Rot3 is a 3D rotation represented as a rotation matrix if the preprocessor symbol GTSAM_USE_QUATERNIONS is not defined, or as a quaternion if it is defined. More...

#include <Rot3.h>

Inheritance diagram for gtsam::Rot3:

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Collaboration diagram for gtsam::Rot3:

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Classes
struct	CayleyChart

struct	ChartAtOrigin

Public Types
enum

typedef OptionalJacobian< N, N >	ChartJacobian

typedef Eigen::Matrix< double, N, N >	Jacobian

typedef Eigen::Matrix< double, N, 1 >	TangentVector

Public Member Functions
const Rot3 &	derived () const

Rot3	compose (const Rot3 &g) const

Rot3	compose (const Rot3 &g, ChartJacobian H1, ChartJacobian H2={}) const

GTSAM_EXPORT SOn	compose (const SOn &g, DynamicJacobian H1, DynamicJacobian H2) const

Rot3	between (const Rot3 &g) const

Rot3	between (const Rot3 &g, ChartJacobian H1, ChartJacobian H2={}) const

GTSAM_EXPORT SOn	between (const SOn &g, DynamicJacobian H1, DynamicJacobian H2) const

Rot3	inverse (ChartJacobian H) const

Rot3	expmap (const TangentVector &v) const

Rot3	expmap (const TangentVector &v, ChartJacobian H1, ChartJacobian H2={}) const
	expmap with optional derivatives

TangentVector	logmap (const Rot3 &g) const

TangentVector	logmap (const Rot3 &g, ChartJacobian H1, ChartJacobian H2={}) const
	logmap with optional derivatives

Rot3	retract (const TangentVector &v) const
	retract as required by manifold concept: applies v at *this

Rot3	retract (const TangentVector &v, ChartJacobian H1, ChartJacobian H2={}) const
	retract with optional derivatives

TangentVector	localCoordinates (const Rot3 &g) const
	localCoordinates as required by manifold concept: finds tangent vector between *this and g

TangentVector	localCoordinates (const Rot3 &g, ChartJacobian H1, ChartJacobian H2={}) const
	localCoordinates with optional derivatives

Testable
void	print (const std::string &s="") const

bool	equals (const Rot3 &p, double tol=1e-9) const

Group Action on Point3
Point3	rotate (const Point3 &p, OptionalJacobian< 3, 3 > H1={}, OptionalJacobian< 3, 3 > H2={}) const

Point3	operator* (const Point3 &p) const
	rotate point from rotated coordinate frame to world = R*p

Point3	unrotate (const Point3 &p, OptionalJacobian< 3, 3 > H1={}, OptionalJacobian< 3, 3 > H2={}) const
	rotate point from world to rotated frame \( p^c = (R_c^w)^T p^w \)

Group Action on Unit3
Unit3	rotate (const Unit3 &p, OptionalJacobian< 2, 3 > HR={}, OptionalJacobian< 2, 2 > Hp={}) const
	rotate 3D direction from rotated coordinate frame to world frame

Unit3	unrotate (const Unit3 &p, OptionalJacobian< 2, 3 > HR={}, OptionalJacobian< 2, 2 > Hp={}) const
	unrotate 3D direction from world frame to rotated coordinate frame

Unit3	operator* (const Unit3 &p) const
	rotate 3D direction from rotated coordinate frame to world frame

Standard Interface
Matrix3	matrix () const

Matrix3	transpose () const

Point3	column (int index) const

Point3	r1 () const
	first column

Point3	r2 () const
	second column

Point3	r3 () const
	third column

Vector3	xyz (OptionalJacobian< 3, 3 > H={}) const

Vector3	ypr (OptionalJacobian< 3, 3 > H={}) const

Vector3	rpy (OptionalJacobian< 3, 3 > H={}) const

double	roll (OptionalJacobian< 1, 3 > H={}) const

double	pitch (OptionalJacobian< 1, 3 > H={}) const

double	yaw (OptionalJacobian< 1, 3 > H={}) const

Static Public Member Functions
static Rot3	Retract (const TangentVector &v)
	Retract at origin: possible in Lie group because it has an identity.

static Rot3	Retract (const TangentVector &v, ChartJacobian H)
	Retract at origin with optional derivative.

static TangentVector	LocalCoordinates (const Rot3 &g)
	LocalCoordinates at origin: possible in Lie group because it has an identity.

static TangentVector	LocalCoordinates (const Rot3 &g, ChartJacobian H)
	LocalCoordinates at origin with optional derivative.

Constructors and named constructors
	Rot3 ()

	Rot3 (const Point3 &col1, const Point3 &col2, const Point3 &col3)

	Rot3 (double R11, double R12, double R13, double R21, double R22, double R23, double R31, double R32, double R33)
	Construct from a rotation matrix, as doubles in row-major order !!!

template<typename Derived >
	Rot3 (const Eigen::MatrixBase< Derived > &R)

	Rot3 (const Matrix3 &R)

	Rot3 (const SO3 &R)

	Rot3 (const Quaternion &q)

	Rot3 (double w, double x, double y, double z)

virtual	~Rot3 ()

Rot3	normalized () const

static Rot3	Random (std::mt19937 &rng)

static Rot3	Rx (double t)
	Rotation around X axis as in http://en.wikipedia.org/wiki/Rotation_matrix, counterclockwise when looking from unchanging axis.

static Rot3	Ry (double t)
	Rotation around Y axis as in http://en.wikipedia.org/wiki/Rotation_matrix, counterclockwise when looking from unchanging axis.

static Rot3	Rz (double t)
	Rotation around Z axis as in http://en.wikipedia.org/wiki/Rotation_matrix, counterclockwise when looking from unchanging axis.

static Rot3	RzRyRx (double x, double y, double z, OptionalJacobian< 3, 1 > Hx={}, OptionalJacobian< 3, 1 > Hy={}, OptionalJacobian< 3, 1 > Hz={})
	Rotations around Z, Y, then X axes as in http://en.wikipedia.org/wiki/Rotation_matrix, counterclockwise when looking from unchanging axis.

static Rot3	RzRyRx (const Vector &xyz, OptionalJacobian< 3, 3 > H={})
	Rotations around Z, Y, then X axes as in http://en.wikipedia.org/wiki/Rotation_matrix, counterclockwise when looking from unchanging axis.

static Rot3	Yaw (double t)
	Positive yaw is to right (as in aircraft heading). See ypr.

static Rot3	Pitch (double t)
	Positive pitch is up (increasing aircraft altitude).See ypr.

static Rot3	Roll (double t)

static Rot3	Ypr (double y, double p, double r, OptionalJacobian< 3, 1 > Hy={}, OptionalJacobian< 3, 1 > Hp={}, OptionalJacobian< 3, 1 > Hr={})

static Rot3	Quaternion (double w, double x, double y, double z)

static Rot3	AxisAngle (const Point3 &axis, double angle)

static Rot3	AxisAngle (const Unit3 &axis, double angle)

static Rot3	Rodrigues (const Vector3 &w)

static Rot3	Rodrigues (double wx, double wy, double wz)

static Rot3	AlignPair (const Unit3 &axis, const Unit3 &a_p, const Unit3 &b_p)
	Determine a rotation to bring two vectors into alignment, using the rotation axis provided.

static Rot3	AlignTwoPairs (const Unit3 &a_p, const Unit3 &b_p, const Unit3 &a_q, const Unit3 &b_q)
	Calculate rotation from two pairs of homogeneous points using two successive rotations.

static Rot3	ClosestTo (const Matrix3 &M)

Group
Rot3	operator* (const Rot3 &R2) const
	Syntatic sugar for composing two rotations.

Rot3	inverse () const
	inverse of a rotation

Rot3	conjugate (const Rot3 &cRb) const

static Rot3	Identity ()
	identity rotation for group operation

Manifold
enum	CoordinatesMode { EXPMAP, CAYLEY }

Rot3	retractCayley (const Vector &omega) const
	Retraction from R^3 to Rot3 manifold using the Cayley transform.

Vector3	localCayley (const Rot3 &other) const
	Inverse of retractCayley.

Lie Group
Matrix3	AdjointMap () const

static Rot3	Expmap (const Vector3 &v, OptionalJacobian< 3, 3 > H={})

static Vector3	Logmap (const Rot3 &R, OptionalJacobian< 3, 3 > H={})

static Matrix3	ExpmapDerivative (const Vector3 &x)
	Derivative of Expmap.

static Matrix3	LogmapDerivative (const Vector3 &x)
	Derivative of Logmap.

Advanced Interface
std::pair< Unit3, double >	axisAngle () const

gtsam::Quaternion	toQuaternion () const

Rot3	slerp (double t, const Rot3 &other) const
	Spherical Linear intERPolation between *this and other. More...

GTSAM_EXPORT friend std::ostream &	operator<< (std::ostream &os, const Rot3 &p)
	Output stream operator.

Detailed Description

Rot3 is a 3D rotation represented as a rotation matrix if the preprocessor symbol GTSAM_USE_QUATERNIONS is not defined, or as a quaternion if it is defined.

Member Enumeration Documentation

◆ CoordinatesMode

enum gtsam::Rot3::CoordinatesMode

The method retract() is used to map from the tangent space back to the manifold. Its inverse, is localCoordinates(). For Lie groups, an obvious retraction is the exponential map, but this can be expensive to compute. The following Enum is used to indicate which method should be used. The default is determined by ROT3_DEFAULT_COORDINATES_MODE, which may be set at compile time, and itself defaults to Rot3::CAYLEY, or if GTSAM_USE_QUATERNIONS is defined, to Rot3::EXPMAP.

Enumerator
EXPMAP	Use the Lie group exponential map to retract.
CAYLEY	Retract and localCoordinates using the Cayley transform.

Constructor & Destructor Documentation

◆ Rot3() [1/6]

gtsam::Rot3::Rot3 ( )

default constructor, unit rotation

◆ Rot3() [2/6]

gtsam::Rot3::Rot3	(	const Point3 &	col1,
		const Point3 &	col2,
		const Point3 &	col3
	)

Constructor from columns

Parameters

r1	X-axis of rotated frame
r2	Y-axis of rotated frame
r3	Z-axis of rotated frame

◆ Rot3() [3/6]

template<typename Derived >

gtsam::Rot3::Rot3 ( const Eigen::MatrixBase< Derived > & R )

inlineexplicit

Constructor from a rotation matrix Version for generic matrices. Need casting to Matrix3 in quaternion mode, since Eigen's quaternion doesn't allow assignment/construction from a generic matrix. See: http://stackoverflow.com/questions/27094132/cannot-understand-if-this-is-circular-dependency-or-clang#tab-top

◆ Rot3() [4/6]

gtsam::Rot3::Rot3 ( const Matrix3 & R )

inlineexplicit

Constructor from a rotation matrix Overload version for Matrix3 to avoid casting in quaternion mode.

◆ Rot3() [5/6]

gtsam::Rot3::Rot3 ( const SO3 & R )

inlineexplicit

Constructor from an SO3 instance

◆ Rot3() [6/6]

gtsam::Rot3::Rot3 ( const Quaternion & q )

Constructor from a quaternion. This can also be called using a plain Vector, due to implicit conversion from Vector to Quaternion

Parameters

q	The quaternion

◆ ~Rot3()

virtual gtsam::Rot3::~Rot3 ( )

inlinevirtual

Virtual destructor

Member Function Documentation

◆ AdjointMap()

Matrix3 gtsam::Rot3::AdjointMap ( ) const

inline

Calculate Adjoint map

◆ AxisAngle() [1/2]

static Rot3 gtsam::Rot3::AxisAngle	(	const Point3 &	axis,
		double	angle
	)

inlinestatic

Convert from axis/angle representation

Parameters

axis	is the rotation axis, unit length
angle	rotation angle

Returns: incremental rotation

◆ AxisAngle() [2/2]

static Rot3 gtsam::Rot3::AxisAngle	(	const Unit3 &	axis,
		double	angle
	)

inlinestatic

Convert from axis/angle representation

Parameters

axis	is the rotation axis
angle	rotation angle

Returns: incremental rotation

◆ axisAngle()

std::pair<Unit3, double> gtsam::Rot3::axisAngle ( ) const

Compute the Euler axis and angle (in radians) representation of this rotation. The angle is in the range [0, π]. If the angle is not in the range, the axis is flipped around accordingly so that the returned angle is within the specified range.

Returns: pair consisting of Unit3 axis and angle in radians

◆ ClosestTo()

static Rot3 gtsam::Rot3::ClosestTo ( const Matrix3 & M )

inlinestatic

Static, named constructor that finds Rot3 element closest to M in Frobenius norm.

Uses Full SVD to compute the orthogonal matrix, thus is highly accurate and robust.

N. J. Higham. Matrix nearness problems and applications. In M. J. C. Gover and S. Barnett, editors, Applications of Matrix Theory, pages 1–27. Oxford University Press, 1989.

◆ column()

Point3 gtsam::Rot3::column ( int index ) const

Deprecated:: , this is base 1, and was just confusing

◆ conjugate()

Rot3 gtsam::Rot3::conjugate ( const Rot3 & cRb ) const

inline

Conjugation: given a rotation acting in frame B, compute rotation c1Rc2 acting in a frame C

Parameters

cRb	rotation from B frame to C frame

Returns: c1Rc2 = cRb * b1Rb2 * cRb'

◆ equals()

bool gtsam::Rot3::equals	(	const Rot3 &	p,
		double	tol = `1e-9`
	)		const

equals with an tolerance

◆ expmap()

Rot3 gtsam::LieGroup< Rot3 , N >::expmap ( const TangentVector & v ) const

inlineinherited

expmap as required by manifold concept Applies exponential map to v and composes with *this

◆ Expmap()

static Rot3 gtsam::Rot3::Expmap	(	const Vector3 &	v,
		OptionalJacobian< 3, 3 >	H = `{}`
	)

inlinestatic

Exponential map at identity - create a rotation from canonical coordinates \( [R_x,R_y,R_z] \) using Rodrigues' formula

◆ logmap()

TangentVector gtsam::LieGroup< Rot3 , N >::logmap ( const Rot3 & g ) const

inlineinherited

logmap as required by manifold concept Applies logarithmic map to group element that takes *this to g

◆ Logmap()

static Vector3 gtsam::Rot3::Logmap	(	const Rot3 &	R,
		OptionalJacobian< 3, 3 >	H = `{}`
	)

static

Log map at identity - returns the canonical coordinates \( [R_x,R_y,R_z] \) of this rotation

◆ matrix()

Matrix3 gtsam::Rot3::matrix ( ) const

return 3*3 rotation matrix

◆ normalized()

Rot3 gtsam::Rot3::normalized ( ) const

Normalize rotation so that its determinant is 1. This means either re-orthogonalizing the Matrix representation or normalizing the quaternion representation.

This method is akin to ClosestTo but uses a computationally cheaper algorithm.

Ref: https://drive.google.com/file/d/0B9rLLz1XQKmaZTlQdV81QjNoZTA/view

◆ pitch()

double gtsam::Rot3::pitch ( OptionalJacobian< 1, 3 > H = {} ) const

Accessor to get to component of angle representations NOTE: these are not efficient to get to multiple separate parts, you should instead use xyz() or ypr() TODO: make this more efficient

◆ print()

void gtsam::Rot3::print ( const std::string & s = "" ) const

print

◆ Quaternion()

static Rot3 gtsam::Rot3::Quaternion	(	double	w,
		double	x,
		double	y,
		double	z
	)

inlinestatic

Create from Quaternion coefficients

◆ Random()

static Rot3 gtsam::Rot3::Random ( std::mt19937 & rng )

static

Random, generates a random axis, then random angle \(\in\) [-pi,pi] Example: std::mt19937 engine(42); Unit3 unit = Unit3::Random(engine);

◆ Rodrigues() [1/2]

static Rot3 gtsam::Rot3::Rodrigues ( const Vector3 & w )

inlinestatic

Rodrigues' formula to compute an incremental rotation

Parameters

w	a vector of incremental roll,pitch,yaw

Returns: incremental rotation

◆ Rodrigues() [2/2]

static Rot3 gtsam::Rot3::Rodrigues	(	double	wx,
		double	wy,
		double	wz
	)

inlinestatic

Rodrigues' formula to compute an incremental rotation

Parameters

wx	Incremental roll (about X)
wy	Incremental pitch (about Y)
wz	Incremental yaw (about Z)

Returns: incremental rotation

◆ roll()

double gtsam::Rot3::roll ( OptionalJacobian< 1, 3 > H = {} ) const

Accessor to get to component of angle representations NOTE: these are not efficient to get to multiple separate parts, you should instead use xyz() or ypr() TODO: make this more efficient

◆ rotate()

Point3 gtsam::Rot3::rotate	(	const Point3 &	p,
		OptionalJacobian< 3, 3 >	H1 = `{}`,
		OptionalJacobian< 3, 3 >	H2 = `{}`
	)		const

rotate point from rotated coordinate frame to world \( p^w = R_c^w p^c \)

◆ rpy()

Vector3 gtsam::Rot3::rpy ( OptionalJacobian< 3, 3 > H = {} ) const

Use RQ to calculate roll-pitch-yaw angle representation

Returns: a vector containing rpy s.t. R = Rot3::Ypr(y,p,r)

◆ slerp()

Rot3 gtsam::Rot3::slerp	(	double	t,
		const Rot3 &	other
	)		const

Spherical Linear intERPolation between *this and other.

Parameters

t	a value between 0 and 1
other	final point of iterpolation geodesic on manifold

◆ toQuaternion()

gtsam::Quaternion gtsam::Rot3::toQuaternion ( ) const

Compute the quaternion representation of this rotation.

Returns: The quaternion

◆ transpose()

Matrix3 gtsam::Rot3::transpose ( ) const

Return 3*3 transpose (inverse) rotation matrix

◆ xyz()

Vector3 gtsam::Rot3::xyz ( OptionalJacobian< 3, 3 > H = {} ) const

Use RQ to calculate xyz angle representation

Returns: a vector containing x,y,z s.t. R = Rot3::RzRyRx(x,y,z)

◆ yaw()

double gtsam::Rot3::yaw ( OptionalJacobian< 1, 3 > H = {} ) const

Accessor to get to component of angle representations NOTE: these are not efficient to get to multiple separate parts, you should instead use xyz() or ypr() TODO: make this more efficient

◆ Ypr()

static Rot3 gtsam::Rot3::Ypr	(	double	y,
		double	p,
		double	r,
		OptionalJacobian< 3, 1 >	Hy = `{}`,
		OptionalJacobian< 3, 1 >	Hp = `{}`,
		OptionalJacobian< 3, 1 >	Hr = `{}`
	)

inlinestatic

Returns rotation nRb from body to nav frame. For vehicle coordinate frame X forward, Y right, Z down: Positive yaw is to right (as in aircraft heading). Positive pitch is up (increasing aircraft altitude). Positive roll is to right (increasing yaw in aircraft). Tait-Bryan system from Spatial Reference Model (SRM) (x,y,z) = (roll,pitch,yaw) as described in http://www.sedris.org/wg8home/Documents/WG80462.pdf.

For vehicle coordinate frame X forward, Y left, Z up: Positive yaw is to left (as in aircraft heading). Positive pitch is down (decreasing aircraft altitude). Positive roll is to right (decreasing yaw in aircraft).

◆ ypr()

Vector3 gtsam::Rot3::ypr ( OptionalJacobian< 3, 3 > H = {} ) const

Use RQ to calculate yaw-pitch-roll angle representation

Returns: a vector containing ypr s.t. R = Rot3::Ypr(y,p,r)

The documentation for this class was generated from the following file:

/home/docs/checkouts/readthedocs.org/user_builds/gtsam-jlblanco-docs/checkouts/latest/gtsam/geometry/Rot3.h

Classes

Public Types

Public Member Functions

Static Public Member Functions

Constructors and named constructors

Group

Manifold

Lie Group

Advanced Interface

Detailed Description

Member Enumeration Documentation

◆ CoordinatesMode

Constructor & Destructor Documentation

◆ Rot3() [1/6]

◆ Rot3() [2/6]

◆ Rot3() [3/6]

◆ Rot3() [4/6]

◆ Rot3() [5/6]

◆ Rot3() [6/6]

◆ ~Rot3()

Member Function Documentation

◆ AdjointMap()

◆ AxisAngle() [1/2]

◆ AxisAngle() [2/2]

◆ axisAngle()

◆ ClosestTo()

◆ column()

◆ conjugate()

◆ equals()

◆ expmap()

◆ Expmap()

◆ logmap()

◆ Logmap()

◆ matrix()

◆ normalized()

◆ pitch()

◆ print()

◆ Quaternion()

◆ Random()

◆ Rodrigues() [1/2]

◆ Rodrigues() [2/2]

◆ roll()

◆ rotate()

◆ rpy()

◆ slerp()

◆ toQuaternion()

◆ transpose()

◆ xyz()

◆ yaw()

◆ Ypr()

◆ ypr()