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mrpt::math::CQuaternion< T > Class Template Reference

## Detailed Description

### template<class T> class mrpt::math::CQuaternion< T >

A quaternion, which can represent a 3D rotation as pair , with a real part "r" and a 3D vector , or alternatively, q = r + ix + jy + kz.

The elements of the quaternion can be accessed by either:

• r(), x(), y(), z(), or
• the operator [] with indices running from 0 (=r) to 3 (=z).

Users will usually employ the typedef "CQuaternionDouble" instead of this template.

mrpt::poses::CPose3D

Definition at line 43 of file CQuaternion.h.

#include <mrpt/math/CQuaternion.h>

Inheritance diagram for mrpt::math::CQuaternion< T >:

## Public Types

typedef T value_type

## Public Member Functions

r () const
Return r coordinate of the quaternion. More...

x () const
Return x coordinate of the quaternion. More...

y () const
Return y coordinate of the quaternion. More...

z () const
Return z coordinate of the quaternion. More...

void r (const T r)
Set r coordinate of the quaternion. More...

void x (const T x)
Set x coordinate of the quaternion. More...

void y (const T y)
Set y coordinate of the quaternion. More...

void z (const T z)
Set z coordinate of the quaternion. More...

template<class ARRAYLIKE3 >
void fromRodriguesVector (const ARRAYLIKE3 &v)
Set this quaternion to the rotation described by a 3D (Rodrigues) rotation vector : If , then the quaternion is , otherwise:

where . More...

void crossProduct (const CQuaternion &q1, const CQuaternion &q2)
Calculate the "cross" product (or "composed rotation") of two quaternion: this = q1 x q2 After the operation, "this" will represent the composed rotations of q1 and q2 (q2 applied "after" q1). More...

void rotatePoint (const double lx, const double ly, const double lz, double &gx, double &gy, double &gz) const
Rotate a 3D point (lx,ly,lz) -> (gx,gy,gz) as described by this quaternion. More...

void inverseRotatePoint (const double lx, const double ly, const double lz, double &gx, double &gy, double &gz) const
Rotate a 3D point (lx,ly,lz) -> (gx,gy,gz) as described by the inverse (conjugate) of this quaternion. More...

double normSqr () const
Return the squared norm of the quaternion. More...

void normalize ()
Normalize this quaternion, so its norm becomes the unitity. More...

template<class MATRIXLIKE >
void normalizationJacobian (MATRIXLIKE &J) const
Calculate the 4x4 Jacobian of the normalization operation of this quaternion. More...

template<class MATRIXLIKE >
void rotationJacobian (MATRIXLIKE &J) const
Compute the Jacobian of the rotation composition operation , that is the 4x4 matrix . More...

template<class MATRIXLIKE >
void rotationMatrix (MATRIXLIKE &M) const
Calculate the 3x3 rotation matrix associated to this quaternion:

template<class MATRIXLIKE >
void rotationMatrixNoResize (MATRIXLIKE &M) const
Fill out the top-left 3x3 block of the given matrix with the rotation matrix associated to this quaternion (does not resize the matrix, for that, see rotationMatrix). More...

void conj (CQuaternion &q_out) const
Return the conjugate quaternion. More...

CQuaternion conj () const
Return the conjugate quaternion. More...

void rpy (T &roll, T &pitch, T &yaw) const
Return the yaw, pitch & roll angles associated to quaternion. More...

template<class MATRIXLIKE >
void rpy_and_jacobian (T &roll, T &pitch, T &yaw, MATRIXLIKE *out_dr_dq=NULL, bool resize_out_dr_dq_to3x4=true) const
Return the yaw, pitch & roll angles associated to quaternion, and (optionally) the 3x4 Jacobian of the transformation. More...

CQuaternion operator* (const T &factor)

CQuaternion (TConstructorFlags_Quaternions)
Can be used with UNINITIALIZED_QUATERNION as argument, does not initialize the 4 elements of the quaternion (use this constructor when speed is critical). More...

CQuaternion ()
Default constructor: construct a (1, (0,0,0) ) quaternion representing no rotation. More...

CQuaternion (const T r, const T x, const T y, const T z)
Construct a quaternion from its parameters 'r', 'x', 'y', 'z', with q = r + ix + jy + kz. More...

## Private Types

typedef CArrayNumeric< T, 4 > Base

## Lie Algebra methods

template<class ARRAYLIKE3 >
void ln (ARRAYLIKE3 &out_ln) const
Logarithm of the 3x3 matrix defined by this pose, generating the corresponding vector in the SO(3) Lie Algebra, which coincides with the so-called "rotation vector" (I don't have space here for the proof ;-). More...

template<class ARRAYLIKE3 >
ARRAYLIKE3 ln () const
overload that returns by value More...

template<class ARRAYLIKE3 >
void ln_noresize (ARRAYLIKE3 &out_ln) const
Like ln() but does not try to resize the output vector. More...

template<class ARRAYLIKE3 >
static CQuaternion< T > exp (const ARRAYLIKE3 &v)
Exponential map from the SO(3) Lie Algebra to unit quaternions. More...

template<class ARRAYLIKE3 >
static void exp (const ARRAYLIKE3 &v, CQuaternion< T > &out_quat)

## ◆ Base

template<class T>
 typedef CArrayNumeric mrpt::math::CQuaternion< T >::Base
private

Definition at line 45 of file CQuaternion.h.

## ◆ value_type

 typedef T mrpt::math::CArrayNumeric< T, N >::value_type
inherited

Definition at line 28 of file CArrayNumeric.h.

## ◆ CQuaternion() [1/3]

template<class T>
 mrpt::math::CQuaternion< T >::CQuaternion ( TConstructorFlags_Quaternions )
inline

Can be used with UNINITIALIZED_QUATERNION as argument, does not initialize the 4 elements of the quaternion (use this constructor when speed is critical).

Definition at line 51 of file CQuaternion.h.

## ◆ CQuaternion() [2/3]

template<class T>
 mrpt::math::CQuaternion< T >::CQuaternion ( )
inline

Default constructor: construct a (1, (0,0,0) ) quaternion representing no rotation.

Definition at line 54 of file CQuaternion.h.

## ◆ CQuaternion() [3/3]

template<class T>
 mrpt::math::CQuaternion< T >::CQuaternion ( const T r, const T x, const T y, const T z )
inline

Construct a quaternion from its parameters 'r', 'x', 'y', 'z', with q = r + ix + jy + kz.

Definition at line 63 of file CQuaternion.h.

## ◆ conj() [1/2]

template<class T>
 void mrpt::math::CQuaternion< T >::conj ( CQuaternion< T > & q_out ) const
inline

Return the conjugate quaternion.

Definition at line 297 of file CQuaternion.h.

## ◆ conj() [2/2]

template<class T>
 CQuaternion mrpt::math::CQuaternion< T >::conj ( ) const
inline

Return the conjugate quaternion.

Definition at line 306 of file CQuaternion.h.

## ◆ crossProduct()

template<class T>
 void mrpt::math::CQuaternion< T >::crossProduct ( const CQuaternion< T > & q1, const CQuaternion< T > & q2 )
inline

Calculate the "cross" product (or "composed rotation") of two quaternion: this = q1 x q2 After the operation, "this" will represent the composed rotations of q1 and q2 (q2 applied "after" q1).

Definition at line 179 of file CQuaternion.h.

## ◆ exp() [1/2]

template<class T>
template<class ARRAYLIKE3 >
 static CQuaternion mrpt::math::CQuaternion< T >::exp ( const ARRAYLIKE3 & v )
inlinestatic

Exponential map from the SO(3) Lie Algebra to unit quaternions.

ln, mrpt::poses::SE_traits

Definition at line 160 of file CQuaternion.h.

References mrpt::math::UNINITIALIZED_QUATERNION.

## ◆ exp() [2/2]

template<class T>
template<class ARRAYLIKE3 >
 static void mrpt::math::CQuaternion< T >::exp ( const ARRAYLIKE3 & v, CQuaternion< T > & out_quat )
inlinestatic

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Definition at line 168 of file CQuaternion.h.

## ◆ fromRodriguesVector()

template<class T>
template<class ARRAYLIKE3 >
 void mrpt::math::CQuaternion< T >::fromRodriguesVector ( const ARRAYLIKE3 & v )
inline

Set this quaternion to the rotation described by a 3D (Rodrigues) rotation vector : If , then the quaternion is , otherwise:

where .

"Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors (2006)", James Diebel.

Definition at line 96 of file CQuaternion.h.

Referenced by mrpt::math::CQuaternion< T >::exp().

## ◆ inverseRotatePoint()

template<class T>
 void mrpt::math::CQuaternion< T >::inverseRotatePoint ( const double lx, const double ly, const double lz, double & gx, double & gy, double & gz ) const
inline

Rotate a 3D point (lx,ly,lz) -> (gx,gy,gz) as described by the inverse (conjugate) of this quaternion.

Definition at line 203 of file CQuaternion.h.

Referenced by mrpt::poses::CPose3DQuat::inverseComposePoint().

## ◆ ln() [1/2]

template<class T>
template<class ARRAYLIKE3 >
 void mrpt::math::CQuaternion< T >::ln ( ARRAYLIKE3 & out_ln ) const
inline

Logarithm of the 3x3 matrix defined by this pose, generating the corresponding vector in the SO(3) Lie Algebra, which coincides with the so-called "rotation vector" (I don't have space here for the proof ;-).

Parameters
 [out] out_ln The target vector, which can be: std::vector<>, or mrpt::math::CVectorDouble or any row or column Eigen::Matrix<>.
exp, mrpt::poses::SE_traits

Definition at line 132 of file CQuaternion.h.

References mrpt::math::CQuaternion< T >::ln_noresize().

## ◆ ln() [2/2]

template<class T>
template<class ARRAYLIKE3 >
 ARRAYLIKE3 mrpt::math::CQuaternion< T >::ln ( ) const
inline

Definition at line 139 of file CQuaternion.h.

## ◆ ln_noresize()

template<class T>
template<class ARRAYLIKE3 >
 void mrpt::math::CQuaternion< T >::ln_noresize ( ARRAYLIKE3 & out_ln ) const
inline

Like ln() but does not try to resize the output vector.

Definition at line 147 of file CQuaternion.h.

Referenced by mrpt::math::CQuaternion< T >::ln().

## ◆ normalizationJacobian()

template<class T>
template<class MATRIXLIKE >
 void mrpt::math::CQuaternion< T >::normalizationJacobian ( MATRIXLIKE & J ) const
inline

Calculate the 4x4 Jacobian of the normalization operation of this quaternion.

The output matrix can be a dynamic or fixed size (4x4) matrix.

Definition at line 231 of file CQuaternion.h.

## ◆ normalize()

template<class T>
 void mrpt::math::CQuaternion< T >::normalize ( )
inline

Normalize this quaternion, so its norm becomes the unitity.

Definition at line 220 of file CQuaternion.h.

References mrpt::math::CQuaternion< T >::normSqr().

## ◆ normSqr()

template<class T>
 double mrpt::math::CQuaternion< T >::normSqr ( ) const
inline

Return the squared norm of the quaternion.

Definition at line 213 of file CQuaternion.h.

## ◆ operator*()

template<class T>
 CQuaternion mrpt::math::CQuaternion< T >::operator* ( const T & factor )
inline

Definition at line 398 of file CQuaternion.h.

## ◆ r() [1/2]

template<class T>
 T mrpt::math::CQuaternion< T >::r ( ) const
inline

## ◆ r() [2/2]

template<class T>
 void mrpt::math::CQuaternion< T >::r ( const T r )
inline

Set r coordinate of the quaternion.

Definition at line 79 of file CQuaternion.h.

## ◆ rotatePoint()

template<class T>
 void mrpt::math::CQuaternion< T >::rotatePoint ( const double lx, const double ly, const double lz, double & gx, double & gy, double & gz ) const
inline

Rotate a 3D point (lx,ly,lz) -> (gx,gy,gz) as described by this quaternion.

Definition at line 192 of file CQuaternion.h.

## ◆ rotationJacobian()

template<class T>
template<class MATRIXLIKE >
 void mrpt::math::CQuaternion< T >::rotationJacobian ( MATRIXLIKE & J ) const
inline

Compute the Jacobian of the rotation composition operation , that is the 4x4 matrix .

The output matrix can be a dynamic or fixed size (4x4) matrix.

Definition at line 261 of file CQuaternion.h.

## ◆ rotationMatrix()

template<class T>
template<class MATRIXLIKE >
 void mrpt::math::CQuaternion< T >::rotationMatrix ( MATRIXLIKE & M ) const
inline

Calculate the 3x3 rotation matrix associated to this quaternion:

.

Definition at line 280 of file CQuaternion.h.

## ◆ rotationMatrixNoResize()

template<class T>
template<class MATRIXLIKE >
 void mrpt::math::CQuaternion< T >::rotationMatrixNoResize ( MATRIXLIKE & M ) const
inline

Fill out the top-left 3x3 block of the given matrix with the rotation matrix associated to this quaternion (does not resize the matrix, for that, see rotationMatrix).

Definition at line 288 of file CQuaternion.h.

## ◆ rpy()

template<class T>
 void mrpt::math::CQuaternion< T >::rpy ( T & roll, T & pitch, T & yaw ) const
inline

Return the yaw, pitch & roll angles associated to quaternion.

For the equations, see The MRPT Book, or see http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToEuler/Quaternions.pdf
rpy_and_jacobian

Definition at line 317 of file CQuaternion.h.

## ◆ rpy_and_jacobian()

template<class T>
template<class MATRIXLIKE >
 void mrpt::math::CQuaternion< T >::rpy_and_jacobian ( T & roll, T & pitch, T & yaw, MATRIXLIKE * out_dr_dq = NULL, bool resize_out_dr_dq_to3x4 = true ) const
inline

Return the yaw, pitch & roll angles associated to quaternion, and (optionally) the 3x4 Jacobian of the transformation.

Note that both the angles and the Jacobian have one set of normal equations, plus other special formulas for the degenerated cases of |pitch|=90 degrees.

For the equations, see The MRPT Book, or http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToEuler/Quaternions.pdf
rpy

Definition at line 328 of file CQuaternion.h.

## ◆ x() [1/2]

template<class T>
 T mrpt::math::CQuaternion< T >::x ( ) const
inline

## ◆ x() [2/2]

template<class T>
 void mrpt::math::CQuaternion< T >::x ( const T x )
inline

Set x coordinate of the quaternion.

Definition at line 80 of file CQuaternion.h.

References mrpt::math::CQuaternion< T >::r().

## ◆ y() [1/2]

template<class T>
 T mrpt::math::CQuaternion< T >::y ( ) const
inline

## ◆ y() [2/2]

template<class T>
 void mrpt::math::CQuaternion< T >::y ( const T y )
inline

Set y coordinate of the quaternion.

Definition at line 81 of file CQuaternion.h.

References mrpt::math::CQuaternion< T >::x().

## ◆ z() [1/2]

template<class T>
 T mrpt::math::CQuaternion< T >::z ( ) const
inline

## ◆ z() [2/2]

template<class T>
 void mrpt::math::CQuaternion< T >::z ( const T z )
inline

Set z coordinate of the quaternion.

Definition at line 82 of file CQuaternion.h.

References mrpt::math::CQuaternion< T >::y().

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