Quaternion operations. More...

Functions
template<typename D1, typename D2>
D1::Scalar	angleBetweenQuaternions (const Eigen::QuaternionBase< D1 > &q1, const Eigen::QuaternionBase< D2 > &q2)
	Compute the minimal angle between q1 and q2.
template<typename D, typename Matrix3>
void	assignQuaternion (Eigen::QuaternionBase< D > &quat, const Eigen::MatrixBase< Matrix3 > &R)
template<typename D1, typename D2>
bool	defineSameRotation (const Eigen::QuaternionBase< D1 > &q1, const Eigen::QuaternionBase< D2 > &q2, const typename D1::RealScalar &prec=Eigen::NumTraits< typename D1::Scalar >::dummy_precision())
	Check if two quaternions define the same rotations.
template<typename Vector3Like>
Eigen::Quaternion< typename Vector3Like::Scalar, Vector3Like::Options >	exp3 (const Eigen::MatrixBase< Vector3Like > &v)
	Exp: so3 -> SO3 (quaternion)
template<typename Vector3Like, typename QuaternionLike>
void	exp3 (const Eigen::MatrixBase< Vector3Like > &v, Eigen::QuaternionBase< QuaternionLike > &quat_out)
	Exp: so3 -> SO3 (quaternion)
template<typename Vector6Like>
Eigen::Matrix< typename Vector6Like::Scalar, 7, 1, Vector6Like::Options >	exp6 (const Eigen::MatrixBase< Vector6Like > &vec6)
	The se3 -> SE3 exponential map, using quaternions to represent the output rotation.
template<typename Vector6Like, typename Config_t>
void	exp6 (const Eigen::MatrixBase< Vector6Like > &vec6, Eigen::MatrixBase< Config_t > &qout)
	The se3 -> SE3 exponential map, using quaternions to represent the output rotation.
template<typename MotionDerived>
Eigen::Matrix< typename MotionDerived::Scalar, 7, 1, typename MotionDerived::Vector3::Options >	exp6 (const MotionDense< MotionDerived > &motion)
	The se3 -> SE3 exponential map, using quaternions to represent the output rotation.
template<typename MotionDerived, typename Config_t>
void	exp6 (const MotionDense< MotionDerived > &motion, Eigen::MatrixBase< Config_t > &qout)
	The se3 -> SE3 exponential map, using quaternions to represent the output rotation.
template<typename D>
void	firstOrderNormalize (const Eigen::QuaternionBase< D > &q)
template<typename Quaternion>
bool	isNormalized (const Eigen::QuaternionBase< Quaternion > &quat)
	Check whether the input quaternion is Normalized within the default precision.
template<typename Quaternion>
bool	isNormalized (const Eigen::QuaternionBase< Quaternion > &quat, const typename Quaternion::Coefficients::RealScalar &prec)
	Check whether the input quaternion is Normalized within the given precision.
template<typename Vector3Like, typename Matrix43Like>
void	Jexp3CoeffWise (const Eigen::MatrixBase< Vector3Like > &v, const Eigen::MatrixBase< Matrix43Like > &Jexp)
	Derivative of $q = \exp{\mathbf{v} + \delta\mathbf{v}}$ where $\delta\mathbf{v}$ is a small perturbation of $\mathbf{v}$ at identity.
template<typename QuaternionLike, typename Matrix3Like>
void	Jlog3 (const Eigen::QuaternionBase< QuaternionLike > &quat, const Eigen::MatrixBase< Matrix3Like > &Jlog)
	Computes the Jacobian of log3 operator for a unit quaternion.
template<typename QuaternionLike>
Eigen::Matrix< typename QuaternionLike::Scalar, 3, 1, typename QuaternionLike::Vector3::Options >	log3 (const Eigen::QuaternionBase< QuaternionLike > &quat)
	Log: SO3 -> so3.
template<typename QuaternionLike>
Eigen::Matrix< typename QuaternionLike::Scalar, 3, 1, typename QuaternionLike::Vector3::Options >	log3 (const Eigen::QuaternionBase< QuaternionLike > &quat, typename QuaternionLike::Scalar &theta)
	Same as log3 but with a unit quaternion as input.
template<typename Quaternion>
void	normalize (const Eigen::QuaternionBase< Quaternion > &quat)
	Normalize the input quaternion.
template<typename Scalar, typename QuaternionIn1, typename QuaternionIn2, typename QuaternionOut>
void	slerp (const Scalar &u, const Eigen::QuaternionBase< QuaternionIn1 > &quat0, const Eigen::QuaternionBase< QuaternionIn2 > &quat1, const Eigen::QuaternionBase< QuaternionOut > &res)
template<typename Derived>
void	uniformRandom (Eigen::QuaternionBase< Derived > &q)
	Uniformly random quaternion sphere.

Detailed Description

Quaternion operations.

Function Documentation

◆ angleBetweenQuaternions()

template<typename D1, typename D2>

D1::Scalar angleBetweenQuaternions	(	const Eigen::QuaternionBase< D1 > &	q1,
		const Eigen::QuaternionBase< D2 > &	q2 )

Compute the minimal angle between q1 and q2.

Parameters

[in]	q1	input quaternion.
[in]	q2	input quaternion.

Returns: angle between the two quaternions

Definition at line 35 of file quaternion.hpp.

◆ assignQuaternion()

template<typename D, typename Matrix3>

void assignQuaternion	(	Eigen::QuaternionBase< D > &	quat,
		const Eigen::MatrixBase< Matrix3 > &	R )

Definition at line 215 of file quaternion.hpp.

◆ defineSameRotation()

template<typename D1, typename D2>

bool defineSameRotation	(	const Eigen::QuaternionBase< D1 > &	q1,
		const Eigen::QuaternionBase< D2 > &	q2,
		const typename D1::RealScalar &	prec = Eigen::NumTraits<typename D1::Scalar>::dummy_precision() )

Check if two quaternions define the same rotations.

Note: Two quaternions define the same rotation iff q1 == q2 OR q1 == -q2.

Parameters

[in]	q1	input quaternion.
[in]	q2	input quaternion.

Returns: Return true if the two input quaternions define the same rotation.

Definition at line 58 of file quaternion.hpp.

◆ exp3() [1/2]

template<typename Vector3Like>

Eigen::Quaternion< typename Vector3Like::Scalar, Vector3Like::Options > exp3 ( const Eigen::MatrixBase< Vector3Like > & v )

Exp: so3 -> SO3 (quaternion)

Returns: the integral of the velocity vector as a quaternion.

Parameters

[in] v The angular velocity vector.

Definition at line 75 of file explog-quaternion.hpp.

◆ exp3() [2/2]

template<typename Vector3Like, typename QuaternionLike>

void exp3	(	const Eigen::MatrixBase< Vector3Like > &	v,
		Eigen::QuaternionBase< QuaternionLike > &	quat_out )

Exp: so3 -> SO3 (quaternion)

Returns: the integral of the velocity vector as a quaternion.

Parameters

[in]	v	The angular velocity vector.
[out]	qout	The quaternion where the result is stored.

Definition at line 27 of file explog-quaternion.hpp.

◆ exp6() [1/4]

template<typename Vector6Like>

Eigen::Matrix< typename Vector6Like::Scalar, 7, 1, Vector6Like::Options > exp6 ( const Eigen::MatrixBase< Vector6Like > & vec6 )

The se3 -> SE3 exponential map, using quaternions to represent the output rotation.

Returns: the integral of the spatial velocity over unit time.

Parameters

[in] vec6 the vector representing the spatial velocity.

Definition at line 184 of file explog-quaternion.hpp.

◆ exp6() [2/4]

template<typename Vector6Like, typename Config_t>

void exp6	(	const Eigen::MatrixBase< Vector6Like > &	vec6,
		Eigen::MatrixBase< Config_t > &	qout )

The se3 -> SE3 exponential map, using quaternions to represent the output rotation.

Returns: the integral of the spatial velocity over unit time.

Parameters

[in]	vec6	the vector representing the spatial velocity.
[out]	qout	the output transform in R^3 x S^3.

Definition at line 170 of file explog-quaternion.hpp.

◆ exp6() [3/4]

template<typename MotionDerived>

Eigen::Matrix< typename MotionDerived::Scalar, 7, 1, typename MotionDerived::Vector3::Options > exp6 ( const MotionDense< MotionDerived > & motion )

The se3 -> SE3 exponential map, using quaternions to represent the output rotation.

Returns: the integral of the twist motion over unit time.

Parameters

[in] motion the spatial motion.

Definition at line 149 of file explog-quaternion.hpp.

◆ exp6() [4/4]

template<typename MotionDerived, typename Config_t>

void exp6	(	const MotionDense< MotionDerived > &	motion,
		Eigen::MatrixBase< Config_t > &	qout )

The se3 -> SE3 exponential map, using quaternions to represent the output rotation.

Returns: the integral of the twist motion over unit time.

Parameters

[in]	motion	the spatial motion.
[out]	q	the output transform in $\mathbb{R}^3 x S^3$ .

Definition at line 92 of file explog-quaternion.hpp.

◆ firstOrderNormalize()

template<typename D>

void firstOrderNormalize ( const Eigen::QuaternionBase< D > & q )

Approximately normalize by applying the first order limited development of the normalization function.

Only additions and multiplications are required. Neither square root nor division are used (except a division by 2). Let $\delta = ||q||^2 - 1$ . Using the following limited development:

$\frac{1}{||q||} = (1 + \delta)^{-\frac{1}{2}} = 1 - \frac{\delta}{2} + \mathcal{O}(\delta^2)$

The output is

$q_{out} = q \times \frac{3 - ||q_{in}||^2}{2}$

The output quaternion is guaranted to statisfy the following:

$| ||q_{out}|| - 1 | \le \frac{M}{2} ||q_{in}|| ( ||q_{in}||^2 - 1 )^2$

where $M = \frac{3}{4} (1 - \epsilon)^{-\frac{5}{2}}$ and $\epsilon$ is the maximum tolerance of $||q_{in}||^2 - 1$ .

Warning: should already be close to zero.

Note: See http://eigen.tuxfamily.org/dox/TopicFunctionTakingEigenTypes.html#title3 to know the reason why the argument is const.

Definition at line 90 of file quaternion.hpp.

◆ isNormalized() [1/2]

template<typename Quaternion>

bool isNormalized ( const Eigen::QuaternionBase< Quaternion > & quat )

inline

Check whether the input quaternion is Normalized within the default precision.

Parameters

[in] quat Input quaternion

Returns: true if quat is normalized within the default precision.

Definition at line 245 of file quaternion.hpp.

◆ isNormalized() [2/2]

template<typename Quaternion>

bool isNormalized	(	const Eigen::QuaternionBase< Quaternion > &	quat,
		const typename Quaternion::Coefficients::RealScalar &	prec )

inline

Check whether the input quaternion is Normalized within the given precision.

Parameters

[in]	quat	Input quaternion
[in]	prec	Required precision

Returns: true if quat is normalized within the precision prec.

Definition at line 230 of file quaternion.hpp.

◆ Jexp3CoeffWise()

template<typename Vector3Like, typename Matrix43Like>

void Jexp3CoeffWise	(	const Eigen::MatrixBase< Vector3Like > &	v,
		const Eigen::MatrixBase< Matrix43Like > &	Jexp )

Derivative of $q = \exp{\mathbf{v} + \delta\mathbf{v}}$ where $\delta\mathbf{v}$ is a small perturbation of $\mathbf{v}$ at identity.

Returns: The Jacobian of the quaternion components variation.

Definition at line 292 of file explog-quaternion.hpp.

◆ Jlog3()

template<typename QuaternionLike, typename Matrix3Like>

void Jlog3	(	const Eigen::QuaternionBase< QuaternionLike > &	quat,
		const Eigen::MatrixBase< Matrix3Like > &	Jlog )

Computes the Jacobian of log3 operator for a unit quaternion.

Parameters

[in]	quat	A unit quaternion representing the input rotation.
[out]	Jlog	The resulting Jacobian of the log operator.

Definition at line 332 of file explog-quaternion.hpp.

◆ log3() [1/2]

template<typename QuaternionLike>

Eigen::Matrix< typename QuaternionLike::Scalar, 3, 1, typename QuaternionLike::Vector3::Options > log3 ( const Eigen::QuaternionBase< QuaternionLike > & quat )

Log: SO3 -> so3.

Pseudo-inverse of log from $SO3 -> { v \in so3, ||v|| \le pi }$ .

Parameters

[in] quat The unit quaternion representing a certain rotation.

Returns: The angular velocity vector associated to the quaternion.

Definition at line 278 of file explog-quaternion.hpp.

◆ log3() [2/2]

template<typename QuaternionLike>

Eigen::Matrix< typename QuaternionLike::Scalar, 3, 1, typename QuaternionLike::Vector3::Options > log3	(	const Eigen::QuaternionBase< QuaternionLike > &	quat,
		typename QuaternionLike::Scalar &	theta )

Same as log3 but with a unit quaternion as input.

Parameters

[in]	quat	the unit quaternion.
[out]	theta	the angle value (resuling from compurations).

Returns: The angular velocity vector associated to the rotation matrix.

Definition at line 211 of file explog-quaternion.hpp.

◆ normalize()

template<typename Quaternion>

void normalize ( const Eigen::QuaternionBase< Quaternion > & quat )

inline

Normalize the input quaternion.

Parameters

[in] quat Input quaternion

Definition at line 258 of file quaternion.hpp.

◆ slerp()

template<typename Scalar, typename QuaternionIn1, typename QuaternionIn2, typename QuaternionOut>

void slerp	(	const Scalar &	u,
		const Eigen::QuaternionBase< QuaternionIn1 > &	quat0,
		const Eigen::QuaternionBase< QuaternionIn2 > &	quat1,
		const Eigen::QuaternionBase< QuaternionOut > &	res )

inline

Returns: the spherical linear interpolation between the two quaternions

Parameters

[in]	u	Interpolation factor
[in]	quat	Input quaternion

Definition at line 274 of file quaternion.hpp.

◆ uniformRandom()

template<typename Derived>

void uniformRandom ( Eigen::QuaternionBase< Derived > & q )

Uniformly random quaternion sphere.

Definition at line 115 of file quaternion.hpp.

Functions

Detailed Description

Function Documentation

◆ angleBetweenQuaternions()

◆ assignQuaternion()

◆ defineSameRotation()

◆ exp3() [1/2]

◆ exp3() [2/2]

◆ exp6() [1/4]

◆ exp6() [2/4]

◆ exp6() [3/4]

◆ exp6() [4/4]

◆ firstOrderNormalize()

◆ isNormalized() [1/2]

◆ isNormalized() [2/2]

◆ Jexp3CoeffWise()

◆ Jlog3()

◆ log3() [1/2]

◆ log3() [2/2]

◆ normalize()

◆ slerp()

◆ uniformRandom()