Dual Quaternion

class DualQuaternion(real=None, dual=None)[source]

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

Warning

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

Seealso: UnitDualQuaternion()

classmethod Pure(x)[source]

Return type: Self

__add__(right)[source]

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)[source]

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)[source]

Product of dual quaternion

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 >

Return type: Self

__sub__(right)[source]

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()[source]

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()[source]

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()[source]

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: numpy.ndarray[Any, numpy.dtype[numpy.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])