pinocchio  2.4.4
A fast and flexible implementation of Rigid Body Dynamics algorithms and their analytical derivatives
pinocchio::quaternion Namespace Reference

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. More...
 
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. More...
 
template<typename Vector3Like , typename QuaternionLike >
void exp3 (const Eigen::MatrixBase< Vector3Like > &v, Eigen::QuaternionBase< QuaternionLike > &quat_out)
 Exp: so3 -> SO3 (quaternion) More...
 
template<typename Vector3Like >
Eigen::Quaternion< typename Vector3Like::Scalar, Vector3Like ::Options > exp3 (const Eigen::MatrixBase< Vector3Like > &v)
 Exp: so3 -> SO3 (quaternion) More...
 
template<typename D >
void firstOrderNormalize (const Eigen::QuaternionBase< D > &q)
 
template<typename Quaternion >
bool isNormalized (const Eigen::QuaternionBase< Quaternion > &quat, const typename Quaternion::Coefficients::RealScalar &prec=Eigen::NumTraits< typename Quaternion::Coefficients::RealScalar >::dummy_precision())
 Check whether the input quaternion is Normalized within the given precision. More...
 
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. More...
 
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. More...
 
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. More...
 
template<typename QuaternionLike >
Eigen::Matrix< typename QuaternionLike::Scalar, 3, 1, typename QuaternionLike::Vector3 ::Options > log3 (const Eigen::QuaternionBase< QuaternionLike > &quat)
 Log: SO3 -> so3. More...
 
template<typename Derived >
void uniformRandom (const Eigen::QuaternionBase< Derived > &q)
 Uniformly random quaternion sphere.
 

Detailed Description

Quaternion operations.

Function Documentation

◆ angleBetweenQuaternions()

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

Compute the minimal angle between q1 and q2.

Parameters
[in]q1input quaternion.
[in]q2input quaternion.
Returns
angle between the two quaternions

Definition at line 35 of file quaternion.hpp.

◆ defineSameRotation()

bool pinocchio::quaternion::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]q1input quaternion.
[in]q2input quaternion.
Returns
Return true if the two input quaternions define the same rotation.

Definition at line 59 of file quaternion.hpp.

◆ exp3() [1/2]

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

Exp: so3 -> SO3 (quaternion)

Returns
the integral of the velocity vector as a queternion.
Parameters
[in]vThe angular velocity vector.
[out]qoutThe quanternion where the result is stored.

Definition at line 26 of file explog-quaternion.hpp.

◆ exp3() [2/2]

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

Exp: so3 -> SO3 (quaternion)

Returns
the integral of the velocity vector as a queternion.
Parameters
[in]vThe angular velocity vector.

Definition at line 67 of file explog-quaternion.hpp.

◆ firstOrderNormalize()

void pinocchio::quaternion::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
\( ||q||^2 - 1 \) 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 88 of file quaternion.hpp.

◆ isNormalized()

bool pinocchio::quaternion::isNormalized ( const Eigen::QuaternionBase< Quaternion > &  quat,
const typename Quaternion::Coefficients::RealScalar &  prec = Eigen::NumTraits< typename Quaternion::Coefficients::RealScalar >::dummy_precision() 
)
inline

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

Parameters
[in]quatInput quaternion
[in]precRequired precision
Returns
true if quat is normalized within the precision prec.

Definition at line 225 of file quaternion.hpp.

◆ Jexp3CoeffWise()

void pinocchio::quaternion::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 146 of file explog-quaternion.hpp.

◆ Jlog3()

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

Computes the Jacobian of log3 operator for a unit quaternion.

 

Parameters
[in]quatA unit quaternion representing the input rotation.
[out]JlogThe resulting Jacobian of the log operator.

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

◆ log3() [1/2]

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

Same as log3 but with a unit quaternion as input.

Parameters
[in]quatthe unit quaternion.
[out]thetathe angle value (resuling from compurations).
Returns
The angular velocity vector associated to the rotation matrix.

Definition at line 84 of file explog-quaternion.hpp.

◆ log3() [2/2]

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

Log: SO3 -> so3.

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

Parameters
[in]quatThe unit quaternion representing a certain rotation.
Returns
The angular velocity vector associated to the quaternion.

Definition at line 133 of file explog-quaternion.hpp.