Factorization of Rational Curves in the
Study Quadric and Revolute Linkages
Gabor Hegedüs∗, Josef Schicho∗, and Hans-Peter Schröcker†
March 21, 2018
Given a generic rational curve C in the group of Euclidean displacements
we construct a linkage such that the constrained motion of one of the links is
exactly C. Our construction is based on the factorization of polynomials over
dual quaternions. Low degree examples include the Bennett mechanisms and
contain new types of overconstrained 6R-chains as sub-mechanisms.
Keywords: Dual quaternions, rational motion, factorization, Bennett mechanism, over-
constrained mechanism, 6R-chain.
MSC 2010: 12D05, 51J15, 68T40
1 Introduction
The research on this paper started with an attempt to understand the geometry of
Bennett linkages [1–6] from the point of view of dual quaternions. The group of Eu-
clidean displacements can be embedded as an open subset of the Study quadric in the
projectivization of the dual quaternions regarded as a real vector space of dimension
eight. Rotation subgroups and and their composition then get an algebraic meaning (see
Hao [7], Selig [8, Chapter 9 and 10]). This was exploited in [4] to devise an algorithm
for the synthesis of a Bennett linkage to three pre-assigned poses. The key observation
there was that the coupler curve is the intersection of a unique 2-plane with the Study
quadric. Here, we translate the synthesis problem entirely into the language of dual
quaternions and we show that the problem is equivalent to the factorization of a left
quadratic polynomial into two linear ones.
∗Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences (RI-
CAM), Altenbergerstrasse 69, 4040 Linz, Austria, e-mail: {gabor.hegedues}{josef.schicho}@oeaw.ac.at
RICAM Linz
†University Innsbruck, Unit Geometry and CAD, Technikerstraße 13, 6020 Innsbruck, Austria, e-mail:
hans-peter.schroecker@uibk.ac.at
1
arXiv:1202.0139v4  [math.RA]  9 May 2012
Factorizations of left polynomials over the quaternions have been studied by Niven in
[9], who was interested in the number of such factorizations, and Gordon and Motzkin
in [10], who proved that this number is either infinite or at most equal to the factorial
of the degree of the polynomial. ([10] studies more generally polynomials over central
simple algebras over commutative fields.) The more recent paper [11] by Huang and So
gives an explicit solution formula for quadratic polynomials. It is not difficult to extend
these results to dual quaternions.
Once the relation between the closure conditions in linkages and the factorizations
of left polynomials over dual quaternions became clear, it also became clear that this
relation holds for arbitrary linkages with rotational joints. Given a generic rational
curve C of degree n on the Study quadric with parametrization P(t), we construct n!
different factorizations P(t) = (t−h1) · · · (t−hn) with rotation quaternions h1, . . . , hn (or
translation quaternions in limiting cases). Each factorization corresponds to the movement
of an open nR-chain that guides the end-effector along C (Section 3). Combining all open
chains yields an overconstrained mechanism whose combinatorial structure is investigated
in Section 4. In Section 5, we discuss the extension to linkages with prismatic joints.
The case n = 2 is just the Bennett case. Our results agree with the recent findings of
[12] and naturally include the limit case of Bennett linkages (RPRP linkages). For n = 3
we obtain linkages that contain new examples of overconstrained 6R-chains. We present
them in more detail in Section 6.
A preliminary short version of this article is [13]. The present article is more complete.
We give strict proofs for all new results, discuss the extension to prismatic joints, present
additional examples and illustrate our results by figures. The Maple source for some
of the examples and additional animations can be found on the accompanying web-site
http://geometrie.uibk.ac.at/schroecker/qf/.
2 Preliminaries
In this section, we recall the well-known and classical description of the group of Euclidean
displacement by dual quaternions. The presentation is already adapted to our needs in
later sections. More general references are [7, 8].
We denote by SE3 the group of direct Euclidean displacements, i.e., the group of maps
from R3 to itself that preserves distances and orientation. It is well-known that SE3 is a
semidirect product of the translation subgroup T and the special orthogonal group SO3,
which may be identified with the stabilizer of a single point.
We denote by D := R + ϵR the ring of dual numbers, with multiplication defined by
ϵ2 = 0. The algebra H is the non-commutative algebra of quaternions, and DH is the
algebra of quaternions with coefficients in D. Every dual quaternion has a primal and a
dual part (both quaternions in H), a scalar part in D and a vectorial part in D3. The
conjugated dual quaternion h of h is obtained by multiplying the vectorial part of h by
−1. The dual numbers hh and h + h are called the norm and trace of h, respectively.
By projectivizing DH as a real 8-dimensional vectorspace, we obtain P7. The condition
that the norm of h is strictly real, i.e. its dual part is zero, is a homogeneous quadratic
2
equation. Its zero set, denoted by S, is called the Study quadric. The linear 3-space
of all dual quaternions with zero primal part is denoted by E. It is contained in the
Study quadric. The complement S −E can be identified with SE3. The primal part
describes SO3. Translations correspond to dual quaternions with primal part ±1 and
strictly vectorial dual part. More precisely, the group isomorphism is given by sending
h = p + ϵq to the map
v ∈R3 7→pvp + qp
pp
∈R3.
The image of this map is strictly vectorial, the map is in SE3, and the above formula
indeed provides a group homomorphism. Its bijectivity follows from the fact that both
groups are connected and of the same dimension.
A nonzero dual quaternion h = p + ϵq represents a rotation if and only if its norm and
trace are strictly real (hh, h + h ∈R) and its primal vectorial part is nonzero (p /∈R). It
represents a translation if and only if its norm and trace are strictly real and its primal
vectorial part is zero (p ∈R). The 1-parameter rotation subgroups with fixed axis and
the 1-parameter translation subgroups with fixed direction can be characterized as lines
on S through the identity element 1. Translations are characterized as those lines that
meet the exceptional 3-plane E.
3 Rational motions and open linkages
In this section we study rational curves in the Study quadric and construct a linkage
with rotational joints such that the last link moves along the prescribed curve. The main
technique is the factorization of polynomials over the dual quaternions.
3.1 Polynomial factorization over dual quaternions
Let C ⊂S be a rational curve of degree n > 0. Then there exists a parametrization of
C by a polynomial (antn + an−1tn−1 + an−2tn−2 + · · · + a0)t∈P1, where a0, . . . , an ∈DH
and P1 denotes the real projective line. It is no loss of generality to assume that this
polynomial is monic (an = 1). This can always be achieved by an appropriate choice of
coordinates.
Conversely, let DH[t] be the set of left polynomials with coefficients over DH. This set
can be given a ring structure by the convention that t commutes with the coefficients.
Let P ∈DH[t] be a polynomial of degree n > 0. We call the map fP : P1 →P7, t 7→P(t)
the map associated to P. The image is a rational curve of degree at most n.
If Q ∈R[t], Q ̸= 0, then the maps fP and fPQ are equal. Conversely, if P has a factor
in R[t] of positive degree, we can divide by it without changing the associated map. Note
that, in general, the set of right factors is different from the set of left factors. However,
a polynomial in R[t] or in D[t] is a left factor if and only if it is a right factor, because it
is in the center of DH[t].
For n > 0, an open linkage with n rotational joints can be described algebraically as
follows. Let h1, . . . , hn be rotations; for each i, the group parametrized by (t −hi)t∈P1 –
the parameter t determines the rotation angle – is the group of the (i + 1)-th link relative
3
to the i-th link. If we choose the same parameter for the n rotations, then the position
of the last link with respect to the first link is given by a product
P = (t −h1)(t −h2) · · · (t −hn)
with
t ∈P1.
(1)
P is a monic polynomial of degree n. Because it describes a rigid motion, it satisfies
the Study condition PP ∈R[t]. In this case we call P a motion polynomial. We ask
the converse question: Given a monic motion polynomial P of degree n, is it possible to
construct a factorization of type (1)? We will see that, in general, the answer is positive
(Theorem 3, below).
For the time being we restrict ourselves to factorizations with rotation quaternions
only. Later, in Section 5, we will show how to incorporate translation quaternions as well.
Lemma 1. If h is a rotation quaternion then M := (t −h)(t −h) is in R[t] and has no
real roots.
Proof. The first claim follows from the expansion M = t2 −t(h + h) + hh and the
observations that h + h ∈R (because the scalar dual part of h is zero) and hh ∈R
(because h lies on the Study quadric).
The discriminant of M equals ∆= (h + h)2 −4hh = (h −h)2. Since h is a rotation
quaternion, ∆is negative and M has no real roots. (The case ∆= 0 characterizes
translations.)
Proposition 2. Let P ∈DH[t] be a motion polynomial of degree n > 0 without strictly
real factors. If there is a factorization P(t) = (t−h1) · · · (t−hn) with rotation quaternions
h1, . . . , hn ∈DH, the polynomial PP ∈R[t] has no real zeroes.
Proof. Assume P = (t −h1) · · · (t −hn) and let Mi := (t −hi)(t −hi) for i = 1, . . . , n. By
Lemma 1, Mi is in R[t] and has no real roots. The same is true for PP = (t −h1) · · · (t −
hn)(t −hn) · · · (t −h1) = M1 · · · Mn.
We call a motion polynomial P ∈DH[t] of degree n > 0 generic if PP has n distinct
quadratic, irreducible factors. This implies that primal(P) has no strictly real factors,
because any such factor would appear with multiplicity ≥2 in the factorization of PP.
Theorem 3. Let P ∈DH[t] be a generic motion polynomial of degree n > 0. Then there
exists a factorization P(t) = (t −h1) · · · (t −hn) with hi ∈DH representing rotations.
We divide the proof of Theorem 3 into several lemmas. A central tool (also for the
computation of the factorization) is polynomial division in DH[t].
Lemma 4 (polynomial division). Let P1, P2 ∈DH[t] and assume P2 is monic. Then there
exists a unique representation P1 = QP2 + R with Q, R ∈DH[t] and deg(R) < deg(P2).
Moreover, if h ∈DH such that P2(h) = 0, then P1(h) = R(h).
Proof. The proof is a generalization of polynomial division to the non-commutative
case. Let n := deg(P1) and m := deg(P2). Existence of the representation is trivial if
n < m (Q = 0, R = P1). Assume inductively that polynomial division is possible for
4
all polynomials of degree less than n. We can write P1 = atn + P3 with deg(P3) < n,
and atn = atn−mP2 + P4 with deg(P4) < n. By the induction hypothesis, we have
P3 = Q3P2 +R3 and P4 = Q4P2 +R4 with deg(R3), deg(R4) < m. Combining, we obtain
P1 = (atn−m + Q4 + Q3)P2 + R3 + R4. This shows existence.
Now, assume that P1 = Q1P2 + R1 = Q2P2 + R2. Then (Q1 −Q2)P2 = R1 −R2. If
Q1 ̸= Q2, the polynomial on the left has degree greater or equal m while the polynomial
on the right has degree less than m. This is impossible, so that Q1 = Q2 and R1 = R2.
This shows uniqueness.
For the last statement, we have to show that P2(h) = 0 implies (QP2)(h) = 0. This is
not trivial, because h does not commute with the coefficients of P2. But the statement is
linear in Q, hence it is enough to prove it for monomials: If P2(h) = 0 and Q = atr, then
(QP2)(h) = aP2(h)hr = 0.
Lemma 5. Let P ∈DH[t] and h ∈DH. Then (t −h) is a right factor of P if and only
if P(h) = 0.
Proof. By Lemma 4, there is a unique representation P = Q(t −h) + R, and R is a
constant equal to P(h).
Lemma 6. Let P ∈DH[t] be a motion polynomial. Let M ∈R[t] be a monic polynomial
of degree two that divides PP but not primal(P). Then there exists a unique h ∈DH
such that P(h) = M(h) = 0.
Proof. By Lemma 4, we may write P = QM + R with Q, R ∈DH[t], deg(R) < 2. Since
M does not divide primal(P), we have primal(R) ̸= 0. On the other hand,
PP = (QM + R)(MQ + R) = (QQM + QR + RQ)M + RR.
Because M divides PP, we conclude RR = cM for some c ∈D, c ̸= 0. Assume that the
primal part of c is zero, that is, RR = ϵkM with k ∈H. This implies primal(R) = 0 and
contradicts our assumption. Hence primal(c) ̸= 0. The leading coefficient r1 of R cannot
have zero primal part because otherwise we get the contradiction r1r1 = 0 = c. Hence,
r1 is invertible in DH and, because of r1r1 = c, so is c. The polynomial R = r1t + r0 is
linear with invertible leading coefficient. Hence it has a unique zero h = −r−1
1 r0 ∈DH.
From R(h) = 0, we obtain cM(h) = 0 and, because c is invertible, M(h) = P(h) = 0.
This shows existence.
In order to show uniqueness, assume there exists h′ ∈DH with P(h′) = M(h′) = 0.
This implies R(h′) = 0. But then h′ = h because the zero of R is unique.
Proof of Theorem 3: We proceed by induction on n. For n = 0, the statement is trivial.
Assume n ≥1. Since the primal part of P has no strictly real factors, P itself has no
strictly real factors. Consequently, PP has no linear factors.
Let M be one of the irreducible quadratic factors of PP. By Lemma 6, there is a
unique h such that M(h) = P(h) = 0. By Lemma 5, there exists Q ∈DH[t] such
that P = Q(t −h). Obviously, Q is monic of degree n −1. Moreover, we have PP =
Q(t−h)(t−h)Q = QQM so that QQ is in R[t]. Furthermore, Q cannot have a strictly real
5
factor: This factor would also be a left factor and hence divide P. For similar reasons, the
primal part of Q cannot have strictly real factor (it would divide primal(P)). By induction
hypothesis, we obtain Q = (t −h1) · · · (t −hn−1) and so P = (t −h1) · · · (t −hn−1)(t −h).
Because of PP ∈R[t], h must be a rotation or translation quaternion. By genericity of
P, it is actually a rotation quaternion.
Remark 7. Theorem 3 is almost a converse to Proposition 2. But there are polynomials
P without real factors such that PP ∈R[t] where the proposition and the theorem do
not say anything. In this case, there exists an irreducible polynomial R ∈R[t] dividing
the primal part of P but not the dual part. Since R then also divides the primal part
of P and PP ∈R[t], the factor R appears with multiplicity two in the factorization of
PP. For instance, if P = t2 + 1 + ϵi, it can be shown that P is not the product of two
linear rotation polynomials. (Here, we assume that (1, i, j, k) is the standard basis of the
quaternion algebra H.) On the other hand, P = t2 + 1 + ϵjt + ϵi = (t −k)(t −k + ϵj) is
a product of two rotation polynomials. A systematic analysis would be good, but it is
probably more difficult. At this place, we only observe that P = t2 + 1 + ϵi is a quadratic
parametrization of a straight line.
3.2 Computing the factorization
Our proof of Theorem 3 is constructive and, with exception of the factorization of
PP, can be implemented in rational arithmetic. We describe this in more detail. Our
considerations will lead us to two new insights:
• The factorization depends on an ordering of the n quadratic factors of the polynomial
PP. Hence, there exist n! different factorizations and Theorem 3 actually admits a
stronger version (Theorem 8, below).
• If one factorization of P is known, the remaining factorizations can be computed
in rational arithmetic. In particular, it is possible to construct completely rational
examples by starting with the polynomial P = (t −h1) · · · (t −hn) where h1, . . . , hn
are rotation quaternions with rational coefficients.
Given is a motion polynomial P ∈DH[t]. We want to compute rotation quaternions
h1, . . . , hn such that P = (t −h1) · · · (t −hn). The pseudo-code for this calculation is
given in Algorithm 1. It returns an n-tuple H = (h1, . . . , hn). Here are some remarks on
the actual calculations:
• In Line 4, the choice of Mi ∈M is arbitrary. Thus, Algorithm 1 is not deterministic.
• The common root hi of Mi and P in Line 5 is found by computing the remainder
R = r1t + r0 of the polynomial division of P by Mi and setting hi = −r−1
1 r0. This
is justified in the proof of Lemma 6.
• The proof of Theorem 3 shows that the polynomial division of P by (t −hi) is
possible without remainder. Moreover, the quotient satisfies all requirements of
Theorem 3 so that it is suitable as input for yet another repeat-loop.
6
Algorithm 1 Compute factorization P = (t −h1) . . . (t −hn)
Require: generic motion polynomial P
1: M ←{M1, . . . , Mn} where PP = M1 · · · Mn.
2: Let H denote the empty tuple.
3: repeat
4:
Choose Mi ∈M and set M ←M \ {Mi}.
5:
Compute hi such that Mi(hi) = P(hi) = 0.
6:
Append hi to H.
7:
P ←P/(t −hi)
8: until deg P = 0
9: return H = (h1, . . . , hn)
Maple source code for Algorithm 1 can be found on the web-site http://geometrie.
uibk.ac.at/schroecker/qf/. The non-deterministic nature of Algorithm 1 implies
existence of different factorizations into products of rotation quaternions.
Theorem 8. Let P ∈DH[t] be a generic motion polynomial of degree n > 0. Then there
is a one-to-one correspondence between factorizations P(t) = (t −h1) · · · (t −hn) into
linear polynomials over DH and permutations of the n distinct quadratic irreducible factors
M1, . . . , Mn of PP. In any of these factorizations we have Mi(hi) = 0 for i = 1, . . . , n.
Proof. Our proof of Theorem 3 can be translated into a construction of a factorization
of P into linear factors over DH. The only non-deterministic step is the choice of a
quadratic factor of P ′P ′, where P ′ is the left factor from which the next right linear
factor is going to be constructed. The construction is also complete in the sense that
every factorization can be obtained by this non-deterministic algorithm.
Example 1. We give an example that illustrates some of the results we obtained so far.
Consider the quadratic polynomial
P = t2 −t(1 + (ϵ −1)i + (1 −ϵ)j + 2(1 + ϵ)k) −1 −2ϵ + i −ϵj + (2 −ϵ)k.
We can write PP = M1M2 where
M1 = t2 + 2
and
M2 = t2 −2t + 3.
In order to find the common root h2 of P and M1, we compute the remainder of the
polynomial division of P by M1. It is R = r1t + r0 where
r1 = −1 + (1 −ϵ)i + (ϵ −1)j −2(1 + ϵ)k,
r0 = −3 −2ϵ + i −ϵj + (2 −ϵ)k.
Now
h2 = −r−1
1 r0 =
−4
7 + 30
49ϵ

i −
1
7 + 3
49ϵ

j +
9
7 + 13
49ϵ

k
and, by polynomial division, we find P = (t −h1)(t −h2) with
h1 = 1 +
−3
7 + 19
49ϵ

i +
8
7 + −46
49 ϵ

j +
5
7 + 85
49ϵ

k.
7
Note that P(h2) = M1(h2) = 0.
Using M2 instead of M1, we can compute a second factorization P = (t −h′
1)(t −h′
2)
where
h′
1 = (1 −ϵ)j + (1 + ϵ)k,
h′
2 = 1 −(1 −ϵ)i + (1 + ϵ)k.
Here, P(h′
2) = M2(h′
2) = 0.
We already showed how to compute the rotation quaternion hi from the factor Mi. But
it is also possible to compute Mi from hi. Clearly, Mi is the unique monic polynomial
of degree two such that Mi(hi) = 0 (the minimal polynomial of hi). It is given by
Mi = t2−(hi+hi)t+hihi. This observation is important, because it allows us to construct
completely rational examples. Setting P = (t −h1) · · · (t −hn) with h1, . . . , hn ∈DH we
can directly compute the minimal polynomials M1, . . . , Mn and also the factorization in
Line 1 of Algorithm 1. The polynomials in our examples were actually obtained in this
way.
3.3 Kinematic interpretation
Now we are going to translate Theorem 3 into the language of kinematics. A precise
formulation takes into account the possibility of neighboring factors t −hi and t −hi+1
that describe rotations about the same axis. In this case, we call hi and hi+1 compatible.
This is the case if and only if hihi+1 = hi+1hi.
Let h ∈DH be a dual quaternion representing a rotation.
The parametrization
(t −h)t∈P1 of the rotation group defined by h is called a linear parametrization. More
generally, let R1, R2 ∈R[t] such that R1 is monic, deg(R1) = n, deg(R2) < n, without
common factor. Then the parametrization (R1(t) −h1R2(t))t∈P1 is called a rational
parametrization of degree n. Higher degree parametrizations of rotation groups may
arise as the product of linear parametrizations, if the two axes coincide, that is, if the
rotation quaternions are compatible.
A rational curve C ⊂S of degree n in the Study quadric admits a parametrization by
a motion polynomial P of degree n. We say that C is a generic rational curve of degree
n in the Study quadric if P is generic (its norm polynomial has n distinct irreducible
quadratic factors). It is straightforward to show that this notion of genericity of C is
well-defined, i.e., it does not depend on the choice of the motion polynomial P.
Corollary 9. Let C ⊂S be a generic rational curve of degree n in the Study quadric,
passing through 1. Then C can be obtained as movement of the last link of an open
kR-linkage, with k ≤n. The rotations in the k joints have a simultaneous rational
parametrization, and the sum of the degree of these parametrizations equals n.
Proof. Let P ∈DH[t] be a motion polynomial of degree n that parametrizes C. The
primal part of P cannot have real factors because C is generic. By Theorem 3, there exists
a factorization P = (t −h1) · · · (t −hn) with rotation quaternions h1, . . . , hn. Assume
that hi, . . . , hi+m−1 are compatible. Because every dual quaternion compatible with hi is
a real linear combination of 1 and hi, the product (t −hi) · · · (t −hi+m−1) can be written
8
as R1 −hiR2 for R1, R2 ∈R[t] such that R1 is monic, deg(R1) = m, deg(R2) < m. This
already implies the corollary’s statement.
Remark 10. It is sometimes advantageous to think of the open kR-chain referred to in
Corollary 9 as an open nR-chain with the possibility of coinciding consecutive axes. See,
for example, Theorem 11, below.
Example 2. We give a kinematic interpretation of the calculation in Example 1. The
two factorizations
P = (t −h1)(t −h2) = (t −h′
1)(t −h′
2)
show that the motion parametrized by P occurs as end-effector motion of two open
2R-chains, parametrized with the same rational parameter t. The axes of the rotation
quaternions h2, h1, k1, k2 in the moving frame form a closed 4R-chain whose coupler
motion is parametrized by P. Hence, we actually presented a method to synthesize a
Bennett mechanism to three given poses. Without loss of generality, we assume that
one of the poses is the identity. From this data, it is easy to compute the quadratic
parametrization of the Bennett motion as in [4]. It serves as input for our factorization
algorithm. Note that in general a cubic algebraic number needs to be introduced for the
factorization of the real quartic polynomial PP. This has also been observed in other
synthesis algorithms for Bennett linkages.
4 Interchanging factors and closed linkages
In this chapter, we investigate the linkage formed by combining the n! open nR-chains
that can be used to generate a generic rational curve C of degree n on the Study quadric.
In order to describe the combinatorial structure of a linkage, we recall the definition of
the link graph. Recall that the rigid parts of a linkage are called links, and the existence
of a rotational joint between two links means that the two links share a fixed line, the
rotation axis. It is possible that m > 2 links share the same rotation axis. In this case,
we have
�m
2

 different joints supported at this axis. The link graph consists of a node for
each link and an edge for each joint connecting two links. This should not be confused
with the axis graph, which is also used in the literature. There, the nodes correspond to
the axes of joints and the edges correspond to links.
If the linkage moves rationally with mobility one, for every link there exists an algebraic
function from P1 to P7, mapping a time parameter t ∈P1 to the pose of the link at time
t. The pose can be represented by an element of SE3 or, equivalently, by a point on the
Study quadric. If two links L1 and L2 are connected by a joint J, the relative motion
of L2 with respect to L1 is parametrized by the quotient ψJ := φ1φ−1
2
of the two pose
functions. This motion is a rotation. Hence it also has parametrization by a motion
polynomial of degree one. The linkage we construct will have the property that ψJ is a
linear polynomial (t −hJ).
Note that when hJ is specified for all joints J, the relative motions of any pair of links
can be obtained by multiplication; the linkage kinematics is fully determined. It is easy
9
to extract the Denavit-Hartenberg parameters from the rotation quaternions hJ. In this
paper we are content with the linkage specification by dual quaternions.
Theorem 11. Let C ⊂S be a generic rational curve of degree n in the Study quadric,
passing through 1. Then C is contained in the motion of a link in a mechanism with
revolute joints, 2n links and n2n−1 joints. The link graph is the 1-skeleton of the n-
dimensional hypercube.
Remark 12. In Theorem 11 we can only state that C is contained in the motion of one
link. It cannot be excluded that the mechanism has more than one degree of freedom.
An extreme case arises from the product P = (t −h1) · · · (t −hn) of compatible rotation
quaternions h1, . . . , hn. With the understanding of Remark 10, the corresponding link
graph can still be considered as 1-skeleton of a hypercube but the linkage has n trivial
degrees of freedom. A sufficient condition that ensures equality of C and the link motion
is that every 4-cycle in the link graph corresponds to a non-degenerate Bennett linkage.
Proof of Theorem 11. Let P(t) be a generic motion polynomial of degree n, parametrizing
C. We denote the n irreducible quadratic factors of PP by M1, . . . , Mn and construct
a linkage whose nodes are labelled by the subsets of G := {1, . . . , n}. Two nodes are
connected by an edge if and only if the corresponding subsets differ by exactly one
element (Figure 1). This graph is known as Hamming graph.
Let F ⊆G be a subset of cardinality m ≤n. By successively dividing out right factors
(t −hi) with Mi = (t −hi)(t −hi) ∈F, we obtain a factorization P = UV with monic
U, V ∈DH[t], and V V is the product of the factors in F. This can be done in m! ways.
We claim that the result is always the same. To show this, we assume that we have two
factorizations P = U1V1 = U2V2 corresponding to two different permutations σ1, σ2 of
M1, . . . , Mn. In our situation, the first m elements in σ1 are a permutation of the first
m elements in σ2, in fact this is the set F, and we assume that the remaining elements
are equal. This is possible because we are free to choose any order in the remaining
factors in G \ F. Now we apply our non-deterministic algorithm of dividing out right
factors to P, in the reverse order of the permutations. We obtain the two factorizations
P = V1U1 = V2U2. But in these two division processes, the first n −m choices are equal,
and it follows that U1 = U2. Consequently, we get U1 = U2 and V1 = V2. Hence F
determines the right factor V (and also the left factor U) uniquely, and we place the link
LF at position V (t), depending on one real parameter t but not on an ordering of F.
This already proves the claim on the linkage graph.
Now we show that linkage consists of revolute joints only. Assume that F1 and F2 differ
by a single element Mi. Without loss of generality we may assume F2 = F1 ∪{Mi}. Let
P = U1V1 = U2V2 be the factorizations obtained as above. Then we have V2 = (t −hi)V1,
where hi is unique common solution of U1 and Mi (see Lemma 6). Hence the relative
position of LF2 with respect to LF1, depending on t, is a rotation group parametrized by
(t −hi)t∈P1. So we see that LF2 is connected with LF1 by a rotational joint.
The two links whose relative movement is the curve C are easy to determine: They
are labelled by the empty set and by G.
10
{}
{1}
{2}
{3}
{1, 2}
{1, 3}
{2, 3}
{1, 2, 3}
h4,1 = h6,1
h2,1 = h5,1
h1,1 = h3,1
h1,2
h2,2
h3,2
h4,2
h5,2
h6,2
h1,3 = h2,3
h3,3 = h4,3
h6,3 = h5,3
Figure 1: Linkage graph obtained by factorizing a cubic curve
Example 3. We present the construction of a linkage to the graph of Figure 1. A
parametrization of a rational cubic curve C ⊂S reads
P = t3 −t2(3 + (4 −ϵ)i + (1 + 3ϵ)j + 2(1 + ϵ)k)−
t(3(1 + ϵ) −3(3 −ϵ)i −(1 + 11ϵ)j −(3 + 5ϵ)k)+
2(3 −(1 −3ϵ)i −(1 + 2ϵ)j −(1 + ϵ)k).
The real polynomial PP factors as
PP = (t2 −2t + 2
|
{z
}
=:M1
)(t2 −2t + 4
|
{z
}
=:M2
)(t2 −2t + 6
|
{z
}
=:M3
).
The polynomial P admits six factorizations P = (t −hl,1)(t −hl,2)(t −hl,3), one for each
permutation of the triple (M1, M2, M3):
(M1, M2, M3):
h1,1 = 1 + (65
31 −814
961ϵ)i + (16
31 + 1373
961 ϵ)j + (18
31 + 1719
961 ϵ)k
h1,2 = 1 + (395
403 −94035
162409ϵ)i + (319
403 + 53380
162409ϵ)j + ( 41995
162409ϵ + 479
403)k
h1,3 = 1 + (12
13 + 72
169ϵ)i + (210
169ϵ −4
13)j + ( 3
13 −
8
169ϵ)k
(M1, M2, M3):
h1,1 = 1 + (65
31 −814
961ϵ)i + (16
31 + 1373
961 ϵ)j + (18
31 + 1719
961 ϵ)k
h1,2 = 1 + (395
403 −94035
162409ϵ)i + (319
403 + 53380
162409ϵ)j + ( 41995
162409ϵ + 479
403)k
h1,3 = 1 + (12
13 + 72
169ϵ)i + (210
169ϵ −4
13)j + ( 3
13 −
8
169ϵ)k
(M1, M3, M2):
h2,1 = 1 + (5
3 −5
9ϵ)i + (1
3 + 11
9 ϵ)j + (14
9 ϵ + 1
3)k
h2,2 = 1 + (55
39 −1324
1521ϵ)i + (38
39 + 814
1521ϵ)j + ( 748
1521ϵ + 56
39)k
11
h2,3 = 1 + (12
13 + 72
169ϵ)i + (210
169ϵ −4
13)j + ( 3
13 −
8
169ϵ)k
(M2, M1, M3):
h3,1 = 1 + (65
31 −814
961ϵ)i + (16
31 + 1373
961 ϵ)j + (1719
961 ϵ + 18
31)k
h3,2 = 1 + ( 72
217 −16813
47089ϵ)i + (136
217 −7695
47089ϵ)j + (14752
47089ϵ + 153
217)k
h3,3 = 1 + (11
7 + 10
49ϵ)i −(1
7 −85
49ϵ)j + (5
7 −5
49ϵ)k
(M2, M3, M1):
h4,1 = 1 + i + ϵj + ϵk
h4,2 = 1 + (10
7 −59
49ϵ)i + (8
7 + 13
49ϵ)j + (54
49ϵ + 9
7)k
h4,3 = 1 + (11
7 + 10
49ϵ)i −(1
7 −85
49ϵ)j + (5
7 −5
49ϵ)k
(M3, M1, M2):
h5,1 = 1 + (5
3 −5
9ϵ)i + (1
3 + 11
9 ϵ)j + (14
9 ϵ + 1
3)k
h5,2 = 1 + (1
3 −4
9ϵ)i + (2
3 −2
9ϵ)j + (2
3 + 4
9ϵ)k
h5,3 = 1 + 2i + 2ϵj + k
(M3, M2, M1):
h6,1 = 1 + i + ϵj + ϵk
h6,2 = 1 + (1 −ϵ)i + j + (1 + ϵ)k
h6,3 = 1 + 2i + 2ϵj + k
The rotation quaternions hl,i correspond to edges of the linkage graph in Figure 1. The
six rotation quaternions hl,2 are all different. The six rotation quaternions hl,1 come in
three pairs, corresponding to permutations of the shape (Mi, Mj, Mk) and (Mj, Mi, Mk).
Similarly, equal rotation quaternions hl,3 come from permutations (Mi, Mj, Mk) and
(Mi, Mk, Mj), respectively.
Remark 13. There exist cases where two consecutive rotation quaternions, like h6,1 and
h6,2 in the above example, are compatible. In this case, the linkage graph of Figure 1 is
still correct but with the understanding that the revolute axes to h6,1 and h6,2 coincide
(compare Remark 10). It is possible to reflect this in the linkage graph by removing
the edge labeled h6,2 and inserting an additional edge connecting {} and {1, 2}. The
relative motion between the the links {} and {1, 2} is then the product of two linearly
parametrized rotations about the same axis, that is, a quadratically parametrized rotation.
From the description of the linkage graph given in Theorem 11 we can draw even more
conclusions:
• Every 4-cycle in the mechanism corresponds to a Bennett linkage (general case), to
a planar or spherical 4-bar linkage with a rational coupler curve, or to four joints
sharing a common revolute axis.
• Any of the n! paths of length n that connects the base {} with the platform
{1, . . . , n} corresponds to an open nR-chain. Just as any two permutations can
be transformed into each other by transpositions, any two of these nR-chains are
connected by a series of Bennett substitutions, where two consecutive revolute axes
Hl,i, Hl,i+1 are replaced by two axes Hm,i, Hm,i+1 that complete the first axis pair
to the axis quadruple of a Bennett mechanism.
• The mechanism’s theoretical degree of freedom, computed according to the formula
of Chebychev-Grübler-Kutzbach, is fn = 6(2n −1) −5n2n−1. Thus, f2 = −2,
f3 = −18, f4 = −70, f5 = −214 etc. with rapid decrease as n goes to ∞.
12
B0
B1
B2
P0
P1
P2
Figure 2: Planar overconstrained linkage
• Our construction stays within the planar motion group SE(2) or the spherical
motion group SO(3). That is, if the polynomial P describes a planar or spherical
motion, all rotation quaternions obtained from different factorizations of P belong
to the same group. Thus, we can construct planar or spherical linkages with
the same combinatorics as in the spatial case but with a theoretical degree of
freedom of gn = 3(2n −1) −n2n, that is g2 = 1, g3 = −3, g4 = −19, g5 = −67
etc. One example is depicted in Figure 2. It can be thought of as a linkage
which is composed of two planar 3RRR-platforms with identical anchor points
on the base (B0, B1, B2) and on the platform (P0, P1, P2). In addition, there
exists a hexagon formed by the middle revolute joints, whose side lengths remain
constant during the motion. In Figure 2, the hexagon sides are drawn as double
lines.
An animation of this linkage can be found on the accompanying web-
site http://geometrie.uibk.ac.at/schroecker/qf/. Its theoretical degree of
freedom is −3.
5 Translation quaternions and prismatic joints
Theorem 11 admits a small but interesting generalization that allows for a mixture of
translational and rotational joints. In fact, the factorization Algorithm 1 also works
under slightly more general assumptions: We can allow that the norm polynomial PP has
a factorization into n distinct monic quadratic polynomials M1, . . . , Mn, each of which
is either irreducible or the square of a linear polynomial. In this case, the assumptions
of Lemma 6 are fulfilled and we again get n! factorizations of P into linear motion
polynomials.
If M is a square factor of the norm polynomial PP, the factor (t −h) satisfying
M = (t −h)(t −h) parametrizes a translational one-parameter subgroup. The primal
part of h is a scalar, say λ, and M = (t −λ)2; the dual part of H is purely vectorial and
specifies the direction of the translation.
13
If the motion polynomial P has a factorization PP ∈R[t] as P = (t −h1) · · · (t −hn)
such that the minimal polynomials M1, . . . , Mn of h1, . . . , hn are all distinct, we can find
this factorization by Algorithm 1, regardless whether h1, . . . , hn represent rotations or
translations. As in the purely rotational case, there will be exactly n! such factorizations.
If the factorization P = (t −h1) · · · (t −hn) contains a translation polynomial hi with
primal part pi, the point P(pi) lies in the exceptional three-space E. Conversely, if the
curve C intersects E and admits a factorized parametrization P = (t −h1) · · · (t −hn)
with rotation or translation polynomials h1, . . . , hn, primal(P) must vanish for some
parameter value p. This implies that one of the factors (p −h1), . . . , (p −hn) has zero
primal part, that is, primal(hi) = p ∈R and hi is a translation polynomial.
We illustrate this at hand of two examples.
Example 4. Assume that P(t) = (t −h1) · · · (t −hn) and all quaternions h1, . . . , hn are
translation quaternions. This means, that the motion is a pure translation which can
be described by the sum of vectors s(t) = (u1 + tv1) + · · · + (un + tvn) with ui, vi ∈R3.
But s(t) also equals the sum of any permutation of the summands ui + tvi. This trivial
statement is a special case of Theorem 8.
Example 5. The polynomial
P = t2 −t(2 + (1 −ϵ)i + (1 + ϵ)j + (1 + 2ϵ)k) + 1 −2ϵ + (1 −ϵ)i + (1 + 2ϵ)j + (1 + ϵ)k
admits the factorizations P = (t −h1)(t −h2) = (t −k1)(t −k2) where
h1 = 1 + 1
3((3 −7ϵ)i + (3 + 2ϵ)j + (3 + 5ϵ)k),
h2 = 1 + ϵ
3(4i + j + k)
k1 = 1 + ϵj + ϵk,
k2 = 1 + (1 −ϵ)i + j + (1 + ϵ)k.
Both, k1 and h2 are translation quaternions. Indeed, P(1) = ϵ(−2 + j + k) ∈E, that is,
the curve C with parametrization P intersects the exceptional three-space E.
The kinematic interpretation of Example 5 is well-known. The motion C is the coupler
motion of an RPRP linkage [14] and can be obtained as the limiting case of the Bennett
motion The observations made in Example 5 immediately generalize to higher degree
polynomials.
6 New overconstrained 6R-chains
For n = 3, the mechanism described in the preceding section contains a number of
overconstrained 6R-chains, some of which are new. This is also true for certain examples
in case of n = 4. Although only a side result, the discovery of new overconstrained
6R-chains is an important contribution of this article. Therefore we present a more
detailed discussion.
14
(a)
F
G
(b)
F
G
(c)
F
G
Figure 3: Different types of closed 6R-chains: Bennett linkage plus dangling link (trivial)
(a), Waldron’s double Bennett hybrid (b), and new (c)
6.1 Coupler curves of degree three
Here, we assume n = 3. The link graph is the 1-skeleton of a three-dimensional cube.
The relative motion between the link labeled by the empty set F = {} (the base) and
the link labeled G = {1, 2, 3} (the platform) is a rational cubic curve C ⊂S in the Study
quadric. In general, there is nothing special about links F and G. We could as well take
any two links which are diagonally opposite in the cube.
Every path of length three that connects F and G corresponds to an open 3R-chain
which is capable of performing the motion C. Combining two of these paths yields
an overconstrained 6R-chain with coupler motion C. There is a total of six paths of
lengths three connecting F and G and a total of 15 possible combinations of two such
paths. They can be classified into three types of different combinatorics (Figure 3) or,
equivalently, into different types of pairs of permutations. We say that two permutations
σ1 and σ2 differ by σ2 ◦σ−1
1 .
• The first type (a) corresponds to permutations which differ by a neighbor transpo-
sition. The resulting 6R-chain is trivial. It consists of a Bennett linkage plus one
dangling link. The complete mechanism contains six sub-linkages of this type.
• The second type (b) corresponds to permutations that differ by a cyclic permutation.
The resulting 6R-chain is Waldron’s double Bennett hybrid [15, pp. 63–65]. There
are six sub-linkages of this type.
• The third type (c) is new.
It corresponds to permutations that differ by the
permutation (3, 2, 1). Three of the 15 6R-chains are of this type.
Currently, overconstrained 6R-chain are classified by relations between their Denavit-
Hartenberg parameters (distance, angle, and offset) on the cyclic sequence of revolute
axes, see for example the list at the end of [16]. The reasons for their mobility are of
geometric nature. Often, they are inferred from the mobility of related structures that are
already known to be flexible. Our newly found examples are mobile for algebraic reasons
or, more precisely, our construction and proof of mobility is algebraic. Therefore, we do
not offer simple relations between their Denavit-Hartenberg parameters. However, it is
possible to compute exact symbolic expressions of these parameters for numeric examples.
15
Below we list the angles, distances and offsets for the three new overconstrained 6R-chains
obtained from Example 3.
Linkage h1,3, h1,2, h1,1, h6,1, h6,2, h6,3
Distances: 16
√
29
377 ,
√
1115179082
63302
, 37
√
854
1586 , 24
√
145
899
,
√
2
2 ,
√
6
6
Offsets:
7945
59218, 38174
√
3
62281 , 545
√
5
3538 ,
7
58, 2
√
3
3 , 11
√
5
58
Angle cosines: 27
√
5
65 , 29
√
3
93 , 41
√
15
195 , 13
√
5
31 ,
√
3
3 ,
√
15
5
Linkage h2,3, h2,2, h2,1, h4,1, h4,2, h4,3
Distances:
√
1115179082
185822
, 8
√
29
87 , 37
√
854
1586 ,
√
2
6 , 12
√
145
203
,
√
6
6
Offsets:
7945
59218, 1765
√
5
3538 , 16
√
3
61 ,
7
58, 31
√
5
58 , 968
√
3
3063
Angle cosines: 151
√
3
273 , 4
√
5
15 , 41
√
15
195 , 5
√
3
9 , 2
√
5
7 ,
√
15
5
Linkage h3,3, h3,2, h3,1, h5,1, h5,2, h5,3
Distances:
√
6
42 ,
√
1115179082
63302
, 12
√
145
203
, 37
√
854
11346 ,
√
2
2 , 8
√
29
87
Offsets: 968
√
3
3063 , 53315
59218, 545
√
5
3538 , 16
√
3
61 , 53
58, 11
√
5
58
Angle cosines: 9
√
15
35 , 29
√
3
93 , 2
√
5
7 , 359
√
15
1395 ,
√
3
3 , 4
√
5
15
6.2 Coupler curves of degree four
Now we consider a second construction of new overconstrained 6R-chains whose coupler
curve is of degree four (n = 4). Let h1, h2, h3, h4 be dual quaternions representing
rotations. Set P = (t −h1)(t −h2)(t −h3)(h −h4) and Mi := (t −hi)(t −hi) for
i = 1, 2, 3, 4. The link graph of the linkage constructed in Theorem 11 has 16 links and
32 joints. Any open chain connecting base and platform has four links, any closed chain
composed of two open chains of this type has eight links. In order to get a 6R-chain,
we assume that (h1, h2) and (h3, h4) are compatible pairs. Now we use Algorithm 1 to
compute another factorization P = (t −g1)(t −g2)(t −g3)(t −g4). This gives rise to an
8R-linkage. But we can remove the link between h1 and h2 and the link between h3 and
h4 in order to get a 6R-linkage. In contrast to the previous examples, the relative motion
of some neighbouring links is rational of degree two.
The second factorization depends on the chosen permutation of the factors Mi, i =
1, . . . , 4. Of course, using the permutation (M1, M2, M3, M4) is prohibited. Because of
compatibility relations, we also have
P = (t −h2)(t −h1)(t −h3)(t −h4) = (t −h1)(t −h2)(t −h4)(t −h3).
Hence, we must avoid common right or left factors with these two factorizations as they
lead to dangling links. In total, there are 16 permutations of this type, because they
start with M1 or M2, or end with M3 or M4, or both.
16
A possible choice of a permutation is (M3, M1, M4, M2). The resulting linkage is known
as “serial Goldberg 6R linkage” (see [15, 4.3.1]). There are four permutations that lead
to this type of linkage.
Another possible choice is (M3, M4, M1, M2). The resulting linkage is of a new type
and, again, can be obtained from four different permutations. There are some apparent
relations between the Denavit-Hartenberg parameters, which can be explained by the fact
that the axes of three joints of the 6R-linkage are axes of a 4-cycle of the full “hypercube”
linkage. But these relations are not sufficient for the mobility.
7 Conclusion
Factorization of motion polynomials is a new and powerful method for synthesizing
linkages with a prescribed rational coupler motion. We gave an explicit construction,
based on dual quaternion factorization, for such linkages.
It allows to incorporate
translation quaternions (prismatic joints) and compatible quaternions (high degree
relative rotations of neighboring links).
We studied the resulting linkages and explained how to obtain new types of closed
6R-chains from them. We provided ideas for their construction, but the method leaves
room for creativity which may lead to more new types. For example, it is possible
to construct overconstrained chains with 2k prismatic and 6 −2k revolute joints by
combining the ideas of Sections 5 and 6. Coinciding consecutive joints will produce
overconstrained chains consisting of five joints (Goldberg linkages).
An important new insight is that flexibility of overconstrained 6R-chains has rather
algebraic than geometric reasons. Currently, the authors work on a systematic description
and computational tools for synthesizing linkages based on this new method.
Acknowledgements
This research was supported by the Austrian Science Fund (FWF): I 408-N13 and
DK W 1214-N15.
References
[1] G. T. Bennett, A New Mechanism, Engineering 76 (1903) 777–778.
[2] G. T. Bennett, The skew isogramm-mechanism, Proc. London Math. Soc. 13 (2nd
Series) (1913–1914) 151–173.
[3] O. Bottema, B. Roth, Theoretical Kinematics, Dover Publications, 1990.
[4] K. Brunnthaler, H.-P. Schröcker, M. Husty, A New Method for the Synthesis of
Bennett Mechanisms, in: Proceedings of CK 2005, International Workshop on
Computational Kinematics, Cassino, 2005.
17
[5] J. Krames, Zur Geometrie des Bennett’schen Mechanismus, Österreich. Akad. Wiss.
Math.-Natur. Kl. S.-B. II 146 (1937) 159–173.
[6] A. J. Perez, J. M. McCarthy, Bennett’s linkage and the cylindroid, Mech. Machine
Theory 37 (11) (2002) 1241–1445.
[7] K. Hao, Dual number method, rank of a screw system and generation of Lie sub-
algebras, Mech. Mach. Theory 33 (7) (1998) 1063–1084.
[8] J. Selig, Geometric Fundamentals of Robotics, Monographs in Computer Science,
Springer, 2 edn., 2005.
[9] I. Niven, Equations in quaternions, Amer. Math. Monthly 48 (10) (1941) 654–661.
[10] B. Gordon, T. S. Motzkin, On the zeros of polynomials over division rings, Trans.
Amer. Math. Soc. 116 (1965) 218–226.
[11] L. Huang, W. So, Quadratic formulas for quaternions, Appl. Math. Lett. 15 (5)
(2002) 533–540.
[12] M. Hamann, Line-symmetric motions with respect to reguli, Mech. Mach. Theory
46 (7) (2011) 960–974.
[13] G. Hegedűs, J. Schicho, H.-P. Schröcker, Construction of Overconstrained Linkages
by Factorization of Rational Motions, Accepted for publication in the proceedings
of ARK, 2012.
[14] A. Perez-Gracia, Synthesis of Spatial RPRP Closed Linkages for a Given Screw
System, ASME J. Mechanisms and Robotics 3 (2).
[15] P. Dietmaier, Einfach übergeschlossene Mechanismen mit Drehgelenken, Habilitation
thesis, Graz University of Technology, 1995.
[16] J. E. Baker, Screw replacements in isomeric variants of Bricard’s line-symmetric
six-bar, Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Science 223 (2009) 2391–2398.
18