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Class inertialsim::geometry::MatrixLieGroup

template <typename Derived, int GroupDimension, int AlgebraDimension>

ClassList > inertialsim > geometry > MatrixLieGroup

Abstract base class for matrix Lie groups using CRTP. More...

  • #include <lie_group.h>

Inherited by the following classes: inertialsim::geometry::RigidTransformBase, inertialsim::geometry::RigidTransformBase, inertialsim::geometry::RigidTransformBase, inertialsim::geometry::RotationBase, inertialsim::geometry::RotationBase, inertialsim::geometry::VectorBase, inertialsim::geometry::VectorBase, inertialsim::geometry::VectorBase

Public Types

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
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
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

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

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

Derived classes will inherit common Lie group operations such as composition, interpolation, etc.

See Reference [14], Reference [15], Reference [16], Reference [17], and Reference [18] for further details on Lie groups and their application to translations, rotations and rigid body motions.

Template parameters:

  • Derived The derived class type (CRTP pattern).
  • GroupDimension Dimension of the group matrix representation (e.g., 3 for SO(3) & T(3), 4 for SE(3)).
  • AlgebraDimension Dimension of the algebra representation (e.g., 3 for so(3) and t(4), 6 for se(3)).

Public Types Documentation

typedef AlgebraArray

Matrix of algebra elements (rows = axes, cols = samples). 

using inertialsim::geometry::MatrixLieGroup< Derived, GroupDimension, AlgebraDimension >::AlgebraArray =  Eigen::Matrix<Scalar, AlgebraDimension, Eigen::Dynamic>;

typedef AlgebraElement

Algebra element type (vector). 

using inertialsim::geometry::MatrixLieGroup< Derived, GroupDimension, AlgebraDimension >::AlgebraElement =  Eigen::Matrix<Scalar, AlgebraDimension, 1>;

typedef GroupArray

Array of group elements. 

using inertialsim::geometry::MatrixLieGroup< Derived, GroupDimension, AlgebraDimension >::GroupArray =  std::vector<GroupElement>;

typedef GroupElement

Group element type (square matrix). 

using inertialsim::geometry::MatrixLieGroup< Derived, GroupDimension, AlgebraDimension >::GroupElement =  Eigen::Matrix<Scalar, GroupDimension, GroupDimension>;

typedef JacobianMatrix

Jacobian matrix (algebra to algebra). 

using inertialsim::geometry::MatrixLieGroup< Derived, GroupDimension, AlgebraDimension >::JacobianMatrix =  Eigen::Matrix<Scalar, AlgebraDimension, AlgebraDimension>;

typedef JacobianMatrixArray

Array of Jacobian matrices. 

using inertialsim::geometry::MatrixLieGroup< Derived, GroupDimension, AlgebraDimension >::JacobianMatrixArray =  std::vector<JacobianMatrix>;

Public Functions Documentation

function Apply

Apply the group action to an array of vectors. 

inline Vector3D inertialsim::geometry::MatrixLieGroup::Apply (
    const Vector3D & vectors
) const

Parameters:

  • vectors Array of vectors to apply the group action to.

Returns:

Array of transformed vectors.

function Interpolate

Interpolate at given times. 

inline Derived inertialsim::geometry::MatrixLieGroup::Interpolate (
    const Timestamps & interp_time,
    Interpolator method=Interpolator::LINEAR
) const

Parameters:

  • interp_time Times at which to interpolate.
  • method Interpolation method (LINEAR or CUBIC).

Returns:

New instance with interpolated values.

function Inverse

Invert an instance. 

inline Derived inertialsim::geometry::MatrixLieGroup::Inverse () const

Returns:

Inverted instance.

function Slice

Get a slice of elements. 

inline Derived inertialsim::geometry::MatrixLieGroup::Slice (
    int start,
    int end
) const

Parameters:

  • start Start index (inclusive).
  • end End index (exclusive).

Returns:

New instance containing the sliced elements.

function TimeDerivative

Compute time derivative using finite differences. 

inline AlgebraArray inertialsim::geometry::MatrixLieGroup::TimeDerivative (
    int order=1,
    int accuracy=4,
    const std::string & side="right"
) const

Parameters:

  • order Order of derivative (default 1).
  • accuracy Accuracy of finite difference (default 4).
  • side "right" for body frame, "left" for world frame.

Returns:

Matrix of algebra derivatives (rows = samples, cols = algebra dim).

function num_elements

Number of elements stored in the instance. 

inline int inertialsim::geometry::MatrixLieGroup::num_elements () const

function operator*

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. 

inline Derived inertialsim::geometry::MatrixLieGroup::operator* (
    const Derived & rhs
) const

Parameters:

  • rhs Right-hand side instance.

Returns:

New instance containing the composed elements.

function operator[]

Get a single element by index. 

inline Derived inertialsim::geometry::MatrixLieGroup::operator[] (
    int index
) const

Parameters:

  • index Index of the element to retrieve.

Returns:

New instance containing a single element.

function time

Time array (or nullopt if not set). 

inline const std::optional< Timestamps > & inertialsim::geometry::MatrixLieGroup::time () const

function ~MatrixLieGroup

virtual inertialsim::geometry::MatrixLieGroup::~MatrixLieGroup () = default

Public Static Functions Documentation

function Compose

Compose a sequence of instances (first to last). 

static inline Derived inertialsim::geometry::MatrixLieGroup::Compose (
    const std::vector< Derived > & instances
)

function FromIdentity

Create identity elements. 

static inline Derived inertialsim::geometry::MatrixLieGroup::FromIdentity (
    int num_elements=1
)

Parameters:

  • num_elements Number of identity elements.

Returns:

Instance with identity elements.

function FromRandom

Create random elements. 

static inline Derived inertialsim::geometry::MatrixLieGroup::FromRandom (
    int num_elements=1,
    std::optional< Eigen::numext::uint64_t > seed=std::nullopt
)

Parameters:

  • num_elements Number of random elements.

Returns:

Instance with random elements.

Protected Functions Documentation

function AsAlgebra

Get the algebra elements. 

inline AlgebraArray inertialsim::geometry::MatrixLieGroup::AsAlgebra () const

Returns:

Array of algebra elements.

function AsGroup

Get the group elements. 

inline GroupArray inertialsim::geometry::MatrixLieGroup::AsGroup () const

Returns:

Array of group elements.

function CubicHermiteInterpolate

Cubic Hermite spline interpolation. 

inline Derived inertialsim::geometry::MatrixLieGroup::CubicHermiteInterpolate (
    const Timestamps & time
) const

Parameters:

  • time Times at which to interpolate.

Returns:

Interpolated instance.

function LinearInterpolate

Linear interpolation implementation. 

inline Derived inertialsim::geometry::MatrixLieGroup::LinearInterpolate (
    const Timestamps & time
) const

Parameters:

  • time Times at which to interpolate.

Returns:

Interpolated instance.

function MatrixLieGroup

Constructor. 

inline explicit inertialsim::geometry::MatrixLieGroup::MatrixLieGroup (
    const std::optional< Timestamps > & time=std::nullopt
)

Parameters:

  • time Optional time array. If provided, will be validated.

function derived [½]

CRTP helper to access derived class. 

inline Derived & inertialsim::geometry::MatrixLieGroup::derived ()

function derived [2/2]

inline const Derived & inertialsim::geometry::MatrixLieGroup::derived () const

function num_times

inline int inertialsim::geometry::MatrixLieGroup::num_times () const

Protected Static Functions Documentation

function Exp

Exponential map from algebra to group. 

static inline GroupElement inertialsim::geometry::MatrixLieGroup::Exp (
    const AlgebraElement & vector
)

Parameters:

  • vector Algebra element.

Returns:

Group element.

function FromAlgebra

Create instance from algebra elements. 

static inline Derived inertialsim::geometry::MatrixLieGroup::FromAlgebra (
    const AlgebraArray & elements,
    const std::optional< Timestamps > & time
)

Parameters:

  • elements Array of algebra elements.
  • time Optional time array.

Returns:

New instance.

function FromGroup

Create instance from group elements. 

static inline Derived inertialsim::geometry::MatrixLieGroup::FromGroup (
    const GroupArray & elements,
    const std::optional< Timestamps > & time
)

Parameters:

  • elements Array of group elements.
  • time Optional time array.

Returns:

New instance.

function Jacobian

Calculate the Jacobian given algebra elements. 

static inline JacobianMatrixArray inertialsim::geometry::MatrixLieGroup::Jacobian (
    const AlgebraArray & algebra,
    const std::string & side="left",
    bool inverse=false
)

Parameters:

  • algebra Algebra elements.
  • side "left" for world frame, "right" for body frame.
  • inverse If true, compute the inverse Jacobian.

Returns:

Jacobian matrix.

function Log

Logarithm map from group to algebra. 

static inline AlgebraElement inertialsim::geometry::MatrixLieGroup::Log (
    const GroupElement & element
)

Parameters:

  • element Group element.

Returns:

Algebra element.

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