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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 $ (r,\mathbf{u}) $, with a real part "r" and a 3D vector $ \mathbf{u} = (x,y,z) $, or alternatively, q = r + ix + jy + kz.

The elements of the quaternion can be accessed by either:

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

For more information about quaternions, see:

See also
mrpt::poses::CPose3D

Definition at line 43 of file CQuaternion.h.

#include <mrpt/math/CQuaternion.h>

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

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 $ \mathbf{v} $: If $ \mathbf{v}=0 $, then the quaternion is $ \mathbf{q} = [1 ~ 0 ~ 0 ~ 0]^\top $, otherwise:

\[ \mathbf{q} = \left[ \begin{array}{c} \cos(\frac{\theta}{2}) \\ v_x \frac{\sin(\frac{\theta}{2})}{\theta} \\ v_y \frac{\sin(\frac{\theta}{2})}{\theta} \\ v_z \frac{\sin(\frac{\theta}{2})}{\theta} \end{array} \right] \]

where $ \theta = |\mathbf{v}| = \sqrt{v_x^2+v_y^2+v_z^2} $. 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 $ p = f(\cdot) = q_{this} \times r $, that is the 4x4 matrix $ \frac{\partial f}{\partial q_{this} } $. More...
 
template<class MATRIXLIKE >
void rotationMatrix (MATRIXLIKE &M) const
 Calculate the 3x3 rotation matrix associated to this quaternion:

\[ \mathbf{R} = \left( \begin{array}{ccc} q_r^2+q_x^2-q_y^2-q_z^2 & 2(q_x q_y - q_r q_z) & 2(q_z q_x+q_r q_y) \\ 2(q_x q_y+q_r q_z) & q_r^2-q_x^2+q_y^2-q_z^2 & 2(q_y q_z-q_r q_x) \\ 2(q_z q_x-q_r q_y) & 2(q_y q_z+q_r q_x) & q_r^2- q_x^2 - q_y^2 + q_z^2 \end{array} \right)\]

. More...

 
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)
 

Member Typedef Documentation

template<class T>
typedef CArrayNumeric<T,4> mrpt::math::CQuaternion< T >::Base
private

Definition at line 45 of file CQuaternion.h.

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

Definition at line 28 of file CArrayNumeric.h.

Constructor & Destructor Documentation

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.

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.

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.

References ASSERTMSG_, mrpt::mrpt::format(), mrpt::math::CQuaternion< T >::normSqr(), mrpt::math::CQuaternion< T >::r(), mrpt::math::CQuaternion< T >::x(), mrpt::math::CQuaternion< T >::y(), and mrpt::math::CQuaternion< T >::z().

Member Function Documentation

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

Return the conjugate quaternion.

Definition at line 306 of file CQuaternion.h.

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.

References mrpt::math::CQuaternion< T >::normalize(), mrpt::math::CQuaternion< T >::r(), mrpt::math::CQuaternion< T >::x(), mrpt::math::CQuaternion< T >::y(), and mrpt::math::CQuaternion< T >::z().

Referenced by mrpt::poses::CPose3DQuat::composeFrom(), mrpt::poses::CPose3DQuat::inverseComposeFrom(), mrpt::poses::CPose3DPDF::jacobiansPoseComposition(), and TEST_F().

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

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

See also
ln, mrpt::poses::SE_traits

Definition at line 160 of file CQuaternion.h.

References mrpt::math::CQuaternion< T >::fromRodriguesVector(), and mrpt::math::UNINITIALIZED_QUATERNION.

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.

References mrpt::math::CQuaternion< T >::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 $ \mathbf{v} $: If $ \mathbf{v}=0 $, then the quaternion is $ \mathbf{q} = [1 ~ 0 ~ 0 ~ 0]^\top $, otherwise:

\[ \mathbf{q} = \left[ \begin{array}{c} \cos(\frac{\theta}{2}) \\ v_x \frac{\sin(\frac{\theta}{2})}{\theta} \\ v_y \frac{\sin(\frac{\theta}{2})}{\theta} \\ v_z \frac{\sin(\frac{\theta}{2})}{\theta} \end{array} \right] \]

where $ \theta = |\mathbf{v}| = \sqrt{v_x^2+v_y^2+v_z^2} $.

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

Definition at line 96 of file CQuaternion.h.

References THROW_EXCEPTION, mrpt::math::CQuaternion< T >::x(), mrpt::math::CQuaternion< T >::y(), and mrpt::math::CQuaternion< T >::z().

Referenced by mrpt::math::CQuaternion< T >::exp(), and mrpt::poses::CPose3DRotVec::toQuatXYZ().

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.

References mrpt::math::CQuaternion< T >::r(), mrpt::math::CQuaternion< T >::x(), mrpt::math::CQuaternion< T >::y(), and mrpt::math::CQuaternion< T >::z().

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

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_lnThe target vector, which can be: std::vector<>, or mrpt::math::CVectorDouble or any row or column Eigen::Matrix<>.
See also
exp, mrpt::poses::SE_traits

Definition at line 132 of file CQuaternion.h.

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

Referenced by QuaternionTests::test_ExpAndLnMatches(), and QuaternionTests::test_lnAndExpMatches().

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

overload that returns by value

Definition at line 139 of file CQuaternion.h.

template<class T>
template<class ARRAYLIKE3 >
void mrpt::math::CQuaternion< T >::ln_noresize ( ARRAYLIKE3 &  out_ln) const
inline
template<class T>
template<class MATRIXLIKE >
void mrpt::math::CQuaternion< T >::normalizationJacobian ( MATRIXLIKE &  J) const
inline
template<class T>
void mrpt::math::CQuaternion< T >::normalize ( )
inline
template<class T>
double mrpt::math::CQuaternion< T >::normSqr ( ) const
inline
template<class T>
CQuaternion mrpt::math::CQuaternion< T >::operator* ( const T &  factor)
inline

Definition at line 398 of file CQuaternion.h.

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

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

Compute the Jacobian of the rotation composition operation $ p = f(\cdot) = q_{this} \times r $, that is the 4x4 matrix $ \frac{\partial f}{\partial q_{this} } $.

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

Definition at line 261 of file CQuaternion.h.

References mrpt::math::CQuaternion< T >::r(), mrpt::math::CQuaternion< T >::x(), mrpt::math::CQuaternion< T >::y(), and mrpt::math::CQuaternion< T >::z().

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:

\[ \mathbf{R} = \left( \begin{array}{ccc} q_r^2+q_x^2-q_y^2-q_z^2 & 2(q_x q_y - q_r q_z) & 2(q_z q_x+q_r q_y) \\ 2(q_x q_y+q_r q_z) & q_r^2-q_x^2+q_y^2-q_z^2 & 2(q_y q_z-q_r q_x) \\ 2(q_z q_x-q_r q_y) & 2(q_y q_z+q_r q_x) & q_r^2- q_x^2 - q_y^2 + q_z^2 \end{array} \right)\]

.

Definition at line 280 of file CQuaternion.h.

References mrpt::math::CQuaternion< T >::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.

References mrpt::math::CQuaternion< T >::r(), mrpt::math::CQuaternion< T >::x(), mrpt::math::CQuaternion< T >::y(), and mrpt::math::CQuaternion< T >::z().

Referenced by mrpt::poses::CPose3D::CPose3D(), mrpt::poses::CPose3DQuat::getHomogeneousMatrix(), mrpt::math::CQuaternion< T >::rotationMatrix(), and mrpt::poses::CPose3D::setFromXYZQ().

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

See also
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.

References M_PI, mrpt::math::CQuaternion< T >::r(), mrpt::mrpt::math::square(), mrpt::math::square(), mrpt::math::CQuaternion< T >::x(), mrpt::math::CQuaternion< T >::y(), and mrpt::math::CQuaternion< T >::z().

Referenced by mrpt::poses::CPose3DPDFGaussian::copyFrom(), mrpt::poses::CPose3DPDF::jacobiansPoseComposition(), and mrpt::math::CQuaternion< T >::rpy().

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

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

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




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