Namespaces
	detail

	suffixes

Classes
struct	RealVectorSpace

struct	SO3Space

struct	SE3Space

struct	MapsToManifoldPoint

struct	constants

struct	HullAngle

class	BoundingBox

class	Random

Typedefs
using	NullSpace = RealVectorSpace< 0u >

using	R1Space = RealVectorSpace< 1u >

using	R2Space = RealVectorSpace< 2u >

using	R3Space = RealVectorSpace< 3u >

using	constantsf = constants< float >

using	constantsd = constants< double >

typedef std::vector< Eigen::Vector3d >	SupportGeometry

typedef common::aligned_vector< Eigen::Vector2d >	SupportPolygon

using	Inertia = Eigen::Matrix6d

using	LinearJacobian = Eigen::Matrix< double, 3, Eigen::Dynamic >

using	AngularJacobian = Eigen::Matrix< double, 3, Eigen::Dynamic >

using	Jacobian = Eigen::Matrix< double, 6, Eigen::Dynamic >

Enumerations
enum	AxisType { AXIS_X = 0 , AXIS_Y = 1 , AXIS_Z = 2 }

enum	IntersectionResult { INTERSECTING = 0 , PARALLEL , BEYOND_ENDPOINTS }
	Intersection_t is returned by the computeIntersection() function to indicate whether there was a valid intersection between the two line segments. More...

Functions
template<typename SpaceT >
SpaceT::Matrix	inverse (const typename SpaceT::Matrix &mat)

template<typename SpaceT >
SpaceT::EuclideanPoint	toEuclideanPoint (const typename SpaceT::Point &point)

template<typename SpaceT >
SpaceT::Point	toManifoldPoint (const typename SpaceT::EuclideanPoint &point)

template<typename SpaceT >
SpaceT::Point	integratePosition (const typename SpaceT::Point &pos, const typename SpaceT::Vector &vel, double dt)

template<typename SpaceT >
SpaceT::Vector	integrateVelocity (const typename SpaceT::Vector &vel, const typename SpaceT::Vector &acc, double dt)

Eigen::Quaterniond	expToQuat (const Eigen::Vector3d &_v)

Eigen::Vector3d	quatToExp (const Eigen::Quaterniond &_q)

Eigen::Vector3d	matrixToEulerXYX (const Eigen::Matrix3d &_R)
	get the Euler XYX angle from R More...

Eigen::Vector3d	matrixToEulerXYZ (const Eigen::Matrix3d &_R)
	get the Euler XYZ angle from R More...

Eigen::Vector3d	matrixToEulerZYX (const Eigen::Matrix3d &_R)
	get the Euler ZYX angle from R More...

Eigen::Vector3d	matrixToEulerXZY (const Eigen::Matrix3d &_R)
	get the Euler XZY angle from R More...

Eigen::Vector3d	matrixToEulerYZX (const Eigen::Matrix3d &_R)
	get the Euler YZX angle from R More...

Eigen::Vector3d	matrixToEulerZXY (const Eigen::Matrix3d &_R)
	get the Euler ZXY angle from R More...

Eigen::Vector3d	matrixToEulerYXZ (const Eigen::Matrix3d &_R)
	get the Euler YXZ angle from R More...

Eigen::Matrix3d	quatDeriv (const Eigen::Quaterniond &_q, int _el)

Eigen::Matrix3d	quatSecondDeriv (const Eigen::Quaterniond &_q, int _el1, int _el2)

Eigen::Vector3d	rotatePoint (const Eigen::Quaterniond &_q, const Eigen::Vector3d &_pt)

Eigen::Vector3d	rotatePoint (const Eigen::Quaterniond &_q, double _x, double _y, double _z)

Eigen::Matrix3d	expMapRot (const Eigen::Vector3d &_expmap)
	Computes the Rotation matrix from a given expmap vector. More...

Eigen::Matrix3d	expMapJac (const Eigen::Vector3d &_expmap)
	Computes the Jacobian of the expmap. More...

Eigen::Matrix3d	expMapJacDot (const Eigen::Vector3d &_expmap, const Eigen::Vector3d &_qdot)
	Computes the time derivative of the expmap Jacobian. More...

Eigen::Matrix3d	expMapJacDeriv (const Eigen::Vector3d &_expmap, int _qi)
	computes the derivative of the Jacobian of the expmap wrt to _qi indexed dof; _qi \( \in \) {0,1,2} More...

Eigen::Vector3d	logMap (const Eigen::Matrix3d &_R)
	Log mapping. More...

Eigen::Vector6d	logMap (const Eigen::Isometry3d &_T)
	Log mapping. More...

Eigen::Vector6d	AdT (const Eigen::Isometry3d &_T, const Eigen::Vector6d &_V)
	Rectify the rotation part so as that it satifies the orthogonality condition. More...

Eigen::Matrix6d	getAdTMatrix (const Eigen::Isometry3d &T)
	Get linear transformation matrix of Adjoint mapping. More...

Eigen::Vector6d	AdR (const Eigen::Isometry3d &_T, const Eigen::Vector6d &_V)
	Fast version of Ad([R 0; 0 1], V) More...

Eigen::Vector6d	AdTAngular (const Eigen::Isometry3d &_T, const Eigen::Vector3d &_w)
	fast version of Ad(T, se3(w, 0)) More...

Eigen::Vector6d	AdTLinear (const Eigen::Isometry3d &_T, const Eigen::Vector3d &_v)
	fast version of Ad(T, se3(0, v)) More...

Eigen::Vector6d	AdInvT (const Eigen::Isometry3d &_T, const Eigen::Vector6d &_V)
	fast version of Ad(Inv(T), V) More...

Eigen::Vector6d	AdInvRLinear (const Eigen::Isometry3d &_T, const Eigen::Vector3d &_v)
	Fast version of Ad(Inv([R 0; 0 1]), se3(0, v)) More...

Eigen::Vector6d	ad (const Eigen::Vector6d &_X, const Eigen::Vector6d &_Y)
	adjoint mapping More...

Eigen::Vector6d	dAdT (const Eigen::Isometry3d &_T, const Eigen::Vector6d &_F)
	dual adjoint mapping More...

Eigen::Vector6d	dAdInvT (const Eigen::Isometry3d &_T, const Eigen::Vector6d &_F)
	fast version of dAd(Inv(T), F) More...

Eigen::Vector6d	dAdInvR (const Eigen::Isometry3d &_T, const Eigen::Vector6d &_F)
	fast version of dAd(Inv([R 0; 0 1]), F) More...

Eigen::Matrix3d	eulerXYXToMatrix (const Eigen::Vector3d &_angle)
	Given Euler XYX angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotY(angle(1)) * RotX(angle(2)). More...

Eigen::Matrix3d	eulerXYZToMatrix (const Eigen::Vector3d &_angle)
	Given EulerXYZ angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotY(angle(1)) * RotZ(angle(2)). More...

Eigen::Matrix3d	eulerXZXToMatrix (const Eigen::Vector3d &_angle)
	Given EulerXZX angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotZ(angle(1)) * RotX(angle(2)). More...

Eigen::Matrix3d	eulerXZYToMatrix (const Eigen::Vector3d &_angle)
	Given EulerXZY angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotZ(angle(1)) * RotY(angle(2)). More...

Eigen::Matrix3d	eulerYXYToMatrix (const Eigen::Vector3d &_angle)
	Given EulerYXY angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotX(angle(1)) * RotY(angle(2)). More...

Eigen::Matrix3d	eulerYXZToMatrix (const Eigen::Vector3d &_angle)
	Given EulerYXZ angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotX(angle(1)) * RotZ(angle(2)). More...

Eigen::Matrix3d	eulerYZXToMatrix (const Eigen::Vector3d &_angle)
	Given EulerYZX angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotZ(angle(1)) * RotX(angle(2)). More...

Eigen::Matrix3d	eulerYZYToMatrix (const Eigen::Vector3d &_angle)
	Given EulerYZY angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotZ(angle(1)) * RotY(angle(2)). More...

Eigen::Matrix3d	eulerZXYToMatrix (const Eigen::Vector3d &_angle)
	Given EulerZXY angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotX(angle(1)) * RotY(angle(2)). More...

Eigen::Matrix3d	eulerZYXToMatrix (const Eigen::Vector3d &_angle)
	Given EulerZYX angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotY(angle(1)) * RotX(angle(2)). More...

Eigen::Matrix3d	eulerZXZToMatrix (const Eigen::Vector3d &_angle)
	Given EulerZXZ angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotX(angle(1)) * RotZ(angle(2)). More...

Eigen::Matrix3d	eulerZYZToMatrix (const Eigen::Vector3d &_angle)
	Given EulerZYZ angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotY(angle(1)) * RotZ(angle(2)). More...

Eigen::Isometry3d	expMap (const Eigen::Vector6d &_S)
	Exponential mapping. More...

Eigen::Isometry3d	expAngular (const Eigen::Vector3d &_s)
	fast version of Exp(se3(s, 0)) More...

Eigen::Vector6d	dad (const Eigen::Vector6d &_s, const Eigen::Vector6d &_t)
	fast version of ad(se3(Eigen_Vec3(0), v), S) More...

Inertia	transformInertia (const Eigen::Isometry3d &_T, const Inertia &_AI)

Eigen::Matrix3d	parallelAxisTheorem (const Eigen::Matrix3d &_original, const Eigen::Vector3d &_comShift, double _mass)
	Use the Parallel Axis Theorem to compute the moment of inertia of a body whose center of mass has been shifted from the origin. More...

bool	verifyRotation (const Eigen::Matrix3d &_R)
	Check if determinant of _R is equat to 1 and all the elements are not NaN values. More...

bool	verifyTransform (const Eigen::Isometry3d &_T)
	Check if determinant of the rotational part of _T is equat to 1 and all the elements are not NaN values. More...

Eigen::Vector3d	fromSkewSymmetric (const Eigen::Matrix3d &_m)

Eigen::Matrix3d	makeSkewSymmetric (const Eigen::Vector3d &_v)

Eigen::Matrix3d	computeRotation (const Eigen::Vector3d &axis, AxisType axisType=AxisType::AXIS_X)
	Compute a rotation matrix from a vector. More...

Eigen::Isometry3d	computeTransform (const Eigen::Vector3d &axis, const Eigen::Vector3d &translation, AxisType axisType=AxisType::AXIS_X)
	Compute a transform from a vector and a position. More...

Eigen::Isometry3d	getFrameOriginAxisZ (const Eigen::Vector3d &_origin, const Eigen::Vector3d &_axisZ)
	Generate frame given origin and z-axis. More...

SupportPolygon	computeSupportPolgyon (const SupportGeometry &_geometry, const Eigen::Vector3d &_axis1=Eigen::Vector3d::UnitX(), const Eigen::Vector3d &_axis2=Eigen::Vector3d::UnitY())
	Project the support geometry points onto a plane with the given axes and then compute their convex hull, which will take the form of a polgyon. More...

SupportPolygon	computeSupportPolgyon (std::vector< std::size_t > &_originalIndices, const SupportGeometry &_geometry, const Eigen::Vector3d &_axis1=Eigen::Vector3d::UnitX(), const Eigen::Vector3d &_axis2=Eigen::Vector3d::UnitY())
	Same as computeSupportPolgyon, except you can pass in a std::vector<std::size_t> which will have the same size as the returned SupportPolygon, and each entry will contain the original index of each point in the SupportPolygon. More...

static bool	HullAngleComparison (const HullAngle &a, const HullAngle &b)

SupportPolygon	computeConvexHull (const SupportPolygon &_points)
	Computes the convex hull of a set of 2D points. More...

Eigen::Vector2d	computeCentroidOfHull (const SupportPolygon &_convexHull)
	Compute the centroid of a polygon, assuming the polygon is a convex hull. More...

static bool	isLeftTurn (const Eigen::Vector2d &p1, const Eigen::Vector2d &p2, const Eigen::Vector2d &p3)

SupportPolygon	computeConvexHull (std::vector< std::size_t > &_originalIndices, const SupportPolygon &_points)
	Computes the convex hull of a set of 2D points and fills in _originalIndices with the original index of each entry in the returned SupportPolygon. More...

IntersectionResult	computeIntersection (Eigen::Vector2d &_intersectionPoint, const Eigen::Vector2d &a1, const Eigen::Vector2d &a2, const Eigen::Vector2d &b1, const Eigen::Vector2d &b2)
	Compute the intersection between a line segment that goes from a1 -> a2 and a line segment that goes from b1 -> b2. More...

double	cross (const Eigen::Vector2d &_v1, const Eigen::Vector2d &_v2)
	Compute a 2D cross product. More...

bool	isInsideSupportPolygon (const Eigen::Vector2d &_p, const SupportPolygon &_support, bool _includeEdge=true)
	Returns true if the point _p is inside the support polygon. More...

Eigen::Vector2d	computeClosestPointOnLineSegment (const Eigen::Vector2d &_p, const Eigen::Vector2d &_s1, const Eigen::Vector2d &_s2)
	Returns the point which is closest to _p that also lays on the line segment that goes from _s1 -> _s2. More...

Eigen::Vector2d	computeClosestPointOnSupportPolygon (const Eigen::Vector2d &_p, const SupportPolygon &_support)
	Returns the point which is closest to _p that also lays on the edge of the support polygon. More...

Eigen::Vector2d	computeClosestPointOnSupportPolygon (std::size_t &_index1, std::size_t &_index2, const Eigen::Vector2d &_p, const SupportPolygon &_support)
	Same as closestPointOnSupportPolygon, but also fills in _index1 and _index2 with the indices of the line segment. More...

template<typename Derived >
Derived::PlainObject	AdTJac (const Eigen::Isometry3d &_T, const Eigen::MatrixBase< Derived > &_J)
	Adjoint mapping for dynamic size Jacobian. More...

template<typename Derived >
Derived::PlainObject	AdTJacFixed (const Eigen::Isometry3d &_T, const Eigen::MatrixBase< Derived > &_J)
	Adjoint mapping for fixed size Jacobian. More...

template<typename Derived >
Derived::PlainObject	AdRJac (const Eigen::Isometry3d &_T, const Eigen::MatrixBase< Derived > &_J)
	Change coordinate Frame of a Jacobian. More...

template<typename Derived >
Derived::PlainObject	AdRInvJac (const Eigen::Isometry3d &_T, const Eigen::MatrixBase< Derived > &_J)

template<typename Derived >
Derived::PlainObject	adJac (const Eigen::Vector6d &_V, const Eigen::MatrixBase< Derived > &_J)

template<typename Derived >
Derived::PlainObject	AdInvTJac (const Eigen::Isometry3d &_T, const Eigen::MatrixBase< Derived > &_J)
	Adjoint mapping for dynamic size Jacobian. More...

template<typename Derived >
Derived::PlainObject	AdInvTJacFixed (const Eigen::Isometry3d &_T, const Eigen::MatrixBase< Derived > &_J)
	Adjoint mapping for fixed size Jacobian. More...

double	wrapToPi (double angle)
	Compute the angle (in the range of -pi to +pi) which ignores any full rotations. More...

template<typename MatrixType , typename ReturnType >
void	extractNullSpace (const Eigen::JacobiSVD< MatrixType > &_SVD, ReturnType &_NS)

template<typename MatrixType , typename ReturnType >
void	computeNullSpace (const MatrixType &_M, ReturnType &_NS)

template<typename T >
constexpr T	toRadian (const T &degree)

template<typename T >
constexpr T	toDegree (const T &radian)

const Eigen::Matrix2d	CR ((Eigen::Matrix2d()<< 0.0, -1.0, 1.0, 0.0).finished())
	a cross b = (CR*a) dot b const Matd CR(2,2,0.0,-1.0,1.0,0.0); More...

int	delta (int _i, int _j)

template<typename T >
constexpr int	sign (T x, std::false_type)

template<typename T >
constexpr int	sign (T x, std::true_type)

template<typename T >
constexpr int	sign (T x)

double	sqr (double _x)

double	Tsinc (double _theta)

bool	isZero (double _theta)

double	asinh (double _X)

double	acosh (double _X)

double	atanh (double _X)

double	asech (double _X)

double	acosech (double _X)

double	acotanh (double _X)

double	round (double _x)

double	round2 (double _x)

template<typename T >
T	clip (const T &val, const T &lower, const T &upper)

template<typename DerivedA , typename DerivedB >
DerivedA::PlainObject	clip (const Eigen::MatrixBase< DerivedA > &val, const Eigen::MatrixBase< DerivedB > &lower, const Eigen::MatrixBase< DerivedB > &upper)

bool	isEqual (double _x, double _y)

bool	isInt (double _x)

bool	isNan (double _v)
	Returns whether _v is a NaN (Not-A-Number) value. More...

bool	isNan (const Eigen::MatrixXd &_m)
	Returns whether _m is a NaN (Not-A-Number) matrix. More...

bool	isInf (double _v)
	Returns whether _v is an infinity value (either positive infinity or negative infinity). More...

bool	isInf (const Eigen::MatrixXd &_m)
	Returns whether _m is an infinity matrix (either positive infinity or negative infinity). More...

bool	isSymmetric (const Eigen::MatrixXd &_m, double _tol=1e-6)
	Returns whether _m is symmetric or not. More...

unsigned	seedRand ()

double	random (double _min, double _max)

template<int N>
Eigen::Matrix< double, N, 1 >	randomVector (double _min, double _max)

template<int N>
Eigen::Matrix< double, N, 1 >	randomVector (double _limit)

Eigen::VectorXd	randomVectorXd (std::size_t size, double min, double max)

Eigen::VectorXd	randomVectorXd (std::size_t size, double limit)

Typedef Documentation

◆ AngularJacobian

using dart::math::AngularJacobian = typedef Eigen::Matrix<double, 3, Eigen::Dynamic>

◆ constantsd

using dart::math::constantsd = typedef constants<double>

◆ constantsf

using dart::math::constantsf = typedef constants<float>

◆ Inertia

using dart::math::Inertia = typedef Eigen::Matrix6d

◆ Jacobian

using dart::math::Jacobian = typedef Eigen::Matrix<double, 6, Eigen::Dynamic>

◆ LinearJacobian

using dart::math::LinearJacobian = typedef Eigen::Matrix<double, 3, Eigen::Dynamic>

◆ NullSpace

using dart::math::NullSpace = typedef RealVectorSpace<0u>

◆ R1Space

using dart::math::R1Space = typedef RealVectorSpace<1u>

◆ R2Space

using dart::math::R2Space = typedef RealVectorSpace<2u>

◆ R3Space

using dart::math::R3Space = typedef RealVectorSpace<3u>

◆ SupportGeometry

typedef std::vector<Eigen::Vector3d> dart::math::SupportGeometry

◆ SupportPolygon

typedef common::aligned_vector<Eigen::Vector2d> dart::math::SupportPolygon

Enumeration Type Documentation

◆ AxisType

enum dart::math::AxisType

Enumerator
AXIS_X
AXIS_Y
AXIS_Z

◆ IntersectionResult

enum dart::math::IntersectionResult

Intersection_t is returned by the computeIntersection() function to indicate whether there was a valid intersection between the two line segments.

Enumerator
INTERSECTING	An intersection was found.
PARALLEL	The line segments are parallel.
BEYOND_ENDPOINTS	There is no intersection because the end points do not expand far enough.

Function Documentation

◆ acosech()

double dart::math::acosech ( double _X )

inline

◆ acosh()

double dart::math::acosh ( double _X )

inline

◆ acotanh()

double dart::math::acotanh ( double _X )

inline

◆ ad()

Eigen::Vector6d dart::math::ad	(	const Eigen::Vector6d &	_X,
		const Eigen::Vector6d &	_Y
	)

adjoint mapping

Note: \(ad_X Y = ( w_X @times w_Y@,,~w_X @times v_Y - w_Y @times v_X),\), where \(X=(w_X,v_X)@in se(3), @quad Y=(w_Y,v_Y)@in se(3) \).

◆ AdInvRLinear()

Eigen::Vector6d dart::math::AdInvRLinear	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector3d &	_v
	)

Fast version of Ad(Inv([R 0; 0 1]), se3(0, v))

◆ AdInvT()

Eigen::Vector6d dart::math::AdInvT	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector6d &	_V
	)

fast version of Ad(Inv(T), V)

◆ AdInvTJac()

template<typename Derived >

Derived::PlainObject dart::math::AdInvTJac	(	const Eigen::Isometry3d &	_T,
		const Eigen::MatrixBase< Derived > &	_J
	)

Adjoint mapping for dynamic size Jacobian.

◆ AdInvTJacFixed()

template<typename Derived >

Derived::PlainObject dart::math::AdInvTJacFixed	(	const Eigen::Isometry3d &	_T,
		const Eigen::MatrixBase< Derived > &	_J
	)

Adjoint mapping for fixed size Jacobian.

◆ adJac()

template<typename Derived >

Derived::PlainObject dart::math::adJac	(	const Eigen::Vector6d &	_V,
		const Eigen::MatrixBase< Derived > &	_J
	)

◆ AdR()

Eigen::Vector6d dart::math::AdR	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector6d &	_V
	)

Fast version of Ad([R 0; 0 1], V)

◆ AdRInvJac()

template<typename Derived >

Derived::PlainObject dart::math::AdRInvJac	(	const Eigen::Isometry3d &	_T,
		const Eigen::MatrixBase< Derived > &	_J
	)

◆ AdRJac()

template<typename Derived >

Derived::PlainObject dart::math::AdRJac	(	const Eigen::Isometry3d &	_T,
		const Eigen::MatrixBase< Derived > &	_J
	)

Change coordinate Frame of a Jacobian.

◆ AdT()

Eigen::Vector6d dart::math::AdT	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector6d &	_V
	)

Rectify the rotation part so as that it satifies the orthogonality condition.

It is one step of \(R_{i_1}=1/2(R_i + R_i^{-T})\). Hence by calling this function iterativley, you can make the rotation part closer to SO(3).

reparameterize such as ||s'|| < M_PI and Exp(s) == Epx(s')

adjoint mapping

Note: \(Ad_TV = ( Rw@,, ~p @times Rw + Rv)\), where \(T=(R,p)@in SE(3), @quad V=(w,v)@in se(3) \).

◆ AdTAngular()

Eigen::Vector6d dart::math::AdTAngular	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector3d &	_w
	)

fast version of Ad(T, se3(w, 0))

◆ AdTJac()

template<typename Derived >

Derived::PlainObject dart::math::AdTJac	(	const Eigen::Isometry3d &	_T,
		const Eigen::MatrixBase< Derived > &	_J
	)

Adjoint mapping for dynamic size Jacobian.

◆ AdTJacFixed()

template<typename Derived >

Derived::PlainObject dart::math::AdTJacFixed	(	const Eigen::Isometry3d &	_T,
		const Eigen::MatrixBase< Derived > &	_J
	)

Adjoint mapping for fixed size Jacobian.

◆ AdTLinear()

Eigen::Vector6d dart::math::AdTLinear	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector3d &	_v
	)

fast version of Ad(T, se3(0, v))

◆ asech()

double dart::math::asech ( double _X )

inline

◆ asinh()

double dart::math::asinh ( double _X )

inline

◆ atanh()

double dart::math::atanh ( double _X )

inline

◆ clip() [1/2]

template<typename DerivedA , typename DerivedB >

DerivedA::PlainObject dart::math::clip	(	const Eigen::MatrixBase< DerivedA > &	val,
		const Eigen::MatrixBase< DerivedB > &	lower,
		const Eigen::MatrixBase< DerivedB > &	upper
	)

inline

◆ clip() [2/2]

template<typename T >

T dart::math::clip	(	const T &	val,
		const T &	lower,
		const T &	upper
	)

inline

◆ computeCentroidOfHull()

Eigen::Vector2d dart::math::computeCentroidOfHull ( const SupportPolygon & _convexHull )

Compute the centroid of a polygon, assuming the polygon is a convex hull.

◆ computeClosestPointOnLineSegment()

Eigen::Vector2d dart::math::computeClosestPointOnLineSegment	(	const Eigen::Vector2d &	_p,
		const Eigen::Vector2d &	_s1,
		const Eigen::Vector2d &	_s2
	)

Returns the point which is closest to _p that also lays on the line segment that goes from _s1 -> _s2.

◆ computeClosestPointOnSupportPolygon() [1/2]

Eigen::Vector2d dart::math::computeClosestPointOnSupportPolygon	(	const Eigen::Vector2d &	_p,
		const SupportPolygon &	_support
	)

Returns the point which is closest to _p that also lays on the edge of the support polygon.

◆ computeClosestPointOnSupportPolygon() [2/2]

Eigen::Vector2d dart::math::computeClosestPointOnSupportPolygon	(	std::size_t &	_index1,
		std::size_t &	_index2,
		const Eigen::Vector2d &	_p,
		const SupportPolygon &	_support
	)

Same as closestPointOnSupportPolygon, but also fills in _index1 and _index2 with the indices of the line segment.

◆ computeConvexHull() [1/2]

SupportPolygon dart::math::computeConvexHull ( const SupportPolygon & _points )

Computes the convex hull of a set of 2D points.

◆ computeConvexHull() [2/2]

SupportPolygon dart::math::computeConvexHull	(	std::vector< std::size_t > &	_originalIndices,
		const SupportPolygon &	_points
	)

Computes the convex hull of a set of 2D points and fills in _originalIndices with the original index of each entry in the returned SupportPolygon.

◆ computeIntersection()

IntersectionResult dart::math::computeIntersection	(	Eigen::Vector2d &	_intersectionPoint,
		const Eigen::Vector2d &	a1,
		const Eigen::Vector2d &	a2,
		const Eigen::Vector2d &	b1,
		const Eigen::Vector2d &	b2
	)

Compute the intersection between a line segment that goes from a1 -> a2 and a line segment that goes from b1 -> b2.

◆ computeNullSpace()

template<typename MatrixType , typename ReturnType >

void dart::math::computeNullSpace	(	const MatrixType &	_M,
		ReturnType &	_NS
	)

◆ computeRotation()

Eigen::Matrix3d dart::math::computeRotation	(	const Eigen::Vector3d &	axis,
		AxisType	axisType = `AxisType::AXIS_X`
	)

Compute a rotation matrix from a vector.

One axis of the rotated coordinates by the rotation matrix matches the input axis where the axis is specified by axisType.

◆ computeSupportPolgyon() [1/2]

SupportPolygon dart::math::computeSupportPolgyon	(	const SupportGeometry &	_geometry,
		const Eigen::Vector3d &	_axis1 = `Eigen::Vector3d::UnitX()`,
		const Eigen::Vector3d &	_axis2 = `Eigen::Vector3d::UnitY()`
	)

Project the support geometry points onto a plane with the given axes and then compute their convex hull, which will take the form of a polgyon.

_axis1 and _axis2 must both have unit length for this function to work correctly.

◆ computeSupportPolgyon() [2/2]

SupportPolygon dart::math::computeSupportPolgyon	(	std::vector< std::size_t > &	_originalIndices,
		const SupportGeometry &	_geometry,
		const Eigen::Vector3d &	_axis1,
		const Eigen::Vector3d &	_axis2
	)

Same as computeSupportPolgyon, except you can pass in a std::vector<std::size_t> which will have the same size as the returned SupportPolygon, and each entry will contain the original index of each point in the SupportPolygon.

◆ computeTransform()

Eigen::Isometry3d dart::math::computeTransform	(	const Eigen::Vector3d &	axis,
		const Eigen::Vector3d &	translation,
		AxisType	axisType = `AxisType::AXIS_X`
	)

Compute a transform from a vector and a position.

The rotation of the result transform is computed by computeRotationMatrix(), and the translation is just the input translation.

◆ CR()

const Eigen::Matrix2d dart::math::CR ( (Eigen::Matrix2d()<< 0.0, -1.0, 1.0, 0.0).finished() )

a cross b = (CR*a) dot b const Matd CR(2,2,0.0,-1.0,1.0,0.0);

◆ cross()

double dart::math::cross	(	const Eigen::Vector2d &	_v1,
		const Eigen::Vector2d &	_v2
	)

Compute a 2D cross product.

◆ dad()

Eigen::Vector6d dart::math::dad	(	const Eigen::Vector6d &	_s,
		const Eigen::Vector6d &	_t
	)

fast version of ad(se3(Eigen_Vec3(0), v), S)

fast version of ad(se3(w, 0), se3(v, 0)) -> check

dual adjoint mapping

Note: \(ad^{@,*}_V F = (m @times w + f @times v@,,~ f @times w),\) , where \(F=(m,f)@in se^{@,*}(3), @quad V=(w,v)@in se(3) \).

◆ dAdInvR()

Eigen::Vector6d dart::math::dAdInvR	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector6d &	_F
	)

fast version of dAd(Inv([R 0; 0 1]), F)

◆ dAdInvT()

Eigen::Vector6d dart::math::dAdInvT	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector6d &	_F
	)

fast version of dAd(Inv(T), F)

◆ dAdT()

Eigen::Vector6d dart::math::dAdT	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector6d &	_F
	)

dual adjoint mapping

Note: \(Ad^{@,*}_TF = ( R^T (m - p@times f)@,,~ R^T f)\), where \(T=(R,p)@in SE(3), F=(m,f)@in se(3)^*\).

◆ delta()

int dart::math::delta	(	int	_i,
		int	_j
	)

inline

◆ eulerXYXToMatrix()

Eigen::Matrix3d dart::math::eulerXYXToMatrix ( const Eigen::Vector3d & _angle )

Given Euler XYX angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotY(angle(1)) * RotX(angle(2)).

◆ eulerXYZToMatrix()

Eigen::Matrix3d dart::math::eulerXYZToMatrix ( const Eigen::Vector3d & _angle )

Given EulerXYZ angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotY(angle(1)) * RotZ(angle(2)).

◆ eulerXZXToMatrix()

Eigen::Matrix3d dart::math::eulerXZXToMatrix ( const Eigen::Vector3d & _angle )

Given EulerXZX angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotZ(angle(1)) * RotX(angle(2)).

◆ eulerXZYToMatrix()

Eigen::Matrix3d dart::math::eulerXZYToMatrix ( const Eigen::Vector3d & _angle )

Given EulerXZY angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotZ(angle(1)) * RotY(angle(2)).

◆ eulerYXYToMatrix()

Eigen::Matrix3d dart::math::eulerYXYToMatrix ( const Eigen::Vector3d & _angle )

Given EulerYXY angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotX(angle(1)) * RotY(angle(2)).

◆ eulerYXZToMatrix()

Eigen::Matrix3d dart::math::eulerYXZToMatrix ( const Eigen::Vector3d & _angle )

Given EulerYXZ angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotX(angle(1)) * RotZ(angle(2)).

◆ eulerYZXToMatrix()

Eigen::Matrix3d dart::math::eulerYZXToMatrix ( const Eigen::Vector3d & _angle )

Given EulerYZX angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotZ(angle(1)) * RotX(angle(2)).

◆ eulerYZYToMatrix()

Eigen::Matrix3d dart::math::eulerYZYToMatrix ( const Eigen::Vector3d & _angle )

Given EulerYZY angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotZ(angle(1)) * RotY(angle(2)).

◆ eulerZXYToMatrix()

Eigen::Matrix3d dart::math::eulerZXYToMatrix ( const Eigen::Vector3d & _angle )

Given EulerZXY angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotX(angle(1)) * RotY(angle(2)).

◆ eulerZXZToMatrix()

Eigen::Matrix3d dart::math::eulerZXZToMatrix ( const Eigen::Vector3d & _angle )

Given EulerZXZ angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotX(angle(1)) * RotZ(angle(2)).

◆ eulerZYXToMatrix()

Eigen::Matrix3d dart::math::eulerZYXToMatrix ( const Eigen::Vector3d & _angle )

Given EulerZYX angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotY(angle(1)) * RotX(angle(2)).

singularity : angle[1] = -+ 0.5*PI

◆ eulerZYZToMatrix()

Eigen::Matrix3d dart::math::eulerZYZToMatrix ( const Eigen::Vector3d & _angle )

Given EulerZYZ angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotY(angle(1)) * RotZ(angle(2)).

singularity : angle[1] = 0, PI

◆ expAngular()

Eigen::Isometry3d dart::math::expAngular ( const Eigen::Vector3d & _s )

fast version of Exp(se3(s, 0))

See: https://github.com/dartsim/dart/issues/88

◆ expMap()

Eigen::Isometry3d dart::math::expMap ( const Eigen::Vector6d & _S )

Exponential mapping.

◆ expMapJac()

Eigen::Matrix3d dart::math::expMapJac ( const Eigen::Vector3d & _q )

Computes the Jacobian of the expmap.

◆ expMapJacDeriv()

Eigen::Matrix3d dart::math::expMapJacDeriv	(	const Eigen::Vector3d &	_q,
		int	_qi
	)

computes the derivative of the Jacobian of the expmap wrt to _qi indexed dof; _qi \( \in \) {0,1,2}

◆ expMapJacDot()

Eigen::Matrix3d dart::math::expMapJacDot	(	const Eigen::Vector3d &	_q,
		const Eigen::Vector3d &	_qdot
	)

Computes the time derivative of the expmap Jacobian.

◆ expMapRot()

Eigen::Matrix3d dart::math::expMapRot ( const Eigen::Vector3d & _q )

Computes the Rotation matrix from a given expmap vector.

◆ expToQuat()

Eigen::Quaterniond dart::math::expToQuat ( const Eigen::Vector3d & _v )

◆ extractNullSpace()

template<typename MatrixType , typename ReturnType >

void dart::math::extractNullSpace	(	const Eigen::JacobiSVD< MatrixType > &	_SVD,
		ReturnType &	_NS
	)

◆ fromSkewSymmetric()

Eigen::Vector3d dart::math::fromSkewSymmetric ( const Eigen::Matrix3d & _m )

◆ getAdTMatrix()

Eigen::Matrix6d dart::math::getAdTMatrix ( const Eigen::Isometry3d & T )

Get linear transformation matrix of Adjoint mapping.

◆ getFrameOriginAxisZ()

Eigen::Isometry3d dart::math::getFrameOriginAxisZ	(	const Eigen::Vector3d &	_origin,
		const Eigen::Vector3d &	_axisZ
	)

Generate frame given origin and z-axis.

◆ HullAngleComparison()

static bool dart::math::HullAngleComparison	(	const HullAngle &	a,
		const HullAngle &	b
	)

static

◆ integratePosition()

template<typename SpaceT >

SpaceT::Point dart::math::integratePosition	(	const typename SpaceT::Point &	pos,
		const typename SpaceT::Vector &	vel,
		double	dt
	)

◆ integrateVelocity()

template<typename SpaceT >

SpaceT::Vector dart::math::integrateVelocity	(	const typename SpaceT::Vector &	vel,
		const typename SpaceT::Vector &	acc,
		double	dt
	)

◆ inverse()

template<typename SpaceT >

SpaceT::Matrix dart::math::inverse ( const typename SpaceT::Matrix & mat )

◆ isEqual()

bool dart::math::isEqual	(	double	_x,
		double	_y
	)

inline

◆ isInf() [1/2]

bool dart::math::isInf ( const Eigen::MatrixXd & _m )

inline

Returns whether _m is an infinity matrix (either positive infinity or negative infinity).

◆ isInf() [2/2]

bool dart::math::isInf ( double _v )

inline

Returns whether _v is an infinity value (either positive infinity or negative infinity).

◆ isInsideSupportPolygon()

bool dart::math::isInsideSupportPolygon	(	const Eigen::Vector2d &	_p,
		const SupportPolygon &	_support,
		bool	_includeEdge
	)

Returns true if the point _p is inside the support polygon.

◆ isInt()

bool dart::math::isInt ( double _x )

inline

◆ isLeftTurn()

static bool dart::math::isLeftTurn	(	const Eigen::Vector2d &	p1,
		const Eigen::Vector2d &	p2,
		const Eigen::Vector2d &	p3
	)

static

◆ isNan() [1/2]

bool dart::math::isNan ( const Eigen::MatrixXd & _m )

inline

Returns whether _m is a NaN (Not-A-Number) matrix.

◆ isNan() [2/2]

bool dart::math::isNan ( double _v )

inline

Returns whether _v is a NaN (Not-A-Number) value.

◆ isSymmetric()

bool dart::math::isSymmetric	(	const Eigen::MatrixXd &	_m,
		double	_tol = `1e-6`
	)

inline

Returns whether _m is symmetric or not.

◆ isZero()

bool dart::math::isZero ( double _theta )

inline

◆ logMap() [1/2]

Eigen::Vector6d dart::math::logMap ( const Eigen::Isometry3d & _T )

Log mapping.

◆ logMap() [2/2]

Eigen::Vector3d dart::math::logMap ( const Eigen::Matrix3d & _R )

Log mapping.

Note: When \(|Log(R)| = @pi\), Exp(LogR(R) = Exp(-Log(R)). The implementation returns only the positive one.

◆ makeSkewSymmetric()

Eigen::Matrix3d dart::math::makeSkewSymmetric ( const Eigen::Vector3d & _v )

◆ matrixToEulerXYX()

Eigen::Vector3d dart::math::matrixToEulerXYX ( const Eigen::Matrix3d & _R )

get the Euler XYX angle from R

◆ matrixToEulerXYZ()

Eigen::Vector3d dart::math::matrixToEulerXYZ ( const Eigen::Matrix3d & _R )

get the Euler XYZ angle from R

◆ matrixToEulerXZY()

Eigen::Vector3d dart::math::matrixToEulerXZY ( const Eigen::Matrix3d & _R )

get the Euler XZY angle from R

◆ matrixToEulerYXZ()

Eigen::Vector3d dart::math::matrixToEulerYXZ ( const Eigen::Matrix3d & _R )

get the Euler YXZ angle from R

◆ matrixToEulerYZX()

Eigen::Vector3d dart::math::matrixToEulerYZX ( const Eigen::Matrix3d & _R )

get the Euler YZX angle from R

◆ matrixToEulerZXY()

Eigen::Vector3d dart::math::matrixToEulerZXY ( const Eigen::Matrix3d & _R )

get the Euler ZXY angle from R

◆ matrixToEulerZYX()

Eigen::Vector3d dart::math::matrixToEulerZYX ( const Eigen::Matrix3d & _R )

get the Euler ZYX angle from R

◆ parallelAxisTheorem()

Eigen::Matrix3d dart::math::parallelAxisTheorem	(	const Eigen::Matrix3d &	_original,
		const Eigen::Vector3d &	_comShift,
		double	_mass
	)

Use the Parallel Axis Theorem to compute the moment of inertia of a body whose center of mass has been shifted from the origin.

◆ quatDeriv()

Eigen::Matrix3d dart::math::quatDeriv	(	const Eigen::Quaterniond &	_q,
		int	_el
	)

◆ quatSecondDeriv()

Eigen::Matrix3d dart::math::quatSecondDeriv	(	const Eigen::Quaterniond &	,
		int	_el1,
		int	_el2
	)

◆ quatToExp()

Eigen::Vector3d dart::math::quatToExp ( const Eigen::Quaterniond & _q )

◆ random()

double dart::math::random	(	double	_min,
		double	_max
	)

inline

Deprecated:: Please use Random::uniform() instead.

◆ randomVector() [1/2]

template<int N>

Eigen::Matrix<double, N, 1> dart::math::randomVector ( double _limit )

Deprecated:: Please use Random::uniform() instead.

◆ randomVector() [2/2]

template<int N>

Eigen::Matrix<double, N, 1> dart::math::randomVector	(	double	_min,
		double	_max
	)

Deprecated:: Please use Random::uniform() instead.

◆ randomVectorXd() [1/2]

Eigen::VectorXd dart::math::randomVectorXd	(	std::size_t	size,
		double	limit
	)

inline

Deprecated:: Please use Random::uniform() instead.

◆ randomVectorXd() [2/2]

Eigen::VectorXd dart::math::randomVectorXd	(	std::size_t	size,
		double	min,
		double	max
	)

inline

Deprecated:: Please use Random::uniform() instead.

◆ rotatePoint() [1/2]

Eigen::Vector3d dart::math::rotatePoint	(	const Eigen::Quaterniond &	_q,
		const Eigen::Vector3d &	_pt
	)

◆ rotatePoint() [2/2]

Eigen::Vector3d dart::math::rotatePoint	(	const Eigen::Quaterniond &	_q,
		double	_x,
		double	_y,
		double	_z
	)

◆ round()

double dart::math::round ( double _x )

inline

◆ round2()

double dart::math::round2 ( double _x )

inline

◆ seedRand()

unsigned dart::math::seedRand ( )

inline

◆ sign() [1/3]

template<typename T >

constexpr int dart::math::sign ( T x )

inlineconstexpr

◆ sign() [2/3]

template<typename T >

constexpr int dart::math::sign	(	T	x,
		std::false_type
	)

inlineconstexpr

◆ sign() [3/3]

template<typename T >

constexpr int dart::math::sign	(	T	x,
		std::true_type
	)

inlineconstexpr

◆ sqr()

double dart::math::sqr ( double _x )

inline

◆ toDegree()

template<typename T >

constexpr T dart::math::toDegree ( const T & radian )

constexpr

◆ toEuclideanPoint()

template<typename SpaceT >

SpaceT::EuclideanPoint dart::math::toEuclideanPoint ( const typename SpaceT::Point & point )

◆ toManifoldPoint()

template<typename SpaceT >

SpaceT::Point dart::math::toManifoldPoint ( const typename SpaceT::EuclideanPoint & point )

◆ toRadian()

template<typename T >

constexpr T dart::math::toRadian ( const T & degree )

constexpr

◆ transformInertia()

Inertia dart::math::transformInertia	(	const Eigen::Isometry3d &	_T,
		const Inertia &	_I
	)

◆ Tsinc()

double dart::math::Tsinc ( double _theta )

inline

◆ verifyRotation()

bool dart::math::verifyRotation ( const Eigen::Matrix3d & _T )

Check if determinant of _R is equat to 1 and all the elements are not NaN values.

◆ verifyTransform()

bool dart::math::verifyTransform ( const Eigen::Isometry3d & _T )

Check if determinant of the rotational part of _T is equat to 1 and all the elements are not NaN values.

◆ wrapToPi()

double dart::math::wrapToPi ( double angle )

inline

Compute the angle (in the range of -pi to +pi) which ignores any full rotations.

Namespaces

Classes

Typedefs

Enumerations

Functions

Typedef Documentation

◆ AngularJacobian

◆ constantsd

◆ constantsf

◆ Inertia

◆ Jacobian

◆ LinearJacobian

◆ NullSpace

◆ R1Space

◆ R2Space

◆ R3Space

◆ SupportGeometry

◆ SupportPolygon

Enumeration Type Documentation

◆ AxisType

◆ IntersectionResult

Function Documentation

◆ acosech()

◆ acosh()

◆ acotanh()

◆ ad()

◆ AdInvRLinear()

◆ AdInvT()

◆ AdInvTJac()

◆ AdInvTJacFixed()

◆ adJac()

◆ AdR()

◆ AdRInvJac()

◆ AdRJac()

◆ AdT()

◆ AdTAngular()

◆ AdTJac()

◆ AdTJacFixed()

◆ AdTLinear()

◆ asech()

◆ asinh()

◆ atanh()

◆ clip() [1/2]

◆ clip() [2/2]

◆ computeCentroidOfHull()

◆ computeClosestPointOnLineSegment()

◆ computeClosestPointOnSupportPolygon() [1/2]

◆ computeClosestPointOnSupportPolygon() [2/2]

◆ computeConvexHull() [1/2]

◆ computeConvexHull() [2/2]

◆ computeIntersection()

◆ computeNullSpace()

◆ computeRotation()

◆ computeSupportPolgyon() [1/2]

◆ computeSupportPolgyon() [2/2]

◆ computeTransform()

◆ CR()

◆ cross()

◆ dad()

◆ dAdInvR()

◆ dAdInvT()

◆ dAdT()

◆ delta()

◆ eulerXYXToMatrix()

◆ eulerXYZToMatrix()

◆ eulerXZXToMatrix()

◆ eulerXZYToMatrix()

◆ eulerYXYToMatrix()

◆ eulerYXZToMatrix()

◆ eulerYZXToMatrix()

◆ eulerYZYToMatrix()

◆ eulerZXYToMatrix()

◆ eulerZXZToMatrix()

◆ eulerZYXToMatrix()

◆ eulerZYZToMatrix()

◆ expAngular()

◆ expMap()

◆ expMapJac()

◆ expMapJacDeriv()

◆ expMapJacDot()