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demo_se_2_3.cpp
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demo_se_2_3.cpp
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/**
* \file demo_se_2_3.cpp
*
* Created on: Feb 5, 2021
* \author: artivis
*
* ---------------------------------------------------------
* This file is:
* (c) 2021 artivis
*
* adapted from the file se_2_3_localization.cpp in manif:
* (c) 2020 Prashanth Ramadoss @ DIC-IIT, Genova, Italy
*
* This file is part of `kalmanif`, a C++ template-only library
* for Kalman filtering on Lie groups targeted at estimation for robotics.
* kalmanif is:
* (c) 2015 mherb
* (c) 2021 artivis
* ---------------------------------------------------------
*
* ---------------------------------------------------------
* Demonstration example:
*
* 3D Robot localization and linear velocity estimation
* based on strap-down IMU model and fixed beacons.
*
* ---------------------------------------------------------
*
* We consider a robot in 3D space surrounded by a small
* number of punctual landmarks or _beacons_.
* The robot is assumed to be mounted with an IMU whose
* measurements are fed as exogeneous inputs to the system.
* The robot is able to measure the location
* of the beacons w.r.t its own reference frame.
* We assume in this example that the IMU frame coincides with the robot frame.
*
* The robot extended pose X is in SE_2(3) and the beacon positions b_k in R^3,
*
* X = | R p v| // position, orientation and linear velocity
* | 1 |
* | 1|
*
* b_k = (bx_k, by_k, bz_k) // lmk coordinates in world frame
*
* alpha_k = (alphax_k, alphay_k, alphaz_k) // linear accelerometer measurements in IMU frame
*
* omega_k = (omegax_k, omegay_k, omegaz_k) // gyroscope measurements in IMU frame
*
* g = (0, 0, -9.80665) // acceleration due to gravity in world frame
*
* Consider robot coordinate frame B and world coordinate frame A.
* - p is the position of the origin of the robot frame B with respect to the world frame A
* - R is the orientation of the robot frame B with respect to the world frame A
* - v is the velocity of the robot frame with respect to the world frame,
* expressed in a frame whose origin coincides with the robot frame, oriented similar to the world frame
* (it is equivalent to p_dot in continuous time. This is usually called mixed-frame representation
* and is denoted as (B[A] v_AB), where B[A] is the mixed frame as described above.
* For reference, please see "Multibody Dynamics Notation" by Silvio Traversaro and Alessandro Saccon.
* Link: https://research.tue.nl/en/publications/multibody-dynamics-notation-version-2)
* - a is the frame acceleration in mixed-representation (equivalent to p_doubledot in continuous time).
* - omega_b as the angular velocity of the robot expressed in the robot frame
*
* The kinematic equations (1) can be written as,
* p <-- p + v dt + 0.5 a dt^2
* R <-- R Exp_SO3(omega_b)
* v <-- v + a dt
*
* However, we would like to express the kinematics equations in the form,
* X <-- X * Exp(u)
* where, X \in SE_2(3), u \in R^9 and u_hat \in se_2(3)
* Note that here input vector u is expressed in the local frame (robot frame).
* This can be seen as a motion integration on a manifold defined by the group SE_2(3).
*
* The exponential mapping of SE_2(3) is defined as,
* for u = [u_p, u_w, u_v]
* Exp(u) = | Exp_SO3(u_w) JlSO3(u_w) u_p JlSO3(u_w) u_v |
* | 0 0 0 1 0 |
* | 0 0 0 0 1 |
* where, JlSO3 is the left Jacobian of the SO(3) group.
*
* Please see the Appendix C of the paper "A micro Lie theory for state estimation in robotics",
* for the definition of the left Jacobian of SO(3).
* Please see the Appendix D of the paper, for the definition of Exp map for SE(3).
* The Exp map of SE_2(3) is a simple extension from the Exp map of SE(3).
* Also, please refer to Example 7 of the paper to understand when and how the left Jacobian of SO(3)
* appears in the definitions of Exp maps. The Example 7 illustrates the scenario for SE(3).
* We use a direct extension here for SE_2(3).
* One can arrive to such a definition by following the convergent Taylor's series expansion
* for the matrix exponential of the Lie algebra element (Equation 16 of the paper).
*
* As a result of X <-- X * Exp(u), we get (2)
* p <-- p + R JlSO3(u_w) u_p
* R <-- R Exp_SO3(u_w)
* v <-- v + R JlSO3(u_w) u_v
*
* It is important to notice the subtle difference between (1) and (2) here,
* which is specifically the influence of the left Jacobian of SO(3) in (2).
* The approach in (1) considers the motion integration is done by defining
* the exponential map in R3xSO(3)xR3 instead of SE_2(3),
* in the sense explored in Example 7 of the Micro Lie theory paper. It must be noted that
* as dt tends to 0, both sets of equations (1) and (2) tend to be the same, since JlSO3 tends to identity.
*
* Since, (2) exploits the algebra of the SE_2(3) group properly,
* we would like to draw a relationship between the sets of equations (2)
* and the IMU measurements which will constitute the exogeneous input vector u \in se_2(3).
*
* Considering R.T as the transpose of R, the IMU measurements are modeled as,
* - linear accelerometer measurements alpha = R.T (a - g) + w_acc
* - gyroscope measurements omega = omega_b + w_omega
* Note that the IMU measurements are expressed in the IMU frame (coincides with the robot frame - assumption).
* The IMU measurements are corrupted by noise,
* - w_omega is the additive white noise affecting the gyroscope measurements
* - w_acc is the additive white noise affecting the linear accelerometer measurements
* It must be noted that we do not consider IMU biases in the IMU measurement model in this example.
*
* Taking into account all of the above considerations, the exogenous input vector u (3) becomes,
* u = (u_p, u_w, u_v) where,
* u_w = omega dt
* u_p = (R.T v dt + 0.5 dt^2 (alpha + R.T g)
* u_v = (alpha + R.T g) dt
*
* This choice of input vector allows us to directly use measurements from the IMU
* for an unified motion integration involving position, orientation and linear velocity of the robot using SE_2(3).
* Equations (2) and (3) lead us to the following evolution equations,
*
* p <-- p + JlSO3 R.T v dt + 0.5 JlSO3 (alpha + R.T g) dt^2
* R <-- R Exp_SO3(omega dt)
* v <-- v + JlSO3 (alpha + R.T g) dt
*
* The system propagation noise covariance matrix becomes,
* U = diagonal(0, 0, 0, sigma_omegax^2, sigma_omegay^2, sigma_omegaz^2, sigma_accx^2, sigma_accy^2, sigma_accz^2).
*
* At the arrival of a exogeneous input u, the robot pose is updated
* with X <-- X * Exp(u) = X + u.
*
* Landmark measurements are of the range and bearing type,
* though they are put in Cartesian form for simplicity.
* Their noise n is zero mean Gaussian, and is specified
* with a covariances matrix R.
* We notice that the SE_2(3) action is the same as a
* rigid motion action of SE(3).
* This is the action of X \in SE_2(3) on a 3-d point b \in R^3 defined as,
* X b = R b + p
*
* Thus, the landmark measurements can be expressed as a group action on 3d points,
* y = h(X,b) = X^-1 * b
*
* y_k = (brx_k, bry_k, brz_k) // lmk coordinates in robot frame
*
* We consider the beacons b_k situated at known positions.
* We define the extended pose to estimate as X in SE_2(3).
* The estimation error dx and its covariance P are expressed
* in the tangent space at X.
*
* All these variables are summarized again as follows
*
* X : robot's extended pose, SE_2(3)
* u : robot control input, u = u(X, y_imu) \in se_2(3) with X as state and y_imu = [alpha, omega] as IMU readings, see Eq. (3)
* U : control perturbation covariance
* b_k : k-th landmark position, R^3
* y : Cartesian landmark measurement in robot frame, R^3
* R : covariance of the measurement noise
*
* The motion and measurement models are
*
* X_(t+1) = f(X_t, u) = X_t * Exp ( u ) // motion equation
* y_k = h(X, b_k) = X^-1 * b_k // measurement equation
*
* The algorithm below comprises first a simulator to
* produce measurements, then uses these measurements
* to estimate the state, using a Lie-based error-state Kalman filter.
*
* This file has plain code with only one main() function.
* There are no function calls other than those involving `manif`.
*
* Printing simulated state and estimated state together
* with an unfiltered state (i.e. without Kalman corrections)
* allows for evaluating the quality of the estimates.
*/
#include <kalmanif/kalmanif.h>
#include <kalmanif/system_models/simple_imu_system_model.h>
#include "utils/rand.h"
#include "utils/plots.h"
#include "utils/utils.h"
#include <vector>
using namespace kalmanif;
using namespace manif;
using State = SE_2_3d;
using StateCovariance = Covariance<State>;
using SystemModel = SimpleImuSystemModel<State::Scalar>;
using Control = SystemModel::Control;
using MeasurementModel = Landmark3DMeasurementModel<State>;
using Landmark = MeasurementModel::Landmark;
using Measurement = MeasurementModel::Measurement;
using Array6d = Eigen::Array<double, 6, 1>;
using Array9d = Eigen::Array<double, 9, 1>;
using Vector3d = Eigen::Matrix<double, 3, 1>;
using Vector6d = Eigen::Matrix<double, 6, 1>;
using Vector9d = Eigen::Matrix<double, 9, 1>;
using Matrix3d = Eigen::Matrix<double, 3, 3>;
using Matrix6d = Eigen::Matrix<double, 6, 6>;
// Filters
using EKF = ExtendedKalmanFilter<State>;
using SEKF = SquareRootExtendedKalmanFilter<State>;
using IEKF = InvariantExtendedKalmanFilter<State>;
using UKFM = UnscentedKalmanFilterManifolds<State>;
// Smoothers
using ERTS = RauchTungStriebelSmoother<EKF>;
using SERTS = RauchTungStriebelSmoother<SEKF>;
using IERTS = RauchTungStriebelSmoother<IEKF>;
using URTSM = RauchTungStriebelSmoother<UKFM>;
int main (int argc, char* argv[]) {
KALMANIF_DEMO_PROCESS_INPUT(argc, argv);
KALMANIF_DEMO_PRETTY_PRINT();
// START CONFIGURATION
constexpr double eot = 60; // s
constexpr double dt = 0.01; // s
constexpr int landmark_freq = 50; // Hz
(void)landmark_freq;
// acceleration due to gravity in world frame
Vector3d g;
g << 0, 0, -9.80665;
State X_simulation = State::Identity(),
X_unfiltered = State::Identity(); // propagation only, for comparison purposes
// IMU measurements in IMU frame
Vector3d alpha, alpha_const, omega, alpha_prev, omega_prev;
alpha_const << 0.1, 0.01, 0.1; // constant acceleration in IMU frame without gravity compensation
omega << 0.01, 0.1, 0; // constant angular velocity about x- and y-direction in IMU frame
// Previous IMU measurements in IMU frame initialized to values expected when stationary
alpha_prev = alpha = alpha_const - (X_simulation.rotation()).transpose() * g;
omega_prev << 0, 0, 0;
// Define a control vector and its noise and covariance
Control u_simu, u_est, u_unfilt;
Vector6d u_nom, u_noisy, u_noise;
Array6d u_sigmas;
Matrix6d U;
u_sigmas << 0.01, 0.01, 0.01, 0.01, 0.01, 0.01;
U = (u_sigmas * u_sigmas).matrix().asDiagonal();
// Define the beacon's measurements
Eigen::Vector3d y, y_noise;
Eigen::Array3d y_sigmas;
Eigen::Matrix3d R;
y_sigmas << 0.01, 0.01, 0.01;
R = (y_sigmas * y_sigmas).matrix().asDiagonal();
std::vector<MeasurementModel> measurement_models = {
MeasurementModel(Landmark(2.0, 0.0, 0.0), R),
MeasurementModel(Landmark(3.0, -1.0, -1.0), R),
MeasurementModel(Landmark(2.0, -1.0, 1.0), R),
MeasurementModel(Landmark(2.0, 1.0, 1.0), R),
MeasurementModel(Landmark(2.0, 1.0, -1.0), R)
};
std::vector<Measurement> measurements(measurement_models.size());
SystemModel system_model;
system_model.setCovariance(U);
StateCovariance state_cov_init = StateCovariance::Zero();
state_cov_init.block<3, 3>(0, 0) = 0.001 * Matrix3d::Identity();
state_cov_init.block<3, 3>(3, 3) = 0.01 * Matrix3d::Identity();
state_cov_init.block<3, 3>(6, 6) = 0.001 * Matrix3d::Identity();
Vector9d n = randn<Array9d>();
Vector9d X_init_coeffs = state_cov_init.cwiseSqrt() * n;
State X_init(
X_init_coeffs(0), X_init_coeffs(1), X_init_coeffs(2),
X_init_coeffs(3), X_init_coeffs(4), X_init_coeffs(5),
X_init_coeffs(6), X_init_coeffs(7), X_init_coeffs(8)
);
EKF ekf;
ekf.setState(X_init);
ekf.setCovariance(state_cov_init);
SEKF sekf(X_init, state_cov_init);
IEKF iekf(X_init, state_cov_init);
UKFM ukfm(X_init, state_cov_init);
ERTS erts(X_init, state_cov_init);
SERTS serts(X_init, state_cov_init);
IERTS ierts(X_init, state_cov_init);
URTSM urtsm(X_init, state_cov_init);
// Store some data for plots
DemoDataCollector<State> collector;
collector.reserve(
eot/dt, "UNFI", "EKF", "SEKF", "IEKF", "UKFM" , "ERTS", "SERTS", "IERTS", "URTSM"
);
// Make 10 steps. Measure up to three landmarks each time.
for (double t = 0; t < eot; t += dt) {
//// I. Simulation
/// input vector
u_nom << alpha_prev, omega_prev;
/// simulate noise
u_noise = randn(u_sigmas); // control noise
u_noisy = u_nom + u_noise; // noisy control
u_simu = u_nom;
u_est = u_noisy;
u_unfilt = u_noisy;
/// first we move - - - - - - - - - - - - - - - - - - - - - - - - - - - -
X_simulation = system_model(X_simulation, u_simu, dt);
/// update expected IMU measurements
alpha = alpha_const - X_simulation.rotation().transpose() * g; // update expected IMU measurement after moving
/// then we measure all landmarks - - - - - - - - - - - - - - - - - - - -
for (std::size_t i = 0; i < measurement_models.size(); ++i) {
auto measurement_model = measurement_models[i];
y = measurement_model(X_simulation); // landmark measurement, before adding noise
/// simulate noise
y_noise = randn(y_sigmas); // measurement noise
y = y + y_noise; // landmark measurement, noisy
measurements[i] = y; // store for the estimator just below
}
//// II. Estimation
/// First we move
ekf.propagate(system_model, u_est, dt);
sekf.propagate(system_model, u_est, dt);
iekf.propagate(system_model, u_est, dt);
ukfm.propagate(system_model, u_est, dt);
erts.propagate(system_model, u_est, dt);
serts.propagate(system_model, u_est, dt);
ierts.propagate(system_model, u_est, dt);
urtsm.propagate(system_model, u_est, dt);
X_unfiltered = system_model(X_unfiltered, u_unfilt, dt);
/// Then we correct using the measurements of each lmk
// if (int(t*100) % int(100./landmark_freq) == 0) {
for (std::size_t i = 0; i < measurement_models.size(); ++i) {
// landmark
auto measurement_model = measurement_models[i];
// measurement
y = measurements[i];
// filter update
ekf.update(measurement_model, y);
sekf.update(measurement_model, y);
iekf.update(measurement_model, y);
ukfm.update(measurement_model, y);
erts.update(measurement_model, y);
serts.update(measurement_model, y);
ierts.update(measurement_model, y);
urtsm.update(measurement_model, y);
}
// }
alpha_prev = alpha;
omega_prev = omega;
//// III. Results
collector.collect(X_simulation, t);
collector.collect("UNFI", X_unfiltered, StateCovariance::Zero(), t);
collector.collect("EKF", ekf.getState(), ekf.getCovariance(), t);
collector.collect("SEKF", sekf.getState(), sekf.getCovariance(), t);
collector.collect("IEKF", iekf.getState(), iekf.getCovariance(), t);
collector.collect("UKFM", ukfm.getState(), ukfm.getCovariance(), t);
}
// END OF TEMPORAL LOOP, forward pass
// Batch backward pass - smoothing
{
erts.smooth();
const auto& Xs_erts = erts.getStates();
const auto& Ps_erts = erts.getCovariances();
serts.smooth();
const auto& Xs_serts = serts.getStates();
const auto& Ps_serts = serts.getCovariances();
ierts.smooth();
const auto& Xs_ierts = ierts.getStates();
const auto& Ps_ierts = ierts.getCovariances();
urtsm.smooth();
const auto& Xs_urtsm = urtsm.getStates();
const auto& Ps_urtsm = urtsm.getCovariances();
double t=0;
for (std::size_t i=0; i<Xs_erts.size(); ++i, t+=dt) {
collector.collect("ERTS", Xs_erts[i], Ps_erts[i], t);
collector.collect("SERTS", Xs_serts[i], Ps_serts[i], t);
collector.collect("IERTS", Xs_ierts[i], Ps_ierts[i], t);
collector.collect("URTSM", Xs_urtsm[i], Ps_urtsm[i], t);
}
}
// print the trajectory
if (!quiet) {
KALMANIF_DEMO_PRINT_TRAJECTORY(collector);
}
// Generate some metrics and print them
DemoDataProcessor<State>().process(collector).print();
// Actually plots only if PLOT_EXAMPLES=ON
DemoTrajPlotter<State>::plot(collector, filename, plot_trajectory);
DemoDataPlotter<State>::plot(collector, filename, plot_error);
return EXIT_SUCCESS;
}