Lie.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
 
 * -------------------------------------------------------------------------- */
 
#pragma once
 
#include <gtsam/base/Manifold.h>
#include <gtsam/base/Group.h>
 
namespace gtsam {
 
template <class Class, int N>
struct LieGroup {
 
  enum { dimension = N };
  typedef OptionalJacobian<N, N> ChartJacobian;
  typedef Eigen::Matrix<double, N, N> Jacobian;
  typedef Eigen::Matrix<double, N, 1> TangentVector;
 
  const Class & derived() const {
    return static_cast<const Class&>(*this);
  }
 
  Class compose(const Class& g) const {
    return derived() * g;
  }
 
  Class between(const Class& g) const {
    return derived().inverse() * g;
  }
 
  Class compose(const Class& g, ChartJacobian H1,
      ChartJacobian H2 = boost::none) const {
    if (H1) *H1 = g.inverse().AdjointMap();
    if (H2) *H2 = Eigen::Matrix<double, N, N>::Identity();
    return derived() * g;
  }
 
  Class between(const Class& g, ChartJacobian H1,
      ChartJacobian H2 = boost::none) const {
    Class result = derived().inverse() * g;
    if (H1) *H1 = - result.inverse().AdjointMap();
    if (H2) *H2 = Eigen::Matrix<double, N, N>::Identity();
    return result;
  }
 
  Class inverse(ChartJacobian H) const {
    if (H) *H = - derived().AdjointMap();
    return derived().inverse();
  }
 
  Class expmap(const TangentVector& v) const {
    return compose(Class::Expmap(v));
  }
 
  TangentVector logmap(const Class& g) const {
    return Class::Logmap(between(g));
  }
 
  Class expmap(const TangentVector& v, //
      ChartJacobian H1, ChartJacobian H2 = boost::none) const {
    Jacobian D_g_v;
    Class g = Class::Expmap(v,H2 ? &D_g_v : 0);
    Class h = compose(g); // derivatives inlined below
    if (H1) *H1 = g.inverse().AdjointMap();
    if (H2) *H2 = D_g_v;
    return h;
  }
 
  TangentVector logmap(const Class& g, //
      ChartJacobian H1, ChartJacobian H2 = boost::none) const {
    Class h = between(g); // derivatives inlined below
    Jacobian D_v_h;
    TangentVector v = Class::Logmap(h, (H1 || H2) ? &D_v_h : 0);
    if (H1) *H1 = - D_v_h * h.inverse().AdjointMap();
    if (H2) *H2 = D_v_h;
    return v;
  }
 
  static Class Retract(const TangentVector& v) {
    return Class::ChartAtOrigin::Retract(v);
  }
 
  static TangentVector LocalCoordinates(const Class& g) {
    return Class::ChartAtOrigin::Local(g);
  }
 
  static Class Retract(const TangentVector& v, ChartJacobian H) {
    return Class::ChartAtOrigin::Retract(v,H);
  }
 
  static TangentVector LocalCoordinates(const Class& g, ChartJacobian H) {
    return Class::ChartAtOrigin::Local(g,H);
  }
 
  Class retract(const TangentVector& v) const {
    return compose(Class::ChartAtOrigin::Retract(v));
  }
 
  TangentVector localCoordinates(const Class& g) const {
    return Class::ChartAtOrigin::Local(between(g));
  }
 
  Class retract(const TangentVector& v, //
      ChartJacobian H1, ChartJacobian H2 = boost::none) const {
    Jacobian D_g_v;
    Class g = Class::ChartAtOrigin::Retract(v, H2 ? &D_g_v : 0);
    Class h = compose(g); // derivatives inlined below
    if (H1) *H1 = g.inverse().AdjointMap();
    if (H2) *H2 = D_g_v;
    return h;
  }
 
  TangentVector localCoordinates(const Class& g, //
      ChartJacobian H1, ChartJacobian H2 = boost::none) const {
    Class h = between(g); // derivatives inlined below
    Jacobian D_v_h;
    TangentVector v = Class::ChartAtOrigin::Local(h, (H1 || H2) ? &D_v_h : 0);
    if (H1) *H1 = - D_v_h * h.inverse().AdjointMap();
    if (H2) *H2 = D_v_h;
    return v;
  }
};
 
struct lie_group_tag: public manifold_tag, public group_tag {};
 
namespace internal {
 
template<class Class>
struct LieGroupTraits: GetDimensionImpl<Class, Class::dimension> {
  typedef lie_group_tag structure_category;
 
  typedef multiplicative_group_tag group_flavor;
  static Class Identity() { return Class::identity();}
 
  typedef Class ManifoldType;
  enum { dimension = Class::dimension };
  typedef Eigen::Matrix<double, dimension, 1> TangentVector;
  typedef OptionalJacobian<dimension, dimension> ChartJacobian;
 
  static TangentVector Local(const Class& origin, const Class& other,
      ChartJacobian Horigin = boost::none, ChartJacobian Hother = boost::none) {
    return origin.localCoordinates(other, Horigin, Hother);
  }
 
  static Class Retract(const Class& origin, const TangentVector& v,
      ChartJacobian Horigin = boost::none, ChartJacobian Hv = boost::none) {
    return origin.retract(v, Horigin, Hv);
  }
 
  static TangentVector Logmap(const Class& m, ChartJacobian Hm = boost::none) {
    return Class::Logmap(m, Hm);
  }
 
  static Class Expmap(const TangentVector& v, ChartJacobian Hv = boost::none) {
    return Class::Expmap(v, Hv);
  }
 
  static Class Compose(const Class& m1, const Class& m2, //
      ChartJacobian H1 = boost::none, ChartJacobian H2 = boost::none) {
    return m1.compose(m2, H1, H2);
  }
 
  static Class Between(const Class& m1, const Class& m2, //
      ChartJacobian H1 = boost::none, ChartJacobian H2 = boost::none) {
    return m1.between(m2, H1, H2);
  }
 
  static Class Inverse(const Class& m, //
      ChartJacobian H = boost::none) {
    return m.inverse(H);
  }
};
 
template<class Class> struct LieGroup: LieGroupTraits<Class>, Testable<Class> {};
 
} // \ namepsace internal
 
template<class Class>
inline Class between_default(const Class& l1, const Class& l2) {
  return l1.inverse().compose(l2);
}
 
template<class Class>
inline Vector logmap_default(const Class& l0, const Class& lp) {
  return Class::Logmap(l0.between(lp));
}
 
template<class Class>
inline Class expmap_default(const Class& t, const Vector& d) {
  return t.compose(Class::Expmap(d));
}
 
template<typename T>
class IsLieGroup: public IsGroup<T>, public IsManifold<T> {
public:
  typedef typename traits<T>::structure_category structure_category_tag;
  typedef typename traits<T>::ManifoldType ManifoldType;
  typedef typename traits<T>::TangentVector TangentVector;
  typedef typename traits<T>::ChartJacobian ChartJacobian;
 
  BOOST_CONCEPT_USAGE(IsLieGroup) {
    BOOST_STATIC_ASSERT_MSG(
        (boost::is_base_of<lie_group_tag, structure_category_tag>::value),
        "This type's trait does not assert it is a Lie group (or derived)");
 
    // group opertations with Jacobians
    g = traits<T>::Compose(g, h, Hg, Hh);
    g = traits<T>::Between(g, h, Hg, Hh);
    g = traits<T>::Inverse(g, Hg);
    // log and exp map without Jacobians
    g = traits<T>::Expmap(v);
    v = traits<T>::Logmap(g);
    // log and exponential map with Jacobians
    g = traits<T>::Expmap(v, Hg);
    v = traits<T>::Logmap(g, Hg);
  }
private:
  T g, h;
  TangentVector v;
  ChartJacobian Hg, Hh;
};
 
template<class T>
T BCH(const T& X, const T& Y) {
  static const double _2 = 1. / 2., _12 = 1. / 12., _24 = 1. / 24.;
  T X_Y = bracket(X, Y);
  return T(X + Y + _2 * X_Y + _12 * bracket(X - Y, X_Y) - _24 * bracket(Y, bracket(X, X_Y)));
}
 
template <class T> Matrix wedge(const Vector& x);
 
template <class T>
T expm(const Vector& x, int K=7) {
  Matrix xhat = wedge<T>(x);
  return T(expm(xhat,K));
}
 
template <typename T>
T interpolate(const T& X, const T& Y, double t) {
  assert(t >= 0 && t <= 1);
  return traits<T>::Compose(X, traits<T>::Expmap(t * traits<T>::Logmap(traits<T>::Between(X, Y))));
}
 
template<class T>
class TransformCovariance
{
private:
  typename T::Jacobian adjointMap_;
public:
  explicit TransformCovariance(const T &X) : adjointMap_{X.AdjointMap()} {}
  typename T::Jacobian operator()(const typename T::Jacobian &covariance)
  { return adjointMap_ * covariance * adjointMap_.transpose(); }
};
 
} // namespace gtsam
 
#define GTSAM_CONCEPT_LIE_INST(T) template class gtsam::IsLieGroup<T>;
#define GTSAM_CONCEPT_LIE_TYPE(T) typedef gtsam::IsLieGroup<T> _gtsam_IsLieGroup_##T;