User:Prof McCarthy/dualquaternion

hesis, Columbia University, 1963.</ref> A dual quaternion is an ordered pair of quaternions  = (A, B) and therefore is constructed from eight real parameters. Because rigid body displacements are defined by six parameters, dual quaternion parameters include two algebraic constraints.

W. R. Hamilton introduced quaternions in 1843 and by 1873 W. K. Clifford obtained a broad generalization of these numbers that he called biquaternions, which is an example of what is now called a Clifford algebra. At the turn of the 20th century, Aleksandr Kotelnikov and E. Study developed dual vectors and dual quaternions for use in the study of mechanics.

In ring theory, dual quaternions are a division ring constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in the form q = q₀ + ε q_ε, where q₀ and q_ε are ordinary quaternions and ε is the dual unit (εε = 0).

Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length. This fact is used in theoretical kinematics (see McCarthy), and in applications to 3D computer graphics and robotics.

In 1891 Eduard Study realized that this associative algebra was ideal for describing the group of motions of three-dimensional space. He further developed the idea in Geometrie der Dynamen in 1901. B. L. van der Waerden called the structure "Study biquaternions", one of three eight-dimensional algebras referred to as biquaternions.

Formulas

In order to describe operations with dual quaternions, it is helpful to first consider quaternions.

A quaternion is a linear combinations of the basis elements 1, i, j, and k. Hamilton's product rule for i, j, and k is often written as

i² = j² = k² = ijk = -1.

Compute i(ijk)=-jk=-i, to obtain jk=i, and (ijk)k=-ij=-k or ij=k. Now because j(jk)=ji=-k, we see that this product yields ij=-ji, which links quaternions to the properties of determinants.

A convenient way to work with the quaternion product is to write a quaternion as the sum of a scalar and a vector, that is A = a₀ + A, where a₀ is a real number and A = A₁i + A₂j + A₃k is a three dimensional vector. The vector dot and cross operations can now be used to define the quaternion product of A=a₀+A and C=c₀+C as

$G=AC=(a_{0}+\mathbf {A} )(c_{0}+\mathbf {C} )=(a_{0}c_{0}-\mathbf {A} \cdot \mathbf {C} )+(c_{0}\mathbf {A} +a_{0}\mathbf {C} +\mathbf {A} \times \mathbf {C} ).$

A dual quaternion is usually described as a quaternion with dual numbers as coefficients. A dual number is an ordered pair â=(a, b). Two dual numbers add componentwise and multiply by the rule âĉ=(a, b)(c, d) = (ab, ad + bc). Dual numbers are often written in the form â=a+εb, where ε is the dual unit that commutes with i, j, k and has the property ε²=0.

The result is that a dual quaternion is the ordered pair of quaternions Â=(A, B). Two dual quaternions add componentwise and multiply by the rule,

${\hat {A}}{\hat {C}}=(A,B)(C,D)=(AC,AD+BC).\!$

It is convenient to write a dual quaternion as the sum of a dual scalar and a dual vector, Â = â₀ + A, where â₀=(a, b) and A=(A, B) is the dual vector that defines a screw. This notation allows us to write the product of two dual quaternions as

${\hat {G}}={\hat {A}}{\hat {C}}=({\hat {a}}_{0}+{\mathsf {A}})({\hat {c}}_{0}+{\mathsf {C}})=({\hat {a}}_{0}{\hat {c}}_{0}-{\mathsf {A}}\cdot {\mathsf {C}})+({\hat {c}}_{0}{\mathsf {A}}+{\hat {a}}_{0}{\mathsf {C}}+{\mathsf {A}}\times {\mathsf {C}}).$

Addition

The addition of dual quaternions is defined componentwise so that given,

${\hat {A}}=(A,B)=a_{0}+a_{1}i+a_{2}j+a_{3}k+b_{0}\epsilon +b_{1}\epsilon i+b_{2}\epsilon j+b_{3}\epsilon k,$

and

${\hat {C}}=(C,D)=c_{0}+c_{1}i+c_{2}j+c_{3}k+d_{0}\epsilon +d_{1}\epsilon i+d_{2}\epsilon j+d_{3}\epsilon k,$

then

${\hat {A}}+{\hat {C}}=(A+B,C+D)=(a_{0}+c_{0})+(a_{1}+c_{1})i+(a_{2}+c_{2})j+(a_{3}+c_{3})k+(b_{0}+d_{0})\epsilon +(b_{1}+d_{1})\epsilon i+(b_{2}+d_{2})\epsilon j+(b_{3}+d_{3})\epsilon k,$

Multiplication

Multiplication of two dual quaternion follows from the multiplication rules for the quaternion units i, j, k and commutative multiplication by the dual unit ε. In particular, given

${\hat {A}}=(A,B)=A+\epsilon B,$

and

${\hat {C}}=(C,D)=C+\epsilon D,$

then

${\hat {A}}{\hat {C}}=(A+\epsilon B)(C+\epsilon D)=AC+\epsilon (AD+BC).\!$

Notice that there is no BD term, because the definition of dual numbers requires that ε² = 0.

This gives us the multiplication table (note the multiplication order is row times column):

Multiplication table for dual quaternion units

*	1	i	j	k	ε	εi	εj	εk
1	1	i	j	k	ε	εi	εj	εk
i	i	-1	k	-j	εi	-ε	εk	-εj
j	j	-k	-1	i	εj	-εk	-ε	εi
k	k	j	-i	-1	εk	εj	-εi	-ε
ε	ε	εi	εj	εk	0	0	0	0
εi	εi	-ε	εk	-εj	0	0	0	0
εj	εj	-εk	-ε	εi	0	0	0	0
εk	εk	εj	-εi	-ε	0	0	0	0

Conjugate

The conjugate of a dual quaternion is the extension of the conjugate of a quaternion, that is

${\hat {A}}^{*}=(A^{*},B^{*})=A^{*}+\epsilon B^{*}.\!$

It is useful to introduce the functions Sc(∗) and Vec(∗) that select the scalar and vector parts of a quaternion, or the dual scalar and dual vector parts of a dual quaternion. In particular, if  = â₀ + A, then

${\mbox{Sc}}({\hat {A}})={\hat {a}}_{0},{\mbox{Vec}}({\hat {A}})={\mathsf {A}}.\!$

This allows the definition of the conjugate of  as

${\hat {A}}^{*}={\mbox{Sc}}({\hat {A}})-{\mbox{Vec}}({\hat {A}}).\!$

or,

$({\hat {a}}_{0}+{\mathsf {A}})*={\hat {a}}_{0}-{\mathsf {A}}.\!$

Using the scalar and vector form of the product of dual quaternions, the product of a dual quaternion with its conjugate yields

${\hat {A}}{\hat {A}}^{*}=({\hat {a}}_{0}+{\mathsf {A}})({\hat {a}}_{0}-{\mathsf {A}})*={\hat {a}}_{0}^{2}+{\mathsf {A}}\cdot {\mathsf {A}}.\!$

This is a dual scalar which is the magnitude squared of the dual quaternion.

Norm

The norm of a dual quaternion |Â| is computed using the conjugate to compute |Â| = √(ÂÂ*). This is a dual number called the magnitude of the dual quaternion. Dual quaternions with |Â| = 1 are unit dual quaternions.

Dual quaternions of magnitude 1 are used to represent spatial Euclidean displacements. Notice that the requirement that ÂÂ* = 1, introduces two algebraic constraints on the components of Â, that is

${\hat {A}}{\hat {A}}^{*}=(A,B)(A^{*},B^{*})=(AA^{*},AB^{*}+BA^{*})=(1,0).\!$

Inverse

The inverse of a dual quaternion is defined as

${\hat {A}}^{-1}={\frac {{\hat {A}}^{*}}{{\hat {A}}{\hat {A}}^{*}}}.$

This is the dual version of the inverse of a quaternion.

Dual quaternions and spatial displacements

A benefit of the dual quaternion formulation of the composition of two spatial displacements D_B=([R_B], b) and D_A=([R_A],a) is that the resulting dual quaternion yields directly the screw axis and dual angle of the composite displacement D_C=D_BD_A.

In general, the dual quaternion associated with a spatial displacement D = ([A],d) is constructed from its screw axis S={S, V) and the dual angle (φ, d) where φ is the rotation about and d the slide along this axis, which defines the displacement D. The associated dual quaternion is given by,

${\hat {S}}=\cos {\frac {\hat {\phi }}{2}}+\sin {\frac {\hat {\phi }}{2}}{\mathsf {S}}.$

Let the composition of the displacement D_B with D_A be the displacement D_C=D_BD_A. The screw axis and dual angle of D_C is obtained from the product of the dual quaternions of D_A and D_B, given by

${\hat {A}}=\cos({\hat {\alpha }}/2)+\sin({\hat {\alpha }}/2){\mathsf {A}}\quad {\text{and}}\quad {\hat {B}}=\cos({\hat {\beta }}/2)+\sin({\hat {\beta }}/2){\mathsf {B}}.$

That is, the composite displacement D_C=D_BD_A has the associated dual quaternion given by

${\hat {C}}=\cos {\frac {\hat {\gamma }}{2}}+\sin {\frac {\hat {\gamma }}{2}}{\mathsf {C}}={\Big (}\cos {\frac {\hat {\beta }}{2}}+\sin {\frac {\hat {\beta }}{2}}{\mathsf {B}}{\Big )}{\Big (}\cos {\frac {\hat {\alpha }}{2}}+\sin {\frac {\hat {\alpha }}{2}}{\mathsf {A}}{\Big )}.$

Expand this product in order to obtain

$\cos {\frac {\hat {\gamma }}{2}}+\sin {\frac {\hat {\gamma }}{2}}{\mathsf {C}}={\Big (}\cos {\frac {\hat {\beta }}{2}}\cos {\frac {\hat {\alpha }}{2}}-\sin {\frac {\hat {\beta }}{2}}\sin {\frac {\hat {\alpha }}{2}}{\mathsf {B}}\cdot {\mathsf {A}}{\Big )}+{\Big (}\sin {\frac {\hat {\beta }}{2}}\cos {\frac {\hat {\alpha }}{2}}{\mathsf {B}}+\sin {\frac {\hat {\alpha }}{2}}\cos {\frac {\hat {\beta }}{2}}{\mathsf {A}}+\sin {\frac {\hat {\beta }}{2}}\sin {\frac {\hat {\alpha }}{2}}{\mathsf {B}}\times {\mathsf {A}}{\Big )}.$

Divide both sides of this equation by the identity

$\cos {\frac {\hat {\gamma }}{2}}=\cos {\frac {\hat {\beta }}{2}}\cos {\frac {\hat {\alpha }}{2}}-\sin {\frac {\hat {\beta }}{2}}\sin {\frac {\hat {\alpha }}{2}}{\mathsf {B}}\cdot {\mathsf {A}}$

to obtain

$\tan {\frac {\hat {\gamma }}{2}}{\mathsf {C}}={\frac {\tan {\frac {\hat {\beta }}{2}}{\mathsf {B}}+\tan {\frac {\hat {\alpha }}{2}}{\mathsf {A}}+\tan {\frac {\hat {\beta }}{2}}\tan {\frac {\hat {\alpha }}{2}}{\mathsf {B}}\times {\mathsf {A}}}{1-\tan {\frac {\hat {\beta }}{2}}\tan {\frac {\hat {\alpha }}{2}}{\mathsf {B}}\cdot {\mathsf {A}}}}.$

This is Rodrigues formula for the screw axis of a composite displacement defined in terms of the screw axes of the two displacements. He derived this formula in 1840.

The three screw axes A, B, and C form a spatial triangle and the dual angles at these vertices between the common normals that form the sides of this triangle are directly related to the dual angles of the three spatial displacements.

Matrix form of dual quaternion multiplication

The matrix representation of the quaternion product is convenient for programming quaternion computations using matrix algebra, which is true for dual quaternion operations as well.

The quaternion product AC is a linear transformation by the operator A of the components of the quaternion C, therefore there is a matrix representation of A operating on the vector formed from the components of C.

Assemble the components of the quaternion C=c₀+C into the array C=(C₁, C₂, C₃, c₀). Notice that the components of the vector part of the quaternion are listed first and the scalar is listed last. This is an arbitrary choice, but once this convention is selected we must abide by it.

The quaternion product AC can now be represented as the matrix product

$[A^{+}]C={\begin{bmatrix}a_{0}&-A_{3}&A_{2}&A_{1}\\A_{3}&a_{0}&-A_{1}&A_{2}\\-A_{2}&A_{1}&a_{0}&A_{3}\\-A_{1}&-A_{2}&-A_{3}&a_{0}\end{bmatrix}}{\begin{Bmatrix}C_{1}\\C_{2}\\C_{3}\\c_{0}\end{Bmatrix}}.$

The product AC can also be viewed as an operation by C on the components of A, in which case we have

$[C^{-}]A={\begin{bmatrix}c_{0}&C_{3}&-C_{2}&C_{1}\\-C_{3}&c_{0}&C_{1}&C_{2}\\C_{2}&-C_{1}&c_{0}&C_{3}\\-C_{1}&-C_{2}&-C_{3}&c_{0}\end{bmatrix}}{\begin{Bmatrix}A_{1}\\A_{2}\\A_{3}\\a_{0}\end{Bmatrix}}.$

The dual quaternion product ÂĈ = (A, B)(C, D) = (AC, AD+BC) can be formulated as a matrix operation as follows. Assemble the components of Ĉ into the eight dimensional array Ĉ = (C₁, C₂, C₃, c₀, D₁, D₂, D₃, c₀), then ÂĈ is given by the 8x8 matrix product

$[{\hat {A}}^{+}]{\hat {C}}={\begin{bmatrix}A^{+}&0\\B^{+}&A^{+}\end{bmatrix}}{\begin{Bmatrix}C\\D\end{Bmatrix}}.$

As we saw for quaternions, the product ÂĈ can be viewed as the operation of Ĉ on the coordinate vector Â, which means ÂĈ can also be formulated as,

$[{\hat {C}}^{-}]{\hat {A}}={\begin{bmatrix}C^{-}&0\\D^{-}&C^{-}\end{bmatrix}}{\begin{Bmatrix}A\\B\end{Bmatrix}}.$

More on spatial displacements

The dual quaternion of a displacement D=([A], d) can be constructed from the quaternion S=cosφ + sinφS that defines the rotation [A] and the vector quaternion constructed from the translation vector d, given by D = d₁i + d₂j + d₃k. Using this notation, the dual quaternion for the displacement D=([A], d) is given by

${\hat {S}}=S+\varepsilon {\frac {1}{2}}DS.$

Let the Plucker coordinates of a line in the direction x through a point p in a moving body and its coordinates in the fixed frame which is in the direction X through the point P be given by,

${\hat {x}}=\mathbf {x} +\varepsilon \mathbf {p} \times \mathbf {x} \quad {\text{and}}\quad {\hat {X}}=\mathbf {X} +\varepsilon \mathbf {P} \times \mathbf {X} .$

Then the dual quaternion of the displacement of this body transforms Plucker coordinates in the moving frame to Plucker coordinates in the fixed frame by the formula

${\hat {X}}={\hat {S}}{\hat {x}}{\hat {S}}^{*}.$

Using the matrix form of the dual quaternion product this becomes,

${\hat {X}}=[{\hat {S}}^{+}][{\hat {S}}^{-}]^{*}{\hat {x}}.$

This calculation is easily managed using matrix operations.