gtsam 4.1.1 gtsam
gtsam::Rot3 Class Reference Inheritance diagram for gtsam::Rot3:

## Manifold

enum  CoordinatesMode { EXPMAP , CAYLEY }
The method retract() is used to map from the tangent space back to the manifold. More...

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.

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

std::pair< Unit3, double > axisAngle () const
Compute the Euler axis and angle (in radians) representation of this rotation. More...

gtsam::Quaternion toQuaternion () const
Compute the quaternion representation of this rotation. More...

Vector quaternion () const
Converts to a generic matrix to allow for use with matlab In format: w x y z.

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

## Constructors and named constructors

Rot3 ()
default constructor, unit rotation

Rot3 (const Point3 &col1, const Point3 &col2, const Point3 &col3)
Constructor from columns More...

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

template<typename Derived >
Rot3 (const Eigen::MatrixBase< Derived > &R)
Constructor from a rotation matrix Version for generic matrices. More...

Rot3 (const Matrix3 &R)
Constructor from a rotation matrix Overload version for Matrix3 to avoid casting in quaternion mode.

Rot3 (const SO3 &R)
Constructor from an SO3 instance.

Rot3 (const Quaternion &q)
Constructor from a quaternion. More...

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

virtual ~Rot3 ()
Virtual destructor.

Rot3 normalized () const
Normalize rotation so that its determinant is 1. More...

static Rot3 Random (std::mt19937 &rng)
Random, generates a random axis, then random angle \in [-p,pi] Example: std::mt19937 engine(42); Unit3 unit = Unit3::Random(engine);.

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=boost::none, OptionalJacobian< 3, 1 > Hy=boost::none, OptionalJacobian< 3, 1 > Hz=boost::none)
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=boost::none)
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=boost::none, OptionalJacobian< 3, 1 > Hp=boost::none, OptionalJacobian< 3, 1 > Hr=boost::none)
Returns rotation nRb from body to nav frame. More...

static Rot3 Quaternion (double w, double x, double y, double z)
Create from Quaternion coefficients.

static Rot3 AxisAngle (const Point3 &axis, double angle)
Convert from axis/angle representation. More...

static Rot3 AxisAngle (const Unit3 &axis, double angle)
Convert from axis/angle representation. More...

static Rot3 Rodrigues (const Vector3 &w)
Rodrigues' formula to compute an incremental rotation. More...

static Rot3 Rodrigues (double wx, double wy, double wz)
Rodrigues' formula to compute an incremental rotation. More...

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

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

static Rot3 identity ()
identity rotation for group operation

## Lie Group

static Rot3 Expmap (const Vector3 &v, OptionalJacobian< 3, 3 > H=boost::none)
Exponential map at identity - create a rotation from canonical coordinates $$[R_x,R_y,R_z]$$ using Rodrigues' formula.

static Vector3 Logmap (const Rot3 &R, OptionalJacobian< 3, 3 > H=boost::none)
Log map at identity - returns the canonical coordinates $$[R_x,R_y,R_z]$$ of this rotation.

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

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

## Public Member Functions

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

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

Group Action on Point3
Point3 rotate (const Point3 &p, OptionalJacobian< 3, 3 > H1=boost::none, OptionalJacobian< 3, 3 > H2=boost::none) const
rotate point from rotated coordinate frame to world $$p^w = R_c^w p^c$$

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=boost::none, OptionalJacobian< 3, 3 > H2=boost::none) 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=boost::none, OptionalJacobian< 2, 2 > Hp=boost::none) const
rotate 3D direction from rotated coordinate frame to world frame

Unit3 unrotate (const Unit3 &p, OptionalJacobian< 2, 3 > HR=boost::none, OptionalJacobian< 2, 2 > Hp=boost::none) 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
return 3*3 rotation matrix

Matrix3 transpose () const
Return 3*3 transpose (inverse) rotation matrix.

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=boost::none) const
Use RQ to calculate xyz angle representation. More...

Vector3 ypr (OptionalJacobian< 3, 3 > H=boost::none) const
Use RQ to calculate yaw-pitch-roll angle representation. More...

Vector3 rpy (OptionalJacobian< 3, 3 > H=boost::none) const
Use RQ to calculate roll-pitch-yaw angle representation. More...

double roll (OptionalJacobian< 1, 3 > H=boost::none) 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.

double pitch (OptionalJacobian< 1, 3 > H=boost::none) 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.

double yaw (OptionalJacobian< 1, 3 > H=boost::none) 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. Public Member Functions inherited from gtsam::LieGroup< Rot3, 3 >
const Rot3derived () const

Rot3 compose (const Rot3 &g) const

Rot3 compose (const Rot3 &g, ChartJacobian H1, ChartJacobian H2=boost::none) const

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

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=boost::none) const

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

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

Rot3 inverse (ChartJacobian H) const

Rot3 expmap (const TangentVector &v) const
expmap as required by manifold concept Applies exponential map to v and composes with *this

Rot3 expmap (const TangentVector &v, ChartJacobian H1, ChartJacobian H2=boost::none) const
expmap with optional derivatives

TangentVector logmap (const Rot3 &g) const
logmap as required by manifold concept Applies logarithmic map to group element that takes *this to g

TangentVector logmap (const Rot3 &g, ChartJacobian H1, ChartJacobian H2=boost::none) 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=boost::none) 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=boost::none) const
localCoordinates with optional derivatives

## Classes

struct  CayleyChart

struct  ChartAtOrigin

## Friends

class boost::serialization::access
Serialization function. Static Public Member Functions inherited from gtsam::LieGroup< Rot3, 3 >
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. Public Types inherited from gtsam::LieGroup< Rot3, 3 >
enum

typedef OptionalJacobian< N, N > ChartJacobian

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

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

## ◆ 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.

## ◆ Rot3() [1/3]

 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() [2/3]

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() [3/3]

 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

## ◆ axisAngle()

 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

## ◆ 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

## ◆ 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'

## ◆ 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.

Implementation from here: https://stackoverflow.com/a/23082112/1236990

Essentially, this computes the orthogonalization error, distributes the error to the x and y rows, and then performs a Taylor expansion to orthogonalize.

## ◆ 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
Returns
incremental rotation

## ◆ rpy()

 Vector3 gtsam::Rot3::rpy ( OptionalJacobian< 3, 3 > H = boost::none ) 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

## ◆ xyz()

 Vector3 gtsam::Rot3::xyz ( OptionalJacobian< 3, 3 > H = boost::none ) const

Use RQ to calculate xyz angle representation.

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

## ◆ Ypr()

 static Rot3 gtsam::Rot3::Ypr ( double y, double p, double r, OptionalJacobian< 3, 1 > Hy = boost::none, OptionalJacobian< 3, 1 > Hp = boost::none, OptionalJacobian< 3, 1 > Hr = boost::none )
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 = boost::none ) 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 files:
• /Users/dellaert/git/github/gtsam/geometry/Rot3.h
• /Users/dellaert/git/github/gtsam/geometry/Rot3.cpp
• /Users/dellaert/git/github/gtsam/geometry/Rot3M.cpp