian surface (that is paramodular in the Brumer-Kramer setting; and they deduce from it an Abelian surface satisfying the Eichler-Shimura conjecture). No other non-trivial example for the Eichler-Shimura conjecture in dimension 2 is present in the literature. This is due to the limitations of the database of genuine weight 2 Bianchi modular forms that was available to the four above-named authors. That database [RS ¸13] did treat only level One Bianchi modular forms, yielding extremely few genuine forms. So in order to systematically build up more evidence for the Brumer-Kramer paramodularity conjecture and the Eichler-Shimura conjecture in dimension 2, it will not be enough to simply continue on a larger database of level One genuine Bianchi modular forms, but rather one should use a database of higher level genuine Bianchi modular forms. The latter type of database is provided with the present paper [RS ¸T16]. Again, as for level One, the higher weight spaces of genuine forms are very rare, but the weight 2 spaces in our database are more abundant, and give hope for the construction of an algorithm that could systematically produce the desired Abelian surfaces.

In order to find the spaces of genuine Bianchi modular forms in the present database, a major task was to establish dimension formulas for the non-genuine forms: For level One, formulas were already established by Finis and Grunewald [START_REF] Finis | The cohomology of lattices in SL(2, C)[END_REF], yielding the dimensions of Langlands Base-Change forms and Complex Multiplication (CM) forms outside the Base-Change space. These dimensions of the non-genuine subspace did, in [RS ¸13], only need to be subtracted from the full dimension of the Bianchi modular forms space, which was computed on the machine from the geometry of the Bianchi modular group. The latter machine computations extend to higher level via the Eckmann-Shapiro lemma, which is implemented for calculating the cohomology that corresponds to the Bianchi modular forms space through the Eichler-Shimura(-Harder) isomorphism. The Base-Change dimension formulas however did require a greater effort for their extension to higher level. What added a particular additional challenge here, is that at higher levels, also twists of Base-Change appear, and have to be taken care of as part of the non-genuine space.

Result. Let K be an imaginary quadratic field of discriminant D K , odd class number h K and ring of integers O K . Let N ≥ 1 be a square-free integer, coprime to D K . Then we provide an explicit formula for the dimension of the space S nG k (N O K ) of non-genuine modular newforms of level N O K and arbitrary weight k in Theorem 3. This formula comes with instructions (Section 6) on how to evaluate it on a computer.

We also provide such formulas for class number 1, K = Q( √ -p) for some prime p ≡ 3 mod 4 and level p or p 2 where p 2 = (p) (see Section 6.3). But our dimension formulas need to be explicit enough to be evaluable on the machine, so they involve a lot of case distinctions, and consequently are provided only for a part of the levels. The authors hope that it should however become clear for experts how to obtain formulas for the missing levels. For instance, for some theorems we assume that the class number is odd, which is equivalent to the discriminant being -4 or -8 or -p, where p is a prime congruent to 3 modulo 4. However, for even discriminants and class numbers, one should proceed analogously (see the remarks in Section 7).

In conclusion, we can say that the formulas and databases established with the present paper constitute a significant progress on the question raised in section 9.1 of [BS ¸V16] on the exhaustion of spaces of newforms by non-genuine forms.

Organization of the paper. In Section 2, we fix the notation and assumptions that we will use throughout the paper. We also derive a first quantitative expression for the dimension of the Base-Change space. In Section 3, we take the twists of Base-Change into account. In Section 4, we establish formulas for the dimension of the space of CM-forms that are not Base-Change. In Section 5, we provide a formula for the total dimension of the space generated by newforms that are non-genuine, i.e. of any of the three types mentioned above. This formula is made explicit in Section 6, in a way that it can be evaluated on the machine. We make some concluding remarks on our formulas in Section 7. In Section 8, we present our machine results, in which the non-genuine dimension is subtracted from the full dimension of the newforms space, the latter being computed from the geometry of the Bianchi modular group. Due to the sparsity of the genuine forms, those results can be considered as a treasure map to the conditions under which they exist.

Setting

Let D > 1 be a square-free integer and let K = Q( √ -D) and denote its ring of integers by O K . For any ideal n of O K and integer k ≥ 2 we let S K k,k (Γ 0 (n)) denote the space of Cuspidal Bianchi modular forms over K of weight k, k, level n and trivial Nebentypus. This space admits an action of a Hecke algebra generated by operators indexed by the prime ideals of O K . Each eigenvector F of this algebra corresponds to an automorphic representation Π F in A 2 (K), the set of all cuspidal automorphic representations of GL(2, A K ). We are interested in the ones appearing in the image of the Base-Change operator defined by Langlands [START_REF] Langlands | Base Change for GL(2)[END_REF],

BC K Q : A 2 (Q) → A 2 (K),
along with their twists. Here we generalise the formula of [START_REF] Finis | The cohomology of lattices in SL(2, C)[END_REF] to arbitrary level n coprime to the discriminant D K of K. More precisely, we provide an algorithm computing the dimension of this space without really computing the spaces in hand. In some simple cases we also provide explicit formulas. We also provide an algorithm for level (p) = p 2 , where p is a rational prime that ramifies in K. We hope that our proposed strategy outlines a complete plan for the interested reader to fill out the missing level cases without much effort. It is clear that without loss of generality it is enough to study the Base-Change image problem on the new subspace of S K k,k (Γ 0 (n)), which we will denote by S k (n) in the rest of this paper. A necessary condition for Base-Change to have non-trivial image is that the level is Galois stable, i.e. an ideal generated by a rational number. Let n = N O K for some N ≥ 1 coprime to D K . The first crucial observation is that one can consider the Base-Change operator locally. Let Π = BC K Q (π) be an automorphic representation in the Base-Change image. Then for any prime ideal p coprime to D K , the conductor of Π p is the extension to O K of the conductor of π p , where p is the prime below p. We therefore get that if π contributes to the Base-Change subspace of S k (n), then its conductor away from D K is exactly N . This leaves us with the primes dividing the discriminant. Let p divide D K . Then Π p is a principal unramified series, since we required that the level is coprime to the discriminant. Let ω p be the quadratic character of conductor p. A complete list of all the possible local components π p of π at p is provided in [START_REF] Finis | The cohomology of lattices in SL(2, C)[END_REF] which we also provide here for convenience:

• unramified Principal Series.

• 1 ⊕ ω p .

• A certain supercuspidal representation ( of conductor p 2 when p > 2).

We now have a complete description of the inverse image of the Base-Change operator in terms of its inertial type at the primes p|N D K : If p|N , then the type can be anything of conductor equal to the power of p dividing N . And if p|D K , then it is one of the types mentioned above. In order to compute the dimension of the Base-Change subspace we would have to keep in mind that the Base-Change map is one-to-one when the component at a ramified prime is of the first kind and two-to-one if it is of the other two. Moreover, if a newform in the preimage spaces has CM by K, then its image is Eisenstein, so it does not contribute to the dimension. The above discussion yields the following formula.

Proposition 1. Let S(d) be the set of prime divisors of d. Let ω d be the quadratic character of conductor d, under the convention that ω d is trivial at a prime p|d if and only if p 2 ||d.

Then we obtain

dim S BC k (N O K ) = d|D 2 K 1 2 |S(d)| dim S d-sc,new k (Γ 0 (N d), ω d ) -dim S d-sc,new k (Γ 0 (N d), ω d ) CMK ,
where for each d, the space S d-sc,new

(Γ 0 (N d), ω d ) CM K is the CM subspace.
Of course many of the spaces appearing on the right hand side are trivial since the parity of their nebentypus does not match that of the weight k. The final key observation is that we have formulas for all of the spaces appearing on the right hand side of the above formula (see [START_REF] Cohen | Dimensions des espaces de formes modulaires[END_REF] or [START_REF] Finis | The cohomology of lattices in SL(2, C)[END_REF]).

Twists of Base-Change

It is fairly easy to see that it is possible to obtain non Base-Change forms just by twisting a Base-Change one by a suitable non Base-Change character. We would like to count these forms out of our genuine forms subspace and therefore we would like to find the dimension of the subspace they span as well. Let us denote by S tBC k (n) the subspace of S k (n) generated by those newforms that are twists of Base-Change. For this purpose, we say that

f ∈ S k (n) is twist of Base-Change, if there exists a classical weight k modular form g of level M ∈ N with M dividing n, such that f = BC K Q (g) ⊗ c
for some non-trivial character c. We first prove the following: Theorem 1. Let K be an imaginary quadratic field of discriminant D K and class number h K . Let N ≥ 1 be a square-free integer, coprime to

D K . Let n = N O K . Then for all k ≥ 2, dim S tBC k (n) = (h K -1) dim S BC k (n).
Proof. Since we are interested in newforms of square-free level, we get that the local type at any prime p is Steinberg or unramified principal series, depending on whether p|n or not. Let ǫ be a non Base-Change character of O * K and assume it has non-trivial conductor c(ǫ). Let p|c(ǫ) be a prime ideal of K. Then f ⊗ ǫ has level divisible by p 2 and its type at p is ǫ ⊗ St or ǫ ⊕ ǫ. In both cases, the type comes from Base-Change if and only if ǫ does. This shows that such an ǫ cannot be used to twist a non Base-Change form to a Base-Change one and vice versa. The only option left for ǫ is to be non Base-Change and to have trivial conductor. There precisely h k -1 many such characters. Indeed each such character has the desired property which gives the desired result.

CM-Forms

Another subspace we would like to exclude as non genuine is the one generated by CM-newforms. They are the ones whose corresponding automorphic representation occurs as the automorphic induction of a suitable Hecke character of a quadratic extension M/K. Let us denote this subspace by

S CM k (n). It is quite often the case that S CM k (n) ⊆ S BC k (n)
, but not always. Nevertheless one can easily prove the following: Theorem 2. Let K and n be as in Theorem 1. Moreover assume that h K is odd. Then for all

k ≥ 2, dim S CM k (n) = 0.
Proof. Let f be a newform that supposedly lies in S CM k (n). Let ψ be the Hecke character of some extension M/K whose automorphic induction is f . We then have that L(f, s) = L(ψ, s).

Equating local components at any prime p dividing the discriminant d M/K we see that p 2 should divide n. Since n is assumed to be square-free no such p should exist and thus the extension M/K is unramified. This provides the desired contradiction since the class number of K is assumed to be odd.

Total non-genuine subspace

In this section we provide a formula for the total dimension of the space generated by newforms that are "non-genuine" in any of the three notions discussed above, i.e. a formula for

dim S BC k (n) + S tBC k (n) + S CM k (n) .
Let us denote this space by S nG k (n). Then combining Theorem 1, Theorem 2 and Proposition 1 yields the following: Theorem 3. Let K be an imaginary quadratic field of discriminant D K and odd class number h K , and N ≥ 1 be a square-free integer, coprime to

D K . Let n = N O K . Then dim S nG k (n) = h K dim S BC k (n) = h K d|D 2 K 1 2 |S(d)| dim S d-sc,new k (Γ 0 (N d), ω d ) -dim S d-sc,new k (Γ 0 (N d), ω d ) CMK .
(1)

Dimension formulas

In this section we provide explicit formulas/algorithm for the right hand side of the formula in Theorem 3. We split the computation into two parts, each involving one of the two main ingredients of each summand in the right hand side of (1). 6.1. Computing dim S d-sc,new k (Γ 0 (N d), ω d ). Our starting point is Proposition 4.18 in [START_REF] Finis | The cohomology of lattices in SL(2, C)[END_REF] which we provide here for convenience. Here, we denote by S k (Γ(N )) the space of weight k elliptic modular forms for the principal congruence subgroup Γ(N ) ⊆ SL(2, Z) of level N .

Proposition 2 (Finis, Grunewald, Tirao). Let N ≥ 1 and k ≥ 2 be integers and σ a representation of G N = SL(2, Z/N Z) such that σ(-I 2 ) is the scalar (-1) k . Let U N ⊆ G N be the subgroup of all upper triangular unipotent elements and S 3 and S 4 the images in G N of elements of SL(2, Z) of order 3 and 4, respectively. Then,

dim Hom GN (S k (Γ(N )), σ) = k -1 12 dim σ - 1 2 dim σ UN + ǫ k tr σ(S 3 ) + µ k tr σ(S 4 ) + δ k,2 dim σ GN .
The constants ǫ k and µ k are explicit functions of k and δ k,2 is the usual Kronecker delta notation. In what follows, given a subspace of newforms B ⊆ S k (Γ(N )), we will say that σ defines B if Hom GN (S k (Γ(N )), σ) ∼ = B. We thus need to identify a suitable σ for each subspace involved in the last line of (1). In fact we only need to compute the following five invariants associated to such a sigma: dim σ, dim σ UN , tr σ(S 3 ), tr σ(S 4 ), dim σ GN .

We will denote them by I i (σ), i ∈ {1, 2, 3, 4, 5} respectively. It is important to notice that σ, and therefore the five invariants associated with it, depends only on the level structure of the subspace that is of interest to us and not the weight. It is also clear that the following properties hold:

I i (σ ⊕ σ ′ ) = I i (σ) + I i (σ ′ )
and

I i (⊗ p σ p ) = p I i (σ p )
for all σ = ⊗ p σ p , σ ′ and for all i ∈ {1, 2, 3, 4, 5}.

Let us fix one of the spaces in the right hand side of the formula in Theorem 3, S d-sc,new k (Γ 0 (N d), ω d ) say, and let σ be the representation defining it. In view of the properties just mentioned, we will compute the I i (σ) by determining the N -part and the d-part separately.

In order to compute the N -part, it is enough to notice that it corresponds to the one defining S new k (Γ 0 (N )). Given an integer N ≥ 1, let σ N = ⊗ p σ N p and σ N,new = ⊗ p σ N,new p be the representation of G N defining S k (Γ 0 (N )) and S new k (Γ 0 (N )) respectively. It is immediate then that σ N p = σ p e and σ N,new p = σ p e ,new , where p e ||N . Moreover it is easy to see that

dim S new k (Γ 0 (p e )) = dim S k (Γ 0 (p e )) -2 dim S k (Γ 0 (p e-1 )) + dim S k (Γ 0 (p e-2 )),
for any e ≥ 0, with the understanding that the dimensions mentioned are 0 if the exponent of p becomes negative. This in turn implies the following formula:

(2)

I i (σ p e ,new ) = I i (σ p e ) -2I i (σ p e-1 ) + I i (σ p e-2 ).
As before, the I i 's involved are 0 if the corresponding exponent of p is negative. Formulas for the right hand side are provided in [START_REF] Cohen | Dimensions des espaces de formes modulaires[END_REF] and we also state them here for convenience:

I 1 (σ p e ) = 1 e = 0 p e-1 (p + 1) e ≥ 1 I 2 (σ p e ) = λ(e, 0, p) =      1 e = 0 2p n e = 2n + 1 p n + p n-1 e = 2n ≥ 2 I 3 (σ p e ) = #{x mod p e |x 2 + x + 1 = 0} =        1 e =
I 5 (σ p e ) = 1
where λ(sr p , s p , p) is the one defined in [START_REF] Cohen | Dimensions des espaces de formes modulaires[END_REF]. Using (2) and the above one easily gets: (5)

I 1 (σ p e ,new ) =          1 e = 0 p -1 e = 1 p 2 -p -1 e = 2 p e-3 (p -1) 2 (p + 1) e ≥ 3 (3) I 2 (σ p e ,new ) =          1 e = 0 0 e = 2n + 1 p -2 e = 2 p n-2 (p -1) 2 e = 2n ≥ 4 (4) I 3 (σ p e ,new ) =                    1 e =
I 4 (σ p e ,new ) =                    1 e = 0 or p e = 2 3 -1 p -1 e = 1 and p = 2 - -1 p e = 2 and p = 2 -1 p e = 2 or 2 2 0 otherwise (6)
I 5 (σ p e ,new ) =      1 e = 0 -1 e = 1 0 e ≥ 2 (7)
Let τ d , where d|D 2 K , be the d-part of the representation D K . The I i 's for each τ d ℓ , with ℓ|d, have already been determined in [START_REF] Finis | The cohomology of lattices in SL(2, C)[END_REF]. We then have that σ

= σ N ⊗ τ d defines S k (Γ 0 (N d), ω d ). Moreover I i (σ N ⊗ τ d ) = I i (σ N )I i (τ d
) and we have explicit formulas for both terms in the right hand side.

dim S k (Γ 0 (N d), ω d ) = k -1 12 I 1 (σ N )I 1 (τ d ) + • • • .
If one lets I i (σ p ) = 1 for all i and all p ∤ D K then one gets the formulas derived in [START_REF] Finis | The cohomology of lattices in SL(2, C)[END_REF].

6.2. Computing dim S d-sc,new k (Γ 0 (N d), ω d ) CMK .
We follow the method used in [START_REF] Tsaknias | A possible generalization of Maeda's conjecture[END_REF] to count CM newforms of a given level, weight and Nebentypus. As explained there, every newform has associated to it a pair of characters (ψ f , ψ ∞ ) that satisfies a certain compatibility condition. Moreover, for every such compatible pair, there are h K many newforms associated with it. The problem is thus reduced to counting compatible pairs that match the prescribed level, weight and Nebentypus. The infinity component is always uniquely determined by the weight, so we only have to count all the ψ f 's that correspond to the given level and Nebentypus and are compatible with the given weight. Finally every such ψ f is determined by its local components at every prime p of O K , so we count the possible choices for each of those and multiply them to get the final answer.

Recall that N d = N(m)|D K |, where m is the conductor of ψ f . Since N and D K are coprime, every prime dividing N should divide N(m) too. Moreover, a prime ℓ|d divides N(m) if and only if ℓ 2 |D K . We will treat primes p separately, depending on their ramification and residue class degree, e p and f p respectively. We start with primes p whose residue characteristic is greater than 2. 6.2.1. p inert. In this case p = p and N(p) = p 2 , so the exponent of p in N should be even. Assume that this is the case: p 2n ||N , for some n ≥ 1. We then get that the conductor of ψ f has to be divisible exactly by p n . As we can see in [START_REF] Ranum | The group of classes of congruent quadratic integers with respect to a composite ideal modulus[END_REF], (O/p n ) * is generated by three independent generators: ξ, 1 + p and 1 + pω of order p 2 -1, p n-1 and p n-1 respectively. The residues of rational integers form a subgroup isomorphic to (Z/p n Z) * which is generated by ξ p+1 and 1 + p.Since the restriction of ψ f on Z * is predetermined by the nebentypus we have unique choices for ξ p+1 and 1 + p. In our case the nebentypus is trivial and therefore ψ f (ξ p+1 ) = 1 and ψ f (1 + p) = 1. If n = 1, then the only generator to consider is ξ and since we need the conductor at p to be p we have to exclude 1 from the possible values of ψ f (ξ), which leaves p choices in total. If however n > 1 then in order for the conductor at p to be p n we need either ψ f (1 + p) or ψ f (1 + pω) to be a primitive p n-1 -th root of unity. Since ψ f (1 + p) = 1 we get that ψ f (1 + pω) must satisfy this condition which leaves ϕ(p n-1 ) = p n-2 (p -1) choices. In this case all p + 1 choices for ψ f (ξ) are permitted, so we get in total p n-2 (p 2 -1) many choices. 6.2.2. p split. In this case p = pp and N(p) = N(p) = p. Let p t ||N for some t ≥ 1. Then the p-part of the conductor of ψ f is apriori of the form p α pβ for any 0 ≤ α, β ≤ t such that α + β = t. We will first show that in our case α = β = n ≥ 1 and therefore t must be even. We have the following group homomorphisms:

(Z/p max{a,b} Z) * ֒→ (O K /p α pβ ) * and (O K /p α pβ ) * ∼ = (O K /p α ) * × (O K /p β ) * .
Notice that 1 + p will map to (1 + p, 1 + p) after composing the two maps above. Since we assume that the p-part of the conductor of ψ f is p α pβ , we get that ψ f (1 + p) must be a primitive p α-1 -th root of unity, as well as a p β-1 -th one. This can of course only happen if α = β = n. Let's go back to counting all possible characters of O * K of conductor p n pn . Having in mind the isomorphism above, any such character is completely determined by the images of the generators of (O K /p α ) * and (O K /p β ) * . Wlog we can pick 1 + p and δ, of order p n-1 and p -1 respectively, to generate both groups. Like before, the restriction of ψ f to Z is determined by the nebentypus. For the p-part we have that the two are in fact equal and since the nebentypus has trivial p-part we get the same for the p-part of ψ f |Z. This means that (1 + p, 1 + p) and (δ, δ) should map to 1 and we therefore have that

ψ f ((1 + p, 1)) = ψ f ((1, 1 + p)) -1 and ψ f ((δ, 1)) = ψ f ((1, δ)) -1 .
Apriori, ψ f ((δ, 1)) has p -1 choices. If however n = 1, then this δ is the only generator and if it has trivial image, the conductor then becomes lower, which leaves p -2 choices. If n > 1, the conductor condition is satisfied by restricting ψ f ((1 + p, 1)) to be a primitive p n-1 -th root of unity. This gives p -1 choices for the image of δ and ϕ(p n-1 ) = p n-2 (p -1) many choices for that of 1 + p.

6.2.3. p ramified. In this case p = p 2 and N(p) = p. Assume that p t ||d (remember that N and D K are coprime). Assuming that p u ||D K (and since p is ramified we also have that u ≥ 1), we get that p t-u || N(m) so the p-part of the conductor of ψ f should be p t-u . If t = 1, we clearly have a unique choice for the p-part of ψ f which happens to match the Nebentypus too. If t = 2, we are looking for non-trivial characters of (O K /p) * , which are apriori p -2 many. The nebentypus condition determines these characters uniquely on (Z/pZ) * , which happens to be isomorphic to (O K /p) * . The unique character that is left is actually non-trivial so we have a unique choice. Summarizing all of the above: Recall that it is easy to determine whether a prime is split, inert or ramified in K simply computing

CM (p t ) =                                  1 t = 0 1 t =
D K p .
Putting everything together we get:

(8) dim S d-sc,new k (Γ 0 (N d), ω d ) CMK = p t ||N d CM (p t ).
Notice that the formula above works for any N coprime to D K , not just square-free. Specializing to square-free N we get:

dim S d-sc,new k (Γ 0 (N d), ω d ) CMK = 1 N = 1 and rad(D K )|d, 0 otherwise.
where rad(D K ) is the product of all primes dividing D K . Hopefully it should be obvious by now that we have sketched an algorithm that computes dim S nG k (n) in an elementary way.

6.3. The case K = Q( √ -p), p ≡ 3 mod 4 and n = (p). Let p ≡ 3 mod 4 be a prime number, and (p

) = p 2 in O K for K = Q( √ -p).
In this section, we essentially describe all the ingredients for a dimension formula in the cases where the level n is p or p 2 . For this purpose, we determine the parameters I i (σ p ) for all the suitable σ p in these two cases, further the twists of Base-Change, and show that there is no CM outside Base-Change. For simplicity, we restrict ourselves to the case where the class number h K of K is 1.

Let us begin with n = p. In this case, the only possible type over K of level p is unramified Steinberg. It is fairly easy to see that the only type over Q that can base-change to this is unramified Steinberg at p. This space over Q coincides with the new Γ 0 (p) space and the σ-parameters can be computed using (3) -(7):

I 1 = p -1, I 2 = 0, I 3 = -3 p -1, I 4 = -2, I 5 = -1.
Notice that the new Γ 0 (p) space over Q contains no CM forms. Using the same arguments as in Theorem 1 we see that there are no twists of Base-Change in Γ 0 (p). We also claim that there are no CM forms: Indeed, assume there exists one of level Γ 0 (p), f say. Then f is automorphic induction of ψ from L to K, where L/K is a quadratic extension and ψ is a Hecke character over L. Since the L-series for f and ψ must be the same, we get that L must be ramified at p only. This is absurd since class field theory for K tells us that it does not have any even degree abelian extensions ramified only at p. Summing everything up:

dim S nG k (p) = dim S new k (Γ 0 (p)).
One can then use the I i parameters given above or use classical dimension formulas to compute the right hand side. We move to the n = p 2 = (p) case. Let ω p be the quadratic character of conductor p and recall that ω p is the quadratic character of conductor p. The following list summarizes the possible types over K of level p 2 and for each one gives the possible types over Q that base-change to them:

• All the spaces listed in the trivial level case: After twisting their Base-Change by a quadratic character of conductor p they become of level p 2 . The type at p after Base-Change is again Principal Series, I(ω p , ω p ). • Principal Series I(χ, χ -1 ), where χ is of conductor p, non quadratic. Their Base-Change (or their twist by a suitable character of conductor p) becomes a newform of level p 2 and trivial nebentypus. The type at p after Base-Change is again Principal Series, I(η, η -1 ) with η = ω p . • Unramified Steinberg Series St(p). As mentioned in the level n = p case, their Base-Change image are the Unramified Steinberg Series St(p) over K. After twisting by a quadratic character of conductor p they become newforms of level p 2 and trivial nebentypus. The type at p after Base-Change is ramified Steinberg, ω p ⊗ St(p). • Finally all Supercuspidal series of level p 2 that are not part of the third case described in the trivial level situation. Their Base-Change image (and their twists by ω p ) are Supercuspidal Series of level p 2 and trivial nebentypus. The type at p after Base-Change is again Supercuspidal Series. These four components comprise all the possible Base-Change as well as twists of it that can occur for level p 2 and trivial nebentypus. Table 1 provides the formulas to compute the parameters I i (σ) in each case, which one needs in order to use Proposition 2 to compute the corresponding dimensions. Here SC 3 (p) and SC 4 (p) are the functions tr S 3 (p) and tr S 4 (p) respectively that are defined in the statement of Lemma 4.19 of [START_REF] Finis | The cohomology of lattices in SL(2, C)[END_REF], and CPS(p) is defines as follows:

CPS(p) = -2 p ≡ 1 mod 3 0 otherwise
Finally we need to account for any CM forms not already present in the Base-Change subspace. In fact one can easily see that there are none. The argument is almost identical to the one given for the level p case: The existence of any such CM form would imply the existence of a quadratic extension of K ramified only at p but no such extension exists. At this point, anyone wanting to compute dim S nG k (p 2 ) has everything needed to do so.

Type dim σ dim σ U N trS3 trS4 dim σ G N I(ωp, ωp) p + 1 2 p -3 1 + 1 2 SC3(p) 0 0 I(η, η -1 ), η = ωp (p -3)(p + 1) 2 1 + hK CPS(p) 1 + 1 2 SC4(p) 0 ωp ⊗ St(p) p -1 0 -3 p -1 -1 p -1 -1 Supercuspidal (p -3)(p -1) 2 p -2 + hK -2 -3 p -CPS(p) -SC3(p) -1 - 1 2 -1 p 0
Table 1. Parameter values for the types contributing to Base-Change of level p 2

Final Remarks

We would like to sum up here the cases for which we provide a complete answer:

• Odd Class Number, square-free level, coprime to the discriminant, trivial nebentypus.

• Class Number 1, K = Q( √ -p) for some prime p ≡ 3 mod 4 and level p or p 2 where p 2 = (p). Many of our arguments however provide partial answers to a broader range of cases:

• We have a complete description in (3) -( 7) of the σ parameters away from the discriminant as long as the nebentypus is trivial for any level, not just square-free ones. The generalization to non-trivial nebentypus should be quite straightforward but even more cumbersome to write down as single formulas. • For primes p dividing the discriminant, the 0 exponent case is the one treated in [START_REF] Finis | The cohomology of lattices in SL(2, C)[END_REF] and we provide an answer for exponents 1 and 2 and trivial nebentypus in the case p ≡ 3 mod 4. • Theorem 1 applies to imaginary quadratic fields of even class number too, not only to the ones with an odd class number. • Finally the formula in (8) allows one to compute the dimension of the classical newforms that base-change to Eisenstein forms for any N coprime to the discriminant and trivial nebentypus, not just for the square-free levels.

Computational results for the spaces of genuine forms

The authors have used the software Bianchi.gp ([Rah11], [START_REF] Rahm | Higher torsion in the Abelianization of the full Bianchi groups[END_REF]) to compute the necessary geometric-topological information about the whole Bianchi group, and then applied a MAGMA [START_REF] Bosma | The Magma algebra system. I. The user language. Computational algebra and number theory[END_REF] implementation by S ¸engün (for which we provide the algorithm in [RS ¸T16]) to deduce the dimension of the relevant cohomology space for the congruence subgroup at level n, passing by the Eckmann-Shapiro lemma. They have then substracted the Eisenstein series space to get the cuspidal cohomology space, which by the Eichler-Shimura(-Harder) isomorphism yields the cuspidal forms space. Then they did substract the oldforms using a well-known recursive formula, to get the dimensions of the newforms spaces S k (n) defined in Section 2. 3. Overview of the sample presented in the Appendix. Under "# spaces", we count the newforms spaces that have been computed at the specified discriminant and varying level, and under "# genuine spaces", we count those of them which admit a non-trivial genuine subspace. provided in Table 2. We note that in [RS ¸13], some further subspaces are tabulated, which are in fact populated by CM-forms (arising through automorphic induction). We need this table in order to spot the twists of genuine level One forms to deeper levels. 8.2. Level: the square of minus the discriminant. We have compared the evaluation of our above formulas against our numerical results for the full space of cuspidal newforms. At discriminant -m, at level (m) (of norm m 2 and Hermite Normal Form (HNF) [m 2 , 0, m]) and its divisors, we have carried out this computation from (automorphic form) weight 2 up to the following upper limits for the weight.

Discriminant -7 -11 -19 -43 -67 weight up to 25 21 11 4 2

The result is that of these 83 cuspidal newforms spaces, all are completely exhausted by (twists of) Base-Change, except for the following. Out of these, the forms at discriminant -7, weight 14 and at discriminant -11, weight 12 are twists of the genuine level One Bianchi modular forms already found by Grunewald [START_REF] Finis | The cohomology of lattices in SL(2, C)[END_REF]. In the other cases, there are no level One forms that could be twisted [RS ¸13]. That there are no CM-forms in the cases m congruent to 3 mod 4, is guaranteed for arbitrary weight by Theorem 1. So the remaining spaces must be genuine, and we print them in boldface.

8.3. Square-free levels. Let n ∈ Z be square-free and coprime to the discriminant of the imaginary quadratic field in question. Consider the level (n) of HNF [n 2 , 0, n]. Then we have the formulas of Section 6 for the dimension of the space of base-changed forms of level (n). We have compared them against the machine computed dimension of the space of cuspidal newforms. At discriminant -19, the range of this machine computation was as follows.

Levels As the level is square-free and the class group of the imaginary quadratic field is trivial, there are no twists of base-change forms. By Theorem 1, there are no CM forms. So the above six spaces must be constituted of genuine forms. The above sample of 154 spaces allows us to guess that genuine forms are more likely to occur at low weights than at high weights (supported by Table 3); and that levels which admit genuine forms at some low weights are rather unlikely to admit more of them at higher weights (supported by the Appendix). 9. Appendix: Detailed results in the square-free level case

In addition to the sample of Section 8.3, we include in the following tables a range of square-free ideals which are not Galois-stable (in our setting of imaginary quadratic fields, not totally real). At each discriminant, we first specify the range of the pertinent machine computation, and then in a separate table the spaces with a non-trivial genuine subspace. The computation usually was run also at the Galois-conjugated level, but in the range tables, we print only one HNF per pair of Galois-conjugate levels. We omit the details for weight 2, and outsource them to [RS ¸T16]. A statistical overview is given in Table 3. The bottleneck for the computations were the memory requirements; processor time aspects were completely eclipsed by them. With the quadratic (in the weight) growth of the coefficient modules, the MAGMA program did grow its memory requirements quadratically.

k(

  Γ 0 (N d), ω d ) is the subspace of S new k (Γ 0 (N d), ω d ) spanned by newforms whose local type at every p 2 ||d is the supercuspidal representation mentioned above. And S d-sc,new k

0 or p e =≥ 2 and p = 3 I 4

 34 (σ p e ) = #{x mod p e |x 2 + 1 = 0} =

0 or p e =

  (p -1) 2 t = 2n ≥ 4 and p split p n (p 2 -1) t = 2n ≥ 4 and p inert

		1 and p ramified
	0	t = 1 and p unramified
	1	t = 2 and p ramified
	p -2 p	t = 2 and p split t = 2 and p inert
	0 p n	t = 2n + 1 ≥ 3

Table 2 .

 2 8.1. Level One. We recall here a dimension computation of 4986 different spaces of cuspidal newforms at level 1, at varying discriminant D (over 186 different imaginary quadratic fields) and varying weight k + 2. The precise scope of our computations is given in [RS ¸13]. In only 22 of these spaces were we able to observe genuine forms. The precise data about these exceptional cases is Level One cases where there are genuine classes

	7 k + 2 14 |D|	11 12	71 3	87 4	91 155 199 223 231 8 6 3 2 6	339 3	344 3
	dim	2	2	2	2	2	2	4	2	2	2	2
	|D| 407 415 455 483 571 571 643 760 1003 1003 1051 k + 2 2 2 2 3 2 3 2 4 2 3 2
	dim	2	2	2	2	2	2	2	2	2	2	2
		Weight 2				Higher weight (at least 3)
	Discriminant # spaces # genuine spaces			# spaces		# genuine spaces
	-7		1174		355				556			17
	-11		1307		353				683			14
	-19		504		151				531			6
	-43		318		61				103			1
	-67		123		17				33			1
	-163		24		4				3			0
	Table

Acknowledgements. This research project was funded by Gabor Wiese's FNR Luxembourg grant INTER/DFG/FNR/12/10/COMFGREP. We would like to thank John Cremona, Lassina Dembélé, Aurel Page and Gabor Wiese for helpful discussions. Our very special thanks go to M. Haluk S ¸engün, for having initiated this research project, contributed helpful advice and having designed the MAGMA source code for computing the relevant cohomology of congruence subgroups via a version of the Eckmann-Shapiro lemma.

That is, the genuine spaces at level (6) = [36, 0, 6] already observed in Section 8.3, as well as two further two-dimensional weight 3 spaces and their Galois conjugates.


Arnaud Poitou

Amine Ammar

Germaine Nefussi

Forming limit curves in blow molding for polymers exhibiting deformation induced crystallization

Keywords: Quiescent and induced crystallization, blow molding, Considere criterion, necking

Forming limit curves in blow molding for a polymer exhibiting deformation induced crystallization

Arnaud Poitou -Amine Ammar -Germaine Nefussi LMT -ENS de Cachan, 61 Avenue du président Wilson 94 235 Cachan France poitou@lmt.ens-cachan.fr, ammar@ lmt.ens-cachan.fr, nefussi@lmt.ens-cachan.fr ABSTRACT: We address in this paper the problem of blow molding for a polymer exhibiting a deformation induced crystallization (e.g. Poly EthyleneTerephthalate or PET). In a first part, we give the equations of a model that couples the mechanics to the crystallization for a class of molecular models (e.g. Maxwell or POM-POM models). In a second part, without a real stabilities analysis, we follow the Swift analysis for biaxial deformations. We conclude this study in showing that it is possible to quantify how crystallization during the process stabilizes it. In enhancing strongly the strain hardening, crystallization is namely a stability factor against necking.

Introduction

Blow molding is a process enabling to manufacture bottles. This process involves two steps. In a first step, a preform is injected molded and thus cooled quickly. For PET bottles this cooling time is too short to allow the material to crystallize. The preform is thus quite completely amorphous. In a second step the preform is heated, usually with infrared lights, at a temperature just above the glass temperature transition. The final shape of the bottle is then obtained with a combination of axial deformation and air blowing (figure 1). During this final stage, the material is subjected to very high deformation [START_REF] Chevalier | Induced crystallinity during stretch-blow molding process and its influence on mechanical strength of poly(ethylene terephthalate) bottles[END_REF] and this deformation induces a crystallization, which varies form point to point with respect of the materials mechanical history (strain, strain rate, temperature …).

Figure 1. Blow molding process

The aim of this paper is not to propose a complete modeling of blow molding. We focus here on the flow induced crystallization phenomenon. In a first part we derive a framework for this modeling, which is illustrated here for a material whose constitutive relation is that of the Pom Pom model parameterized by the degree of crystallinity. In a second part, making a parallel between what is classically done for deep drawing, we analyze the occurrence of necking in terms of Considere and Swift Criterion.

Equations

Details of our model can be found in [START_REF] Poitou | Polymer crystallization induced by flow : a thermodynamic approach[END_REF][START_REF] Ammar | Modélisation numérique de la cristallisation induite par l'écoulement d'un thermoplastique. Application à l'injection[END_REF][START_REF] Poitou | A molecular model for flow induced crystallization of polymers » to appear in macromolecular symposia[END_REF]. We give in the following the mathematical formulation of this model.

Rheology

The constitutive relation is chosen to be the Pom-Pom model, which exhibits two relaxation mechanisms: tube orientation S and backbone stretch λ (Figure 2). In its differential form, it can be written:

S G b 2 2 0 4 15 λ φ τ = [1] ) 1 3 1 ( 1 ) : ( 2 . . - - - + = S S D S v grad S S v grad t d S d b T θ [2] ) 1 ( 1 ) : ( - - = λ θ λ λ s D S t d d [3] Figure 2.
Branched polymer in a tube

Quiescent crystallization

Following Avrami model written in a differential form for non isothermal conditions, we assume that the relative degree of crystallinity α (ratio between the actual degree of crystallization and the ultimate or maximum degree of crystallization allowed by the polymer structure), in quiescent conditions (i.e. without any deformation of the material), increases with respect to time as:

= - - = - ) ( . ) ( 1 / 1 1 T f n dt df f Exp n λ α [4]

Deformation induced crystallization

In our formalism, strain (or flow) induced crystallization is directly deduced from equations [START_REF] Chevalier | Induced crystallinity during stretch-blow molding process and its influence on mechanical strength of poly(ethylene terephthalate) bottles[END_REF][START_REF] Poitou | Polymer crystallization induced by flow : a thermodynamic approach[END_REF][START_REF] Ammar | Modélisation numérique de la cristallisation induite par l'écoulement d'un thermoplastique. Application à l'injection[END_REF][START_REF] Poitou | A molecular model for flow induced crystallization of polymers » to appear in macromolecular symposia[END_REF]. If we assume indeed that the rheological parameters depend on the degree of crystallinity, a model of flow-induced crystallization is "naturally" associated to the rheology as soon as one writes the rheological model into a thermodynamical form. In our formalism, for the Pom Pom model, this strain induced crystallization model writes:

n S b f T n dm H m H m H y H y H f 1 1 0 2 ) ( ) ( 1 1 ) ( 1 2 ) ( Tr 3 1 2 lndet 6 ) ( 1 - + + + - Λ Λ ∂ ∂ - - - - - Λ Λ ∂ ∂ - = λ θ θ μ [5] 2 0 4 15 b G H φ = [6] 1 6 1 - - = ∂ ∂ = S H S y ψ [7] ) 1 ( λ λ λ ψ μ - = ∂ ∂ = H [8] ) exp( ) 1 ( f T T x L f m - - ≈ ∂ ∂ = Λ ∞ ψ [9]
Equation [START_REF] Doufas | A continuum model for flow induced crystallization of polymer melts[END_REF] means that the crystallization is induced by three different sources. The last one is the classical thermal activation. The first one means that crystallization occurs as soon as (i) the molecules are oriented and (ii) the orientation relaxation time increases with respect to the degree of crystallinity. In a very similar way, the second term means that crystallization occurs as soon as (i) the molecules are stretched and (ii) the stretch relaxation time increases with respect to the degree of crystallinity.

Tensile test

A tensile test can be simulated in solving simultaneously equations [1-3] and [5-9] for a constant tensile velocity. Figure 3 shows a typical tensile curve without any strain-induced crystallization for increasing tensile velocities. The effect of velocity is classically to increase the force and to allow for a larger deformation without necking. Figure 4 shows the same tensile curves for the same rheological parameters but with strain induced crystallization. This situation evidences a strain hardening and a shift of the limit deformation before necking, that increase both with crystallization and strain rate. This strain hardening is effectively due to crystallization as can be seen on figure 5, where are plotted the relative degree of crystallinity during the test.

Bidimensionnal test

In order to see the influence of bidimensionnal deformations, we have solved the equations for constant tensile velocities along two directions. We have then plotted the "limit curve" associated to the maximum of the force along one direction. Figures 6 and7 show these limit curves for different velocities, without and with strain induced crystallization. In comparison with the Swift criterion, if we associate 0.2 0.4 0.6 0.8 1 4 3 2 1 these limit curves to a forming limit curve, we see the positive role of crystallization, which acts as a necking stabilizer.

Conclusion

We have shown in this paper a model for flow-induced crystallization associated to a molecular model of the viscoelastic constitutive equation. This model is to be compared to McHugh one [START_REF] Doufas | A continuum model for flow induced crystallization of polymer melts[END_REF]. It allows us to describe both a strain hardening during plane deformation and qualitative forming limit curves in two dimensions that outline the stabilizing role of strain-induced crystallization. This model remains qualitative for blow molding because (i) it does not contain any yield stress (ii) a precise stability analysis has not been done yet. However these new development are not out of reach.

Figure 3 .Figure 4 .

 34 Figure 3. Tensile test for increasing tensile velocities, without strain induced crystallization (Relaxation times θ b =911 s, θ s =150 s, initial length L = 0.1 m, tensile velocity V=10 -4 , 2 10 -4 , 4 10 -4 , 8 10 -4 , 1.6 10 -3, 3.2 10 -3 , 6.4 10 -3 m/s)

Figure 5 .

 5 Figure 5. Relative degree of crystallinity during the tests illustrated in figure 4

Figure 6 .

 6 Figure 6. "Limit"curve without crystallization


Keywords:

This is the first part of a three-part tutorial on dual quaternion algebra applied to robotics, whose goal is to help readers grasp the main ideas behind dual quaternions and their application to robot kinematic modeling and control. Part I presents the fundamentals of dual quaternion algebra, starting from the basic definitions of complex numbers and rotations in the plane and then extending the idea to rigid motions in the tridimensional case by means of quaternions and dual quaternions. In Part II, the dual quaternion algebra is applied to the kinematic modeling of different types of robots, such as mobile robots, robot manipulators, cooperative systems, mobile manipulators, and humanoids. Finally, Part III presents several control laws that are useful to control robots such as the ones modeled in Part II. Along with a relatively formal presentation, all three parts present several examples and exercises that may help readers in the comprehension of this increasingly important topic.

CHAPTER 1

Introduction

In robotics textbooks, one of the first chapters is typically dedicated to the presentation of the theory of rigid body motion. In general, the representations of translation and rotation are introduced separately, and then they are grouped together to form the homogeneous transformation.

Typically, rotation matrices are used to represent rotation, whereas translation is represented by the Cartesian position. Grouping them together leads to homogeneous transformation matrices (HTM). Even in cases where different parameterizations of rigid motions are shown for the case of completeness, the homogenous transformation matrices are used throughout the text. Examples of this kind of exposition are found in the textbooks of [START_REF] Paul | Robot manipulators: mathematics, programming, and control: the computer control of robot manipulators[END_REF], [START_REF] Spong | Robot Modeling and Control[END_REF], [START_REF] Dombre | Robot Manipulators: Modeling, Performance Analysis and Control[END_REF] and [START_REF] Siciliano | Robotics: Modelling, Planning and Control[END_REF].

An alternative is presented by [START_REF] Murray | A mathematical introduction to robotic manipulation[END_REF]. In their textbook, a complete presentation of robot modeling and control is made in the light of screw theory. However, this kind of presentation seems to be much more an exception than the rule.

Although HTM are quite common to represent kinematic motion, they impose some additional work to control the end-effector. More specifically, a very common technique is kinematic control in task space. Such control techniques take into consideration the relationship between the task space variables and the joint variables. This relationship is typically given by ẋ = J q, (1.0.1) where x is the vector of the task space variables, q is the vector of the joint variables, and J is the Jacobian matrix relating the joints velocities and the end-effector velocities.

The parametrization of the task space variables is usually given by a vector and not directly by the HTM. Hence, it is necessary to choose the parametrization and also to extract these variables from the HTM that represents the end-effector pose.

Although one can consider these problems as secondary issues, the better theory is usually the one that explains more using fewer-and preferably simpler-arguments. Hence, a theory capable of eliminating intermediate steps between modeling and control would be better according to the aforementioned criterion. Moreover, from an engineering point of view, fewer intermediate steps between modeling and control can potentially reduce errors in implementation and development, leading to safer robots.

1. INTRODUCTION [START_REF] Murray | A mathematical introduction to robotic manipulation[END_REF] present a more mathematical approach to robot modeling and control based on the screw theory and matrix exponentials, whereas J. [START_REF] Mccarthy | Introduction to theoretical kinematics[END_REF] presents an approach based on dual quaternions. Despite the fact that the latter is not a robotics textbook, McCarthy and his collaborators consistently developed a theory for robot modeling using dual quaternion theory (Dooley and J. McCarthy, 1993;[START_REF] Perez | Dual Quaternion Synthesis of Constrained Robotic Systems[END_REF]. Also, as they are more focused on mechanism analysis, they have not provided a complete treatment on robot control.

This tutorial presents another point of view on kinematic modeling and control based on dual quaternions. The advantage of using unit dual quaternions is that they are more compact than HTM, as the former has eight elements whereas the latter has sixteen. In addition, dual quaternions have strong algebraic properties and can be used to represent rigid motions, twists, wrenches and several geometrical primitives-e.g., Plücker lines, planes, etc.-in a very straightforward way. Moreover, it is easy to extract geometric parameters from a given unit dual quaternion (e.g., translation, axis of rotation, angle of rotation) and dual quaternions multiplications are less expensive than HTM multiplications (Adorno, 2011, p. 42). Also, unit dual quaternions do not have representational singularities (although this feature is also present in HTM) and are easily mapped into a vector structure, which can be particularly convenient when controlling a robot, as there is no need to extract parameters from the dual quaternion to perform such task. Thanks to the aforementioned advantages, dual quaternion algebra will be used throughout the tutorial as the main mathematical tool for both robot modeling and control.

Although this material is far from being complete, it is comprehensive and mature enough for those wishing to obtain an initial background on dual quaternion algebra and the main techniques of robot kinematic modeling and control. Should readers have any suggestions and/or constructive criticisms, please email them to adorno@ufmg.br.

CHAPTER 2

Rigid motions: from complex numbers to dual quaternions Before entering into the world of quaternions and dual quaternions, first we are going to take a look at complex numbers and how they can be very useful in the representation of rigid motions in the plane. For instance, let us consider the complex plane in figure 2.0.1. A point (x, y) is represented by the complex number p = x + îy, where î is the standard imaginary unit; that is, î2 = -1. Since x + îy = x (1 + î • 0) + y (0 + î • 1), therefore 1 and î constitute an orthogonal basis in the complex plane. If the complex plane is rotated by θ, as shown in figure 2.0.2, then 1 and î are mapped into a new orthonormal basis:

1 → cos θ + î sin θ î → -sin θ + î cos θ.
As the basis has changed, a point p = x + îy-given initially in the rotated complex plane-when expressed in the fixed complex plane is given by p = x (cos θ + î sin θ) + y (-sin θ + î cos θ) = (x cos θ -y sin θ) + î (x sin θ + y cos θ) . Remarkably, if r cos θ + î sin θ, then we can verify by direct calculation that (2.0.1) p = rp .

Example 2.0.1 Consider a point p = 1 + î rigidly attached to a complex plane F (that is, p never changes with respect to F ). This complex plane is rotated by π/4 and consequently p rotates accordingly with respect to the fixed frame F. The new coordinates of the rotated point p with respect to the fixed frame is

p = rp = cos π 4 + î sin π 4 (1 + î) = cos π 4 -sin π 4 + î sin π 4 + cos π 4 = î√ 2.
It is interesting to note that two successive rotations given by angles θ 1 and θ 2 are represented by the multiplication r 1 r 2 . This is easily shown if we use the Euler formula e îθ = cos θ + î sin θ, Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno since r = e îθ and hence

r 1 r 2 = e îθ 1 e îθ 2 = e î(θ 1 +θ 2 ) = cos (θ 1 + θ 2 ) + î sin (θ 1 + θ 2 ) .
Example 2.0.2 Consider the rotated point of example 2.0.1. If another rotation of π/4 rad is performed, the final rotation is π/2. Thus, after the second rotation the rotated point is given by

cos π 4 + î sin π 4 î√ 2 = -1 + î.
If we rotate the original complex plane by π/2, the point p = 1 + î will be rotated accordingly and the final rotated point, with respect to the fixed frame, is

cos π 2 + î sin π 2 (1 + î) = -1 + i.
It should be clear by now that two rotations of π/4 correspond to a single rotation of π/2, as expected.

A rotation of -θ is given by e -îθ ; that is,

e -îθ = cos (-θ) + î sin (-θ) = cos θ -î sin θ = r * ,
where r * is the complex conjugate of r. Since r * r = rr * = 1, we conclude that r * is the inverse operation of r. This is true only for complex numbers r such that r = 1, which is the case of complex numbers representing rotations in the plane.

Let us recall that a rotation r relates the coordinates of the point p in the rotated complex plane F to the coordinates of the point given in the fixed frame F by means of the equation p = rp . If a point is given in the fixed frame F and we want to find the coordinates of this point in the rotated frame F , we can exploit the fact that r * is the inverse of r to find p . More specifically, Example 2.0.3 Consider a rotated frame F , which suffered a rotation of π/2 with respect to the fixed frame F, and a point p = 1 + î given with respect to F. This point, represented in the rotated frame F , is given by

p = r * p = cos π 2 -î sin π 2 (1 + î) = 1 -î.
From example 2.0.3, we realize that the complex number r = cos θ + î sin θ also provides a mapping of the same point between two coordinate systems. If we denote the fixed frame as F 0 and the rotated frame as F 1 , the complex number r represents the rotation of F 1 with respect to F 0 , and it is appropriate to explicitly indicate this rotation as r 0 1 . In this notation, the superscript always represents the reference frame, and hence p 0 and p 1 represent the coordinates of point p with respect to frames F 0 and F 1 , respectively. This way, (2.0.3)

p 0 = r 0 1 p 1
provides the mapping of point p from frame F 1 to frame F 0 . The inverse mapping is given by (2.0.4)

p 1 = r 1 0 p 0 ,
where r 1 0 = (r 0 1 ) * .

Basic facts and definitions of quaternions

Quaternions were introduced by Hamilton in the nineteenth century and can be regarded as an extension of complex numbers, where the three imaginary (also called quaternionic) units î, , k are defined and have the following properties [START_REF] Hamilton | Kinematic control of free rigid bodies using dual quaternions[END_REF]:

î2 = 2 = k2 = î k = -1. (2.1.1)
This way, the set H of quaternions is defined formally as

(2.1.2) H h 1 + îh 2 + h 3 + kh 4 : h 1 , h 2 , h 3 , h 4 ∈ R .
From (2.1.2) we see that C ⊂ H, that is, a complex number is a particular case of a quaternion where the coefficients related to the imaginary units  and k are both equal to zero.

Using an analogy with standard complex numbers, quaternions can be divided in real and imaginary parts. The real part of h, denoted by Re (h), is the scalar h 1 . The imaginary part, Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno denoted by Im (h) is the one containing the imaginary components; that is, Im (h) îh 2 +h 3 + kh 4 . Hence, h = Re (h) + Im (h).

The addition/subtraction and multiplication of quaternions use the properties of the imaginary units and are given by the next two definitions.

Definition 2.1.1 (Quaternion addition/subtraction) Let h = h 1 + îh 2 + h 3 + kh 4 and h = h 1 + îh 2 + h 3 + kh 4 be two quaternions. The quaternion sum (the subtraction is analogous) is

h + h = h 1 + h 1 + î (h 2 + h 2 ) +  (h 3 + h 3 ) + k (h 4 + h 4 )
Definition 2.1.2 (Quaternion multiplication) Let h = h 1 + îh 2 + h 3 + kh 4 and h = h 1 + îh 2 + h 3 + kh 4 be two quaternions. The quaternion multiplication is

hh = (h 1 + îh 2 + h 3 + kh 4 )(h 1 + îh 2 + h 3 + kh 4 ) = (h 1 h 1 -h 2 h 2 -h 3 h 3 -h 4 h 4 )+ î(h 1 h 2 + h 2 h 1 + h 3 h 4 -h 4 h 3 )+ (h 1 h 3 -h 2 h 4 + h 3 h 1 + h 4 h 2 )+ k(h 1 h 4 + h 2 h 3 -h 3 h 2 + h 4 h 1 ).
The set of quaternions H forms a group under quaternion multiplication [START_REF] Murray | A mathematical introduction to robotic manipulation[END_REF]. It is easy to show that quaternions are associative and distributive, but noncommutative. The next two definitions refer to the conjugate and norm of quaternions.

Definition 2.1.3 (Quaternion conjugate) The conjugate h * of a quaternion h = h 1 + îh 2 + h 3 + kh 4 is analogous to complex numbers; that is,

h * = Re (h) -Im (h) = h 1 -îh 2 + h 3 + kh 4 . Definition 2.1.4 (Quaternion norm)
The norm of a quaternion h is defined as

h = √ hh * = √ h * h.
Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno

It is easy to show that the norm of a quaternion is a non-negative real number (see exercise 2.8.4).

Fact 2.1.1 For a non-zero quaternion h, its inverse is given by

h -1 = h * h 2 . Proof. By definition, if h -1 is the inverse of h, then h -1 h = 1 = hh -1 .
Thus, starting from the first equality we have

h -1 h = 1 h -1 hh * = h * h -1 h 2 = h * .
Since h 2 is a scalar, then h -1 = h * / h 2 . The reasoning for the second equality is the same and yields, as expected, the same result.

Sometimes it is useful to perform the multiplication between matrices and quaternions. Some authors prefer to use an implicit notation for this operation; that is, they implicitly consider the parametrization of the quaternion into a vector before doing the multiplication. However, this type of notation can lead to confusion, mainly if several complex operations are performed mixing quaternions and matrices. In the sequel, the vec 4 operator is introduced, followed by the definition of the multiplication between four-by-four matrices and quaternions.

Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno Definition 2.1.5 Given a quaternion h = h 1 +îh 2 + h 3 + kh 4 , the vec 4 operator performs the one-by-one mapping vec 4 : H → R 4 ; that is,

vec 4 h      h 1 h 2 h 3 h 4      .

The inverse operation is given by vec

4 : R 4 → H; that is, let u = u 1 u 2 u 3 u 4 T , h = vec 4 u, Re (h) = u 1 , Im (h) = u 2 î + u 3  + u 4 k.
The vec 4 operator just takes the coefficients of the quaternion and stacks them in a vector. The inverse operation just takes a four-dimensional vector and maps its elements to the coefficients of a quaternion.

Example 2.1.1 Let h = a + îb + c + kd, then vec 4 h = a b c d T . Let v = 0 a 0 -b T , then vec 4 v = îa -kb.
Definition 2.1.5 leads to an important result in terms of quaternion multiplication, shown next.

Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno Definition 2.1.6 Using definition 2.1.2, it is easy to verify by direct calculation that, for h, h ∈ H, The term Hamilton operator is not commonly used, at least in the robotics literature. But since it seemed appropriate, I borrowed the term from [START_REF] Akyar | Dual quaternions in spatial kinematics in an algebraic sense[END_REF].

vec 4 (hh ) = + H 4 (h) vec 4 h = - H 4 (h ) vec 4 h, where + H 4 (h) =      h 1 -h 2 -h 3 -h 4 h 2 h 1 -h 4 h 3 h 3 h 4 h 1 -h 2 h 4 -h 3 h 2 h 1      , - H 4 (h ) =      h 1 -h 2 -h 3 -h 4 h 2 h 1 h 4 -h 3 h 3 -h 4 h 1 h 2 h 4 h 3 -h 2 h 1      ,
This latter definition states that, even though the quaternion multiplication is not commutative, the Hamilton operators commute between them. As it will become evident in the next chapters, this property is quite useful. For a more complete account on the properties of Hamilton operators, see the works of [START_REF] Chou | Quaternion kinematic and dynamic differential equations[END_REF] andAkyar (2008). 1 2.1.1. Pure imaginary quaternions. Quaternions with only imaginary part are called pure imaginary quaternions or simply pure quaternions. They form the set

H p {p ∈ H : Re (p) = 0} (2.1.3)
and are particularly interesting in the sense that they have close connections to R 3 . For example, given two pure quaternions we can define the cross product and dot product using only quaternion operations.

Definition 2.1.7 Let u and v be two pure quaternions. The cross product is defined as

u × v uv -vu 2 .
It is easy to show that the previous definition corresponds to the standard cross product between two vectors in R 3 (see exercise 2.8.5).

Definition 2.1.8 Let u and v be two pure quaternions. The dot product is defined as

u • v - (uv + vu) 2 .
As in the case of the cross product, it is easy to show that the previous definition corresponds to the standard dot product between two vectors in R 3 (see exercise 2.8.6).

If u and v are pure quaternions, using directly the definitions of dot product and cross product we can verify that

uv = -u • v + u × v. (2.1.4)
This is particularly interesting because if u and v are orthogonal, then uv = u × v.

Analogously to complex numbers, which represent rigid motions in the plane, quaternions may be used to represent rigid motions in three dimensions.

2.1.2. Translation. Since pure quaternions are directly related to vectors in R 3 , they represent translations in three dimensions. For instance, consider figure 2.1.1. The translation quaternion p = p x î + p y  + p z k has each one of its coordinates along the orthogonal axes represented by the imaginary units. In addition, analogously to complex numbers, the imaginary units and the real part form an orthogonal basis but since we can visualize only three coordinates, the real axis is omitted. Now we are ready to show that a unit quaternion r = cos (φ/2)+u sin (φ/2) (see exercise 2.8.10) represents a rotation of φ around the rotation axis u. Let us first consider the transformation rur * :

rur * = cos φ 2 + u sin φ 2 u cos φ 2 -u sin φ 2 = cos φ 2 u -sin φ 2 cos φ 2 -u sin φ 2 = cos 2 φ 2 u + cos φ 2 sin φ 2 -sin φ 2 cos φ 2 + u sin 2 φ 2 = u. (2.1.5)
If r is really a rotation around axis u, then the projection of u in the new rotated coordinate system should be unchanged, as expected (see figure 2.1.2b). Now, let us proceed with the transformations of v and w:

Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno

rwr * = cos φ 2 + u sin φ 2 w cos φ 2 -u sin φ 2 = cos φ 2 w -v sin φ 2 cos φ 2 -u sin φ 2 = cos 2 φ 2 -sin 2 φ 2 w -2 cos φ 2 sin φ 2 v = w cos φ -v sin φ. (2.1.6)
The projection of the rotated w axis onto the fixed coordinate system is given by (2.1.6), as shown in figure 2.1.2b. The same reasoning is applied to v to obtain (2.1.7)

rvr * = v cos φ + w sin φ,
which represents the projection of the rotated v axis onto the fixed coordinate system, as shown in figure 2.1.2b. Thus, we conclude that r = cos (φ/2) + u sin (φ/2) is a rotation of φ around rotation axis u, which is equivalent to say that the plane normal to u is rotated by φ. In this sense, a general rotation φ around an arbitrary unit norm rotation axis

n = n x î + n y  + n z k (see figure 2.1.3) is given by r = cos φ 2 + sin φ 2 n. (2.1.8)
Furthermore, as shown by (2.1.5)-(2.1.7), a point p i in frame F i may be projected onto another frame F j by means of (2.1.9)

p j = r j i p i r j i * ,
where the same notation for complex numbers was used; that is, the quaternion r i j represents the rotation of F j with respect to F i and, conversely, r j i represents the rotation of F i with respect to F j .

It is easy to see that, since r has unit norm, its inverse is given by

r * = cos φ 2 -sin φ 2 n = cos -φ 2 + sin -φ 2 n,
which means that r * represents the rotation of -φ around axis n. Hence, r i j * = r j i . Also, since r * is the inverse of r, then rr * = r * r = 1. It is important to note that r = 1 represents absence of rotation, because cos 0 -n sin 0 = 1.

Example 2.1.2 The point p 1 in coordinate system F 1 projected onto the coordinate system F 0 is given by

p 0 = r 0 1 p 1 r 0 1 * . (2.1.10)
Conversely, the projection of p 0 onto coordinate system F 1 yields (2.1.11)

p 1 = r 1 0 p 0 r 1 0 * .
Alternatively, (2.1.11) can be found from (2.1.10) using the fact that the inverse of r 0 1 is (r 0 1 ) * :

r 0 1 p 1 r 0 1 * = p 0 r 0 1 * r 0 1 p 1 r 0 1 * r 0 1 = r 0 1 * p 0 r 0 1 p 1 = r 0 1 * p 0 r 0 1 = r 1 0 p 0 r 1 0 * .
Both projections are shown in figure 2.1.4.

Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno

x 0 y 0 z 0 p 0 x 0 y 0 z 0 x 1 y 1 z 1 p 1 Figure 2.1.4.
The point p 0 in frame F 0 (left) may be projected onto the rotated frame F 1 (right) by means of p 1 = r 1 0 p 0 (r 1 0 )

* .

An important fact is that the product of two unit quaternions is also a unit quaternion, and hence a rotation. This is easy to see if we consider three frames F i , F j , and F k . Thus,

p j = r j i p i r j i * , (2.1.12) p k = r k j p j r k j * . (2.1.13) Replacing (2.1.12) in (2.1.13) yields p k = r k j r j i p i r j i * r k j * . As (ab) * = b * a * , then r j i * r k j * = r k j r j i * , thus p k = r k j r j i p i r k j r j i * .
But we already know that the projection of p i on F k is

p k = r k i p i r k i * . Consequently, we conclude that r k i = r k j r j i . Conversely, r i k = r k i * = r k j r j i * = r j i * r k j * = r i j r j k .
Example 2.1.3 Let a unit quaternion r 0 0 = 1 represent the initial orientation of a reference frame F 0 . After n rotations, the final orientation is given by r 0 n = r 0 1 . . . r n-1 n . Note that the superscript and subscript represent the original and final frames, respectively.

Given transformations such as (2.1.9), there is an equivalent-but more compact-representation given by the adjoint transformation, as shown in the next definition.

Definition 2.1.9 Given a unit quaternion r and a pure quaternion p, the adjoint transformation is defined as Ad (r) p rpr * .

It is easy to see, by direct calculation, that Ad (r 1 r 2 ) p = Ad (r 1 ) Ad (r 2 ) p.

Unit norm quaternions form, under multiplication, the algebraic group Spin (3) (Selig, 2005).

ṙa b = 1 2 ω a ab r a b .

Dual quaternions

The previous section showed how quaternions are used to provide a complete description of rigid motions. However, as we can note from (2.1.14), complete rigid motions-i.e., rotations and translations-are represented by using ring operations on set H; that is, we must use both Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno quaternion additions and quaternion multiplications. Moreover, operations are done on pairs of quaternions, (p, r), which is not a problem per se, but it may be desireable to have a more compact representation. One such representation is the unit dual quaternion, which requires only group operations on set H-i.e., only dual quaternion multiplications-and which are done on single elements. Such compact representation leads to a cleaner presentation and the algebra involved turns out to be simpler, allowing straightforward algebraic manipulations.

In order to define dual quaternions, in addition to the imaginary units î, , and k, we must add a new algebraic unit-namely the dual unit ε-to form a new algebraic structure. The algebra induced by the dual unit was introduced by Clifford (1871) and is called algebra of dual numbers. In this algebra, ε is nilpotent and has the following properties:

ε = 0, ε 2 = 0.
For a dual number a = a + εa , the number a is the primary part whereas a is the dual part. Also, the primary and dual parts can be extracted by using the operators P (a) and D (a) , respectively. Hence, a = P (a) + ε D (a) .

Typically, the primary and dual parts are composed of the same type of elements, that is, scalars, complex numbers or quaternions. In the more general case, when the primary and dual parts consist of quaternions, dual numbers are usually called dual quaternions and form the set H, which is formally defined as follows:

(2.2.1)

H h 1 + εh 2 : h 1 , h 2 ∈ H, ε = 0, ε 2 = 0 .
The attentive reader will notice that R ⊂ C ⊂ H ⊂ H. More specifically, a real number is a complex number with imaginary part equal to zero (e.g., a = a + 0î); a complex number is a quaternion in which the coefficients related to the imaginary numbers  and k are equal to zero (e.g., a + bî = a + bî + 0 + 0 k); finally, a quaternion is a dual quaternion with dual part equal to zero (e.g., a

+ bî + c + d k = a + bî + c + d k + ε(0 + 0î + 0 + 0 k)).
The usual operations-sum/subtraction, multiplication-of dual quaternions take into account the ε operator and are defined bellow. Definition 2.2.1 Let a 1 and a 2 be dual quaternions. The sum/subtraction between them is a 1 ± a 2 = P (a 1 ) ± P (a 2 ) + ε D (a 1 ) ± D (a 2 ) .

Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno Definition 2.2.2 Let a 1 and a 2 be dual quaternions. The multiplication between them is a 1 a 2 = (P (a 1 ) + ε D (a 1 )) (P (a 2 ) + ε D (a 2 )) = P (a 1 ) P (a 2 ) + ε (P (a 1 ) D (a 2 ) + D (a 1 ) P (a 2 )) .

Note that the nilpotent property of the ε operator is used in the multiplication operation. The operators and properties derived for quaternions in section 2.1 are easily extended to dual quaternions. For instance, the real and imaginary parts of a dual quaternion are given by Re (h) Re (P (h)) + ε Re (D (h)) ,

Im (h) Im (P (h)) + ε Im (D (h)) .

This way, given h ∈ H, it is straightforward to show that h = Re (h) + Im (h), since

Re (h) + Im (h) = Re (P (h)) + ε Re (D (h)) + Im (P (h)) + ε Im (D (h))

= Re (P (h)) + Im (P (h))

P(h) + ε(Re (D (h)) + Im (D (h))) D(h) = P (h) + ε D (h) = h. Definition 2.2.3
The conjugate of the dual quaternion h = Re (h) + Im (h) is

h * Re (h) -Im (h) .
It is easy to show that h * = P (h) * + ε D (h) * (see exercise 2.8.18). The set H p of pure dual quaternions is defined formally as (2.2.2)

H p {h ∈ H : Re (h) = 0} .
The cross product and dot product are also defined for pure dual quaternions; that is, given a, b ∈ H p , the cross product is defined as

(2.2.3) a × b ab -ba 2 ,
and the dot product is defined as

(2.2.4) a • b - ab + ba 2 .
When the primary and dual parts of a dual number are composed of real scalars, usually we call it a real dual number or, when there is no ambiguity, simply dual number. The set D of real dual numbers is formally defined as (2.2.5) D {h ∈ H : Im (h) = 0} .

Fact (Taylor expansion). The Taylor expansion of the function f (x) = f P (x) +ε D (x) at the point x 0 = P (x), where x ∈ D, is [START_REF] Gu | Dual-number transformation and its applications to robotics[END_REF]:

f (x) = ∞ n=0 f (n) (x 0 ) n! (x -x 0 ) n = f (x 0 ) + f (x 0 ) (x -x 0 ) + f (x 0 ) 2 (x -x 0 ) 2 + • • • = f P (x) + f P (x) ε D (x) + f P (x) 2 ε D (x) 2 + • • • = f P (x) + εf P (x) D (x) , (2.2.6)
where f (n) (x) denotes the n-th derivative of the function f at x. Definition 2.2.4 Analogously to quaternions, the norm of the dual quaternion h is given by

h = hh * = h * h. Fact 2.2.1
The inverse of the dual number a ∈ D is

a -1 = 1 P (a)
-ε D (a) P (a) 2 , P (a) = 0.

Proof. Let f (x) = x -1 . Since f (P (x)) = 1 P(x) and f (P (x)) = -1 P(x) 2 , using (2.2.6) with x = a leads directly to the expression of a -1 . Fact 2.2.2 For a non-zero dual quaternion h, its inverse is given by

h -1 = h * h 2 .
Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno

Proof. By definition, if h -1 is the inverse of h, then

h -1 h = 1 = hh -1 .
Thus, starting from the first equality we have

h -1 h = 1 h -1 hh * = h * h -1 h 2 = h * .
Since h 2 is a dual scalar, then h -1 = h * / h 2 . The reasoning for the second equality is the same and yields, as expected, the same result.

The vec operator and its inverse can be extended to dual quaternions, analogously to quaternions, as shown below.

Definition 2.2.5 Given a dual quaternion h ∈ H, the vec 8 operator performs the one-by-one mapping vec 8 : H → R 8 ; that is,

vec 8 h = vec 4 (P (h)) vec 4 (D (h)) .
The inverse operation is given by vec 8 : R 8 → H; that is, let

u = u 1 • • • u 8 T , h = vec 8 u, Re (h) = u 1 + εu 5 , Im (h) = îu 2 + u 3 + ku 4 + ε îu 6 + u 7 + ku 8 .
Example 2.2.1 Let u = 0 2 0 0 1 2 3 4

Unit dual quaternions

Unit dual quaternions (i.e., dual quaternions possessing unit norm) play a very important role in the representation of rigid motions, and they belong to the set

S {h ∈ H : h = 1} .
Elements of S equipped with the multiplication operation represent elements of Spin(3) R 3 , the group of rigid motions that double covers SE(3) (Selig, 2005). Considering r ∈ Spin(3) and p ∈ H p , the unit dual quaternion corresponding to the translation p followed by the rotation r is given by

x = r + ε 1 2 pr. (2.3.1) First, it is straightforward to see that x ∈ S because xx * = r + ε 1 2 pr r * + ε 1 2 r * p * = rr * + ε 1 2 rr * p * + 1 2 prr * = 1,
where we used the fact p * = -p and rr * = 1. In addition, P (x) = r and D (x) = (1/2)pr, so the primary part of x represents the rotation and the dual part of x contains the information of translation. This way, for any x ∈ S the rotation is given by r = P (x) and the translation is given by

p = 2 D (x) P (x) * . (2.3.2)
Remarkably, the composition of rigid transformations is given by a sequence of dual quaternion multiplications. More specifically, consider x 1 , x 2 ∈ S such that x 1 = r 1 + ε (1/2) p 1 r 1 and

x 2 = r 2 + ε (1/2) p 2 r 2 , thus (2.3.3) x 3 r 1 + ε 1 2 p 1 r 1 r 2 + ε 1 2 p 2 r 2 = r 1 r 2 + ε 1 2 (r 1 p 2 r 2 + p 1 r 1 r 2 ) .
Clearly, x 3 ∈ S because

x 3 x * 3 = r 1 r 2 + ε 1 2 (r 1 p 2 r 2 + p 1 r 1 r 2 ) r * 2 r * 1 + ε 1 2 (r * 2 p * 2 r * 1 + r * 2 r * 1 p * 1 ) = r 1 r 2 r * 2 r * 1 + ε 1 2 (r 1 r 2 r * 2 p * 2 r * 1 + r 1 r 2 r * 2 r * 1 p * 1 + r 1 p 2 r 2 r * 2 r * 1 + p 1 r 1 r 2 r * 2 r * 1 ) = 1 + ε 1 2 (r 1 p * 2 r * 1 + p * 1 + r 1 p 2 r * 1 + p 1 ) = 1 + ε 1 2 [r 1 (p * 2 + p 2 ) r * 1 + p * 1 + p 1 ] = 1.
In addition,

P (x 3 ) = r 1 r 2 (2.3.4) and p 3 = 2 D (x 3 ) P (x 3 ) * = (r 1 p 2 r 2 + p 1 r 1 r 2 ) r * 2 r * 1 = p 1 + Ad (r 1 ) p 2 . (2.3.5)
Comparing (2.3.4) and (2.3.5) to (2.1.14) we conclude that multiplications of unit dual quaternions represent indeed the composition of rigid transformations.

Last, given x ∈ S the inverse transformation is given by the dual quaternion conjugate x * , which corresponds to the group inverse of Spin(3) R 3 because x * x = xx * = 1 (Selig, 2005). (Consequently, 1 is the group identity of Spin(3) R 3 because x1 = 1x = x.) In order to see that x * is the inverse rigid transformation of x, first one must realize that if p = 0 and r = 1, then x = r + ε (1/2) pr = 1, which can be regarded as "absence" of rigid transformation as there is no translation (p = 0) and no rotation (i.e., r = 1 = cos (φ/2) + n sin (φ/2), which implies φ = 0). By the own definition of S, its elements have unit norm and hence ∀x ∈ S =⇒ xx * = x * x = 1, which implies that the transformation given by x is "canceled" by x * and vice-versa. Hence x * corresponds to the inverse transformation of x.

The next three statements are related to unit dual quaternions and will be useful in the next sections.

Proposition 2.3.1 Let h = r + ε 1 2 pr be a unit dual quaternion with r = cos (φ/2) + sin (φ/2) n, where n = în x + n y + kn z and p = îp x + p y + kp z . The logarithm of h is

log h = φn 2 + ε p 2 .
Proof. See (Han, Wei, and Z.-X. Li, 2008).

Note that log h ∈ H and Re (log h) = 0; hence, log h ∈ H p .

Proposition 2.3.2 Let g ∈ H P , the exponential of g is

exp g = P exp g + ε D g P exp g (2.3.6) P exp g =    cos P g + sin P(g) P(g) P g if P g = 0 1
otherwise.

(2.3.7)

Proof. Eq. (2.3.6) is a direct consequence of (2.2.6) (the extension of the Taylor expansion to H is left as an exercise to the reader) and (2.3.7) is shown by M.-J. Kim, M.-S. [START_REF] Kim | A Compact Differential Formula for the First Derivative of a Unit Quaternion Curve[END_REF].

Definition 2.3.1 Given propositions 2.3.1 and 2.3.2, the geometrical exponentiation of the unit dual quaternion h-i.e., h raised to the λ-th power-is

h {λ} exp(λ log h).
If the unit dual quaternion h is written in terms of its geometrical components, that is, h = r + ε (1/2) pr, where r = cos (φ/2) + n sin (φ/2), then

h {λ} = r {λ} + ε 1 2 λpr {λ} ,
where r {λ} = cos (λφ/2) + n sin (λφ/2).

Plücker lines

Section 2.1.3 has shown that quaternions are used to perform point transformations; that is, given a rotation quaternion r a b between frames F a and F b , a point p b is expressed in frame F a by means of the transformation p a = Ad (r a b ) p b . As we shall see in the following sections, dual quaternions perform line transformations, that is, transformations of lines between different frames. In order to proceed with this idea, first we must learn how lines are represented in a three dimensional space by means of Plücker lines [START_REF] Bottema | Theoretical kinematics[END_REF].

Consider figure 2.4.1, where l ∈ H p with l = 1 represents the direction of the line and p ∈ H p is a point in the line. A Plücker line is represented by the dual quaternion [START_REF] Daniilidis | Hand-Eye Calibration Using Dual Quaternions[END_REF]

l = l + εm,
where m is the line moment; that is,

m = p × l.
Whereas l represents the direction of the line, the quaternion m determines its spatial location independently of p. To see that, consider an arbitrary point p λ = p + λl, where λ ∈ R, which is clearly on l. The new line moment, m λ , is given by

m λ = p λ × l = p λ l -lp λ 2 = (p + λl) l -l (p + λl) 2 = pl + λll -lp -λll 2 = pl -lp 2 = m.
In addition, since Re (l) = Re (m) = 0, then Re (l) = 0; this way, Plücker lines are pure dual quaternions and form the set H p ⊂ H. Furthermore, since l = 1 and l • m = 0, an arbitrary Plücker line has four degrees of freedom [START_REF] Daniilidis | Hand-Eye Calibration Using Dual Quaternions[END_REF].

2.4.1. Transformation of Plücker lines. In the next paragraphs, first a Plücker line is represented with respect to two different frames; then, the result is used to show the existence of a unit dual quaternion (i.e., a dual quaternion with unit norm) that corresponds to a transformation between these two frames.

Consider two frames F a and F b , a reference frame F, and the Plücker line l = l + εm shown in figure 2.4.2. The Plücker line with respect to F a is l a = l a + εm Using the transformation Since

l b = r b a l a r a b , m b b = p b ba × l b + r b a p a a r a b r b a l a r a b -r b a l a r a b r b a p a a r a b 2 = p b ba × l b + r b a (p a a l a -l a p a a ) r a b 2 = p b ba × l b + r b a m a a r a b = p b ba × l b + m b a . (2.
h b a = P h b a + ε D h b a , hence h b a = r b a + ε 1 2 p b ba r b a . (2.4.7)
It is easy to see that h b a = 1. From the dual quaternion norm, if h b a = 1, hence h b a h b * a = 1. This is verified by direct calculation:

h b a h b * a = r b a + ε 1 2 p b ba r b a r a b -ε 1 2 r a b p b ba = r b a r a b + ε 1 2 p b ba r b a r a b -r b a r a b p b ba = 1.

Planes in dual quaternion space

A plane can be completely described by the unit norm vector normal to the plane and the perpendicular distance from the origin of a given coordinate system [START_REF] Selig | Clifford algebra of points, lines and planes[END_REF], as shown in Fig. 2.5.1. When written in dual quaternion form, the unit norm vector is given by n ∈ H p , such that n = 1 and the perpendicular distance d is given by d = q • n, where q ∈ H p is an arbitrary point in the plane; thus, the plane π is given by (2.5.1)

π n + εd = n + εq • n.
Proposition 2.5.1 Let the plane be represented in frame F a by π a = n a + εd a , where d a = q a • n a . The same plane, represented in frame F b , is given by

t b • n b = - t b n b + n b t b 2 = - p b ba + q b n b + n b p b ba + q b 2 = - p b ba n b + n b p b ba 2 - q b n b + n b q b 2 = p b ba • n b + q b • n b .
Since q b • n b = q a • n a (see exercise 2.8.15), hence (2.5.6) Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno From (2.5.1), (2.5.3), and (2.5.4), we obtain

d b = t b • n b = p b ba • n b + d a .
π b = Ad x b a π a = x b a π a x b a * = r b a -ε 1 2 p b ba r b a (n a + εd a ) r b * a -ε 1 2 r b * a p b ba = n b + ε d a + p b ba • n b . (2.5.7)
Comparing (2.5.7) to (2.5.5) and (2.5.6), we conclude the proof. (2.6.2) Expanding (2.6.1) we obtain 2.6.1. Numerical integration of unit dual quaternions. Consider the unit dual quaternion x (t) = r (t) + ε (1/2) p (t) r (t)-where r (t) = cos (φ/2) + n sin (φ/2), with φ φ (t) and n = n (t)-and its corresponding time derivative ẋ (t). When performing the numerical integration of ẋ, one may be tempted to use the simple first-order approximation

ẋa b = 1 2 P ξ a ab + ε D ξ a ab r a b + ε 1 2 p a ab r a b = 1 2 P ξ a ab r a b + ε D ξ a ab r a b + 1 2 P ξ a ab p a ab r a b = 1 2 P ξ a ab r a b + ε 1 2 D ξ a
ẋ ≈ x k -x k-1 T ,
to obtain

(2.6.4)

x k = x k-1 + T ẋ,
where T is the integration step. The problem with such approach is that x has unit norm and belongs to a multiplicative group, but x k usually does not have unit norm due to the approximation error of (2.6.4) and the additive operation. This is particularly troubling, since rigid motions are represented by unit dual quaternions, thus (2.6.4) not only has intrinsic approximation errors, but from an algebraic point of view, does not make sense. Approximative methods should at least respect the group operation of unit dual quaternions. In order to derive such method first consider the angular velocity ω = n φ and the linear velocity v = ṗ associated to ẋ, thus (2.6.6) where ∆φ = φ k -φ k-1 and ∆p = p k -p k-1 . These approximations related to the linear velocity and angular are embedded into the twist ξ = ω + ε (v + p × ω)-where ω ω (t), p p (t), and v v (t)-in the following manner:

φ ≈ φ k -φ k-1 T =⇒ ∆φ ≈ T φ (2.6.5) and ṗ ≈ p k -p k-1 T =⇒ ∆p ≈ T ṗ,
T ξ 2 = ωT 2 + ε v + p × ω 2 T = nT φ 2 + ε ṗT + p × nT φ 2 .
Using (2.6.5) and (2.6.6) we obtain

T ξ 2 = n ∆φ 2 + ε ∆p + p × n∆φ 2 =⇒ exp T ξ 2 = r ∆ + ε 1 2 p ∆ r ∆ ,
where p ∆ = ∆p + p × n∆φ and r ∆ = cos (∆φ/2) + n sin (∆φ/2). Since ξ is given with respect to the inertial frame, thus the associated transformation must be performed using the inertial frame as reference; that is, (2.6.7)

x k = exp T ξ (t) 2 x k-1 .
Example 2.6.1 Consider the time-varying trajectory given by

x (t) = r (t) + ε 1 2 p (t) r (t) ,
where p (t) = î cos t +  sin t + kt and

r (t) = cos φ (t) 2 + n (t) sin φ (t) 2
with φ (t) = 2t and n (t) = î cos (t) / √ 2 +  sin t + k cos (t) / √ 2. (Note that n = 1, as expected.) Figure 2.6.1 shows the trajectory for the time interval t = [0, 5]s. In order to illustrate the numerical integration given by (2.6.7), we must first find the twist ξ (t) associated to ẋ (t). Thus, v (t) = ṗ (t) = -sin (t) î + cos (t)  + k.

From the quaternion propagation equation (2.1.15),

ω (t) = 2 ṙ (t) r (t) * = 2 n (t) + ṅ (t) sin t cos t -ṅ (t) n (t) sin 2 t .
The resultant twist is given by

ξ (t) = ω (t) + ε (v (t) + p (t) × ω (t)) .
The trajectory obtained by numerical integration is given by

x I (t + T ) = exp T ξ (t) 2 x I (t) , (2.6.8) x I (0) = x (0) .
The effect of the integration step T in the trajectory approximation is shown in figure 2.6.2. For T = 0.01 s, the numerical approximation is very accurate as the trajectory generated by (2.6.8) (black curve) is very close to the trajectory generated analytically (red curve with red circles). On the other hand, for increasing values of T the numerical approximation starts to have large deviations from the analytical trajectory, which is expected, since the larger the value of the integration step, the more inaccurate is the numerical approximation.

Additional facts and properties of quaternions and dual quaternions

This section presents additional facts and properties of quaternions and dual quaternions which were ommited from the main text in order to ensure a smoother reading. Exercise 2.8.14. Show that, if r = cos (φ/2)+n sin (φ/2) and s = cos (π -φ/2)-n sin (π -φ/2), then s = -r. Exercise 2.8.17. Show that, given two frames F a and F b , and a rotation r a b , if the angular velocity is expressed with respect to F b , then the quaternion propagation equation is given by ṙa b = (1/2) r a b ω b ab .

Exercise 2.8.18. Show that Re (h) -Im (h) = P (h) * + ε D (h) * .

Exercise 2.8.19. Given a constant λ ∈ R and a unit dual quaternion x = r + ε(1/2)pr, such that ẋ = (1/2)ξx with ξ = ω + ε ( ṗ + p × ω) and ṙ = (1/2) ωr, show that the twist ξ λ related to dx {λ} /dt is given by ξ λ = λ [ω + ε ( ṗ + λp × ω)].

Exercise 2.8.20. Given the unit quaternion r = cos (φ/2) + n sin (φ/2), the unit dual quaternion x = r + ε (1/2) pr, and s, t ∈ R, show that r {s} r {t} = r {s+t} and x {s} x {t} = x {s+t} .

Index

Figure 2

 2 Figure 2.0.1. A point in the complex plane.

  Figure 2.0.2. Rotated complex plane.

  rp = p r * rp = r * p p = r * p. (2.0.2) Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno

  and + H 4 (•) and -H 4 (•) are called Hamilton operators. a a

Figure 2

 2 Figure 2.1.1. Translation quaternion p = p x î + p y  + p z k. Note that both the real and the imaginary axes form an orthogonal basis, but the real axis is omitted.

Figure 2

 2 Figure 2.1.2. Rotation φ around axis u.

Figure 2

 2 Figure 2.1.3. Rotation angle φ around the unit rotation axis n, represented by r = cos (φ/2) + n sin (φ/2).

  Figure 2.1.5. Rigid motion represented by quaternions: first a translation p a ab is performed, followed by a rotation r a b .

  Figure 2.4.1. Plücker line.

  Figure 2.4.2. Plücker line with respect to frames F a and F b .

d

  Figure 2.5.1. Plane with respect to frames F a and F b .

  Figure 2.5.2. Plane with respect to frames F a and F b .

Figure 2 Figure 2

 22 Figure 2.6.1. Time varying trajectory in Spin (3) R 3 of example 2.6.1.

  Exercise 2.8.15. Given p a , t a ∈ H p and r b a ∈ Spin (3), show that p a • t a = p b • t b , where p b = Ad r b a p a and t b = Ad r b a t a .Exercise 2.8.16. Given the unit quaternion r a b

  For h, h ∈ H, using definitions 2.2.5 and 2.1.6 it is easy to verify, by direct calculation, that

							+			-
				vec 8 (hh ) =	H 8 (h) vec 8 h =	H 8 (h ) vec 8 h,
	where
				+								-	
		+ H 8 (h) =		H 4 (P (h)) +	+	0 4	 ,	-H 8 (h ) =		H 4 (P (h )) -	-	0 4	 ,
			H 4 (D (h))	H 4 (P (h))				H 4 (D (h ))	H 4 (P (h ))
		+
	and	H 8 and
								T	, then vec 8 u = î2+ε 1 + î2 + 3 + k4 .

Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno Definition 2.2.6 -H 8 are the Hamilton operators extended to dual quaternions.

  Consider h 1 , h 2 ∈ H. From definition 2.1.2 it is easy to verify, by direct calculation, that Re (h 1 h 2 ) = (vec 4 h 1 ) T vec 4 h * 2 .

	2.7.1. Quaternions.
	Fact 2.7.1

Fact 2.7.2 Let h be a quaternion with unit norm; hence Re ḣh * = Re h ḣ * = 0.

Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno

RIGID MOTIONS: FROM COMPLEX NUMBERS TO DUAL QUATERNIONS

It is important to note, however, that Akyar's definition of Hamilton operators is somewhat different from the one presented here.Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno

Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno

Note that, sometimes, it is convenient in a sequence of rigid motions to represent the original and final frames in the subscript of translation quaternions; for instance, p a ab is a translation from F a to F b , represented in F a . On the other hand, p b ab is the same translation from F a to F b , but represented in F b .

Proof. Consider r = cos (φ/2) + n sin (φ/2), then 3

From the quaternion progapation equation,

Proposition 2.7.2 Given λ ∈ R and a unit quaternion r = cos(φ/2) + n sin(φ/2), then r {-λ} = r {λ} * = (r * ) {λ} .

Proof. The proof can be verified by direct calculation.

Proposition 2.7.3 Given λ ∈ R and a unit quaternion r = cos(φ/2) + n sin(φ/2),

Proof. From propositions 2.7.1 and 2.7.2, and the fact that r {t} r {s} = r {t+s} (see exercise 2.8.20)

= λ ṙr {λ-1} .

3 Note that the geometrical exponentiation, as presented in Definition 2.3.1, is defined only for unit dual quaternions.

This way, the notation ṙ{λ} is ambiguous and thus we use dr {λ} /dt instead.

Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno

Exercises

Exercise 2.8.1. Show that a complex number r = cos θ + î sin θ has unit norm.

Exercise 2.8.2. Show that rotations in the plane are commutative.

Exercise 2.8.3. Show that the following relations

are a direct consequence of (2.1.1).

Exercise 2.8.4. Given a quaternion

Exercise 2.8.5. Given the pure quaternions u = u 1 î + u 2  + u 3 k and v = v 1 î + v 2  + v 3 k, show that the cross product defined in definition 2.1.7 corresponds to the cross product between

Exercise 2.8.6. Given the pure quaternions u = u 1 î + u 2  + u 3 k and v = v 1 î + v 2  + v 3 k, show that the dot product defined in definition 2.1.8 corresponds to the dot product between two

Exercise 2.8.7. Show that H p is isomorphic to R 3 under the addition operation.

Exercise 2.8.8. Show that if {u, v, w} is an orthonormal basis, then uv = w, wu = v, and vw = u.

Exercise 2.8.9. Show that, for a, b ∈ H, the following equality holds: (ab) * = b * a * .

Exercise 2.8.10. Show that if n is a pure quaternion with unit norm, then quaternions of the form r = cos (φ/2) + sin(φ/2)n also have unit norm.

Exercise 2.8.11. Show that if r 1 and r 2 are rotation quaternions, then r = r 1 r 2 has unit norm.

Exercise 2.8.12. Show that if r is a rotation quaternion, then -r represents the same rotation.

Exercise 2.8.13. Show, by means of quaternion operations, that a rotation angle φ around rotation axis n is equivalent to the rotation angle -φ around rotation axis -n.

Robot Kinematic Modeling and Control Based on Dual Quaternion Algebra -Part I: Fundamentals, by Bruno V. Adorno 2.8. EXERCISES