Class inertialsim::geometry::CoordinateRotation¶

ClassList > inertialsim > geometry > CoordinateRotation

3-dimensional coordinate frame rotations (passive convention). More...

#include <coordinate_rotation.h>

Inherits the following classes: inertialsim::geometry::RotationBase

Public Types inherited from inertialsim::geometry::RotationBase¶

Type	Name
typedef MatrixLieGroup< Derived, 3, 3 >	Base

Public Types inherited from inertialsim::geometry::MatrixLieGroup¶

See inertialsim::geometry::MatrixLieGroup

Type	Name
typedef Eigen::Matrix< Scalar, AlgebraDimension, Eigen::Dynamic >	AlgebraArray Matrix of algebra elements (rows = axes, cols = samples).
typedef Eigen::Matrix< Scalar, AlgebraDimension, 1 >	AlgebraElement Algebra element type (vector).
typedef std::vector< GroupElement >	GroupArray Array of group elements.
typedef Eigen::Matrix< Scalar, GroupDimension, GroupDimension >	GroupElement Group element type (square matrix).
typedef Eigen::Matrix< Scalar, AlgebraDimension, AlgebraDimension >	JacobianMatrix Jacobian matrix (algebra to algebra).
typedef std::vector< JacobianMatrix >	JacobianMatrixArray Array of Jacobian matrices.

Public Functions¶

Type	Name
AxisAngle	AsAxisAngle () const Return axis and angle representation (passive).
EulerAngles	AsEuler (const std::string & sequence) const Return Euler angle representation (intrinsic).
RotationVector	AsRotationVector () const Return rotation vector representation (passive).

Public Functions inherited from inertialsim::geometry::RotationBase¶

See inertialsim::geometry::RotationBase

Type	Name
AxisAngle	AsAxisAngle () const Return axis and angle representation.
EulerAngles	AsEuler (const std::string & sequence) const Return the Euler angle representation.
const RotationMatrixArray &	AsMatrix () const Return the rotation matrix representation.
Quaternion	AsQuaternion () const Return the quaternion representation.
RotationVector	AsRotationVector () const Return the rotation vector representation.
RotationVector	Error (const RotationBase< OtherDerived > & reference) const Calculate error compared to a reference rotation.
int	num_rotations () const Number of rotations stored.

Public Functions inherited from inertialsim::geometry::MatrixLieGroup¶

See inertialsim::geometry::MatrixLieGroup

Type	Name
Vector3D	Apply (const Vector3D & vectors) const Apply the group action to an array of vectors.
Derived	Interpolate (const Timestamps & interp_time, Interpolator method=Interpolator::LINEAR) const Interpolate at given times.
Derived	Inverse () const Invert an instance.
Derived	Slice (int start, int end) const Get a slice of elements.
AlgebraArray	TimeDerivative (int order=1, int accuracy=4, const std::string & side="right") const Compute time derivative using finite differences.
int	num_elements () const Number of elements stored in the instance.
Derived	operator* (const Derived & rhs) const Composition operator on the matrix group. Supports broadcasting: if either lhs or rhs has size 1, it will be broadcast against all elements of the other operand. Derived classes may implement more efficient methods.
Derived	operator[] (int index) const Get a single element by index.
const std::optional< Timestamps > &	time () const Time array (or nullopt if not set).
virtual	~MatrixLieGroup () = default

Public Static Functions¶

Type	Name
CoordinateRotation	FromAxisAngle (const Vector3D & axis, const Scalar1D & angle, const std::optional< Timestamps > & time=std::nullopt) Construct from axis and angle (passive).
CoordinateRotation	FromEuler (const EulerAngles & euler, const std::string & sequence, const std::optional< Timestamps > & time=std::nullopt) Construct from Euler angles (intrinsic convention).
CoordinateRotation	FromRotationVector (const RotationVector & rotation_vectors, const std::optional< Timestamps > & time=std::nullopt) Construct from rotation vectors (passive).
JacobianMatrixArray	JacobianImpl (const AlgebraArray & rotation_vectors, const std::string & side, bool inverse) Jacobian of SO(3) exponential map (passive convention).

Public Static Functions inherited from inertialsim::geometry::RotationBase¶

See inertialsim::geometry::RotationBase

Type	Name
RotationMatrixArray	DoubleIntegralFactor (const RotationVector & rotation_vectors) Double integral factor.
Derived	FromAxisAngle (const Vector3D & axis, const Scalar1D & angle, const std::optional< Timestamps > & time=std::nullopt) Construct a Rotation from an axis and angle.
Derived	FromEuler (const EulerAngles & euler, const std::string & sequence, const std::optional< Timestamps > & time=std::nullopt) Construct a Rotation from a sequence of Euler angles.
Derived	FromMatrix (const RotationMatrixArray & matrices, const std::optional< Timestamps > & time=std::nullopt, bool orthonormalize=false) Construct a Rotation from rotation matrices.
Derived	FromQuaternion (const Quaternion & quaternions, const std::optional< Timestamps > & time=std::nullopt, bool scalar_last=false) Construct a Rotation from quaternions.
Derived	FromRotationVector (const RotationVector & rotation_vectors, const std::optional< Timestamps > & time=std::nullopt) Construct a Rotation from rotation vectors.
RotationMatrixArray	IntegralFactor (const RotationVector & rotation_vectors) First integral factor.

Public Static Functions inherited from inertialsim::geometry::MatrixLieGroup¶

See inertialsim::geometry::MatrixLieGroup

Type	Name
Derived	Compose (const std::vector< Derived > & instances) Compose a sequence of instances (first to last).
Derived	FromIdentity (int num_elements=1) Create identity elements.
Derived	FromRandom (int num_elements=1, std::optional< Eigen::numext::uint64_t > seed=std::nullopt) Create random elements.

Protected Functions inherited from inertialsim::geometry::RotationBase¶

See inertialsim::geometry::RotationBase

Type	Name
Vector3D	ApplyImpl (const Vector3D & vectors) const Apply rotation to vectors.
AlgebraArray	AsAlgebraImpl () const Return algebra representation.
GroupArray	AsGroupImpl () const Return as group elements (for CRTP interface).
Derived	InverseImpl () const Return the inverse rotation.
	RotationBase () Default constructor.
	RotationBase (const RotationMatrixArray & matrices, const std::optional< Timestamps > & time=std::nullopt) Construct from rotation matrices.
int	num_elements_impl () const Number of rotations stored.

Protected Functions inherited from inertialsim::geometry::MatrixLieGroup¶

See inertialsim::geometry::MatrixLieGroup

Type	Name
AlgebraArray	AsAlgebra () const Get the algebra elements.
GroupArray	AsGroup () const Get the group elements.
Derived	CubicHermiteInterpolate (const Timestamps & time) const Cubic Hermite spline interpolation.
Derived	LinearInterpolate (const Timestamps & time) const Linear interpolation implementation.
	MatrixLieGroup (const std::optional< Timestamps > & time=std::nullopt) Constructor.
Derived &	derived () CRTP helper to access derived class.
const Derived &	derived () const
int	num_times () const

Protected Static Functions inherited from inertialsim::geometry::RotationBase¶

See inertialsim::geometry::RotationBase

Type	Name
GroupElement	ExpImpl (const AlgebraElement & rotation_vector) Exponential map: rotation vector to rotation matrix.
Derived	FromAlgebraImpl (const AlgebraArray & rotation_vectors, const std::optional< Timestamps > & time=std::nullopt) Construct from algebra elements (for CRTP interface).
Derived	FromGroupImpl (const GroupArray & elements, const std::optional< Timestamps > & time) Construct from group elements (for CRTP interface).
Derived	FromIdentityImpl (int num_elements) Construct identity rotations.
Derived	FromRandomImpl (int num_rotations, std::optional< uint64_t > seed=std::nullopt) Construct uniformly distributed random rotations.
JacobianMatrixArray	JacobianImpl (const AlgebraArray & rotation_vectors, const std::string & side, bool inverse) Jacobian of SO(3) exponential map.
AlgebraElement	LogImpl (const GroupElement & matrix) Logarithm map: rotation matrix to rotation vector.

Protected Static Functions inherited from inertialsim::geometry::MatrixLieGroup¶

See inertialsim::geometry::MatrixLieGroup

Type	Name
GroupElement	Exp (const AlgebraElement & vector) Exponential map from algebra to group.
Derived	FromAlgebra (const AlgebraArray & elements, const std::optional< Timestamps > & time) Create instance from algebra elements.
Derived	FromGroup (const GroupArray & elements, const std::optional< Timestamps > & time) Create instance from group elements.
JacobianMatrixArray	Jacobian (const AlgebraArray & algebra, const std::string & side="left", bool inverse=false) Calculate the Jacobian given algebra elements.
AlgebraElement	Log (const GroupElement & element) Logarithm map from group to algebra.

Detailed Description¶

CoordinateRotation is a subclass of RotationBase that uses the passive convention. Passive rotations act to transform vector components between coordinate systems. The passive convention is the standard for describing orientation in inertial navigation and aerospace applications.

Coordinate rotations are also commonly known as direction cosine matrices (DCM) or coordinate transformation matrices.

** **

In the passive convention (used by this class), rotations act to transform the components of physical vectors to different coordinate systems. This contrasts with the active convention used by the Rotation class, where rotations act to rotate an object relative to a fixed coordinate system.

A passive coordinate rotation matrix is the transpose of the equivalent active rotation matrix: \(C_{i}^{b} = R_{i \rightarrow b}^{T}\)

Some references use the terminology alibi (active) and alias (passive). Reference [02], Reference [05], and Reference [11] discuss active vs. passive transforms in greater detail.

See also: Rotation for active rotations.

See also: CoordinateTransform for combined rotation and translation transforms.

Public Functions Documentation¶

function AsAxisAngle¶

Return axis and angle representation (passive).

inline AxisAngle inertialsim::geometry::CoordinateRotation::AsAxisAngle () const

Returns:

Tuple of axes (3xN) and angles (N).

function AsEuler¶

Return Euler angle representation (intrinsic).

inline EulerAngles inertialsim::geometry::CoordinateRotation::AsEuler (
    const std::string & sequence
) const

Parameters:

sequence Rotation sequence (e.g., "XYZ").

Returns:

Euler angles (3xN matrix).

function AsRotationVector¶

Return rotation vector representation (passive).

inline RotationVector inertialsim::geometry::CoordinateRotation::AsRotationVector () const

Returns:

Rotation vectors (3xN matrix).

Public Static Functions Documentation¶

function FromAxisAngle¶

Construct from axis and angle (passive).

static inline CoordinateRotation inertialsim::geometry::CoordinateRotation::FromAxisAngle (
    const Vector3D & axis,
    const Scalar1D & angle,
    const std::optional< Timestamps > & time=std::nullopt
)

Parameters:

axis Unit axes of rotation (3xN matrix).
angle Angles of rotation (N vector).
time Optional time array.

Returns:

CoordinateRotation instance.

function FromEuler¶

Construct from Euler angles (intrinsic convention).

static inline CoordinateRotation inertialsim::geometry::CoordinateRotation::FromEuler (
    const EulerAngles & euler,
    const std::string & sequence,
    const std::optional< Timestamps > & time=std::nullopt
)

Parameters:

euler Euler angles (3xN matrix).
sequence Rotation sequence (e.g., "XYZ").
time Optional time array.

Returns:

CoordinateRotation instance.

function FromRotationVector¶

Construct from rotation vectors (passive).

static inline CoordinateRotation inertialsim::geometry::CoordinateRotation::FromRotationVector (
    const RotationVector & rotation_vectors,
    const std::optional< Timestamps > & time=std::nullopt
)

Parameters:

rotation_vectors Rotation vectors (3xN matrix).
time Optional time array.

Returns:

CoordinateRotation instance.

function JacobianImpl¶

Jacobian of SO(3) exponential map (passive convention).

static inline JacobianMatrixArray inertialsim::geometry::CoordinateRotation::JacobianImpl (
    const AlgebraArray & rotation_vectors,
    const std::string & side,
    bool inverse
)

Returns the transposed Jacobian due to the passive convention.

Parameters:

rotation_vectors Rotation vectors (3xN).
side "left" or "right" to specify which Jacobian.
inverse If true, return inverse Jacobian.

Returns:

Vector of 3x3 Jacobian matrices.

The documentation for this class was generated from the following file cpp/include/inertialsim/geometry/coordinate_rotation.h