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/Group.h>
#include <gtsam/base/Manifold.h>
 
#include <type_traits>
#include <utility>
 
namespace gtsam {
 
namespace internal {

template <class C, class TangentVector, class ChartJacobian>
using ExpmapWithJacobian = decltype(C::Expmap(
    std::declval<const TangentVector&>(), std::declval<ChartJacobian>()));

template <class C, class ChartJacobian>
using LogmapWithJacobian = decltype(
    C::Logmap(std::declval<const C&>(), std::declval<ChartJacobian>()));

constexpr int dimensionSum(int n, int m) {
  return n == Eigen::Dynamic || m == Eigen::Dynamic ? Eigen::Dynamic : n + m;
}

constexpr int dimensionProduct(int n, int m) {
  return n == Eigen::Dynamic || m == Eigen::Dynamic ? Eigen::Dynamic : n * m;
}

template <class Class, int N>
struct ChartAtIdentity {
  using TangentVector = Eigen::Matrix<double, N, 1>;
  using ChartJacobian = OptionalJacobian<N, N>;
 
  static Class Retract(const TangentVector& v) {
    if constexpr (N == Eigen::Dynamic) {
      return Class::Expmap(v);
    } else {
      return Class::Identity().retract(v);
    }
  }
 
  template <class C = Class,
            class = ExpmapWithJacobian<C, TangentVector, ChartJacobian>>
  static Class Retract(const TangentVector& v, ChartJacobian H) {
    if constexpr (N == Eigen::Dynamic) {
      return Class::Expmap(v, H);
    } else {
      return Class::Identity().retract(v, {}, H);
    }
  }
 
  static TangentVector Local(const Class& value) {
    if constexpr (N == Eigen::Dynamic) {
      return Class::Logmap(value);
    } else {
      return Class::Identity().localCoordinates(value);
    }
  }
 
  template <class C = Class,
            class = LogmapWithJacobian<C, ChartJacobian>>
  static TangentVector Local(const Class& value, ChartJacobian H) {
    if constexpr (N == Eigen::Dynamic) {
      return Class::Logmap(value, H);
    } else {
      return Class::Identity().localCoordinates(value, {}, H);
    }
  }
};
 
}  // namespace internal

template <class Class, int N>
struct LieGroup {
  inline constexpr static auto dimension = N;
  typedef OptionalJacobian<N, N> ChartJacobian;
  typedef Eigen::Matrix<double, N, N> Jacobian;
  typedef Eigen::Matrix<double, N, 1> TangentVector;

  static constexpr int Dim() { return N; }

  template <int M = N>
  std::enable_if_t<M != Eigen::Dynamic, int> dim() const {
    return N;
  }
 
  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 = {}) const {
    if (H1) *H1 = g.inverse().AdjointMap();
    if (H2) *H2 = identityMatrix();
    return derived() * g;
  }
 
  Class between(const Class& g, ChartJacobian H1, ChartJacobian H2 = {}) const {
    Class result = derived().inverse() * g;
    if (H1) *H1 = -result.inverse().AdjointMap();
    if (H2) *H2 = identityMatrix();
    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));
  }

  template <class C = Class,
            class = internal::ExpmapWithJacobian<C, TangentVector,
                                                 ChartJacobian>>
  Class expmap(const TangentVector& v,  //
               ChartJacobian H1, ChartJacobian H2 = {}) 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;
  }

  template <class C = Class,
            class = internal::LogmapWithJacobian<C, ChartJacobian>>
  TangentVector logmap(const Class& g,  //
                       ChartJacobian H1, ChartJacobian H2 = {}) 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);
  }

  template <class C = Class, class = decltype(C::ChartAtOrigin::Retract(
                                 std::declval<const TangentVector&>(),
                                 std::declval<ChartJacobian>()))>
  static Class Retract(const TangentVector& v, ChartJacobian H) {
    return Class::ChartAtOrigin::Retract(v, H);
  }

  template <class C = Class,
            class = decltype(C::ChartAtOrigin::Local(
                std::declval<const C&>(), std::declval<ChartJacobian>()))>
  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));
  }

  template <class C = Class, class = decltype(C::ChartAtOrigin::Retract(
                                 std::declval<const TangentVector&>(),
                                 std::declval<ChartJacobian>()))>
  Class retract(const TangentVector& v,  //
                ChartJacobian H1, ChartJacobian H2 = {}) 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;
  }

  template <class C = Class,
            class = decltype(C::ChartAtOrigin::Local(
                std::declval<const C&>(), std::declval<ChartJacobian>()))>
  TangentVector localCoordinates(const Class& g,  //
                                 ChartJacobian H1,
                                 ChartJacobian H2 = {}) 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;
  }
 
 private:
  // Helper to get identity matrix of correct size for static or dynamic N
  Jacobian identityMatrix() const {
    if constexpr (N == Eigen::Dynamic) {
      return Jacobian::Identity(derived().dim(), derived().dim());
    } else {
      return Jacobian::Identity();
    }
  }
};

struct lie_group_tag : public manifold_tag, public group_tag {};
 
namespace internal {

template <class Class>
struct LieGroupTraits : ManifoldTraits<Class> {
  using Manifold = ManifoldTraits<Class>;
  using structure_category = lie_group_tag;

  using group_flavor = multiplicative_group_tag;
  static Class Identity() { return Class::Identity(); }

  using typename Manifold::ChartJacobian;
  using typename Manifold::ManifoldType;
  using typename Manifold::TangentVector;
  inline constexpr static auto dimension = Manifold::dimension;
  using Jacobian = Eigen::Matrix<double, dimension, dimension>;

  static TangentVector Logmap(const Class& m) { return Class::Logmap(m); }
 
  template <class C = Class,
            class = LogmapWithJacobian<C, ChartJacobian>>
  static TangentVector Logmap(const Class& m, ChartJacobian Hm) {
    return Class::Logmap(m, Hm);
  }
 
  static Class Expmap(const TangentVector& v) { return Class::Expmap(v); }
 
  template <class C = Class,
            class = ExpmapWithJacobian<C, TangentVector, ChartJacobian>>
  static Class Expmap(const TangentVector& v, ChartJacobian Hv) {
    return Class::Expmap(v, Hv);
  }
 
  static Class Compose(const Class& m1, const Class& m2,  //
                       ChartJacobian H1 = {}, ChartJacobian H2 = {}) {
    return m1.compose(m2, H1, H2);
  }
 
  static Class Between(const Class& m1, const Class& m2,  //
                       ChartJacobian H1 = {}, ChartJacobian H2 = {}) {
    return m1.between(m2, H1, H2);
  }
 
  static Class Inverse(const Class& m,  //
                       ChartJacobian H = {}) {
    return m.inverse(H);
  }
 
  static Eigen::Matrix<double, dimension, dimension> AdjointMap(
      const Class& m) {
    // This assumes that the Class itself provides a member function
    // `AdjointMap()` For dynamically-sized types (dimension == Eigen::Dynamic),
    // m.AdjointMap() must return a gtsam::Matrix of the correct runtime
    // dimensions.
    return m.AdjointMap();
  }
};

template <class Class>
struct LieGroup : LieGroupTraits<Class>, Testable<Class> {};
 
}  // namespace 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:
  // Concept marker: allows checking IsLieGroup<T>::value in templates
  static constexpr bool value =
      std::is_base_of_v<lie_group_tag, typename traits<T>::structure_category>;
 
  using structure_category_tag = typename traits<T>::structure_category;
  using ManifoldType = typename traits<T>::ManifoldType;
  using TangentVector = typename traits<T>::TangentVector;
  using ChartJacobian = typename traits<T>::ChartJacobian;
 
  GTSAM_CONCEPT_USAGE(IsLieGroup) {
    static_assert(
        value,
        "This type's trait does not assert it is a Lie group (or derived)");
 
    // group operations 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);
    // AdjointMap
    *Hg = traits<T>::AdjointMap(g);
  }
 
 private:
  T g, h;
  TangentVector v;
  ChartJacobian Hg, Hh;
};

template <typename T>
T interpolate(const T& X, const T& Y, double t,
              typename MakeOptionalJacobian<T, T>::type Hx = {},
              typename MakeOptionalJacobian<T, T>::type Hy = {},
              typename MakeOptionalJacobian<T, double>::type Ht = {}) {
  if (Hx || Hy || Ht) {
    typename MakeJacobian<T, T>::type between_H_x, log_H, exp_H, compose_H_x;
    const T between =
        traits<T>::Between(X, Y, between_H_x);  // between_H_y = identity
    typename traits<T>::TangentVector delta = traits<T>::Logmap(between, log_H);
    const T Delta = traits<T>::Expmap(t * delta, exp_H);
    const T result = traits<T>::Compose(
        X, Delta, compose_H_x);  // compose_H_xinv_y = identity
 
    if (Hx) *Hx = compose_H_x + t * exp_H * log_H * between_H_x;
    if (Hy) *Hy = t * exp_H * log_H;
    if (Ht) *Ht = delta;
    return result;
  }
  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) \
  using _gtsam_IsLieGroup_##T = gtsam::IsLieGroup<T>;