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gtsam/gtsam/geometry/Rot3.h

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/* ----------------------------------------------------------------------------
* GTSAM Copyright 2010, Georgia Tech Research Corporation,
* Atlanta, Georgia 30332-0415
* All Rights Reserved
* Authors: Frank Dellaert, et al. (see THANKS for the full author list)
* See LICENSE for the license information
* -------------------------------------------------------------------------- */
/**
* @file Rot3.h
* @brief 3D rotation represented as a rotation matrix or quaternion
* @author Alireza Fathi
* @author Christian Potthast
* @author Frank Dellaert
* @author Richard Roberts
*/
// \callgraph
#pragma once
#include <gtsam/config.h> // Get GTSAM_USE_QUATERNIONS macro
// You can override the default coordinate mode using this flag
#ifndef ROT3_DEFAULT_COORDINATES_MODE
#ifdef GTSAM_USE_QUATERNIONS
// Exponential map is very cheap for quaternions
#define ROT3_DEFAULT_COORDINATES_MODE Rot3::EXPMAP
#else
// If user doesn't require GTSAM_ROT3_EXPMAP in cmake when building
#ifndef GTSAM_ROT3_EXPMAP
// For rotation matrices, the Cayley transform is a fast retract alternative
#define ROT3_DEFAULT_COORDINATES_MODE Rot3::CAYLEY
#else
#define ROT3_DEFAULT_COORDINATES_MODE Rot3::EXPMAP
#endif
#endif
#endif
#include <gtsam/base/DerivedValue.h>
#include <gtsam/base/Matrix.h>
#include <gtsam/geometry/Point3.h>
#include <gtsam/geometry/Unit3.h>
namespace gtsam {
/// Typedef to an Eigen Quaternion<double>, we disable alignment because
/// geometry objects are stored in boost pool allocators, in Values
/// containers, and and these pool allocators do not support alignment.
typedef Eigen::Quaternion<double, Eigen::DontAlign> Quaternion;
/**
* @brief 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.
* @addtogroup geometry
* \nosubgrouping
*/
class GTSAM_EXPORT Rot3 : public DerivedValue<Rot3> {
public:
static const size_t dimension = 3;
private:
#ifdef GTSAM_USE_QUATERNIONS
/** Internal Eigen Quaternion */
Quaternion quaternion_;
#else
Matrix3 rot_;
#endif
public:
/// @name Constructors and named constructors
/// @{
/** default constructor, unit rotation */
Rot3();
/**
* Constructor from *columns*
* @param r1 X-axis of rotated frame
* @param r2 Y-axis of rotated frame
* @param r3 Z-axis of rotated frame
*/
Rot3(const Point3& col1, const Point3& col2, const Point3& col3);
/** constructor from a rotation matrix, as doubles in *row-major* order !!! */
Rot3(double R11, double R12, double R13,
double R21, double R22, double R23,
double R31, double R32, double R33);
/** constructor from a rotation matrix */
Rot3(const Matrix3& R);
/** constructor from a rotation matrix */
Rot3(const Matrix& R);
/** Constructor from a quaternion. This can also be called using a plain
* Vector, due to implicit conversion from Vector to Quaternion
* @param q The quaternion
*/
Rot3(const Quaternion& q);
/// Random, generates a random axis, then random angle \in [-p,pi]
static Rot3 Random(boost::mt19937 & rng);
/** Virtual destructor */
virtual ~Rot3() {}
/* Static member function to generate some well known rotations */
/// Rotation around X axis as in http://en.wikipedia.org/wiki/Rotation_matrix, counterclockwise when looking from unchanging axis.
static Rot3 Rx(double t);
/// Rotation around Y axis as in http://en.wikipedia.org/wiki/Rotation_matrix, counterclockwise when looking from unchanging axis.
static Rot3 Ry(double t);
/// Rotation around Z axis as in http://en.wikipedia.org/wiki/Rotation_matrix, counterclockwise when looking from unchanging axis.
static Rot3 Rz(double t);
/// 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(double x, double y, double z);
/// Rotations around Z, Y, then X axes as in http://en.wikipedia.org/wiki/Rotation_matrix, counterclockwise when looking from unchanging axis.
inline static Rot3 RzRyRx(const Vector& xyz) {
assert(xyz.size() == 3);
return RzRyRx(xyz(0), xyz(1), xyz(2));
}
/// Positive yaw is to right (as in aircraft heading). See ypr
static Rot3 yaw (double t) { return Rz(t); }
/// Positive pitch is up (increasing aircraft altitude).See ypr
static Rot3 pitch(double t) { return Ry(t); }
//// Positive roll is to right (increasing yaw in aircraft).
static Rot3 roll (double t) { return Rx(t); }
/**
* Returns rotation nRb from body to nav frame.
* 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.
* Assumes vehicle coordinate frame X forward, Y right, Z down.
*/
static Rot3 ypr (double y, double p, double r) { return RzRyRx(r,p,y);}
/** Create from Quaternion coefficients */
static Rot3 quaternion(double w, double x, double y, double z) {
Quaternion q(w, x, y, z);
return Rot3(q);
}
/**
* Rodriguez' formula to compute an incremental rotation matrix
* @param w is the rotation axis, unit length
* @param theta rotation angle
* @return incremental rotation matrix
*/
static Rot3 rodriguez(const Vector& w, double theta);
/**
* Rodriguez' formula to compute an incremental rotation matrix
* @param w is the rotation axis, unit length
* @param theta rotation angle
* @return incremental rotation matrix
*/
static Rot3 rodriguez(const Point3& w, double theta);
/**
* Rodriguez' formula to compute an incremental rotation matrix
* @param w is the rotation axis
* @param theta rotation angle
* @return incremental rotation matrix
*/
static Rot3 rodriguez(const Unit3& w, double theta);
/**
* Rodriguez' formula to compute an incremental rotation matrix
* @param v a vector of incremental roll,pitch,yaw
* @return incremental rotation matrix
*/
static Rot3 rodriguez(const Vector& v);
/**
* Rodriguez' formula to compute an incremental rotation matrix
* @param wx Incremental roll (about X)
* @param wy Incremental pitch (about Y)
* @param wz Incremental yaw (about Z)
* @return incremental rotation matrix
*/
static Rot3 rodriguez(double wx, double wy, double wz)
{ return rodriguez((Vector(3) << wx, wy, wz));}
/// @}
/// @name Testable
/// @{
/** print */
void print(const std::string& s="R") const;
/** equals with an tolerance */
bool equals(const Rot3& p, double tol = 1e-9) const;
/// @}
/// @name Group
/// @{
/// identity rotation for group operation
inline static Rot3 identity() {
return Rot3();
}
/// derivative of inverse rotation R^T s.t. inverse(R)*R = identity
Rot3 inverse(boost::optional<Matrix&> H1=boost::none) const;
/// Compose two rotations i.e., R= (*this) * R2
Rot3 compose(const Rot3& R2,
boost::optional<Matrix&> H1=boost::none, boost::optional<Matrix&> H2=boost::none) const;
/** compose two rotations */
Rot3 operator*(const Rot3& R2) const;
/**
* Conjugation: given a rotation acting in frame B, compute rotation c1Rc2 acting in a frame C
* @param cRb rotation from B frame to C frame
* @return c1Rc2 = cRb * b1Rb2 * cRb'
*/
Rot3 conjugate(const Rot3& cRb) const {
// TODO: do more efficiently by using Eigen or quaternion properties
return cRb * (*this) * cRb.inverse();
}
/**
* Return relative rotation D s.t. R2=D*R1, i.e. D=R2*R1'
*/
Rot3 between(const Rot3& R2,
boost::optional<Matrix&> H1=boost::none,
boost::optional<Matrix&> H2=boost::none) const;
/// @}
/// @name Manifold
/// @{
/// dimension of the variable - used to autodetect sizes
static size_t Dim() { return dimension; }
/// return dimensionality of tangent space, DOF = 3
size_t dim() const { return dimension; }
/**
* 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.
*/
enum CoordinatesMode {
EXPMAP, ///< Use the Lie group exponential map to retract
#ifndef GTSAM_USE_QUATERNIONS
CAYLEY, ///< Retract and localCoordinates using the Cayley transform.
SLOW_CAYLEY ///< Slow matrix implementation of Cayley transform (for tests only).
#endif
};
#ifndef GTSAM_USE_QUATERNIONS
/// Retraction from R^3 to Rot3 manifold using the Cayley transform
Rot3 retractCayley(const Vector& omega) const;
#endif
/// Retraction from R^3 \f$ [R_x,R_y,R_z] \f$ to Rot3 manifold neighborhood around current rotation
Rot3 retract(const Vector& omega, Rot3::CoordinatesMode mode = ROT3_DEFAULT_COORDINATES_MODE) const;
/// Returns local retract coordinates \f$ [R_x,R_y,R_z] \f$ in neighborhood around current rotation
Vector3 localCoordinates(const Rot3& t2, Rot3::CoordinatesMode mode = ROT3_DEFAULT_COORDINATES_MODE) const;
/// @}
/// @name Lie Group
/// @{
/**
* Exponential map at identity - create a rotation from canonical coordinates
* \f$ [R_x,R_y,R_z] \f$ using Rodriguez' formula
*/
static Rot3 Expmap(const Vector& v) {
if(zero(v)) return Rot3();
else return rodriguez(v);
}
/**
* Log map at identity - return the canonical coordinates \f$ [R_x,R_y,R_z] \f$ of this rotation
*/
static Vector3 Logmap(const Rot3& R);
/// Left-trivialized derivative of the exponential map
static Matrix3 dexpL(const Vector3& v);
/// Left-trivialized derivative inverse of the exponential map
static Matrix3 dexpInvL(const Vector3& v);
/**
* Right Jacobian for Exponential map in SO(3) - equation (10.86) and following equations in
* G.S. Chirikjian, "Stochastic Models, Information Theory, and Lie Groups", Volume 2, 2008.
*/
static Matrix3 rightJacobianExpMapSO3(const Vector3& x);
/** Right Jacobian for Log map in SO(3) - equation (10.86) and following equations in
* G.S. Chirikjian, "Stochastic Models, Information Theory, and Lie Groups", Volume 2, 2008.
*/
static Matrix3 rightJacobianExpMapSO3inverse(const Vector3& x);
/// @}
/// @name Group Action on Point3
/// @{
/**
* rotate point from rotated coordinate frame to world \f$ p^w = R_c^w p^c \f$
*/
Point3 rotate(const Point3& p, boost::optional<Matrix&> H1 = boost::none,
boost::optional<Matrix&> H2 = boost::none) const;
/// rotate point from rotated coordinate frame to world = R*p
Point3 operator*(const Point3& p) const;
/**
* rotate point from world to rotated frame \f$ p^c = (R_c^w)^T p^w \f$
*/
Point3 unrotate(const Point3& p, boost::optional<Matrix&> H1 = boost::none,
boost::optional<Matrix&> H2 = boost::none) const;
/// @}
/// @name Group Action on Unit3
/// @{
/// rotate 3D direction from rotated coordinate frame to world frame
Unit3 rotate(const Unit3& p, boost::optional<Matrix&> HR = boost::none,
boost::optional<Matrix&> Hp = boost::none) const;
/// unrotate 3D direction from world frame to rotated coordinate frame
Unit3 unrotate(const Unit3& p, boost::optional<Matrix&> HR = boost::none,
boost::optional<Matrix&> Hp = boost::none) const;
/// rotate 3D direction from rotated coordinate frame to world frame
Unit3 operator*(const Unit3& p) const;
/// @}
/// @name Standard Interface
/// @{
/** return 3*3 rotation matrix */
Matrix3 matrix() const;
/** return 3*3 transpose (inverse) rotation matrix */
Matrix3 transpose() const;
/// @deprecated, this is base 1, and was just confusing
Point3 column(int index) const;
Point3 r1() const; ///< first column
Point3 r2() const; ///< second column
Point3 r3() const; ///< third column
/**
* Use RQ to calculate xyz angle representation
* @return a vector containing x,y,z s.t. R = Rot3::RzRyRx(x,y,z)
*/
Vector3 xyz() const;
/**
* Use RQ to calculate yaw-pitch-roll angle representation
* @return a vector containing ypr s.t. R = Rot3::ypr(y,p,r)
*/
Vector3 ypr() const;
/**
* Use RQ to calculate roll-pitch-yaw angle representation
* @return a vector containing ypr s.t. R = Rot3::ypr(y,p,r)
*/
Vector3 rpy() 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
*/
inline double roll() const { return ypr()(2); }
/**
* 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
*/
inline double pitch() const { return ypr()(1); }
/**
* 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
*/
inline double yaw() const { return ypr()(0); }
/// @}
/// @name Advanced Interface
/// @{
/** Compute the quaternion representation of this rotation.
* @return The quaternion
*/
Quaternion toQuaternion() const;
/**
* Converts to a generic matrix to allow for use with matlab
* In format: w x y z
*/
Vector quaternion() const;
/// Output stream operator
GTSAM_EXPORT friend std::ostream &operator<<(std::ostream &os, const Rot3& p);
private:
/** Serialization function */
friend class boost::serialization::access;
template<class ARCHIVE>
void serialize(ARCHIVE & ar, const unsigned int version)
{
ar & boost::serialization::make_nvp("Rot3",
boost::serialization::base_object<Value>(*this));
#ifndef GTSAM_USE_QUATERNIONS
ar & boost::serialization::make_nvp("rot11", rot_(0,0));
ar & boost::serialization::make_nvp("rot12", rot_(0,1));
ar & boost::serialization::make_nvp("rot13", rot_(0,2));
ar & boost::serialization::make_nvp("rot21", rot_(1,0));
ar & boost::serialization::make_nvp("rot22", rot_(1,1));
ar & boost::serialization::make_nvp("rot23", rot_(1,2));
ar & boost::serialization::make_nvp("rot31", rot_(2,0));
ar & boost::serialization::make_nvp("rot32", rot_(2,1));
ar & boost::serialization::make_nvp("rot33", rot_(2,2));
#else
ar & boost::serialization::make_nvp("w", quaternion_.w());
ar & boost::serialization::make_nvp("x", quaternion_.x());
ar & boost::serialization::make_nvp("y", quaternion_.y());
ar & boost::serialization::make_nvp("z", quaternion_.z());
#endif
}
};
/// @}
/**
* [RQ] receives a 3 by 3 matrix and returns an upper triangular matrix R
* and 3 rotation angles corresponding to the rotation matrix Q=Qz'*Qy'*Qx'
* such that A = R*Q = R*Qz'*Qy'*Qx'. When A is a rotation matrix, R will
* be the identity and Q is a yaw-pitch-roll decomposition of A.
* The implementation uses Givens rotations and is based on Hartley-Zisserman.
* @param A 3 by 3 matrix A=RQ
* @return an upper triangular matrix R
* @return a vector [thetax, thetay, thetaz] in radians.
*/
GTSAM_EXPORT std::pair<Matrix3,Vector3> RQ(const Matrix3& A);
}
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