401 lines
12 KiB
C++
401 lines
12 KiB
C++
/* ----------------------------------------------------------------------------
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* GTSAM Copyright 2010-2019, Georgia Tech Research Corporation,
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* Atlanta, Georgia 30332-0415
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* All Rights Reserved
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* Authors: Frank Dellaert, et al. (see THANKS for the full author list)
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* See LICENSE for the license information
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* -------------------------------------------------------------------------- */
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/**
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* @file SOn.h
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* @brief N*N matrix representation of SO(N). N can be Eigen::Dynamic
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* @author Frank Dellaert
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* @date March 2019
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*/
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#pragma once
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#include <gtsam/base/Lie.h>
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#include <gtsam/base/Manifold.h>
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#include <gtsam/base/make_shared.h>
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#include <gtsam/dllexport.h>
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#include <Eigen/Core>
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#include <boost/serialization/nvp.hpp>
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#include <iostream> // TODO(frank): how to avoid?
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#include <string>
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#include <type_traits>
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#include <vector>
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#include <random>
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namespace gtsam {
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namespace internal {
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/// Calculate dimensionality of SO<N> manifold, or return Dynamic if so
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constexpr int DimensionSO(int N) {
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return (N < 0) ? Eigen::Dynamic : N * (N - 1) / 2;
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}
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// Calculate N^2 at compile time, or return Dynamic if so
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constexpr int NSquaredSO(int N) { return (N < 0) ? Eigen::Dynamic : N * N; }
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} // namespace internal
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/**
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* Manifold of special orthogonal rotation matrices SO<N>.
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* Template paramater N can be a fixed integer or can be Eigen::Dynamic
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*/
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template <int N>
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class SO : public LieGroup<SO<N>, internal::DimensionSO(N)> {
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public:
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enum { dimension = internal::DimensionSO(N) };
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using MatrixNN = Eigen::Matrix<double, N, N>;
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using VectorN2 = Eigen::Matrix<double, internal::NSquaredSO(N), 1>;
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using MatrixDD = Eigen::Matrix<double, dimension, dimension>;
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GTSAM_MAKE_ALIGNED_OPERATOR_NEW_IF(true)
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protected:
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MatrixNN matrix_; ///< Rotation matrix
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// enable_if_t aliases, used to specialize constructors/methods, see
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// https://www.fluentcpp.com/2018/05/18/make-sfinae-pretty-2-hidden-beauty-sfinae/
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template <int N_>
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using IsDynamic = typename std::enable_if<N_ == Eigen::Dynamic, void>::type;
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template <int N_>
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using IsFixed = typename std::enable_if<N_ >= 2, void>::type;
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template <int N_>
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using IsSO3 = typename std::enable_if<N_ == 3, void>::type;
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public:
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/// @name Constructors
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/// @{
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/// Construct SO<N> identity for N >= 2
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template <int N_ = N, typename = IsFixed<N_>>
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SO() : matrix_(MatrixNN::Identity()) {}
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/// Construct SO<N> identity for N == Eigen::Dynamic
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template <int N_ = N, typename = IsDynamic<N_>>
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explicit SO(size_t n = 0) {
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// We allow for n=0 as the default constructor, needed for serialization,
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// wrappers etc.
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matrix_ = Eigen::MatrixXd::Identity(n, n);
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}
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/// Constructor from Eigen Matrix, dynamic version
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template <typename Derived>
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explicit SO(const Eigen::MatrixBase<Derived>& R) : matrix_(R.eval()) {}
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/// Named constructor from Eigen Matrix
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template <typename Derived>
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static SO FromMatrix(const Eigen::MatrixBase<Derived>& R) {
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return SO(R);
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}
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/// Named constructor from lower dimensional matrix
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template <typename Derived, int N_ = N, typename = IsDynamic<N_>>
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static SO Lift(size_t n, const Eigen::MatrixBase<Derived> &R) {
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Matrix Q = Matrix::Identity(n, n);
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const int p = R.rows();
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assert(p >= 0 && p <= static_cast<int>(n) && R.cols() == p);
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Q.topLeftCorner(p, p) = R;
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return SO(Q);
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}
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/// Construct dynamic SO(n) from Fixed SO<M>
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template <int M, int N_ = N, typename = IsDynamic<N_>>
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explicit SO(const SO<M>& R) : matrix_(R.matrix()) {}
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/// Constructor from AngleAxisd
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template <int N_ = N, typename = IsSO3<N_>>
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explicit SO(const Eigen::AngleAxisd& angleAxis) : matrix_(angleAxis) {}
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/// Constructor from axis and angle. Only defined for SO3
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static SO AxisAngle(const Vector3& axis, double theta);
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/// Named constructor that finds SO(n) matrix closest to M in Frobenius norm,
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/// currently only defined for SO3.
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static SO ClosestTo(const MatrixNN& M);
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/// Named constructor that finds chordal mean
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/// \f$ mu = argmin_R \sum sqr(|R-R_i|_F) \f$,
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/// currently only defined for SO3.
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static SO ChordalMean(const std::vector<SO>& rotations);
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/// Random SO(n) element (no big claims about uniformity). SO(3) is specialized in SO3.cpp
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template <int N_ = N, typename = IsDynamic<N_>>
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static SO Random(std::mt19937& rng, size_t n = 0) {
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if (n == 0) throw std::runtime_error("SO: Dimensionality not known.");
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// TODO(frank): this might need to be re-thought
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static std::uniform_real_distribution<double> randomAngle(-M_PI, M_PI);
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const size_t d = SO::Dimension(n);
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Vector xi(d);
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for (size_t j = 0; j < d; j++) {
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xi(j) = randomAngle(rng);
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}
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return SO::Retract(xi);
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}
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/// Random SO(N) element (no big claims about uniformity)
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template <int N_ = N, typename = IsFixed<N_>>
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static SO Random(std::mt19937& rng) {
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// By default, use dynamic implementation above. Specialized for SO(3).
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return SO(SO<Eigen::Dynamic>::Random(rng, N).matrix());
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}
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/// @}
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/// @name Standard methods
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/// @{
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/// Return matrix
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const MatrixNN& matrix() const { return matrix_; }
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size_t rows() const { return matrix_.rows(); }
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size_t cols() const { return matrix_.cols(); }
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/// @}
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/// @name Testable
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/// @{
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void print(const std::string& s = std::string()) const;
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bool equals(const SO& other, double tol) const {
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return equal_with_abs_tol(matrix_, other.matrix_, tol);
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}
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/// @}
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/// @name Group
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/// @{
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/// Multiplication
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SO operator*(const SO& other) const {
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assert(dim() == other.dim());
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return SO(matrix_ * other.matrix_);
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}
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/// SO<N> identity for N >= 2
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template <int N_ = N, typename = IsFixed<N_>>
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static SO Identity() {
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return SO();
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}
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/// SO<N> identity for N == Eigen::Dynamic
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template <int N_ = N, typename = IsDynamic<N_>>
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static SO Identity(size_t n = 0) {
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return SO(n);
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}
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/// inverse of a rotation = transpose
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SO inverse() const { return SO(matrix_.transpose()); }
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/// @}
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/// @name Manifold
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/// @{
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using TangentVector = Eigen::Matrix<double, dimension, 1>;
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using ChartJacobian = OptionalJacobian<dimension, dimension>;
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/// Return compile-time dimensionality: fixed size N or Eigen::Dynamic
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static int Dim() { return dimension; }
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// Calculate manifold dimensionality for SO(n).
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// Available as dimension or Dim() for fixed N.
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static size_t Dimension(size_t n) { return n * (n - 1) / 2; }
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// Calculate ambient dimension n from manifold dimensionality d.
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static size_t AmbientDim(size_t d) { return (1 + std::sqrt(1 + 8 * d)) / 2; }
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// Calculate run-time dimensionality of manifold.
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// Available as dimension or Dim() for fixed N.
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size_t dim() const { return Dimension(static_cast<size_t>(matrix_.rows())); }
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/**
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* Hat operator creates Lie algebra element corresponding to d-vector, where d
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* is the dimensionality of the manifold. This function is implemented
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* recursively, and the d-vector is assumed to laid out such that the last
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* element corresponds to so(2), the last 3 to so(3), the last 6 to so(4)
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* etc... For example, the vector-space isomorphic to so(5) is laid out as:
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* a b c d | u v w | x y | z
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* where the latter elements correspond to "telescoping" sub-algebras:
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* 0 -z y w -d
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* z 0 -x -v c
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* -y x 0 u -b
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* -w v -u 0 a
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* d -c b -a 0
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* This scheme behaves exactly as expected for SO(2) and SO(3).
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*/
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static MatrixNN Hat(const TangentVector& xi);
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/// In-place version of Hat (see details there), implements recursion.
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static void Hat(const Vector &xi, Eigen::Ref<MatrixNN> X);
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/// Inverse of Hat. See note about xi element order in Hat.
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static TangentVector Vee(const MatrixNN& X);
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// Chart at origin
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struct ChartAtOrigin {
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/**
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* Retract uses Cayley map. See note about xi element order in Hat.
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* Deafault implementation has no Jacobian implemented
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*/
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static SO Retract(const TangentVector& xi, ChartJacobian H = boost::none);
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/**
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* Inverse of Retract. See note about xi element order in Hat.
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*/
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static TangentVector Local(const SO& R, ChartJacobian H = boost::none);
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};
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// Return dynamic identity DxD Jacobian for given SO(n)
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template <int N_ = N, typename = IsDynamic<N_>>
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static MatrixDD IdentityJacobian(size_t n) {
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const size_t d = Dimension(n);
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return MatrixDD::Identity(d, d);
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}
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/// @}
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/// @name Lie Group
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/// @{
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/// Adjoint map
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MatrixDD AdjointMap() const;
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/**
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* Exponential map at identity - create a rotation from canonical coordinates
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*/
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static SO Expmap(const TangentVector& omega, ChartJacobian H = boost::none);
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/// Derivative of Expmap, currently only defined for SO3
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static MatrixDD ExpmapDerivative(const TangentVector& omega);
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/**
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* Log map at identity - returns the canonical coordinates of this rotation
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*/
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static TangentVector Logmap(const SO& R, ChartJacobian H = boost::none);
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/// Derivative of Logmap, currently only defined for SO3
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static MatrixDD LogmapDerivative(const TangentVector& omega);
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// inverse with optional derivative
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using LieGroup<SO<N>, internal::DimensionSO(N)>::inverse;
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/// @}
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/// @name Other methods
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/// @{
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/**
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* Return vectorized rotation matrix in column order.
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* Will use dynamic matrices as intermediate results, but returns a fixed size
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* X and fixed-size Jacobian if dimension is known at compile time.
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* */
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VectorN2 vec(OptionalJacobian<internal::NSquaredSO(N), dimension> H =
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boost::none) const;
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/// Calculate N^2 x dim matrix of vectorized Lie algebra generators for SO(N)
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template <int N_ = N, typename = IsFixed<N_>>
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static Matrix VectorizedGenerators() {
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constexpr size_t N2 = static_cast<size_t>(N * N);
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Eigen::Matrix<double, N2, dimension> G;
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for (size_t j = 0; j < dimension; j++) {
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const auto X = Hat(Vector::Unit(dimension, j));
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G.col(j) = Eigen::Map<const VectorN2>(X.data());
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}
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return G;
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}
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/// Calculate n^2 x dim matrix of vectorized Lie algebra generators for SO(n)
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template <int N_ = N, typename = IsDynamic<N_>>
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static Matrix VectorizedGenerators(size_t n = 0) {
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const size_t n2 = n * n, dim = Dimension(n);
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Matrix G(n2, dim);
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for (size_t j = 0; j < dim; j++) {
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const auto X = Hat(Vector::Unit(dim, j));
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G.col(j) = Eigen::Map<const Matrix>(X.data(), n2, 1);
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}
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return G;
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}
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/// @{
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/// @name Serialization
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/// @{
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template <class Archive>
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friend void save(Archive&, SO&, const unsigned int);
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template <class Archive>
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friend void load(Archive&, SO&, const unsigned int);
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template <class Archive>
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friend void serialize(Archive&, SO&, const unsigned int);
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friend class boost::serialization::access;
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friend class Rot3; // for serialize
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/// @}
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};
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using SOn = SO<Eigen::Dynamic>;
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/*
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* Specialize dynamic Hat and Vee, because recursion depends on dynamic nature.
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* The definition is in SOn.cpp. Fixed-size SO3 and SO4 have their own version,
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* and implementation for other fixed N is in SOn-inl.h.
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*/
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template <>
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GTSAM_EXPORT
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Matrix SOn::Hat(const Vector& xi);
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template <>
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GTSAM_EXPORT
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Vector SOn::Vee(const Matrix& X);
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/*
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* Specialize dynamic compose and between, because the derivative is unknowable
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* by the LieGroup implementations, who return a fixed-size matrix for H2.
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*/
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using DynamicJacobian = OptionalJacobian<Eigen::Dynamic, Eigen::Dynamic>;
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template <>
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GTSAM_EXPORT
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SOn LieGroup<SOn, Eigen::Dynamic>::compose(const SOn& g, DynamicJacobian H1,
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DynamicJacobian H2) const;
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template <>
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GTSAM_EXPORT
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SOn LieGroup<SOn, Eigen::Dynamic>::between(const SOn& g, DynamicJacobian H1,
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DynamicJacobian H2) const;
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/*
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* Specialize dynamic vec.
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*/
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template <>
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GTSAM_EXPORT
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typename SOn::VectorN2 SOn::vec(DynamicJacobian H) const;
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/** Serialization function */
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template<class Archive>
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void serialize(
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Archive& ar, SOn& Q,
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const unsigned int file_version
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) {
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Matrix& M = Q.matrix_;
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ar& BOOST_SERIALIZATION_NVP(M);
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}
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/*
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* Define the traits. internal::LieGroup provides both Lie group and Testable
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*/
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template <int N>
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struct traits<SO<N>> : public internal::LieGroup<SO<N>> {};
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template <int N>
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struct traits<const SO<N>> : public internal::LieGroup<SO<N>> {};
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} // namespace gtsam
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#include "SOn-inl.h"
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