Dual Quaternion

class DualQuaternion(real=None, dual=None)

Bases: object

A dual number is an ordered pair \(\hat{a} = (a, b)\) or written as \(a + \epsilon b\) where \(\epsilon^2 = 0\).

A dual quaternion can be considered as either:

• a quaternion with dual numbers as coefficients

• a dual of quaternions, written as an ordered pair of quaternions

The latter form is used here.

References:

• http://web.cs.iastate.edu/~cs577/handouts/dual-quaternion.pdf

• https://en.wikipedia.org/wiki/Dual_quaternion

Warning

Unlike the other spatial math classes, this class does not (yet) support multiple values per object.

Seealso:

UnitDualQuaternion()

classmethod Pure(x)

Return type:

Self

__add__(right)

Sum of two dual quaternions

Returns:

Product

Return type:

DualQuaternion

Example:

>>> from spatialmath import DualQuaternion, Quaternion
>>> d = DualQuaternion(Quaternion([1,2,3,4]), Quaternion([5,6,7,8]))
>>> d + d
 2.0000 <  4.0000,  6.0000,  8.0000 > + ε  10.0000 <  12.0000,  14.0000,  16.0000 >

__init__(real=None, dual=None)

Construct a new dual quaternion

Parameters:

• real (Quaternion or UnitQuaternion) – real quaternion

• dual (Quaternion or UnitQuaternion) – dual quaternion

Raises:

ValueError – incorrect parameters

Example:

>>> from spatialmath import DualQuaternion, Quaternion
>>> d = DualQuaternion(Quaternion([1,2,3,4]), Quaternion([5,6,7,8]))
>>> print(d)
 1.0000 <  2.0000,  3.0000,  4.0000 > + ε  5.0000 <  6.0000,  7.0000,  8.0000 >
>>> d = DualQuaternion([1, 2, 3, 4,  5, 6, 7, 8])
>>> print(d)
 1.0000 <  2.0000,  3.0000,  4.0000 > + ε  5.0000 <  6.0000,  7.0000,  8.0000 >

The dual number is stored internally as two quaternion, respectively called real and dual.

__mul__(right)

Product of dual quaternion :rtype: Self

• dq1 * dq2 is a dual quaternion representing the product of dq1 and dq2. If both are unit dual quaternions, the product will be a unit dual quaternion.

• dq * p transforms the point p (3) by the unit dual quaternion dq.

Example:

>>> from spatialmath import DualQuaternion, Quaternion
>>> d = DualQuaternion(Quaternion([1,2,3,4]), Quaternion([5,6,7,8]))
>>> d * d
-28.0000 <  4.0000,  6.0000,  8.0000 > + ε -120.0000 <  32.0000,  44.0000,  56.0000 >

__sub__(right)

Difference of two dual quaternions

Returns:

Product

Return type:

DualQuaternion

Example:

>>> from spatialmath import DualQuaternion, Quaternion
>>> d = DualQuaternion(Quaternion([1,2,3,4]), Quaternion([5,6,7,8]))
>>> d - d
 0.0000 <  0.0000,  0.0000,  0.0000 > + ε  0.0000 <  0.0000,  0.0000,  0.0000 >

conj()

Conjugate of dual quaternion

Returns:

Conjugate

Return type:

DualQuaternion

There are several conjugates defined for a dual quaternion. This one mirrors conjugation for a regular quaternion. For the dual quaternion \((p, q)\) it returns \((p^*, q^*)\).

Example:

>>> from spatialmath import DualQuaternion, Quaternion
>>> d = DualQuaternion(Quaternion([1,2,3,4]), Quaternion([5,6,7,8]))
>>> d.conj()
 1.0000 < -2.0000, -3.0000, -4.0000 > + ε  5.0000 < -6.0000, -7.0000, -8.0000 >

matrix()

Dual quaternion as a matrix

Returns:

Matrix represensation

Return type:

ndarray(8,8)

Dual quaternion multiplication can also be written as a matrix-vector product.

Example:

>>> from spatialmath import DualQuaternion, Quaternion
>>> d = DualQuaternion(Quaternion([1,2,3,4]), Quaternion([5,6,7,8]))
>>> d.matrix()
array([[ 1., -2., -3., -4.,  0.,  0.,  0.,  0.],
       [ 2.,  1., -4.,  3.,  0.,  0.,  0.,  0.],
       [ 3.,  4.,  1., -2.,  0.,  0.,  0.,  0.],
       [ 4., -3.,  2.,  1.,  0.,  0.,  0.,  0.],
       [ 5., -6., -7., -8.,  1., -2., -3., -4.],
       [ 6.,  5., -8.,  7.,  2.,  1., -4.,  3.],
       [ 7.,  8.,  5., -6.,  3.,  4.,  1., -2.],
       [ 8., -7.,  6.,  5.,  4., -3.,  2.,  1.]])
>>> d.matrix() @ d.vec
array([ -28.,    4.,    6.,    8., -120.,   32.,   44.,   56.])
>>> d * d
-28.0000 <  4.0000,  6.0000,  8.0000 > + ε -120.0000 <  32.0000,  44.0000,  56.0000 >

norm()

Norm of a dual quaternion

Returns:

Norm as a dual number

Return type:

2-tuple

The norm of a UnitDualQuaternion is unity, represented by the dual number (1,0).

Example:

>>> from spatialmath import DualQuaternion, Quaternion
>>> d = DualQuaternion(Quaternion([1,2,3,4]), Quaternion([5,6,7,8]))
>>> d.norm()  # norm is a dual number
(5.477225575051661, 11.832159566199232)

property vec: ndarray[Any, dtype[floating]]

Dual quaternion as a vector

Returns:

Vector represensation

Return type:

ndarray(8)

Example:

>>> from spatialmath import DualQuaternion, Quaternion
>>> d = DualQuaternion(Quaternion([1,2,3,4]), Quaternion([5,6,7,8]))
>>> d.vec
array([1, 2, 3, 4, 5, 6, 7, 8])