ukf-nonlinear-complex-example.cpp

Example of a complex non-linear use-case of the Unscented Kalman Filter (UKF). The system we are interested in is a 4-wheel robot, moving at a low velocity. As such, it can be modeled using a bicycle model.

The state vector of the UKF is:

where is the heading of the robot.

The measurement corresponds to the distance and relative orientation of the robot with different landmarks. Be and the position of the landmark along the x and y axis, the measurement vector can be written as:

Some noise is added to the measurement vector to simulate measurements which are not perfect.

The mean of several angles must be computed in the Unscented Transform. The definition we chose to use is the following:

As the Unscented Transform uses a weighted mean, the actual implementation of the weighted mean of several angles is the following:

where is the weight associated to the measurements for the weighted mean.

Additionally, the addition and subtraction of angles must be carefully done, as the result must stay in the interval or . We decided to use the interval .

/*
 * ViSP, open source Visual Servoing Platform software.
 * Copyright (C) 2005 - 2024 by Inria. All rights reserved.
 *
 * This software is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation; either version 2 of the License, or
 * (at your option) any later version.
 * See the file LICENSE.txt at the root directory of this source
 * distribution for additional information about the GNU GPL.
 *
 * For using ViSP with software that can not be combined with the GNU
 * GPL, please contact Inria about acquiring a ViSP Professional
 * Edition License.
 *
 * See https://visp.inria.fr for more information.
 *
 * This software was developed at:
 * Inria Rennes - Bretagne Atlantique
 * Campus Universitaire de Beaulieu
 * 35042 Rennes Cedex
 * France
 *
 * If you have questions regarding the use of this file, please contact
 * Inria at visp@inria.fr
 *
 * This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
 * WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
 */
 
// UKF includes
#include <visp3/core/vpUKSigmaDrawerMerwe.h>
#include <visp3/core/vpUnscentedKalman.h>
 
// ViSP includes
#include <visp3/core/vpConfig.h>
#include <visp3/core/vpGaussRand.h>
#ifdef VISP_HAVE_DISPLAY
#include <visp3/gui/vpPlot.h>
#endif
 
#if (VISP_CXX_STANDARD >= VISP_CXX_STANDARD_11)
 
#ifdef ENABLE_VISP_NAMESPACE
using namespace VISP_NAMESPACE_NAME;
#endif
 
namespace
{
double normalizeAngle(const double &angle)
{
  double angleIn0to2pi = vpMath::modulo(angle, 2. * M_PI);
  double angleInMinPiPi = angleIn0to2pi;
  if (angleInMinPiPi > M_PI) {
    // Substract 2 PI to be in interval [-Pi; Pi]
    angleInMinPiPi -= 2. * M_PI;
  }
  return angleInMinPiPi;
}
 
vpColVector measurementMean(const std::vector<vpColVector> &measurements, const std::vector<double> &wm)
{
  const unsigned int nbPoints = static_cast<unsigned int>(measurements.size());
  const unsigned int sizeMeasurement = measurements[0].size();
  const unsigned int nbLandmarks = sizeMeasurement / 2;
  vpColVector mean(sizeMeasurement, 0.);
  std::vector<double> sumCos(nbLandmarks, 0.);
  std::vector<double> sumSin(nbLandmarks, 0.);
  for (unsigned int i = 0; i < nbPoints; ++i) {
    for (unsigned int j = 0; j < nbLandmarks; ++j) {
      mean[2*j] += wm[i] * measurements[i][2*j];
      sumCos[j] += wm[i] * std::cos(measurements[i][(2*j)+1]);
      sumSin[j] += wm[i] * std::sin(measurements[i][(2*j)+1]);
    }
  }
  for (unsigned int j = 0; j < nbLandmarks; ++j) {
    mean[(2*j)+1] = std::atan2(sumSin[j], sumCos[j]);
  }
  return mean;
}
 
vpColVector measurementResidual(const vpColVector &meas, const vpColVector &toSubtract)
{
  vpColVector res = meas - toSubtract;
  unsigned int nbMeasures = res.size();
  for (unsigned int i = 1; i < nbMeasures; i += 2) {
    res[i] = normalizeAngle(res[i]);
  }
  return res;
}
 
vpColVector stateAdd(const vpColVector &state, const vpColVector &toAdd)
{
  vpColVector add = state + toAdd;
  add[2] = normalizeAngle(add[2]);
  return add;
}
 
vpColVector stateMean(const std::vector<vpColVector> &states, const std::vector<double> &wm)
{
  vpColVector mean(3, 0.);
  unsigned int nbPoints = static_cast<unsigned int>(states.size());
  double sumCos = 0.;
  double sumSin = 0.;
  for (unsigned int i = 0; i < nbPoints; ++i) {
    mean[0] += wm[i] * states[i][0];
    mean[1] += wm[i] * states[i][1];
    sumCos += wm[i] * std::cos(states[i][2]);
    sumSin += wm[i] * std::sin(states[i][2]);
  }
  mean[2] = std::atan2(sumSin, sumCos);
  return mean;
}
 
vpColVector stateResidual(const vpColVector &state, const vpColVector &toSubtract)
{
  vpColVector res = state - toSubtract;
  res[2] = normalizeAngle(res[2]);
  return res;
}
 
vpColVector fx(const vpColVector &x, const double & /*dt*/)
{
  return x;
}
 
std::vector<vpColVector> generateTurnCommands(const double &v, const double &angleStart, const double &angleStop, const unsigned int &nbSteps)
{
  std::vector<vpColVector> cmds;
  double dTheta = vpMath::rad(angleStop - angleStart) / static_cast<double>(nbSteps - 1);
  for (unsigned int i = 0; i < nbSteps; ++i) {
    double theta = vpMath::rad(angleStart) + dTheta * static_cast<double>(i);
    vpColVector cmd(2);
    cmd[0] = v;
    cmd[1] = theta;
    cmds.push_back(cmd);
  }
  return cmds;
}
 
std::vector<vpColVector> generateCommands()
{
  std::vector<vpColVector> cmds;
  // Starting by an straight line acceleration
  unsigned int nbSteps = 30;
  double dv = (1.1 - 0.001) / static_cast<double>(nbSteps - 1);
  for (unsigned int i = 0; i < nbSteps; ++i) {
    vpColVector cmd(2);
    cmd[0] = 0.001 + static_cast<double>(i) * dv;
    cmd[1] = 0.;
    cmds.push_back(cmd);
  }
 
  // Left turn
  double lastLinearVelocity = cmds[cmds.size() -1][0];
  std::vector<vpColVector> leftTurnCmds = generateTurnCommands(lastLinearVelocity, 0, 2, 15);
  cmds.insert(cmds.end(), leftTurnCmds.begin(), leftTurnCmds.end());
  for (unsigned int i = 0; i < 100; ++i) {
    cmds.push_back(cmds[cmds.size() -1]);
  }
 
  // Right turn
  lastLinearVelocity = cmds[cmds.size() -1][0];
  std::vector<vpColVector> rightTurnCmds = generateTurnCommands(lastLinearVelocity, 2, -2, 15);
  cmds.insert(cmds.end(), rightTurnCmds.begin(), rightTurnCmds.end());
  for (unsigned int i = 0; i < 200; ++i) {
    cmds.push_back(cmds[cmds.size() -1]);
  }
 
  // Left  turn again
  lastLinearVelocity = cmds[cmds.size() -1][0];
  leftTurnCmds = generateTurnCommands(lastLinearVelocity, -2, 0, 15);
  cmds.insert(cmds.end(), leftTurnCmds.begin(), leftTurnCmds.end());
  for (unsigned int i = 0; i < 150; ++i) {
    cmds.push_back(cmds[cmds.size() -1]);
  }
 
  lastLinearVelocity = cmds[cmds.size() -1][0];
  leftTurnCmds = generateTurnCommands(lastLinearVelocity, 0, 1, 25);
  cmds.insert(cmds.end(), leftTurnCmds.begin(), leftTurnCmds.end());
  for (unsigned int i = 0; i < 150; ++i) {
    cmds.push_back(cmds[cmds.size() -1]);
  }
 
  return cmds;
}
}
 
class vpBicycleModel
{
public:
  vpBicycleModel(const double &w)
    : m_w(w)
  { }
 
  vpColVector computeMotion(const vpColVector &u, const vpColVector &x, const double &dt)
  {
    double heading = x[2];
    double vel = u[0];
    double steeringAngle = u[1];
    double distance = vel * dt;
 
    if (std::abs(steeringAngle) > 0.001) {
      // The robot is turning
      double beta = (distance / m_w) * std::tan(steeringAngle);
      double radius = m_w / std::tan(steeringAngle);
      double sinh = std::sin(heading);
      double sinhb = std::sin(heading + beta);
      double cosh = std::cos(heading);
      double coshb = std::cos(heading + beta);
      vpColVector motion(3);
      motion[0] = -radius * sinh + radius * sinhb;
      motion[1] = radius * cosh - radius * coshb;
      motion[2] = beta;
      return motion;
    }
    else {
      // The robot is moving in straight line
      vpColVector motion(3);
      motion[0] = distance * std::cos(heading);
      motion[1] = distance * std::sin(heading);
      motion[2] = 0.;
      return motion;
    }
  }
 
  vpColVector move(const vpColVector &u, const vpColVector &x, const double &dt)
  {
    vpColVector motion = computeMotion(u, x, dt);
    vpColVector newX = x + motion;
    newX[2] = normalizeAngle(newX[2]);
    return newX;
  }
private:
  double m_w; // The length of the wheelbase.
};
 
class vpLandmarkMeasurements
{
public:
  vpLandmarkMeasurements(const double &x, const double &y, const double &range_std, const double &rel_angle_std)
    : m_x(x)
    , m_y(y)
    , m_rngRange(range_std, 0., 4224)
    , m_rngRelativeAngle(rel_angle_std, 0., 2112)
  { }
 
  vpColVector state_to_measurement(const vpColVector &chi)
  {
    vpColVector meas(2);
    double dx = m_x - chi[0];
    double dy = m_y - chi[1];
    meas[0] = std::sqrt(dx * dx + dy * dy);
    meas[1] = normalizeAngle(std::atan2(dy, dx) - chi[2]);
    return meas;
  }
 
  vpColVector measureGT(const vpColVector &pos)
  {
    double dx = m_x - pos[0];
    double dy = m_y - pos[1];
    double range = std::sqrt(dx * dx + dy * dy);
    double orientation = normalizeAngle(std::atan2(dy, dx) - pos[2]);
    vpColVector measurements(2);
    measurements[0] = range;
    measurements[1] = orientation;
    return measurements;
  }
 
  vpColVector measureWithNoise(const vpColVector &pos)
  {
    vpColVector measurementsGT = measureGT(pos);
    vpColVector measurementsNoisy = measurementsGT;
    measurementsNoisy[0] += m_rngRange();
    measurementsNoisy[1] += m_rngRelativeAngle();
    measurementsNoisy[1] = normalizeAngle(measurementsNoisy[1]);
    return measurementsNoisy;
  }
 
private:
  double m_x; // The position along the x-axis of the landmark
  double m_y; // The position along the y-axis of the landmark
  vpGaussRand m_rngRange; // Noise simulator for the range measurement
  vpGaussRand m_rngRelativeAngle; // Noise simulator for the relative angle measurement
};
 
class vpLandmarksGrid
{
public:
  vpLandmarksGrid(const std::vector<vpLandmarkMeasurements> &landmarks)
    : m_landmarks(landmarks)
  { }
 
  vpColVector state_to_measurement(const vpColVector &chi)
  {
    unsigned int nbLandmarks = static_cast<unsigned int>(m_landmarks.size());
    vpColVector measurements(2*nbLandmarks);
    for (unsigned int i = 0; i < nbLandmarks; ++i) {
      vpColVector landmarkMeas = m_landmarks[i].state_to_measurement(chi);
      measurements[2*i] = landmarkMeas[0];
      measurements[(2*i) + 1] = landmarkMeas[1];
    }
    return measurements;
  }
 
  vpColVector measureGT(const vpColVector &pos)
  {
    unsigned int nbLandmarks = static_cast<unsigned int>(m_landmarks.size());
    vpColVector measurements(2*nbLandmarks);
    for (unsigned int i = 0; i < nbLandmarks; ++i) {
      vpColVector landmarkMeas = m_landmarks[i].measureGT(pos);
      measurements[2*i] = landmarkMeas[0];
      measurements[(2*i) + 1] = landmarkMeas[1];
    }
    return measurements;
  }
 
  vpColVector measureWithNoise(const vpColVector &pos)
  {
    unsigned int nbLandmarks = static_cast<unsigned int>(m_landmarks.size());
    vpColVector measurements(2*nbLandmarks);
    for (unsigned int i = 0; i < nbLandmarks; ++i) {
      vpColVector landmarkMeas = m_landmarks[i].measureWithNoise(pos);
      measurements[2*i] = landmarkMeas[0];
      measurements[(2*i) + 1] = landmarkMeas[1];
    }
    return measurements;
  }
 
private:
  std::vector<vpLandmarkMeasurements> m_landmarks; 
};
 
int main(const int argc, const char *argv[])
{
  bool opt_useDisplay = true;
  for (int i = 1; i < argc; ++i) {
    std::string arg(argv[i]);
    if (arg == "-d") {
      opt_useDisplay = false;
    }
    else if ((arg == "-h") || (arg == "--help")) {
      std::cout << "SYNOPSIS" << std::endl;
      std::cout << "  " << argv[0] << " [-d][-h]" << std::endl;
      std::cout << std::endl << std::endl;
      std::cout << "DETAILS" << std::endl;
      std::cout << "  -d" << std::endl;
      std::cout << "    Deactivate display." << std::endl;
      std::cout << std::endl;
      std::cout << "  -h, --help" << std::endl;
      return 0;
    }
  }
 
  const double dt = 0.1; // Period of 0.1s
  const double step = 1.; // Number of update of the robot position between two UKF filtering
  const double sigmaRange = 0.3; // Standard deviation of the range measurement: 0.3m
  const double sigmaBearing = vpMath::rad(0.5); // Standard deviation of the bearing angle: 0.5deg
  const double wheelbase = 0.5; // Wheelbase of 0.5m
  const std::vector<vpLandmarkMeasurements> landmarks = { vpLandmarkMeasurements(5, 10, sigmaRange, sigmaBearing)
                                                        , vpLandmarkMeasurements(10, 5, sigmaRange, sigmaBearing)
                                                        , vpLandmarkMeasurements(15, 15, sigmaRange, sigmaBearing)
                                                        , vpLandmarkMeasurements(20, 5, sigmaRange, sigmaBearing)
                                                        , vpLandmarkMeasurements(0, 30, sigmaRange, sigmaBearing)
                                                        , vpLandmarkMeasurements(50, 30, sigmaRange, sigmaBearing)
                                                        , vpLandmarkMeasurements(40, 10, sigmaRange, sigmaBearing) }; // Vector of landmarks constituting the grid
  const unsigned int nbLandmarks = static_cast<unsigned int>(landmarks.size()); // Number of landmarks constituting the grid
  std::vector<vpColVector> cmds = generateCommands();
  const unsigned int nbCmds = static_cast<unsigned int>(cmds.size());
 
  // Initialize the attributes of the UKF
  std::shared_ptr<vpUKSigmaDrawerAbstract> drawer = std::make_shared<vpUKSigmaDrawerMerwe>(3, 0.1, 2., 0, stateResidual, stateAdd);
 
  vpMatrix R1landmark(2, 2, 0.); // The covariance of the noise introduced by the measurement with 1 landmark
  R1landmark[0][0] = sigmaRange*sigmaRange;
  R1landmark[1][1] = sigmaBearing*sigmaBearing;
  vpMatrix R(2*nbLandmarks, 2 * nbLandmarks);
  for (unsigned int i = 0; i < nbLandmarks; ++i) {
    R.insert(R1landmark, 2*i, 2*i);
  }
 
  const double processVariance = 0.0001;
  vpMatrix Q; // The covariance of the process
  Q.eye(3);
  Q = Q * processVariance;
 
  vpMatrix P0(3, 3); //  The initial guess of the process covariance
  P0.eye(3);
  P0[0][0] = 0.1;
  P0[1][1] = 0.1;
  P0[2][2] = 0.05;
 
  vpColVector X0(3);
  X0[0] = 2.; // x = 2m
  X0[1] = 6.; // y = 6m
  X0[2] = 0.3; // robot orientation = 0.3 rad
 
  vpUnscentedKalman::vpProcessFunction f = fx;
  vpLandmarksGrid grid(landmarks);
  vpBicycleModel robot(wheelbase);
  using std::placeholders::_1;
  using std::placeholders::_2;
  using std::placeholders::_3;
  vpUnscentedKalman::vpMeasurementFunction h = std::bind(&vpLandmarksGrid::state_to_measurement, &grid, _1);
  vpUnscentedKalman::vpCommandStateFunction bx = std::bind(&vpBicycleModel::computeMotion, &robot, _1, _2, _3);
 
  // Initialize the UKF
  vpUnscentedKalman ukf(Q, R, drawer, f, h);
  ukf.init(X0, P0);
  ukf.setCommandStateFunction(bx);
  ukf.setMeasurementMeanFunction(measurementMean);
  ukf.setMeasurementResidualFunction(measurementResidual);
  ukf.setStateAddFunction(stateAdd);
  ukf.setStateMeanFunction(stateMean);
  ukf.setStateResidualFunction(stateResidual);
 
#ifdef VISP_HAVE_DISPLAY
  vpPlot *plot = nullptr;
  if (opt_useDisplay) {
  // Initialize the plot
    plot = new vpPlot(1);
    plot->initGraph(0, 2);
    plot->setTitle(0, "Position of the robot");
    plot->setUnitX(0, "Position along x(m)");
    plot->setUnitY(0, "Position along y (m)");
    plot->setLegend(0, 0, "GT");
    plot->setLegend(0, 1, "Filtered");
  }
#endif
 
  // Initialize the simulation
  vpColVector robot_pos = X0;
 
  for (unsigned int i = 0; i < nbCmds; ++i) {
    robot_pos = robot.move(cmds[i], robot_pos, dt / step);
    if (i % static_cast<int>(step) == 0) {
      // Perform the measurement
      vpColVector z = grid.measureWithNoise(robot_pos);
 
      // Use the UKF to filter the measurement
      ukf.filter(z, dt, cmds[i]);
 
#ifdef VISP_HAVE_DISPLAY
      if (opt_useDisplay) {
        // Plot the filtered state
        vpColVector Xest = ukf.getXest();
        plot->plot(0, 1, Xest[0], Xest[1]);
      }
#endif
    }
 
#ifdef VISP_HAVE_DISPLAY
    if (opt_useDisplay) {
    // Plot the ground truth
      plot->plot(0, 0, robot_pos[0], robot_pos[1]);
    }
#endif
  }
 
  if (opt_useDisplay) {
    std::cout << "Press Enter to quit..." << std::endl;
    std::cin.get();
  }
 
#ifdef VISP_HAVE_DISPLAY
  if (opt_useDisplay) {
    delete plot;
  }
#endif
 
  vpColVector finalError = grid.state_to_measurement(ukf.getXest()) - grid.measureGT(robot_pos);
  const double maxError = 0.3;
  if (finalError.frobeniusNorm() > maxError) {
    std::cerr << "Error: max tolerated error = " << maxError << ", final error = " << finalError.frobeniusNorm() << std::endl;
    return -1;
  }
  return 0;
}
#else
int main()
{
  std::cout << "This example is only available if you compile ViSP in C++11 standard or higher." << std::endl;
  return 0;
}
#endif