testExtendedKalmanFilter.cpp

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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

 * -------------------------------------------------------------------------- */

#include <gtsam/nonlinear/PriorFactor.h>
#include <gtsam/slam/BetweenFactor.h>
#include <gtsam/nonlinear/ExtendedKalmanFilter-inl.h>
#include <gtsam/nonlinear/NonlinearFactor.h>
#include <gtsam/inference/Symbol.h>
#include <gtsam/geometry/Point2.h>

#include <CppUnitLite/TestHarness.h>

using namespace gtsam;

// Convenience for named keys
using symbol_shorthand::X;
using symbol_shorthand::L;

/* ************************************************************************* */
TEST( ExtendedKalmanFilter, linear ) {

  // Create the TestKeys for our example
  Symbol x0('x',0), x1('x',1), x2('x',2), x3('x',3);

  // Create the Kalman Filter initialization point
  Point2 x_initial(0.0, 0.0);
  SharedDiagonal P_initial = noiseModel::Diagonal::Sigmas(Vector2(0.1, 0.1));

  // Create an ExtendedKalmanFilter object
  ExtendedKalmanFilter<Point2> ekf(x0, x_initial, P_initial);

  // Create the controls and measurement properties for our example
  double dt = 1.0;
  Vector u = Vector2(1.0, 0.0);
  Point2 difference(u*dt);
  SharedDiagonal Q = noiseModel::Diagonal::Sigmas(Vector2(0.1, 0.1), true);
  Point2 z1(1.0, 0.0);
  Point2 z2(2.0, 0.0);
  Point2 z3(3.0, 0.0);
  SharedDiagonal R = noiseModel::Diagonal::Sigmas(Vector2(0.25, 0.25), true);

  // Create the set of expected output TestValues
  Point2 expected1(1.0, 0.0);
  Point2 expected2(2.0, 0.0);
  Point2 expected3(3.0, 0.0);

  // Run iteration 1
  // Create motion factor
  BetweenFactor<Point2> factor1(x0, x1, difference, Q);
  Point2 predict1 = ekf.predict(factor1);
  EXPECT(assert_equal(expected1,predict1));

  // Create the measurement factor
  PriorFactor<Point2> factor2(x1, z1, R);
  Point2 update1 = ekf.update(factor2);
  EXPECT(assert_equal(expected1,update1));

  // Run iteration 2
  BetweenFactor<Point2> factor3(x1, x2, difference, Q);
  Point2 predict2 = ekf.predict(factor3);
  EXPECT(assert_equal(expected2,predict2));

  PriorFactor<Point2> factor4(x2, z2, R);
  Point2 update2 = ekf.update(factor4);
  EXPECT(assert_equal(expected2,update2));

  // Run iteration 3
  BetweenFactor<Point2> factor5(x2, x3, difference, Q);
  Point2 predict3 = ekf.predict(factor5);
  EXPECT(assert_equal(expected3,predict3));

  PriorFactor<Point2> factor6(x3, z3, R);
  Point2 update3 = ekf.update(factor6);
  EXPECT(assert_equal(expected3,update3));

  return;
}


// Create Motion Model Factor
class NonlinearMotionModel : public NoiseModelFactor2<Point2,Point2> {

  typedef NoiseModelFactor2<Point2, Point2> Base;
  typedef NonlinearMotionModel This;

protected:
  Matrix Q_;
  Matrix Q_invsqrt_;

public:
  NonlinearMotionModel(){}

  NonlinearMotionModel(const Symbol& TestKey1, const Symbol& TestKey2) :
    Base(noiseModel::Diagonal::Sigmas(Vector2(1.0, 1.0)), TestKey1, TestKey2), Q_(2,2) {

    // Initialize motion model parameters:
    // w is vector of motion noise sigmas. The final covariance is calculated as G*w*w'*G'
    // In this model, Q is fixed, so it may be calculated in the constructor
    Matrix G(2,2);
    Matrix w(2,2);

    G << 1.0, 0.0, 0.0, 1.0;
    w << 1.0, 0.0, 0.0, 1.0;

    w = G*w;
    Q_ = w*w.transpose();
    Q_invsqrt_ = inverse_square_root(Q_);
  }

  ~NonlinearMotionModel() override {}

  // Calculate the next state prediction using the nonlinear function f()
  Point2 f(const Point2& x_t0) const {

    // Calculate f(x)
    double x = x_t0.x() * x_t0.x();
    double y = x_t0.x() * x_t0.y();
    Point2 x_t1(x,y);

    // In this example, the noiseModel remains constant
    return x_t1;
  }

  // Calculate the Jacobian of the nonlinear function f()
  Matrix F(const Point2& x_t0) const {
    // Calculate Jacobian of f()
    Matrix F = Matrix(2,2);
    F(0,0) = 2*x_t0.x();
    F(0,1) = 0.0;
    F(1,0) = x_t0.y();
    F(1,1) = x_t0.x();

    return F;
  }

  // Calculate the inverse square root of the motion model covariance, Q
  Matrix QInvSqrt(const Point2& x_t0) const {
    // This motion model has a constant covariance
    return Q_invsqrt_;
  }

  /* print */
  void print(const std::string& s = "", const KeyFormatter& keyFormatter = DefaultKeyFormatter) const override {
    std::cout << s << ": NonlinearMotionModel\n";
    std::cout << "  TestKey1: " << keyFormatter(key1()) << std::endl;
    std::cout << "  TestKey2: " << keyFormatter(key2()) << std::endl;
  }

  bool equals(const NonlinearFactor& f, double tol = 1e-9) const override {
    const This *e = dynamic_cast<const This*> (&f);
    return (e != nullptr) && (key1() == e->key1()) && (key2() == e->key2());
  }

  double error(const Values& c) const override {
    Vector w = whitenedError(c);
    return 0.5 * w.dot(w);
  }

  size_t dim() const override {
    return 2;
  }

  Vector whitenedError(const Values& c) const {
    return QInvSqrt(c.at<Point2>(key1()))*unwhitenedError(c);
  }

  boost::shared_ptr<GaussianFactor> linearize(const Values& c) const override {
    const X1& x1 = c.at<X1>(key1());
    const X2& x2 = c.at<X2>(key2());
    Matrix A1, A2;
    Vector b = -evaluateError(x1, x2, A1, A2);
    SharedDiagonal constrained =
        boost::dynamic_pointer_cast<noiseModel::Constrained>(this->noiseModel_);
    if (constrained.get() != nullptr) {
      return JacobianFactor::shared_ptr(new JacobianFactor(key1(), A1, key2(),
          A2, b, constrained));
    }
    // "Whiten" the system before converting to a Gaussian Factor
    Matrix Qinvsqrt = QInvSqrt(x1);
    A1 = Qinvsqrt*A1;
    A2 = Qinvsqrt*A2;
    b = Qinvsqrt*b;
    return GaussianFactor::shared_ptr(new JacobianFactor(key1(), A1, key2(),
        A2, b, noiseModel::Unit::Create(b.size())));
  }

  Vector evaluateError(const Point2& p1, const Point2& p2,
      boost::optional<Matrix&> H1 = boost::none, boost::optional<Matrix&> H2 =
          boost::none) const override {

    // error = p2 - f(p1)
    // H1 = d error / d p1 = -d f/ d p1 = -F
    // H2 = d error / d p2 = I
    Point2 prediction = f(p1);

    if(H1)
      *H1 = -F(p1);

    if(H2)
      *H2 = Matrix::Identity(dim(),dim());

    // Return the error between the prediction and the supplied value of p2
    return (p2 - prediction);
  }

};

// Create Measurement Model Factor
class NonlinearMeasurementModel : public NoiseModelFactor1<Point2> {

  typedef NoiseModelFactor1<Point2> Base;
  typedef NonlinearMeasurementModel This;

protected:
  Vector z_; 
  Matrix R_; 
  Matrix R_invsqrt_; 
public:
  NonlinearMeasurementModel(){}

  NonlinearMeasurementModel(const Symbol& TestKey, Vector z) :
    Base(noiseModel::Unit::Create(z.size()), TestKey), z_(z), R_(1,1) {

    // Initialize nonlinear measurement model parameters:
    // z(t) = H*x(t) + v
    // where v ~ N(0, noiseModel_)
    R_ << 1.0;
    R_invsqrt_ = inverse_square_root(R_);
  }

  ~NonlinearMeasurementModel() override {}

  // Calculate the predicted measurement using the nonlinear function h()
  // Byproduct: updates Jacobian H, and noiseModel (R)
  Vector h(const Point2& x_t1) const {

    // Calculate prediction
    Vector z_hat(1);
    z_hat(0) = 2*x_t1.x()*x_t1.x() - x_t1.x()*x_t1.y() - 2.5*x_t1.x() + 7*x_t1.y();

    return z_hat;
  }

  Matrix H(const Point2& x_t1) const {
    // Update Jacobian
    Matrix H(1,2);
    H(0,0) =  4*x_t1.x() - x_t1.y() - 2.5;
    H(0,1) = -1*x_t1.x() + 7;

    return H;
  }

  // Calculate the inverse square root of the motion model covariance, Q
  Matrix RInvSqrt(const Point2& x_t0) const {
    // This motion model has a constant covariance
    return R_invsqrt_;
  }

  /* print */
  void print(const std::string& s = "", const KeyFormatter& keyFormatter = DefaultKeyFormatter) const override {
    std::cout << s << ": NonlinearMeasurementModel\n";
    std::cout << "  TestKey: " << keyFormatter(key()) << std::endl;
  }

  bool equals(const NonlinearFactor& f, double tol = 1e-9) const override {
    const This *e = dynamic_cast<const This*> (&f);
    return (e != nullptr) && Base::equals(f);
  }

  double error(const Values& c) const override {
    Vector w = whitenedError(c);
    return 0.5 * w.dot(w);
  }

  size_t dim() const override {
    return 1;
  }

  Vector whitenedError(const Values& c) const {
    return RInvSqrt(c.at<Point2>(key()))*unwhitenedError(c);
  }

  boost::shared_ptr<GaussianFactor> linearize(const Values& c) const override {
    const X& x1 = c.at<X>(key());
    Matrix A1;
    Vector b = -evaluateError(x1, A1);
    SharedDiagonal constrained =
        boost::dynamic_pointer_cast<noiseModel::Constrained>(this->noiseModel_);
    if (constrained.get() != nullptr) {
      return JacobianFactor::shared_ptr(new JacobianFactor(key(), A1, b, constrained));
    }
    // "Whiten" the system before converting to a Gaussian Factor
    Matrix Rinvsqrt = RInvSqrt(x1);
    A1 = Rinvsqrt*A1;
    b = Rinvsqrt*b;
    return GaussianFactor::shared_ptr(new JacobianFactor(key(), A1, b, noiseModel::Unit::Create(b.size())));
  }

  Vector evaluateError(const Point2& p, boost::optional<Matrix&> H1 = boost::none) const override {
    // error = z - h(p)
    // H = d error / d p = -d h/ d p = -H
    Vector z_hat = h(p);

    if(H1){
      *H1 = -H(p);
    }

    // Return the error between the prediction and the supplied value of p2
    return z_ - z_hat;
  }

};


/* ************************************************************************* */
TEST( ExtendedKalmanFilter, nonlinear ) {

  // Create the set of expected output TestValues (generated using Matlab Kalman Filter)
  Point2 expected_predict[10];
  Point2 expected_update[10];
  expected_predict[0] = Point2(0.81,0.99);
  expected_update[0] =  Point2(0.824926197027,0.29509808);
  expected_predict[1] = Point2(0.680503230541,0.24343413);
  expected_update[1] =  Point2(0.773360464065,0.43518355);
  expected_predict[2] = Point2(0.598086407378,0.33655375);
  expected_update[2] =  Point2(0.908781566884,0.60766713);
  expected_predict[3] = Point2(0.825883936308,0.55223668);
  expected_update[3] =  Point2(1.23221370495,0.74372849);
  expected_predict[4] = Point2(1.51835061468,0.91643243);
  expected_update[4] =  Point2(1.32823362774,0.855855);
  expected_predict[5] = Point2(1.76420456986,1.1367754);
  expected_update[5] =  Point2(1.36378040243,1.0623832);
  expected_predict[6] = Point2(1.85989698605,1.4488574);
  expected_update[6] =  Point2(1.24068063304,1.3431964);
  expected_predict[7] = Point2(1.53928843321,1.6664778);
  expected_update[7] =  Point2(1.04229257957,1.4856408);
  expected_predict[8] = Point2(1.08637382142,1.5484724);
  expected_update[8] =  Point2(1.13201933157,1.5795507);
  expected_predict[9] = Point2(1.28146776705,1.7880819);
  expected_update[9] =  Point2(1.22159588179,1.7434982);

  // Measurements
  double z[10];
  z[0] =  1.0;
  z[1] =  2.0;
  z[2] =  3.0;
  z[3] =  4.0;
  z[4] =  5.0;
  z[5] =  6.0;
  z[6] =  7.0;
  z[7] =  8.0;
  z[8] =  9.0;
  z[9] = 10.0;

  // Create the Kalman Filter initialization point
  Point2 x_initial(0.90, 1.10);
  SharedDiagonal P_initial = noiseModel::Diagonal::Sigmas(Vector2(0.1, 0.1));

  // Create an ExtendedKalmanFilter object
  ExtendedKalmanFilter<Point2> ekf(X(0), x_initial, P_initial);

  // Enter Predict-Update Loop
  Point2 x_predict(0,0), x_update(0,0);
  for(unsigned int i = 0; i < 10; ++i){
    // Create motion factor
    NonlinearMotionModel motionFactor(X(i), X(i+1));
    x_predict = ekf.predict(motionFactor);

    // Create a measurement factor
    NonlinearMeasurementModel measurementFactor(X(i+1), (Vector(1) << z[i]).finished());
    x_update = ekf.update(measurementFactor);

    EXPECT(assert_equal(expected_predict[i],x_predict, 1e-6));
    EXPECT(assert_equal(expected_update[i], x_update,  1e-6));
  }

  return;
}


/* ************************************************************************* */
int main() { TestResult tr; return TestRegistry::runAllTests(tr); }
/* ************************************************************************* */