ROBOOP, A Robotics Object Oriented Package in C++: Quaternion Class Reference

Definition at line 92 of file quaternion.h.


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

Constructor & Destructor Documentation

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.

References EPSILON, i(), R(), s_, and v_.

Member Function Documentation

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.

References s_, and v_.

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.

References s_, and v_.

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.

References s_, and v_.

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.

References s_, and v_.

Referenced by i().

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.

References EPSILON, s_, and v_.

Referenced by power().

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.

References EPSILON, s_, and v_.

Referenced by power().

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