Factorisation of Rational Motions:
A Survey with Examples and Applications
Z. Li
RICAM
Austrian Academy of
Sciences
Linz, Austria
T.-D. Rad
Unit Geometry and CAD
University of Innsbruck
Innsbruck, Austria
J. Schicho
RISC
Johannes Kepler University
Linz, Austria
H.-P. Schröcker
Unit Geometry and CAD
University of Innsbruck
Innsbruck, Austria
October 15, 2018
Abstract
Since its introduction in 2012, the factorization theory for rational motions quickly evolved and found applications in
theoretical and applied mechanism science. We provide an accessible introduction to motion factorization with many
examples, summarize recent developments and hint at some new applications. In particular, we provide pseudo-code for the
generic factorization algorithm, demonstrate how to find a replacement linkage for a special case in the synthesis of Bennett
mechanisms and, as an example of non-generic factorization, synthesize open chains for circular and elliptic translations.
Keywords: motion polynomial, dual quaternions, mechanism synthesis, Bennett flip, multiplication trick
1 Introduction
The factorization of rational motions was introduced in [1,
2] and has ever since proved its usefulness in mechanism
synthesis. So far, it has been used for the construction of
new linkages [2–8] with attractive properties for applications,
including rational parameterisations of all relative motions
in the linkage, computable (sometimes even linear) input-
output relations, and the availability of simple symbolic and
numeric synthesis algorithms.
The fundamental factorization theorem of [2] is a state-
ment about “generic” rational motions. Current research
focuses on non-generic cases. Preliminary results are quite
promising and will also be addressed in this survey article.
Related results can be found in [8] (these proceedings).
We continue this paper with a brief introduction to dual
quaternions and motion polynomials in Section 2. It is fol-
lowed by algorithms and examples obtained from factoriza-
tion in generic cases (Section 3). In particular, we discuss a
special case in the synthesis of Bennett mechanisms where
the two 2R dyads coincide and no valid 4R linkage can be
constructed. It is, however, possible to find a 5R or 6R
linkage that generates the same motion.
In Section 4 we discuss circular and elliptic translations
as an exceptional case of motion factorization. Here, the
generic factorization algorithm fails but straightforward com-
putations give a result. By means of a recently invented
“multiplication trick” we can generate non-planar 4R chains
for elliptic translations and construct linkages from them.
2 Dual Quaternions and Motion Polynomials
Quaternions and dual quaternions already proved their use-
fulness in computational kinematics and mechanism science.
See [9], [10, Ch. 4] or [11, Ch. 9] for an introduction. Here,
we briefly summarize some basic definitions and concepts
that will be needed in later sections.
2.1 Quaternions and dual quaternions
A quaternion can be written as q = q0 +q1i+q2j+q3k with
real numbers q0, q1, q2, q3. The set of all quaternions is
denoted by H. Quaternions are added component-wise and
their multiplication is determined by the rules
i2 = j2 = k2 = ijk = −1.
It is associative but not commutative. We have, for example
(i+j)(2−k) = 2i+2j−ik−jk = 2i+2j+j−i = i+3j but
(2−k)(i+j) = 2i+2j−ki−kj = 2i+2j−j+i = 3i+j.
The conjugate quaternion to q = q0 + q1i + q2j + q3k is
q = q0−q1i−q2j−q3k. Depending on the individual author,
the quaternion norm is either qq = q2
0 + q2
1 + q2
2 + q2
3 ∈R
or the square root of qq. We follow the first convention.
Quaternions with unit norm qq = 1 can model rotations in
SO(3) via
x = x1i+x2j+x3k 7→y1i+y2j+y3k = qxq.
(1)
Here, a vector (x1,x2,x3) ∈R3 is identified with the vector
quaternion x1i+x2j+x3k ∈H. The oriented axis direction
1
arXiv:1501.06862v2  [cs.RO]  29 Jun 2015
of the displacement (1) is (q1,q2,q3) and the rotation angle
ϕ satisfies tan(ϕ/2) = q0(q2
1 +q2
2 +q2
3)−1/2. Note that q and
−q describe the same rotation.
A dual quaternion can be written as h = p + εq with p,
q ∈H. The dual unit ε satisfies ε2 = 0 and commutes with i,
j, k. For example, (i+εj)(j−εk) = ij+ε(j2 −ik)+ε2jk =
k−ε(1−j). The set of dual quaternions is denoted by DH,
the conjugate dual quaternion is h = p + εq, and the dual
quaternion norm is hh = pp+ε(pq+qp). Because of pp,
pq+qp ∈R, it is a dual number. An important difference
between quaternions and dual quaternions is that h ̸= 0 does
not imply existence of h−1 ∈DH such that hh−1 = 1. Dual
quaternions different from the set {εd | d ∈H} do not pos-
sess such a multiplicative inverse. If h is not from that set,
its multiplicative inverse is h−1 = (hh)−1h. It equals the
conjugate up to a scalar dual factor.
Unit dual quaternions model SE(3), the group of rigid
body displacements. Identifying the point (x1,x2,x3) with
the dual quaternion x = x1i+x2j+x3k, the action of the unit
dual quaternion h = p+εq is
x 7→pxp+ pq−qp = pxp+2pq.
(2)
We see that the spherical component of the rigid body dis-
placement is described by the primal part p while the trans-
lation is given by the vector quaternion pq−qp = 2pq.
Again, h and −h model the same rigid body displacement.
This suggests, to projectivise the space of dual quaternions
by identifying h and λh for any λ ∈R\{0}. This turns DH
into the real projective space P7 of dimension seven. All
points (classes of proportional dual quaternions) that can
be normalized to unity describe rigid body displacements.
These are precisely the points of the quadric hypersurface
S ⊂P7 given by pq+qp = 0 (the Study quadric) minus the
three-space given by p = 0 (the exceptional three space E).
2.2 Polynomials and Motion Polynomials
The set of polynomials in t with coefficients in DH is de-
noted by DH[t]. Polynomial multiplication is defined by
the convention that the indeterminate t commutes with all
coefficients. For h, k ∈DH we have, for example
C = (t −h)(t −k) = t2 −(h+k)t +hk.
(3)
The value of the polynomial C = ∑n
i=0 citi ∈DH[t] at h ∈
DH is defined as C(h) := ∑n
i=0 cihi. Note that with this
convention we have for the polynomial C in (3)
C(h) = h2 −(h+k)h+hk = hk −kh,
C(k) = k2 −(h+k)k +hk = −hk +hk = 0.
This seemingly strange behavior is a consequence of the
definition for multiplication and evaluation. Linear factors
of polynomials over DH are not directly related to zeros.
There is, however, an important exception (Lemma 2 of [2]):
Proposition 1 The polynomial C ∈DH[t] can be factored
as C = C′(t −h) with C′ ∈DH[t] if and only if C(h) = 0.
In other words, the zeros of C correspond to linear right
factors of C.
The conjugate of the polynomial C = P+εQ = ∑n
i=0 citi
is defined as C = ∑n
i=0 citi and the norm polynomial is
CC = PP+ε(PQ+QP). Its coefficients are dual numbers.
A polynomial C = P+εQ ∈DH[t] is called a motion poly-
nomial, if its leading coefficient is invertible and the norm
polynomial is even real. This important condition ensures
that the curve parameterised by C(t) for t ∈R lies in the
Study quadric. Hence motion polynomials act via (2) on
points x = x1i+x2j+x3k and yield rational parametric equa-
tions
t 7→PxP+PQ−QP
PP
= PxP+2PQ
PP
,
t ∈R∪{∞}
for all trajectories. The extension of the parameter range to
R∪{∞} via C(∞) = cn is necessary for obtaining complete
curves. Summarizing, we can say that motion polynomials
parameterise rational motions. Conversely, it follows from
[12] that any motion with only rational trajectories is given
by a suitable motion polynomial.
Lets look at linear motion polynomials C = c1t +c0. Be-
cause c1 is invertible, we may transform it to C = t −h by
left multiplying with c−1
1 . This amounts to a change in the
fixed coordinate frame and is no loss of generality. The
norm polynomial is (t −h)(t −h) = t2 −(h+h)t +hh and
the conditions for a linear motion polynomial are
h+h ∈R,
hh ∈R.
The second condition implies that h itself describes a rigid
body displacement, the first condition implies vanishing of
the dual scalar part. In coordinates we have h = h0 +h1i+
h2j+h3k+ε(h5i+h6j+h7k) with h1h5 +h2h6 +h3h7 = 0.
It is known that the rigid body displacements C(t), t ∈R∪
{∞} are either all rotations about the fixed axis with Plücker
coordinates [h1,h2,h3,−h5,−h6,−h7] or, if h1 = h2 = h3 =
0, translations in direction (h5,h6,h7).
Remark 2 In case of a rotation, the rotation angle ϕ satisfies
tan(ϕ/2) = (t −h0)(h2
1 + h2
2 + h2
3)−1/2. Using the coeffi-
cients a = −h−h and b = hh of the norm polynomial, this
becomes tan(ϕ/2) = (2t +a)(4b−a2)−1/2. The rotation an-
gle at the parameter value t is hence already determined by
the norm polynomial of t −h. A similar statement holds true
for the translation distance in case of h1 = h2 = h3 = 0. Also
note that h is a zero of the norm polynomial (t −h)(t −h).
3 Generic Factorization
A motion polynomial C = P + εD is called generic, if its
primal part P has no real factors. If P has a real factor with
2
(possibly complex) zero t0, the point C(t0) lies in the excep-
tional generator E of the Study quadric S. Hence, C is not
generic, if it intersects E (possibly over the complex num-
bers) or, equivalently, if the degree of the spherical motion
component, which is parameterised by P, can be reduced
by dividing off a real factor. The fundamental theorem in
motion factorization is
Proposition 3 ([2], Theorem 1) If C is a monic, generic
motion polynomial of degree n, there exist rotation poly-
nomials t −h1, ..., t −hn such that
C = (t −h1)···(t −hn).
(4)
We call (4) a factorization of C. If C is not generic, fac-
torizations of the shape (4) may or may not exist. It is also
noteworthy and very important for applications in mecha-
nism science that the factorization (4) is, in general, not
unique (compare Algorithm 2 below). It can be viewed as
a decomposition of a rational motion into the product of ro-
tations, whose angles are linked via the common parameter
t. In other words, every factorization of the motion polyno-
mial C corresponds to an open nR chain whose end-effector
can follow the motion parameterised by C. Combining sev-
eral of these open chains will in general result in an over-
constrained linkage with end-effector motion parameterised
by C.
3.1 Computation
Computing a factorization (4) of a generic motion polyno-
mial C = P+εQ is not much harder than the factorization
of real polynomials over the complex numbers. An impor-
tant ingredient is right division for polynomials in DH[t].
Given A and B ∈DH[t] such that the leading coefficient
of B is invertible, there exist unique polynomials Q and R,
called quotient and remainder, such that A = QB + R and
degR < degB. Algorithm 1 computes Q and R under the
additional assumption that B is monic. This is no real loss of
generality and is usually fulfilled in our applications. Note
that there is no essential difference between Algorithm 1
and a standard algorithm for polynomial division in R[t].
In particular, it does not require any quaternion division.
Because of non-commutativity it is, however, important to
strictly adhere to the given order of factors (for example, Bc
in Line 6 must not be replaced by cB).
Algorithm 2 can be used to compute a factorization of a
generic, monic motion polynomial C. (If the leading coeffi-
cient cn of C is different from one, we can compute a factor-
ization of c−1
n C and left-multiply it with cn.) By genericity,
the norm polynomial CC has no real zeros. Hence, there
exist irreducible quadratic polynomials M1,...,Mn ∈R[t]
such that CC = M1 ···Mn. The factorizations of C are la-
beled by the permutations of these factors. The functions
REMAINDER(C,Mn) in Line 3 and QUOTIENT(C,t −hi) in
Line 6 are derived from Algorithm 1. In Line 4 we have to
Algorithm 1 Polynomial division
Require: A,B ∈DH[t], B monic.
Ensure: Q,R such that A = QB+R and degR < degB.
1: Q ←0, R ←A
2: while deg(R) ≥deg(B) do
3:
c ←leading coefficient of R
4:
n ←deg(R)−deg(B)
5:
Q ←Q+ctn
6:
R ←R−Bctn
7: end while
8: return Q,R
compute the zero r−1
1 r0 of a linear polynomial R = r1t +r0.
It exists and is unique if r1 is invertible. Again, this is guar-
anteed by genericity of C.
From Algorithm 2 we infer further properties of motion
factorization. They follow from the labeling of factorizations
by quadratic factors of CC and Remark 2.
Proposition 4 In general, a motion polynomial of degree
n admits n! different factorizations. In any factorization,
the factors labeled by the same quadratic factor Mi of the
norm polynomial give rise to rotations with the same angular
velocity.
Algorithm 2 Factorization of motion polynomials
Require: Generic, monic motion polynomial C ∈DH[t]
of degree n, tuple F = (M1,...,Mn) of quadratic, irre-
ducible real factors of CC.
Ensure: List H = [h1,...,hn] such that C = (t −h1)···(t −
hn) and Mi = (t −hi)(t −hi) for i = 1,...,n; initially
H = [] is the empty list.
1: repeat
2:
F ←(M1,...,Mn−1)
3:
R ←REMAINDER(C,Mn)
4:
h ←unique zero of R
5:
H ←[h,H]
▷prepend h to list
6:
C ←QUOTIENT(C,t −hi)
7: until degC = 0
8: return H.
Even if most examples of motion factorization in literature
are symbolic, it is important to remark that both Algorithm 1
and Algorithm 2 can easily be implemented with floating
point numbers. From a numerical point of view, the most
challenging step in the computation is the factorization of
the norm polynomial CC = M1 ···Mn. But this is a standard
problem of numerical mathematics with a variety of fast and
stable algorithms available.
3
3.2 Synthesis of Bennett linkages
The most important application of motion factorization is
linkage synthesis. As one particular example, we present
the synthesis of spatial four-bar linkages (Bennett linkages).
We are not content with the generic case alone but also
discuss special cases that might occur. Synthesis algorithms
for Bennett linkages are known for a while, see [13] or
[14] and the references therein. The approach via motion
factorization gives additional insights and, more importantly,
allows generalizations as in [5].
We wish to construct a Bennett linkage that visits three
given poses p0, p1, p2 which we view as points in S \ E.
A crucial observation of [13] is that the coupler motion of
a Bennett linkage is parameterised by a quadratic motion
polynomial that can easily be computed. Lets consider the
example
p2 = 1−i−j−k+ε(1+k),
p1 = 3−i−2j−k+ε(1−i+j+2k),
p2 = 1.
Setting C = t2 + (λ p1 −1 −µ p0)t + µ p0 with yet unde-
termined λ, µ ∈R \ {0} we ensure C(0) = µ p0 ≡p0,
C(1)λ p1 ≡p1, C(∞) ≡p2. Here, the symbol “≡” denotes
equality in projective sense, that is, up to scalar factors. Now
we have to select λ and µ such that the dual part of the norm
polynomial CC vanishes. Comparing coefficients, we obtain
the equations
λ −µ = 2µ −λµ −λ = µ(λ −1) = 0
with the unique solution λ = µ = 1. Thus, the coupler
motion of the sought Bennett mechanism is given by the
quadratic motion polynomial
C = t2 +(1−j)t +1−i−j−k−ε((i−j−k)t −1−k).
It is rather typical for this approach to determine the cou-
pler motion before the dimension and position of the actual
mechanism (“mechanism from motion”). This can be done
by computing the two factorizations of C. In order to use
Algorithm 2, we need the quadratic, irreducible factors M1
and M2 of CC. In our example, we obtain
M1 = t2 +2,
M2 = t2 +2t +2.
Running Algorithm 2 twice with F = (M1,M2) and F =
(M2,M1) yields the two factorizations C = (t −h1)(t −h2) =
(t −k1)(t −k2) where
h1 = j+k+ε(i+j−k),
h2 = −1−k−2εj,
k1 = −1−i−εk,
k2 = i+j+ε(i−j).
(5)
The quaternions h2 and k2 describe rotations about the mov-
ing axes in the reference configuration C(∞) = 1 (the iden-
tity). The quaternions h1 and k1 give the position of the fixed
axes in that configuration. The linkage’s Denavit-Hartenberg
parameters can easily be computed from the Plücker coor-
dinates of the moving and fixed revolute axes. The loop
closure equation is
1 = (t −h1)(t −h2)(t −k2)−1(t −k1)−1
≡(t −h1)(t −h2)(t −k2)(t −k1)
= (t2 +2)(t2 +2t +2).
where “≡” denotes equality up to real polynomial factors.
We conclude this section with a hint to [5]. This paper
presents a synthesis method for over-constrained 6R link-
ages to four prescribed poses. It is based on rational cubic
interpolation on the Study quadric S and subsequent motion
factorization and can be viewed as direct generalization of
the synthesis of Bennett linkages we just presented. The
interpolation procedure is, however, more involved: Inter-
polants exist only if the three-space spanned by the given
poses intersects the Study quadric S in a ruled quadric (a
hyperboloid, not an ellipsoid). Then, two one-parametric
families of rational interpolating cubic curves exist, each
giving rise, via motion factorization, to families of solution
linkages.
3.3 What Can Go Wrong?
Above example presented an ideal case for synthesis of Ben-
nett linkages. But things can go wrong in special situations.
Usually, literature on synthesis of Bennett linkages ignores
these exceptional cases but a robust synthesis procedure
should be aware of them.
To begin with, the curve parameterized by C may actually
lie on a straight line. In this case, the end-effector motion is
either a rotation about a fixed axis or a straight line. Situa-
tions like this are easy to detect and we exclude them in the
following. Should C parameterise a planar or spherical mo-
tion, the factorization algorithm will work and yield a planar
or spherical anti-parallelogram linkage. Note, however, that
in these cases a two-parametric set of quadratic interpolants
C exists, because the dual part of CC vanishes identically in
λ and µ.
The conditions for using Algorithm 2 may not be met, that
is, the primal part P of C = P+εQ may have a real factor. If
P is even real, the motion is a curvilinear translation along a
rational curve of degree two. We do not intend to give a full
discussion here. In Section 4 we will deal in detail with the
cases of elliptic and circular translations. If P has precisely
one real factor of degree one, there exists a parameter value
t0 with P(t0) = 0 and hence C(t0) = εQ(t0). Should Q(t0) be
zero, the curve lies on a straight line. Hence Q(t0) ̸= 0 and
C(t0) lies in the exceptional generator of the Study quadric
S. This case has been discussed in [15]. It leads to the
coupler motion of RPRP linkages [16] and is still amenable
to Algorithm 2 — even if not all prerequisites are met. An
4
example is C = (t −h1)(t −h2) = (t −k1)(t −k2) where
h1 = i+j+ε(i−j),
h2 = 1+ε(i+3j−k),
k1 = 1+ε(3i+j−k),
k2 = i+j−ε(i−j).
Note that h1 and k2 are rotation quaternions with (necessar-
ily) parallel axes while h2 and k1 are translation quaternions.
Finally, we also would like to discuss a situation that was
pointed out by M. Hamann (Dresden University of Applied
Sciences). It is possible that Algorithm 2 is applicable but
does not yield a proper linkage because the two synthesized
2R dyads coincide. Using motion factorization, we can
easily characterize this situation and construct a replacement
mechanism.
It is known [17] that the coupler motion of Bennett link-
ages is obtained by reflecting a fixed system in the rulings
of one family on a hyperboloid of revolution (it is line sym-
metric with respect to that system of rulings). Using the hy-
perboloid with equation x2b2c2 +y2a2c2 −z2a2b2 = a2b2c2
and a,b,c ∈R\{0}, a possible motion polynomial reads
C = (b2 +c2)t2
+(2a(cj+bk)−2ε(a2(bj−ck)−bc(bk+cj)))t
+2bci−b2 +c2 −2aε((b2 −c2)i+2bc)
(6)
Two factorizations C = (b2 + c2)(t −h1)(t −h2) = (b2 +
c2)(t −k1)(t −k2) can only coincide if the norm polynomial
factors as CC = M2 with a quadratic, irreducible polynomial
M ∈R[t]. From the discriminant
4096(b2 +c2)8(a2 +c2)2(a+b)2(a−b)2
(7)
of CC we read off that this can only happen if a = b. Indeed,
we then have CC = (a2 +c2)2(t2 +1)2. Thus, we have
Proposition 5 A line symmetric motion with respect to one
family of lines on a hyperboloid H can be generated by only
one 2R dyad if and only if H is a hyperboloid of revolution.
Otherwise, there exist two 2R dyads.
In other words, synthesis of a Bennett linkage fails if the
three input poses are sampled from a line symmetric motion
with respect to a hyperboloid of revolution.
3.4 Bennett flips
Motion factorization not only reveals a caveat of existing
Bennett synthesis methods but can also be used for the con-
struction of a replacement linkage with five or six joints.
This construction is based on an important technique for
rational linkages which we call Bennett flip. Consider the
motion polynomial C = (t −h1)(t −h2), pick an arbitrary ro-
tation quaternion p and use Algorithm 2 to compute rotation
quaternions f, g, r, q such that
(t −h1)(t −p) = (t −f)(t −g),
(t −h2)(t −p) = (t −q)(t −r).
h1
h2
p
f
g
r
q
Figure 1: Link graph for construction of replacement mechanisms
The link diagram of this construction is depicted in Figure 1.
The vertices indicate links and the arrows with label x con-
nect links such that the relative motion of the second (at
the arrow tip) with respect to the first (at the arrow base) is
parameterised by t −x. We have the loop closure equations
1 ≡(t −h1)(t −p)(t −g)(t −f)
≡(t −q)(t −r)(t −p)(t −h2).
(8)
The second closure equation implies (t −q)(t −r) ≡(t −
h2)(t −p) so that
(t −h1)(t −h2)(t −q)(t −r)(t −g)(t −f)
≡(t −h1)(t −h2)(t −h2)(t −p)(t −g)(t −f)
≡(t −h1)(t −p)(t −g)(t −f)
≡1.
(9)
The last equality follows from the first closure equation in
(8). From (9) we infer that the rotation quaternions h1, h2,
q, r, g, f can be assembled, in that order, to form an over-
constrained 6R linkage. As long as p is chosen sufficiently
general, this construction works and the resulting 6R linkage
(of type Waldron’s double Bennett hybrid) has precisely one
degree of freedom. The link connecting h2 and q performs
the motion parameterized by C. Note that this construction
does not hinge on the vanishing of the discriminant (7) but
is valid for arbitrary Bennett motions.
A special choice of p may even cause the axis of g and
r to coincide. This then yields an over-constrained 5R link-
age (a Goldberg mechanism), for which (t −h1)(t −h2)
parameterises the motion of one link. Let’s look at the con-
crete example obtained by plugging a = b = 1 and c = 2
into the motion polynomial (6). It has the factorization
C = (t −h1)(t −h2) where
h1 = −1
5(4j−3k+2ε(3j+4k)),
h2 = −k.
Setting p = j−k+ 5
2ε(j+k) we compute the rotation quater-
nions f, g, r, and q via Bennett flips:
f = 1
5(j−7k)+ 9
10ε(7j+k),
g = k−5εj = r,
q = j+k−5
2ε(j−k).
Because g = r, two consecutive axes coincide and the corre-
sponding linkage has only five revolute joints.
5
4 Exceptional Factorizations
Motion polynomials whose primal part has no real factor
can always be factorized by Algorithm 2. However, many
rational motions of practical and theoretical importance are
not generic. For those motions, computation of the unique
zero hi of a linear remainder polynomial in Line 4 of Al-
gorithm 2 may fail. Note that motion polynomials in H[t]
are always projectively equal to generic motion polynomials
and Algorithm 2 always works. This is an example for the
failure of naive transfer of results of spherical kinematics
to spatial kinematics via wrong application of Kotelnikov’s
and Study’s principle of transfer.
Failure of Algorithm 2 does not imply anything on exis-
tence or non-existence of factorizations. There exist rational
motions with no factorization and rational motions with
infinitely many factorization.
4.1 Elliptic translations
Let us discuss, as an example, the motion polynomial
C = t2 +1+ε(bjt +ai),
a ≥b > 0.
(10)
It parameterizes an elliptic translation. The primal part of C
is real so that Algorithm 2 is not applicable. But we can try
to manually solve the equation
C = (t −h1)(t −h2)
(11)
for h1 = p1 + εd1, h2 = p2 + εd2. Comparing coefficients
of the primal part, we immediately find the conditions p1 +
p2 = 0, p1p2 = 1 and the solution
p1 = xi+yj+zk, p2 = p1
where
x2 +y2 +z2 = 1.
The dual parts are di = uii+vij+wik for i = 1,2. Comparing
coefficients gives the additional equations
(u1 −u2)x+(v1 −v2)y+(w1 −w2)z = 0
(v1 +v2)z−(w1 +w2)y+a = 0
(u1 +u2)z−(w1 +w2)x = 0
(u1 +u2)y−(v1 +v2)x = 0
u1 +u2 = b+v1 +v2 = w1 +w2 = 0.
(12)
Th system (12) has the solution
x = 0, z = a
b, u1 = λ, u2 = −λ, v1 = µ −b, v2 = −µ,
w1 = yb(b−2µ)
2a
, w2 = −yb(b−2µ)
2a
,
with λ, µ ∈R. From x2 + y2 + z2 = 1 and a ≥b > 0 we
infer a = b and y = 0 whence the solution becomes
x = y = 0, z = 1, u1 = λ, u2 = −λ,
v1 = µ −a, v2 = −µ, w1 = 0, w2 = 0.
In summary, we have shown:
h1
h2
k1
k2
ℓ1ℓ1
ℓ2
Figure 2: Three different factorizations of a circular translation
Proposition 6 Elliptic translations do not admit factoriza-
tions of the shape (t −h1)(t −h2) but circular translations
admit infinitely many. If the circular translation is parame-
terized as in (10) with a = b, these factorizations are given
by h1 = k+ε(λi+(µ −a)j), h2 = −k−ε(λi+ µj).
In fact, it is well-known that a circular translation occurs
in infinitely many ways as coupler motion of a parallelogram
linkage. Each factorization corresponds to the motion of one
2R chain in such a linkage. This is illustrated in Figure 2.
4.2 The Multiplication Trick
The elliptic translation (10) admits no factorization with two
linear motion polynomials. But there is a way to obtain
factorizations with four motion polynomials. The basic idea
is to find a real, irreducible polynomial Q such that QC
admits a factorization. Note that C and QC are equal in
projective sense, that is, they parameterise the same motion.
By a result of [18], this is always possible and the degree
of the real polynomial Q is bounded by the degree of the
greatest real factor in the primal part of C.
Returning to the elliptic translation (10) and look at one
particular example. Assuming a ̸= b, we let Q = t2 +1 and
compute QC = (t −h1)(t −h2)(t −h3)(t −h4) where
h1 =
k−a−b
a+bεi,
h2 = −k+ (2(a−b)i−(a+b)2j)ε
2(a+b)
,
h3 = −k−(i−1
2(a−b)j)ε,
h4 =
k+iε.
(13)
By a construction similar to the Bennett flip in Section 3.4,
we may “brace” each link hi with an anti-parallelogram
linkage with joints hi, mi, ki mi−1. Combing these anti-
parallelograms gives a linkage where the link connecting m4
and h4 performs an elliptic translation (Figure 3). The con-
struction is initialized by picking a generic planar rotation
6
h1
h1
h2
h2
h3
h3
h4
h4
k1
k1
k2
k2
k3
k3
k4
k4
m0
m0
m1
m1
m2
m2
m3
m3
m4
m4
h1
h2
h3
h4
k1
k2
k3
k4
m0
m1
m2
m3
m4
Figure 3: Linkage to generate an elliptic translation and correspond-
ing link graph
quaternion m0 and recursively computing ki and mi from
(t −m−1
i−1)(t −hi) = (t −ki)(t −m−1
i ),
i = 1,2,3,4.
The joints triplets hi, mi, hi+1 and ki, mi, ki+1, respectively,
belong to the same link. This is indicated by circular arcs in
Figure 3. Constructions of this type are at the core of [19]
where bounded rational curves of arbitrary degree are drawn
by planar linkages.
The factorization (13) is by no means unique. We also
have the strange factorization QC = (t −k1)(t −k2)(t −
k3)(t −k4) where
k1 = −(a2 −b2)i−2abk
a2 +b2
−(a2 −b2)εj
a2 +b2
,
k2 = (a2 −b2)i−2abk
a2 +b2
−((a2 +b2)2 −2b(a2 −b2))εj
2b(a2 +b2)
,
k3 = −i+ a2 −b2 +2b
2b
εj,
k4 = i−εj.
In contrast to (13), this factorization is not planar. It gives
rise to a spatial open chain with parallel first and second and
parallel third and fourth axis that can follow a curvilinear
translation along an elliptic path. We may combine the four
m0
h1
k1
m1
k2
k2
m2
h2
k3
m3
h3
Figure 4: Spatial linkage to generate a Darboux motion.
revolute axes of the latter factorization with two prismatic
joints to obtain an an over-constrained RRRRPP mechanism
one of whose links performs an elliptic translation.
If we abandon the requirement to generate an elliptic trans-
lation, we may also make the motion polynomial C of (10)
factorizable by right-multiplying it with a suitable quater-
nion polynomial H ∈H[t]. This changes the motion but not
the trajectory of the origin in the moving coordinate frame. If
H is linear, the motion polynomial CH parameterizes a Dar-
boux motion (a spatial motion with only planar trajectories)
and our Bennett flip construction produces a spatial link-
age to draw an ellipse (Figure 4). The anti-parallelograms
of Figure 3 are replaced by Bennett linkages and only ten
joints and eight links are required. While the linkage has
theoretical full-cycle mobility, it is, of course, a demanding
engineering task to realize this in a practical design. For
more information about Darboux linkages we refer to [8].
There, also considerable refinements of the above construc-
tion can be found.
5 Conclusions
This article gives an overview of algorithms and techniques
associated with the factorization of motion polynomials. It
addresses several special but yet instructive cases and gives
concrete applications. We complemented known methods
for the synthesis of Bennett linkages by a discussion of a
special case that so far has skipped attention of researchers
and we showed how to use Bennett flips to produce replace-
ment linkages. We also hinted at a new method for the
construction of rational over-constrained 6R linkages to four
prescribed poses. The attentive reader may have noticed a
7
gap in the chain of synthesis procedures: We did not talk
about 5R linkages. In fact, it is possible to synthesis 5R
linkages to three poses plus “some additional constraints”.
An efficient algorithm to do this hinges on a better under-
standing of three-spaces generated by 2R dyads and will be
addressed in future work.
In a second part, we gave a preview on very recent meth-
ods to factor exceptional motion polynomials that evade the
algorithms for generic situations. The “multiplication trick”
is actually a powerful tool to factor all motion polynomials
and, together with Bennett flips, belongs to a toolbox for the
construction of linkages to arbitrary rational motions or to
arbitrary rational trajectories. The general theory and algo-
rithmic treatment of the “multiplication trick” is currently
being worked out. As these theoretical constructions do not
necessarily require an excessive number of links, there is
hope for useful engineering applications.
Acknowledgments
This research was supported by the Austrian Science Fund
(FWF): P 26607 (Algebraic Methods in Kinematics: Motion
Factorization and Bond Theory).
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