200 lines
7.4 KiB
C++
200 lines
7.4 KiB
C++
/*
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* @file SimpleHelicopter.h
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* @brief Implement SimpleHelicopter discrete dynamics model and variational integrator,
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* following [Kobilarov09siggraph]
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* @author Duy-Nguyen Ta
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*/
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#pragma once
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#include <gtsam/nonlinear/NonlinearFactor.h>
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#include <gtsam/geometry/Pose3.h>
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#include <gtsam/base/numericalDerivative.h>
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#include <cmath>
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namespace gtsam {
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/**
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* Implement the Reconstruction equation: \f$ g_{k+1} = g_k \exp (h\xi_k) \f$, where
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* \f$ h \f$: timestep (parameter)
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* \f$ g_{k+1}, g_{k} \f$: poses at the current and the next timestep
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* \f$ \xi_k \f$: the body-fixed velocity (Lie algebra)
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* It is somewhat similar to BetweenFactor, but treats the body-fixed velocity
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* \f$ \xi_k \f$ as a variable. So it is a three-way factor.
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* Note: this factor is necessary if one needs to smooth the entire graph. It's not needed
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* in sequential update method.
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*/
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class Reconstruction : public NoiseModelFactor3<Pose3, Pose3, Vector6> {
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double h_; // time step
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typedef NoiseModelFactor3<Pose3, Pose3, Vector6> Base;
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public:
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Reconstruction(Key gKey1, Key gKey, Key xiKey, double h, double mu = 1000.0) :
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Base(noiseModel::Constrained::All(6, std::abs(mu)), gKey1, gKey,
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xiKey), h_(h) {
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}
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virtual ~Reconstruction() {}
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/// @return a deep copy of this factor
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virtual gtsam::NonlinearFactor::shared_ptr clone() const {
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return boost::static_pointer_cast<gtsam::NonlinearFactor>(
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gtsam::NonlinearFactor::shared_ptr(new Reconstruction(*this))); }
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/** \f$ log((g_k\exp(h\xi_k))^{-1}g_{k+1}) = 0, with optional derivatives */
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Vector evaluateError(const Pose3& gk1, const Pose3& gk, const Vector6& xik,
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boost::optional<Matrix&> H1 = boost::none,
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boost::optional<Matrix&> H2 = boost::none,
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boost::optional<Matrix&> H3 = boost::none) const {
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Matrix6 D_exphxi_xi;
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Pose3 exphxi = Pose3::Expmap(h_ * xik, H3 ? &D_exphxi_xi : 0);
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Matrix6 D_gkxi_gk, D_gkxi_exphxi;
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Pose3 gkxi = gk.compose(exphxi, D_gkxi_gk, H3 ? &D_gkxi_exphxi : 0);
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Matrix6 D_hx_gk1, D_hx_gkxi;
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Pose3 hx = gkxi.between(gk1, (H2 || H3) ? &D_hx_gkxi : 0, H1 ? &D_hx_gk1 : 0);
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Matrix6 D_log_hx;
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Vector error = Pose3::Logmap(hx, D_log_hx);
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if (H1) *H1 = D_log_hx * D_hx_gk1;
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if (H2 || H3) {
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Matrix6 D_log_gkxi = D_log_hx * D_hx_gkxi;
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if (H2) *H2 = D_log_gkxi * D_gkxi_gk;
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if (H3) *H3 = D_log_gkxi * D_gkxi_exphxi * D_exphxi_xi * h_;
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}
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return error;
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}
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};
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/**
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* Implement the Discrete Euler-Poincare' equation:
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*/
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class DiscreteEulerPoincareHelicopter : public NoiseModelFactor3<Vector6, Vector6, Pose3> {
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double h_; /// time step
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Matrix Inertia_; /// Inertia tensors Inertia = [ J 0; 0 M ]
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Vector Fu_; /// F is the 6xc Control matrix, where c is the number of control variables uk, which directly change the vehicle pose (e.g., gas/brake/speed)
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/// F(.) is actually a function of the shape variables, which do not change the pose, but affect the vehicle's shape, e.g. steering wheel.
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/// Fu_ encodes everything we need to know about the vehicle's dynamics.
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double m_; /// mass. For gravity external force f_ext, which has a fixed formula in this case.
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// TODO: Fk_ and f_ext should be generalized as functions (factor nodes) on control signals and poses/velocities.
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// This might be needed in control or system identification problems.
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// We treat them as constant here, since the control inputs are to specify.
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typedef NoiseModelFactor3<Vector6, Vector6, Pose3> Base;
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public:
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DiscreteEulerPoincareHelicopter(Key xiKey1, Key xiKey_1, Key gKey,
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double h, const Matrix& Inertia, const Vector& Fu, double m,
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double mu = 1000.0) :
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Base(noiseModel::Constrained::All(6, std::abs(mu)), xiKey1, xiKey_1, gKey),
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h_(h), Inertia_(Inertia), Fu_(Fu), m_(m) {
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}
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virtual ~DiscreteEulerPoincareHelicopter() {}
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/// @return a deep copy of this factor
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virtual gtsam::NonlinearFactor::shared_ptr clone() const {
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return boost::static_pointer_cast<gtsam::NonlinearFactor>(
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gtsam::NonlinearFactor::shared_ptr(new DiscreteEulerPoincareHelicopter(*this))); }
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/** DEP, with optional derivatives
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* pk - pk_1 - h_*Fu_ - h_*f_ext = 0
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* where pk = CT_TLN(h*xi_k)*Inertia*xi_k
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* pk_1 = CT_TLN(-h*xi_k_1)*Inertia*xi_k_1
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* */
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Vector evaluateError(const Vector6& xik, const Vector6& xik_1, const Pose3& gk,
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boost::optional<Matrix&> H1 = boost::none,
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boost::optional<Matrix&> H2 = boost::none,
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boost::optional<Matrix&> H3 = boost::none) const {
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Vector muk = Inertia_*xik;
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Vector muk_1 = Inertia_*xik_1;
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// Apply the inverse right-trivialized tangent (derivative) map of the exponential map,
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// using the trapezoidal Lie-Newmark (TLN) scheme, to a vector.
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// TLN is just a first order approximation of the dExpInv_exp above, detailed in [Kobilarov09siggraph]
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// C_TLN formula: I6 - 1/2 ad[xi].
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Matrix D_adjThxik_muk, D_adjThxik1_muk1;
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Vector pk = muk - 0.5*Pose3::adjointTranspose(h_*xik, muk, D_adjThxik_muk);
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Vector pk_1 = muk_1 - 0.5*Pose3::adjointTranspose(-h_*xik_1, muk_1, D_adjThxik1_muk1);
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Matrix D_gravityBody_gk;
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Point3 gravityBody = gk.rotation().unrotate(Point3(0.0, 0.0, -9.81*m_), D_gravityBody_gk, boost::none);
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Vector f_ext = (Vector(6) << 0.0, 0.0, 0.0, gravityBody.x(), gravityBody.y(), gravityBody.z()).finished();
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Vector hx = pk - pk_1 - h_*Fu_ - h_*f_ext;
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if (H1) {
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Matrix D_pik_xi = Inertia_-0.5*(h_*D_adjThxik_muk + Pose3::adjointMap(h_*xik).transpose()*Inertia_);
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*H1 = D_pik_xi;
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}
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if (H2) {
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Matrix D_pik1_xik1 = Inertia_-0.5*(-h_*D_adjThxik1_muk1 + Pose3::adjointMap(-h_*xik_1).transpose()*Inertia_);
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*H2 = -D_pik1_xik1;
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}
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if (H3) {
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*H3 = zeros(6,6);
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insertSub(*H3, -h_*D_gravityBody_gk, 3, 0);
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}
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return hx;
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}
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#if 0
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Vector computeError(const Vector6& xik, const Vector6& xik_1, const Pose3& gk) const {
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Vector pk = Pose3::dExpInv_exp(h_*xik).transpose()*Inertia_*xik;
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Vector pk_1 = Pose3::dExpInv_exp(-h_*xik_1).transpose()*Inertia_*xik_1;
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Point3 gravityBody = gk.rotation().unrotate(Point3(0.0, 0.0, -9.81*m_));
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Vector f_ext = (Vector(6) << 0.0, 0.0, 0.0, gravityBody.x(), gravityBody.y(), gravityBody.z());
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Vector hx = pk - pk_1 - h_*Fu_ - h_*f_ext;
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return hx;
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}
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Vector evaluateError(const Vector6& xik, const Vector6& xik_1, const Pose3& gk,
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boost::optional<Matrix&> H1 = boost::none,
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boost::optional<Matrix&> H2 = boost::none,
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boost::optional<Matrix&> H3 = boost::none) const {
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if (H1) {
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(*H1) = numericalDerivative31(
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boost::function<Vector(const Vector6&, const Vector6&, const Pose3&)>(
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boost::bind(&DiscreteEulerPoincareHelicopter::computeError, *this, _1, _2, _3)
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),
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xik, xik_1, gk, 1e-5
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);
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}
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if (H2) {
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(*H2) = numericalDerivative32(
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boost::function<Vector(const Vector6&, const Vector6&, const Pose3&)>(
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boost::bind(&DiscreteEulerPoincareHelicopter::computeError, *this, _1, _2, _3)
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),
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xik, xik_1, gk, 1e-5
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);
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}
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if (H3) {
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(*H3) = numericalDerivative33(
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boost::function<Vector(const Vector6&, const Vector6&, const Pose3&)>(
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boost::bind(&DiscreteEulerPoincareHelicopter::computeError, *this, _1, _2, _3)
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),
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xik, xik_1, gk, 1e-5
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);
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}
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return computeError(xik, xik_1, gk);
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}
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#endif
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};
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} /* namespace gtsam */
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