Quaternion Class Reference

#include <quaternion.h>

Public Member Functions
Quaternion	conjugate () const
	Conjugate.
Quaternion	dot (const ColumnVector &w, const short sign) const
	Quaternion time derivative.
Real	dot_prod (const Quaternion &q) const
	Quaternion dot product.
ReturnMatrix	E (const short sign) const
	Matrix E.
Quaternion	exp () const
	Exponential of a quaternion.
Quaternion	i () const
	Quaternion inverse. $q^{-1} = \frac{q^{}}{N(q)}$ where $q^{}$ and are the quaternion conjugate and the quaternion norm respectively.
Quaternion	Log () const
	Logarithm of a unit quaternion.
Real	norm () const
	Return the quaternion norm.
Quaternion	operator* (const Quaternion &q) const
	Overload * operator.
Quaternion	operator+ (const Quaternion &q) const
	Overload + operator.
Quaternion	operator- (const Quaternion &q) const
	Overload - operator.
Quaternion	operator/ (const Quaternion &q) const
	Overload / operator.
Quaternion	power (const Real t) const
	Quaternion (const Matrix &R)
	Constructor.
	Quaternion (const Real s, const Real v1, const Real v2, const Real v3)
	Constructor.
	Quaternion (const Real angle_in_rad, const ColumnVector &axis)
	Constructor.
	Quaternion ()
	Constructor.
ReturnMatrix	R () const
	Rotation matrix from a unit quaternion.
Real	s () const
	Return scalar part.
void	set_s (const Real s)
	Set scalar part.
void	set_v (const ColumnVector &v)
	Set vector part.
ReturnMatrix	T () const
	Transformation matrix from a quaternion.
Quaternion &	unit ()
	Normalize a quaternion.
ReturnMatrix	v () const
	Return vector part.
Private Attributes
Real	s_
	Quaternion scalar part.
ColumnVector	v_
	Quaternion vector part.

Detailed Description

Quaternion class definition.

Definition at line 92 of file quaternion.h.

Constructor & Destructor Documentation

Quaternion::Quaternion ( )

Constructor.

Definition at line 77 of file quaternion.cpp.

Quaternion::Quaternion	(	const Real	angle_in_rad,
		const ColumnVector &	axis
	)

Constructor.

Definition at line 85 of file quaternion.cpp.

Quaternion::Quaternion	(	const Real	s,
		const Real	v1,
		const Real	v2,
		const Real	v3
	)

Constructor.

Definition at line 109 of file quaternion.cpp.

Quaternion::Quaternion ( const Matrix & R )

Constructor.

Cite_: Dam. The unit quaternion obtained from a matrix (see Quaternion::R())

$R(s,v) = \left[ \begin{array}{ccc} s^2+v_1^2-v_2^2-v_3^2 & 2v_1v_2+2sv_3 & 2v_1v_3-2sv_2 \\ 2v_1v_2-2sv_3 & s^2-v_1^2+v_2^2-v_3^2 & 2v_2v_3+2sv_1 \\ 2v_1v_3+2sv_2 &2v_2v_3-2sv_1 & s^2-v_1^2-v_2^2+v_3^2 \end{array} \right]$

First we find $s$ :

$R_{11} + R_{22} + R_{33} + R_{44} = 4s^2$

Now the other values are:

$s = \pm \frac{1}{2}\sqrt{R_{11} + R_{22} + R_{33} + R_{44}}$

$v_1 = \frac{R_{32}-R_{23}}{4s}$

$v_2 = \frac{R_{13}-R_{31}}{4s}$

$v_3 = \frac{R_{21}-R_{12}}{4s}$

The sign of $s$ cannot be determined. Depending on the choice of the sign for s the sign of $v$ change as well. Thus the quaternions $q$ and $-q$ represent the same rotation, but the interpolation curve changed with the choice of the sign. A positive sign has been chosen.

Definition at line 120 of file quaternion.cpp.

Member Function Documentation

Quaternion Quaternion::conjugate ( ) const

Conjugate.

The conjugate of a quaternion $q = [s, v]$ is $q^{*} = [s, -v]$

Definition at line 283 of file quaternion.cpp.

Quaternion Quaternion::dot	(	const ColumnVector &	w,
		const short	sign
	)			const

Quaternion time derivative.

The quaternion time derivative, quaternion propagation equation, is

$\dot{s} = - \frac{1}{2}v^Tw_{_0}$

$\dot{v} = \frac{1}{2}E(s,v)w_{_0}$

$E = sI - S(v)$

where $w_{_0}$ is the angular velocity vector expressed in the base frame. If the vector is expressed in the object frame, $w_{_b}$ , the time derivative becomes

$\dot{s} = - \frac{1}{2}v^Tw_{_b}$

$\dot{v} = \frac{1}{2}E(s,v)w_{_b}$

$E = sI + S(v)$

Definition at line 388 of file quaternion.cpp.

Real Quaternion::dot_prod ( const Quaternion & q ) const

Quaternion dot product.

The dot product of quaternion is defined by

$q_1\cdot q_2 = s_1s_2 + v_1 \cdot v_2$

Definition at line 445 of file quaternion.cpp.

ReturnMatrix Quaternion::E ( const short sign ) const

Matrix E.

See Quaternion::dot for explanation.

Definition at line 426 of file quaternion.cpp.

Quaternion Quaternion::exp ( ) const

Exponential of a quaternion.

Let a quaternion of the form $q = [0, \theta v]$ , q is not necessarily a unit quaternion. Then the exponential function is defined by $q = [\cos(\theta),v \sin(\theta)]$ .

Definition at line 336 of file quaternion.cpp.

Quaternion Quaternion::i ( ) const

Quaternion inverse.

$q^{-1} = \frac{q^{*}}{N(q)}$

where $q^{*}$ and $N(q)$ are the quaternion conjugate and the quaternion norm respectively.

Definition at line 323 of file quaternion.cpp.

Quaternion Quaternion::Log ( ) const

Logarithm of a unit quaternion.

The logarithm function of a unit quaternion $q = [\cos(\theta), v \sin(\theta)]$ is defined as $log(q) = [0, v\theta]$ . The result is not necessary a unit quaternion.

Definition at line 365 of file quaternion.cpp.

Real Quaternion::norm ( ) const

Return the quaternion norm.

The norm of quaternion is defined by

$N(q) = s^2 + v\cdot v$

Definition at line 298 of file quaternion.cpp.

Quaternion Quaternion::operator* ( const Quaternion & rhs ) const

Overload * operator.

The multiplication of two quaternions is

$q = q_1q_2 = [s_1s_2 - v_1\cdot v_2, v_1 \times v_2 + s_1v_2 + s_2v_1]$

where $\cdot$ and $\times$ denote the scalar and vector product in $R^3$ respectively.

If $q_1$ and $q_2$ are unit quaternions, then q will also be a unit quaternion.

Definition at line 243 of file quaternion.cpp.

Quaternion Quaternion::operator+ ( const Quaternion & rhs ) const

Overload + operator.

The quaternion addition is

$q_1 + q_2 = [s_1, v_1] + [s_2, v_2] = [s_1+s_2, v_1+v_2]$

The result is not necessarily a unit quaternion even if $q_1$ and $q_2$ are unit quaternions.

Definition at line 203 of file quaternion.cpp.

Quaternion Quaternion::operator- ( const Quaternion & rhs ) const

Overload - operator.

The quaternion soustraction is

$q_1 - q_2 = [s_1, v_1] - [s_2, v_2] = [s_1-s_2, v_1-v_2]$

The result is not necessarily a unit quaternion even if $q_1$ and $q_2$ are unit quaternions.

Definition at line 223 of file quaternion.cpp.

Quaternion Quaternion::operator/ ( const Quaternion & q ) const

Overload / operator.

Definition at line 267 of file quaternion.cpp.

Quaternion Quaternion::power ( const Real t ) const

Definition at line 358 of file quaternion.cpp.

ReturnMatrix Quaternion::R ( ) const

Rotation matrix from a unit quaternion.

$p'=qpq^{-1} = Rp$ where $p$ is a vector, $R$ a rotation matrix and $q$ q quaternion. The rotation matrix obtained from a quaternion is then

$R(s,v) = (s^2 - v^Tv)I + 2vv^T - 2s S(v)$

$R(s,v) = \left[ \begin{array}{ccc} s^2+v_1^2-v_2^2-v_3^2 & 2v_1v_2+2sv_3 & 2v_1v_3-2sv_2 \\ 2v_1v_2-2sv_3 & s^2-v_1^2+v_2^2-v_3^2 & 2v_2v_3+2sv_1 \\ 2v_1v_3+2sv_2 &2v_2v_3-2sv_1 & s^2-v_1^2-v_2^2+v_3^2 \end{array} \right]$

where $S(\cdot)$ is the cross product matrix defined by

$S(u) = \left[ \begin{array}{ccc} 0 & -u_3 & u_2 \\ u_3 &0 & -u_1 \\ -u_2 & u_1 & 0 \\ \end{array} \right]$

Definition at line 458 of file quaternion.cpp.

Real Quaternion::s ( ) const [inline]

Return scalar part.

Definition at line 119 of file quaternion.h.

void Quaternion::set_s ( const Real s ) [inline]

Set scalar part.

Definition at line 120 of file quaternion.h.

void Quaternion::set_v ( const ColumnVector & v )

Set vector part.

Set quaternion vector part.

Definition at line 274 of file quaternion.cpp.

ReturnMatrix Quaternion::T ( ) const

Transformation matrix from a quaternion.

See Quaternion::R() for equations.

Definition at line 499 of file quaternion.cpp.

Quaternion & Quaternion::unit ( )

Normalize a quaternion.

Definition at line 311 of file quaternion.cpp.

ReturnMatrix Quaternion::v ( ) const [inline]

Return vector part.

Definition at line 121 of file quaternion.h.

Member Data Documentation

Real Quaternion::s_ [private]

Quaternion scalar part.

Definition at line 127 of file quaternion.h.

ColumnVector Quaternion::v_ [private]

Quaternion vector part.

Definition at line 128 of file quaternion.h.

The documentation for this class was generated from the following files:

Quaternion Class Reference

Public Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation