Approaching Dual Quaternions From Matrix Algebra
IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 5, OCTOBER 2014
1037
Approaching Dual Quaternions
From Matrix Algebra
Federico Thomas, Member, IEEE
Abstract—Dual quaternions give a neat and succinct way to en- capsulate both translations and rotations into a unified represen- tation that can easily be concatenated and interpolated. Unfortu- nately, the combination of quaternions and dual numbers seems quite abstract and somewhat arbitrary when approached for the first time. Actually, the use of quaternions or dual numbers sepa- rately is already seen as a break in mainstream robot kinematics, which is based on homogeneous transformations. This paper shows how dual quaternions arise in a natural way when approximating 3-D homogeneous transformations by 4-D rotation matrices. This results in a seamless presentation of rigid-body transformations based on matrices and dual quaternions, which permits building intuition about the use of quaternions and their generalizations.Index Terms—Biquaternions, Cayley factorization, double quaternions, dual quaternions, spatial kinematics, quaternions.
1) The order of quaternion multiplication: Quaternions are sometimes multiplied in the opposite order than rotation matrices, as in [4]. The origin of this can be found in the way vector coordinates are represented. For example, in [5], a celebrated book on Computer Graphics, point coor- dinates are represented by row vectors instead of column vectors, as is the common practice in Robotics. Then, transformation matrices postmultiply a point vector to produce a new point vector. The result can be confus- ing for anyone approaching quaternions for the first time. For more details on this matter, see [6]. 2) The way quaternions operate on vectors: Quaternions have been used to rotate vectors in 3-D by essentially sand- wiching a vector in 3-D between a unit quaternion and its conjugate [7, Ch. 17], [8]. Nevertheless, strictly speaking, quaternions cannot operate on vectors. The word vector was introduced by Hamilton to denote the imaginary part of the quaternion, which is different from today’s meaning [9]. 3) The nature of the quaternion imaginary units [6], [8]:
Surprisingly, despite their long life, the use of quaternions in engineering is not free from confusions that mainly concern the following.
puter Graphics .
transformations. Nowadays, quaternions play a fundamental role in the representation of spatial rotations and a chapter de- voted to them can be found in nearly every advanced textbook on Computer Vision, Robot Kinematics and Dynamics, or Com-
I. I NTRODUCTION
2
= ijk = −1, when seeking a new kind of number that would extend the idea of complex numbers [1].
Quaternions were developed independently of their needs for any particular application. The main use of quaternions in the 19th century consisted in expressing physical theories in the no- tation of quaternions. In this context, during the end of the 19th century, researchers working on electromagnetic theory debated about the choice of quaternion or vector notation in their for- mulations. This generated a fierce dispute from about 1880 to 1900, reaching its climax in a series of letters in the journal
= k
value discredited, having been replaced by the simpler algebra of matrices and vectors. Later on, in the mid-20th century, the development of computing machinery made necessary a reex- amination of quaternions from the standpoint of their utility in computer simulations. The need for efficient simulations of air- craft and missile motions was responsible to a large extent for sparking the renewed interest in quaternions [3]. It was rapidly realized that quaternion algebra yields more efficient algo- rithms than matrix algebra for applications involving rigid-body
Manuscript received March 6, 2013; revised July 5, 2013; accepted November 26, 2013. Date of publication August 22, 2014; date of current version September 30, 2014. This paper was recommended for publication by Associate Editor M. C. Cavusoglu upon evaluation of the reviewers’ comments. This work was supported by the Spanish Ministry of Economy and Competitiveness through the Explora program under Contract DPI2011-13208-E.
The author is with the Institut de Rob`otica i Inform`atica Industrial, 08028 Barcelona, Spain (e-mail: fthomas@iri.upc.edu). Digital Object Identifier 10.1109/TRO.2014.2341312
2
= j
2
form a + bi + cj + dk, where i
Hamilton himself contributed to this confusion as he al- ways identified the quaternion units with quadrantal rota- tions, as he called the rotations by π/2 [10, p. 64, art. 71]. Nevertheless, they represent rotations by π [9]. All these confusions are seriously affecting the progress of quaternions in engineering because, as a result, they are used in recipes for manipulating sequences of rotations without a pre- cise understanding of their meaning. The situation just worsens when working with dual quaternions, an extension of ordinary quaternions that permits encapsulating rotations and translations in a unified representation. Thus, it is not strange that many prac- titioners are still averse to using them despite their undeniable value.
This paper shows how quaternions do naturally emerge from 4-D rotation matrices and how dual quaternions are then derived when approximating 3-D homogeneous transformations by 4-D rotations. As a consequence, all common misunderstand- ings concerning quaternions are cleared up because the derived expressions may be interpreted both as matrix expressions and as quaternions.
I N 1843, Hamilton defined quaternions as quadruples of the
Nature [2]. Then, quaternions disappeared from view, and their
IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 5, OCTOBER 2014
4 Soon after Hamilton introduced quaternions, he tried to use
A. Quaternions and Rotations in
Yang and Freudenstein introduced the use of dual quaternions for the analysis of spatial mechanisms [26]. Since then, dual quaternions have been used by several authors in the kinematic analysis and synthesis of mechanisms, and in computer graphics (see, for instance, the works of McCarthy [27], Angeles [28], and Perez-Gracia [29]).
4
C. Quaternions and Matrix Algebra
Matrix algebra was developed in the beginning of 1858 by Cayley and Sylvester. Soon it was realized that matrices could be used to represent the imaginary units used in the definition of quaternions. Actually, a set of 4 × 4 matrices, sometimes called Dirac–Eddington–Conway matrices, with real values can realize every algebraic requirement of quaternions. Alternatively, a set of 2 × 2 matrices, usually called Pauli matrices, with complex values can play the same role (see [30, pp. 143–144] for details). Therefore, there are sets of matrices, which all produce valid matrix representations of quaternions. The choice of one set over other has been driven by esthetic preferences, but we will show how Cayley’s factorization leads to a matrix representation that attenuates this sense of arbitrariness.
= 0 so that a biquaternion has the form q
2
[19] for a review and summary of these papers). Clifford adopted the word biquaternion, previously used by Hamilton to refer to a quaternion with complex coefficients, to denote a combination of two quaternions algebraically combined via a new symbol ω defined to have the property ω
biquaternions, ” and “Further note on biquaternions” [18] (see
In 1882, Clifford introduced the idea of a biquaternion in three papers: “Preliminary sketch of biquaternions,” “Notes on
is a product of rotations in a pair of orthogonal 2-D subspaces [15]. This factorization, known as Cayley’s factoring of 4-D rotations, was also proved using matrix algebra by Van Elfrinkhof in 1897 in a paper [16] rescued from oblivion by Mebius in [17]. Cayley’s factorization plays a central role in what follows as it provides a bridge between homogeneous transformations and quaternions that remained unnoticed in the past.
[14]. Cayley’s results can be used to prove that any rotation in R
While double quaternions have found direct application to represent 4-D rotations, dual quaternions found application to encapsulate both translation and rotation into a unified represen- tation. Then, if 3-D spatial displacements are approximated by 4-D rotations, a beautiful connection between double and dual quaternions can be established.
4
. Nevertheless, it seems that he was not aware of Rodrigues’ work and his use of quaternions as a description of rotations was wrong. He believed that the expression for a rotated vector was linear in the quaternion rather than quadratic. This passage of the history of quaternions is actually a matter of controversy (see [9], [11], and [12] for details). It is Cayley whom we must thank for the correct development of quaternions as a representation of rotations and for establishing the connection with the results published by Rodrigues three years before the discovery of quaternions [13]. Cayley is also credited to be the first to discover that quaternions could also be used to represent rotations in R
2
in the same way as complex numbers can be used to represent rotations in R
3
them to represent rotations in R
R
3 and
R
1038
B. Quaternions and Their Generalizations
- ωq
are both ordinary quater- nions. The use of the term biquaternion is confusing. As ob- served in [19], even Clifford contributed to this confusion by using the symbol
- eq
- eq
- q
- q
1
2
) + η(q
1 − q
2
). Since ξ
2
= ξ, η
2
= η, and ξη = 0, the terms (q
1
2
) and (q
2
1 − q
2
) operate inde- pendently in the double quaternion product, which has been found quite convenient when manipulating kinematic equations expressed in terms of double quaternions [23]. A third pos- sible representation for double quaternions consists in having two quaternions expressed in different bases of imaginary units whose product is commutative. This also leads to couples of quaternions that operate independently when multiplied. Nowa- days, the algebras of ordinary, double, and dual quaternions are grouped under the umbrella of Clifford algebras, also known as geometric algebras (see [24, Ch. 9] or [25] for an introduction).
2
and q
1
, where q
While in most textbooks, the matrix representation of quater- nions is considered as an advanced topic, if ever mentioned, in this paper, matrix algebra is used as the doorway to quaternions. This would probably be the usual practice if matrix algebra had been developed before quaternions.
D. Organization of the Paper
This paper is organized as follows. Section II reviews the con- nection between 4-D rotations and double quaternions in terms of matrix algebra. Section III presents a digression, that can be skipped on a first reading, in which the expressive power of matrix algebra is explored to derive different systems of hyper- complex numbers associated with 4-D rotations. In Section IV, the results presented in Section II are specialized to the 3-D case. Section V deals with the problem of approximating 3-D transformations in homogeneous coordinates by 4-D rotations. The results obtained in Sections IV and V constitute the ba- sic building blocks of the proposed twofold matrix-quaternion formalism for the representation of rigid-body transformations. The reinterpretation of kinematic equations expressed as prod- ucts of transformations in homogeneous coordinates using this formalism is treated in Section VI. Section VII presents some
2
1
= ξ(q
1
ω in several different contexts. For example, in his paper “Preliminary sketch of biquaternions,” it is also used with the multiplication rule
= 1, are called double quaternions. This denomination derives from the fact that the symbols ε and e designate the dual and the double units, respectively [20]. Thus, we have three imaginary units, which can be equal either to the complex unit i (i
ω
2
= 1. Nowadays, in the area of robot kinematics, biquaternions of the form q
1 + εq
2
, where ε
2
= 0, are called dual quaternions, while those of the form q
1 + eq
2
, where e
2
2
Then, q
2
= 0), or to the double unit e (e
2
= 1). These units define the basis of the so-called hypercomplex numbers [21]. The double quaternion q
1
2
can be reformulated by introducing the symbols ξ =
1+e
2
and η =
1−e
2 [18], [22].
= −1), to the dual unit ε (ε
THOMAS: APPROACHING DUAL QUATERNIONS FROM MATRIX ALGEBRA
1039
examples, and, finally, we conclude in Section VIII with a sum- Then, a 4-D rotation matrix, say R, can be expressed as mary of the main points.
L R R L
R = R R = R R (6)
OUR
IMENSIONAL OTATIONS
II. F -D R where
AND OUBLE UATERNIONS
D Q
L
After a proper change in the orientation of the reference frame, R = l I + l A + l A + l A (7)
1
1
2
2
3
3
an arbitrary 4-D rotation matrix (i.e., an orthogonal matrix with and determinant +1) can be expressed as [31, Th. 4]:
⎛ ⎞
R
cos α
1 − sin α
1 R = r
I + r B + r B + r B (8)
1
1
2
2
3
3
⎜ ⎟ sin α cos α
1
1
⎜ ⎟ (1)
⎝ ⎠. where I stands for the cos α
2 2 4 × 4 identity matrix and
− sin α sin α
2 cos α
2
⎛ ⎞ ⎛ ⎞
1 −1
Thus, a 4-D rotation is defined by two mutually orthogonal ⎜ ⎟ ⎜ ⎟ 0 −1 0 −1 ⎜ ⎟ ⎜ ⎟
A A = =
1
2
planes of rotation, each of which is fixed in the sense that points ⎝ ⎠, ⎝ ⎠
1 −1 0 0 in each plane stay within the planes. Then, a 4-D rotation has
1
1 two angles of rotation, α and α , one for each plane of rotation,
1
2
⎛ ⎞ ⎛ ⎞
1 −1 0 through which points in the planes rotate. All points not in the
⎜ ⎟ ⎜ ⎟
1 planes rotate through an angle between and . See [32] −1 0
α α
1
2
⎜ ⎟ ⎜ ⎟ A B
3 = 1 =
⎝ ⎠, ⎝ ⎠
1 for details on the geometric interpretation of rotations in four −1
1 −1 0 dimensions.
⎛ ⎞ ⎛ ⎞ If , the rotation is called an isoclinic rotation. An
α
1
2
= ±α 1 0 −1 isoclinic rotation can be left- or right-isoclinic (depending on
⎜ ⎟ ⎜ ⎟
1
1 ⎜ ⎟ ⎜ ⎟
B = B = whether or , respectively), which can be
2
3
α
1 = α 2 α
1
2
= −α ⎝ ⎠, ⎝ ⎠.
1 −1 represented by a rotation matrix of the form
−1 0 0 −1 0 ⎛ ⎞ l l
−l
3 2 −l
1 Therefore, , A , A , B , B
{I, A
1
2 3 } and {I, B
1
2 3 } can be seen,
⎜ ⎟ l l
L 3 −l 1 −l
2
⎜ ⎟ R = (2) respectively, as bases for left- and right-isoclinic rotations.
⎝ ⎠ l l −l
2 1 −l
3 The details on how to compute Cayley’s factorization (6) can
l l l l
1
2
3 be found in the Appendix.
and Now, it can be verified that
⎛ ⎞ r r r −r
3
2
1
2
2
2 A = A = A = A A A (9)
1
2 3 = −I
1
2
3
⎜ ⎟ r r r
R 3 −r
1
2
⎜ ⎟ R = (3)
⎝ ⎠ r r r −r
2
1
3
and r −r
1 −r 2 −r
3
2
2
2 B = B = B = B B B (10)
1
2 3 = −I.
1
2
3
respectively. Since (2) and (3) are rotation matrices, their rows and columns are unit vectors. As a consequence We can recognize in these two expressions the quaternion def-
2
2
2
2
l + l + l + l = 1 (4) inition. Actually, (9) and (10) reproduce the celebrated formula
1
2
3 that Hamilton carved into the stone of Brougham bridge.
and Expression (9) determines all the possible products of A ,
1
2
2
2
2 A
, and A resulting in
2
3
r + r + r + r = 1. (5)
3 Without loss of generality, we have introduced some changes A A A A A A
= A , = A , = A
1
2
3
2
3
1
3
1
2
in the signs and indices of (2) and (3) with respect to the notation A A A A A A
, , . (11)
2 1 = −A
3
3 2 = −A
1
1 3 = −A
2
used by Cayley [14], [33] to ease the treatment given below and to provide a neat connection with the standard use of quaternions Likewise, all the possible products of B , B , and B can be
1
2
3 for representing rotations in three dimensions.
derived from expression (10). All these products can be sum- Isoclinic rotation matrices have three important properties. marized in the following product tables:
1) The product of two right- (left-) isoclinic matrices is a right- (left-) isoclinic matrix. 2) The product of a right- and a left-isoclinic matrix is com-
I A
1 A
2 A
3 mutative.
I I A A A
1
2
3
3) Any 4-D rotation matrix, according to Cayley’s factoriza- A A A (12)
1 1 −I 3 −A
2
tion, can be decomposed into the product of a right- and A A A
2 2 −A 3 −I
1 a left-isoclinic matrix.
A
3 A
3 A
2
1
−A −I
1040
IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 5, OCTOBER 2014
I D D D
1
2
1
2
3 I B B B
3 I
I D
1 D
2 D
I B
1 B
2 B
3 I
3 D
1 D
3 D 2 (17)
B
1 B
1 B
3 2 (13)
−I −B D
1 I D
2 D
2 D
2 B
2
3 B
1
−B −I D
3 I D
1 B
3 D
3 D
2 D
B B B
3
3
2
1
−B −I
1 I .
−1
Then, clearly D = D , i = 1, 2, 3, and, as a consequence
i
Moreover, it can be verified that
i
B A B A and B A = D , , = D .
1
1
1 2 = −D
2
2
3
3
3 A B A
= B (14)
i j j i
Substituting the above expressions for B ,
i i = 1, 2, 3, in (8),
which is actually a consequence of the commutativity of left- multiplying the result by (7), and factoring out D ,
i i = 1, 2, 3,
and right-isoclinic rotations. Then, in the composition of two we conclude that (6) can be rewritten as
4-D rotations, we have
R = IQ + D Q + D Q + D Q (18)
1
1
2
2
3
3 L R L R L L R R
R R R R R R (15)
1 2 = (R )(R ) = (R )(R ).
1
1
2
2
1
2
1
2
where
L R
It can be concluded that R and R can be seen either as
i i
Q = r (l I + l A + l A + l A )
1
1
2
2
3
3
4 × 4 rotation matrices or, when expressed as in (7) and (8), respectively, as unit quaternions and their product, as a double Q = r (l A I + l A A )
1
1 1 − l
1
2 3 − l
3
2
quaternion because they operate independently in the product Q = r A + l A + l
I A )
2 2 (−l
2
1
3 2 − l
3
1
of two 4-D rotations. It is said that they are unit quaternions because their coefficients satisfy (4) and (5).
Q = r (l A + l A A I ) .
3
3
3
1 2 − l
2 1 − l
3 Next, in Section IV, the above twofold matrix-quaternion rep-
Now, we can shift from the basis , D , D
{I, D
1
2 3 } to the basis
resentation of 4-D rotations is specialized to 3-D rotations and, , E , E , E
{E
1
2
3 4 } defined as
in Section V, generalized to represent 3-D translations. Nev- ⎛ ⎞ ertheless, let us fist explore this twofold representation a bit
1 0 0 0 further. ⎜ ⎟
1 0 0 0 0 ⎜ ⎟
E
1 = 1 + D
2 3 ) =
(I − D − D ⎝ ⎠ (19) 0 0 0 0
4 III. D
IGRESSION
0 0 0 0 One of the multiple advantages of the proposed matrix-
⎛ ⎞ 0 0 0 0 quaternion formulation is that the involved imaginary units have 1 ⎜ ⎟
0 1 0 0 a clear algebraic interpretation. We can operate with these units ⎜ ⎟
E = (I + D ) =
2 1 − D 2 − D
3
⎝ ⎠ (20) 0 0 0 0 4 to obtain different representations of 4-D rotations that would
0 0 0 0 otherwise be quite abstract and difficult to derive. To see this, let us start by defining
⎛ ⎞ 0 0 0 0 ⎛ ⎞
−1 0 0 ⎜ ⎟
1 0 0 0 0 ⎜ ⎟
E = (I + D + D + D ) =
⎜ ⎟
3
1
2
3
1 0
−1
⎝ ⎠ (21) 0 0 1 0 ⎜ ⎟
4 D B = A =
1
1
1
⎝ ⎠ 0 1 0 0 0 0 0 0
−1 ⎛ ⎞
⎛ ⎞
1 ⎜ ⎟
1 ⎜ ⎟
−1 0
−1
⎜ ⎟ E = + D ) =
⎜ ⎟
4 (I − D 1 − D
2
2 2 =
3 D B
= −A
2
⎝ ⎠. (22)
4 ⎝ ⎠
1
1 −1
⎛ ⎞ The elements of this basis are distinguished by the fact that
−1 their multiplication table is the simplest possible for a basis ⎜ ⎟
−1 0 0
−1
⎜ ⎟ D = A B =
3
3
3 ⎝ ⎠.
1 E E E E
1
1
2
3
4 E
1 E
1 Then, it can be verified that
E
2 E
2
2
2
2 E
3 E
3 D = D = D = D D D = I (16)
1
2
3
1
2
3 E E
4 4 .
which allow us to define a kind of quaternion whose imaginary By inverting the system of equations defined by (19)–(22), we units are double units. As with ordinary quaternions, (16) deter- obtain mines all the possible products of D , D , and D , which can
1
2
3 I
be summarized in the following product table: = E + E + E + E
1
2
3
4
THOMAS: APPROACHING DUAL QUATERNIONS FROM MATRIX ALGEBRA
- E
- E
4 D
- E
- E
- E
- E
- E
- E
- Q
- r
- r
1 √ t
(27) We can perform the same factorization for rotations about the y- and the z-axes. Table I compiles the results.
1 ) .
(I + tB
2
1 √ t
)
1
(I + tA
2
2
to obtain more amenable expressions. Then R x (φ) =
x (φ
φ
(26) respectively, where we have introduced the change of variable t = tan
1 ⎞ ⎟ ⎟ ⎠
1 −t 0
1 −t 0 t
⎛ ⎜ ⎜ ⎝ 1 t
2
1 √ t
Now, let us suppose that we want to represent a general rota- tion using the XYZ Cardanian angles [35, p. 28]. Then R
)R y (φ
1
)
2
1 t
)
1
1 B
(I + t
1
2
1 t
1
R x
1 A
(I + t
1
2
1 t
) =
3
)R z (φ
2
(φ) =
1 t
(25) and R
- 1
- (r l
- r
- (r l
- r
- (r l
- r
- 1
- r
- (r l
- r
- r
- (r l
- r
- (r l
- r
- 1
- 1
- Q
- Q
- (r l
- (r l
- (r l
- 1
- 1
2 i
reader interested in further exploring the connections between
j = 0 if i = j. The
E
i
, for i = 1, 2, 3, 4, and E
i
= E
l )A
It is thus concluded that a 4-D rotation can be expressed as a linear combination of four quaternions. In the product of two 4-D rotations, these four quaternions operate independently because E
3 .
IMENSIONAL
3
1
l
2
2 − r
HREE
R
1 ⎞ ⎟ ⎟ ⎠
4
(I + t
1 −t 1 −t t
⎛ ⎜ ⎜ ⎝
2
1 √ t
(φ) =
L x
R
. Then, its Cayley’s factorization into a right- and a left-isoclinic rotation, using the procedure given in the Appendix, yields
(24) which can be readily interpreted as a rotation in R
OTATIONS AND
1 ⎞ ⎟ ⎟ ⎠
1 cos(φ) − sin(φ) 0 sin(φ) cos(φ)
⎛ ⎜ ⎜ ⎝
x (φ) =
The homogenous matrix transformation representing a rota- tion by φ about the x axis is R
We have seen how double quaternions naturally emerge from Cayley’s factorization of 4-D rotations into isoclinic rotations. Now, we specialize this result to 3-D rotations.
UATERNIONS
Q
RDINARY
O
2
1 t
2 A
- 1
- 1
- (r l
- (r l
- r
- r
- 1
- 1
- (r l
- r
1 (t
1
[(1 − t
3
2
2
2
1
2
2
=
2
] and Q
3
)A
2
t
1
3
2
t
t
3
3
3 ].
)B
2
t
1
3
2
)B
t
3
1
2 − t
1
)B
3
t
2
1
)I
)A
t
2
3
3 B
(I + t
3
2
1 t
)
3
3 A
(I + t
2
Therefore, using the commutativity of left- and right-isoclinic rotations, and the product tables (12) and (13), we conclude that R
1 t
)
2
2 B
(I + t
2
2
1
)
3 ) .
x (φ 1 )R y (φ 2 )R z (φ 3 ) = Q
1
3
2 − t
1
)A
3
t
2
1
1 t 2 t 3 )I
[(1 − t
2
1 Q 2 = Q
2
2
1
2
1 (t
1 =
(28) where Q
1
2 Q
l
)A
3 − r
3 − r
3 − r
2
)A
1
l
3
l − r
2
l
l
1
2
1
)A
2
l
3
3
l
1
2
l − r
1
3
l
3
2
l
2
1 − r
l
= (r l − r
2
3
1 − Q 2 − Q
= Q + Q
2
3 K
l )A
3
1 − r
l
2
1
2
3 − E
1 K
, for i = 1, . . . , 4, we obtain R = E
i
Substituting these expressions into (18) and factoring out E
4 .
3
2
4 D 3 = −E 1 − E
2
2 K
1 − E
= E
2
3 − E
2
1
1 = −E
D
1041
1
2
1 − r
1
)I
3
l
3
2
l
2
1
l
= (r l + r
3 K
3
2 − Q
1
= Q − Q
1
(23) where K
4
4 K
3
)I
1
1
− Q
− r
1
1 l
= (r l + r
3
2 + Q
− Q
1
4 = Q
2
and K
3
3 + r 1 l 2 + r 2 l 1 + r 3 l )A
2
2 l + r 3 l 1 )A
− r
3
1 l
− r
2 l
− r
1
1
1
l
3
l + r
2
3
l
1
2
)A
3 l 3 )I
2
l
3
3 − r
l
2
l + r
1
1 − r
l + r
2
3 l 2 )A
l
2
)A
1
l
3
l − r
2
3
1
− r
2 − r
1
)A
2
l
3
3
l
2
3
1
l
3
3
2 l
− r
1 + r 1 l
3 l 3 )I
− r
2
1 l 1 + r 2 l
2 − r
= (r l − r
2
1
= Q + Q
3
3 K
l )A
3
1 − r
l
2
4-D rotations and different sets of imaginary units is referred to [33] and [34].
- 1)(t
- 1)(t
- 1)
- (t
- t
- (t
- (t
- t
IV. T
- D
- 1)(t
- 1)(t
- 1)
- (t
- t
- (t
- (t
- t
IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 5, OCTOBER 2014
⎛ ⎜ ⎝
1 1 t −t 1 ⎞
1 −t t
⎛ ⎜ ⎝
1
√
t 2 + 11 ⎞ ⎟ ⎠ ·
1 −t t
1
1 −t 0 t
1 √ t 2 + 1
1 √ t 2 + 1
=
1 ⎞ ⎟ ⎟ ⎟ ⎠
1
2 t t 2 + 1 1 −t 2 t 2 + 1
1 −t 2 t 2 + 1 −2 t t 2 + 1
⎜ ⎜ ⎜ ⎝
2 ) R z (φ) = ⎛
(I + tB
⎟ ⎠ =
(I + tA 3 ) ·
(I + tA 2 ) ·
1 ·
⎜ ⎜ ⎝
⎜ ⎜ ⎝
1 d 4 δ
1
1 d 2 δ
⎜ ⎜ ⎝
1 − d 4 δ d 4 δ
A
1
1 √ t 2 + 1
1 d δ
⎜ ⎜ ⎝
I
IGHT -I SOCLINIC R OTATIONS Transformation Left-Isoclinic Right-Isoclinic Double Quaternion T x d δ
I NTO L EFT - AND R
I NFINITESIMAL 3-D T RANSLATIONS IN H OMOGENEOUS C OORDINATES A PPROXIMATED BY 4-D R OTATIONS AND T HEIR F ACTORIZATIONS
TABLE II
3 )
(I + tB
1 √ t 2 + 1
1 √ t 2 + 1
1 d 4 δ
−2 t t 2 + 1 2 t t 2 + 1
⎟ ⎠ ·
1 t 1 ⎞
1 −t 1 −t t
⎛ ⎜ ⎝
1 √ t 2 + 1
=
⎟ ⎟ ⎟ ⎠
1 −t 2 t 2 + 1 1 ⎞
1 1 −t 2 t 2 + 1
⎛ ⎜ ⎝ 1 t
Transformation Left-Isoclinic Right-Isoclinic Double Quaternion R x (φ) = ⎛ ⎜ ⎜ ⎜ ⎝
2 ))
IGHT -I SOCLINIC R OTATIONS ( t = tan( φ
I NTO L EFT - AND R
I NTERPRETED AS 4-D R OTATIONS AND T HEIR F ACTORIZATIONS
IMENSIONAL R OTATIONS IN H OMOGENEOUS C OORDINATES
TABLE I
T HREE -D1042
B
1
√
t 2 + 11 −t 0 t
⎞ ⎟ ⎠ =
⎟ ⎟ ⎟ ⎠
1 t −t 1 −t 0 1
⎛ ⎜ ⎝ 1 t
1
√
t 2 + 11 ⎞ ⎟ ⎠ ·
1 −t −t 0 1 t
⎛ ⎜ ⎝ 1 t
1 √ t 2 + 1
=
1 −t 2 t 2 + 1 1 ⎞
1 −t 0 1 ⎞
1 −2 t t 2 + 1
1 −t 2 t 2 + 1 −2 t t 2 + 1
⎜ ⎜ ⎜ ⎝
1 ) R y (φ) = ⎛
(I + tB
1 √ t 2 + 1
(I + tA 1 ) ·
1 √ t 2 + 1
⎟ ⎠ =
1 d 4 δ
- d 4 δ
− d 4 δ
- d 4 δ
- p
- p
- p
- p
- p
- p
- p
- p
- n
- n
2
p
2
3
2p
1
p
2 − 2p
p
3
2p p
2
1
3
2p
2
2p p
3
1
p
2 1 − 2p
2
1
− 2p
2
3
− 2p
2
3 − 2p
3
2
p
3
p ⎞ ⎟ ⎟ ⎠
⎛ ⎜ ⎜ ⎝ p
−p
3
p
2
p
1
p
p −p
= ⎛ ⎜ ⎜ ⎝ 1 − 2p
1
p
2
−p
2
p
1
p p
3
−p
1 −p 2 −p
3
p ⎞ ⎟ ⎟ ⎠
p
p
1
) cos θ
2 I
θ
2 (n
x
A
1
y
A
2
z
A
3
2 I
n (θ) = cos
θ
2 (n
x
B
1
y
B
2
z
B
3 ) .
(30)
θ
R
1
Now, if we substitute into (29) the following values: p = cos θ
2p
1 p
3
− 2p p
2 2p p 1 + 2p 2 p
3
1 − p
2 1 − p
2
2
1 ⎞ ⎟ ⎟ ⎠.
(29) Observe that this is the well-known formula that permits pass- ing from a quaternion representation to the corresponding rota- tion matrix [7, p. 85].
2 , p
= (n x , n y , n z ) (see, for example, [36, p. 47]). In sum
1 = n x sin
θ
2 p
2 = n y sin
θ
2 , p
3 = n z sin
θ
2 with n
2 x
2 y
2 z
= 1, the result can be recognized as the rota- tion through an angle θ about an axis that passes through the ori- gin and has direction given by the unit vector n
p
p
1 − d 4 δ
1 ⎞ ⎟ ⎟ ⎠ d →0
− d 4 δ
1 − d 4 δ
1 ⎞ ⎟ ⎟ ⎠ =
I − d 4 δ
A 2 ·
I + d 4 δ
B
2 T z d δ δ →∞
= ⎛ ⎜ ⎜ ⎝
1
1
1 d δ
≃ ⎛ ⎜ ⎜ ⎝
1 d 4 δ
1
1
1 d 2 δ
− d 2 δ
1 ⎞ ⎟ ⎟ ⎠ = ⎛
⎜ ⎜ ⎝
1 d 4 δ
− d 4 δ
1
1 d 4 δ
− d 4 δ
1 ⎞ ⎟ ⎟ ⎠ · ⎛
1 d 4 δ
⎜ ⎜ ⎝
3
1
1 ⎞ ⎟ ⎟ ⎠ · ⎛
1 ⎞ ⎟ ⎟ ⎠ = ⎛
1 − d 2 δ
δ →∞ ≃ ⎛
1 − d 4 δ
1 ⎞ ⎟ ⎟ ⎠ =
I − d 4 δ
1 ⎞ ⎟ ⎟ ⎠
1
δ →∞ = ⎛
1 T y d δ δ →∞
= ⎛ ⎜ ⎜ ⎝
1 d δ
1 ⎞ ⎟ ⎟ ⎠ · ⎛
1
1 ⎞ ⎟ ⎟ ⎠
δ →∞ ≃ ⎛
⎜ ⎜ ⎝
1
1 d 2 δ
1 − d 2 δ
1 ⎞ ⎟ ⎟ ⎠ = ⎛