gtsam/gtsam/navigation/tests/testImuFactor.cpp

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

 * 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

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

/**
 * @file    testImuFactor.cpp
 * @brief   Unit test for ImuFactor
 * @author  Luca Carlone, Stephen Williams, Richard Roberts
 */

#include <gtsam/navigation/ImuFactor.h>
#include <gtsam/nonlinear/Values.h>
#include <gtsam/inference/Symbol.h>
#include <gtsam/navigation/ImuBias.h>
#include <gtsam/geometry/Pose3.h>
#include <gtsam/base/TestableAssertions.h>
#include <gtsam/base/numericalDerivative.h>
#include <CppUnitLite/TestHarness.h>

#include <boost/bind.hpp>
#include <list>

using namespace std;
using namespace gtsam;

// Convenience for named keys
using symbol_shorthand::X;
using symbol_shorthand::V;
using symbol_shorthand::B;

/* ************************************************************************* */
namespace {
// Auxiliary functions to test evaluate error in ImuFactor
/* ************************************************************************* */
Vector callEvaluateError(const ImuFactor& factor,
    const Pose3& pose_i, const Vector3& vel_i, const Pose3& pose_j, const Vector3& vel_j,
    const imuBias::ConstantBias& bias){
  return factor.evaluateError(pose_i, vel_i, pose_j, vel_j, bias);
}

Rot3 evaluateRotationError(const ImuFactor& factor,
    const Pose3& pose_i, const Vector3& vel_i, const Pose3& pose_j, const Vector3& vel_j,
    const imuBias::ConstantBias& bias){
  return Rot3::Expmap(factor.evaluateError(pose_i, vel_i, pose_j, vel_j, bias).tail(3) ) ;
}

// Auxiliary functions to test Jacobians F and G used for
// covariance propagation during preintegration
/* ************************************************************************* */
Vector updatePreintegratedPosVel(
    const Vector3 deltaPij_old, const Vector3& deltaVij_old, const Rot3& deltaRij_old,
    const Vector3& correctedAcc, const Vector3& correctedOmega, const double deltaT,
    const bool use2ndOrderIntegration_) {

  Matrix3 dRij = deltaRij_old.matrix();
  Vector3 temp = dRij * correctedAcc * deltaT;
  Vector3 deltaPij_new;
  if(!use2ndOrderIntegration_){
    deltaPij_new = deltaPij_old + deltaVij_old * deltaT;
  }else{
    deltaPij_new = deltaPij_old + deltaVij_old * deltaT + 0.5 * temp * deltaT;
  }
  Vector3 deltaVij_new = deltaVij_old + temp;

  Vector result(6);  result << deltaPij_new,  deltaVij_new;
  return result;
}

Rot3 updatePreintegratedRot(const Rot3& deltaRij_old,
    const Vector3& correctedOmega, const double deltaT) {
  Rot3 deltaRij_new = deltaRij_old * Rot3::Expmap(correctedOmega * deltaT);
  return deltaRij_new;
}

// Auxiliary functions to test preintegrated Jacobians
// delPdelBiasAcc_ delPdelBiasOmega_ delVdelBiasAcc_ delVdelBiasOmega_ delRdelBiasOmega_
/* ************************************************************************* */
double accNoiseVar = 0.01;
double omegaNoiseVar = 0.03;
double intNoiseVar = 0.0001;
ImuFactor::PreintegratedMeasurements evaluatePreintegratedMeasurements(
    const imuBias::ConstantBias& bias,
    const list<Vector3>& measuredAccs,
    const list<Vector3>& measuredOmegas,
    const list<double>& deltaTs,
    const bool use2ndOrderIntegration = false){
  ImuFactor::PreintegratedMeasurements result(bias, accNoiseVar * Matrix3::Identity(),
      omegaNoiseVar *Matrix3::Identity(), intNoiseVar * Matrix3::Identity(),use2ndOrderIntegration);

  list<Vector3>::const_iterator itAcc = measuredAccs.begin();
  list<Vector3>::const_iterator itOmega = measuredOmegas.begin();
  list<double>::const_iterator itDeltaT = deltaTs.begin();
  for( ; itAcc != measuredAccs.end(); ++itAcc, ++itOmega, ++itDeltaT) {
    result.integrateMeasurement(*itAcc, *itOmega, *itDeltaT);
  }
  return result;
}

Vector3 evaluatePreintegratedMeasurementsPosition(
    const imuBias::ConstantBias& bias,
    const list<Vector3>& measuredAccs,
    const list<Vector3>& measuredOmegas,
    const list<double>& deltaTs){
  return evaluatePreintegratedMeasurements(bias,
      measuredAccs, measuredOmegas, deltaTs).deltaPij();
}

Vector3 evaluatePreintegratedMeasurementsVelocity(
    const imuBias::ConstantBias& bias,
    const list<Vector3>& measuredAccs,
    const list<Vector3>& measuredOmegas,
    const list<double>& deltaTs)
{
  return evaluatePreintegratedMeasurements(bias,
      measuredAccs, measuredOmegas, deltaTs).deltaVij();
}

Rot3 evaluatePreintegratedMeasurementsRotation(
    const imuBias::ConstantBias& bias,
    const list<Vector3>& measuredAccs,
    const list<Vector3>& measuredOmegas,
    const list<double>& deltaTs){
  return Rot3(evaluatePreintegratedMeasurements(bias,
      measuredAccs, measuredOmegas, deltaTs).deltaRij());
}

Rot3 evaluateRotation(const Vector3 measuredOmega, const Vector3 biasOmega, const double deltaT){
  return Rot3::Expmap((measuredOmega - biasOmega) * deltaT);
}

Vector3 evaluateLogRotation(const Vector3 thetahat, const Vector3 deltatheta){
  return Rot3::Logmap( Rot3::Expmap(thetahat).compose( Rot3::Expmap(deltatheta) ) );
}

}

/* ************************************************************************* */
TEST( ImuFactor, PreintegratedMeasurements )
{
  // Linearization point
  imuBias::ConstantBias bias(Vector3(0,0,0), Vector3(0,0,0)); ///< Current estimate of acceleration and angular rate biases

  // Measurements
  Vector3 measuredAcc(0.1, 0.0, 0.0);
  Vector3 measuredOmega(M_PI/100.0, 0.0, 0.0);
  double deltaT = 0.5;

  // Expected preintegrated values
  Vector3 expectedDeltaP1; expectedDeltaP1 << 0.5*0.1*0.5*0.5, 0, 0;
  Vector3 expectedDeltaV1(0.05, 0.0, 0.0);
  Rot3 expectedDeltaR1 = Rot3::RzRyRx(0.5 * M_PI/100.0, 0.0, 0.0);
  double expectedDeltaT1(0.5);

  bool use2ndOrderIntegration = true;
  // Actual preintegrated values
  ImuFactor::PreintegratedMeasurements actual1(bias, Matrix3::Zero(), Matrix3::Zero(), Matrix3::Zero(), use2ndOrderIntegration);
  actual1.integrateMeasurement(measuredAcc, measuredOmega, deltaT);

  EXPECT(assert_equal(Vector(expectedDeltaP1), Vector(actual1.deltaPij()), 1e-6));
  EXPECT(assert_equal(Vector(expectedDeltaV1), Vector(actual1.deltaVij()), 1e-6));
  EXPECT(assert_equal(expectedDeltaR1, Rot3(actual1.deltaRij()), 1e-6));
  DOUBLES_EQUAL(expectedDeltaT1, actual1.deltaTij(), 1e-6);

  // Integrate again
  Vector3 expectedDeltaP2; expectedDeltaP2 << 0.025 + expectedDeltaP1(0) + 0.5*0.1*0.5*0.5, 0, 0;
  Vector3 expectedDeltaV2 = Vector3(0.05, 0.0, 0.0) + expectedDeltaR1.matrix() * measuredAcc * 0.5;
  Rot3 expectedDeltaR2 = Rot3::RzRyRx(2.0 * 0.5 * M_PI/100.0, 0.0, 0.0);
  double expectedDeltaT2(1);

  // Actual preintegrated values
  ImuFactor::PreintegratedMeasurements actual2 = actual1;
  actual2.integrateMeasurement(measuredAcc, measuredOmega, deltaT);

  EXPECT(assert_equal(Vector(expectedDeltaP2), Vector(actual2.deltaPij()), 1e-6));
  EXPECT(assert_equal(Vector(expectedDeltaV2), Vector(actual2.deltaVij()), 1e-6));
  EXPECT(assert_equal(expectedDeltaR2, Rot3(actual2.deltaRij()), 1e-6));
  DOUBLES_EQUAL(expectedDeltaT2, actual2.deltaTij(), 1e-6);
}

/* ************************************************************************* */
TEST( ImuFactor, ErrorAndJacobians )
{
  // Linearization point
  imuBias::ConstantBias bias; // Bias
  Pose3 x1(Rot3::RzRyRx(M_PI/12.0, M_PI/6.0, M_PI/4.0), Point3(5.0, 1.0, -50.0));
  Vector3 v1(Vector3(0.5, 0.0, 0.0));
  Pose3 x2(Rot3::RzRyRx(M_PI/12.0 + M_PI/100.0, M_PI/6.0, M_PI/4.0), Point3(5.5, 1.0, -50.0));
  Vector3 v2(Vector3(0.5, 0.0, 0.0));

  // Measurements
  Vector3 gravity; gravity << 0, 0, 9.81;
  Vector3 omegaCoriolis; omegaCoriolis << 0, 0, 0;
  Vector3 measuredOmega; measuredOmega << M_PI/100, 0, 0;
  Vector3 measuredAcc = x1.rotation().unrotate(-Point3(gravity)).vector();
  double deltaT = 1.0;
  bool use2ndOrderIntegration = true;
  ImuFactor::PreintegratedMeasurements pre_int_data(bias, Matrix3::Zero(), Matrix3::Zero(), Matrix3::Zero(), use2ndOrderIntegration);
  pre_int_data.integrateMeasurement(measuredAcc, measuredOmega, deltaT);

  // Create factor
  ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pre_int_data, gravity, omegaCoriolis);

  Vector errorActual = factor.evaluateError(x1, v1, x2, v2, bias);

  // Expected error
  Vector errorExpected(9); errorExpected << 0, 0, 0, 0, 0, 0, 0, 0, 0;
  EXPECT(assert_equal(errorExpected, errorActual, 1e-6));

  // Actual Jacobians
  Matrix H1a, H2a, H3a, H4a, H5a;
  (void) factor.evaluateError(x1, v1, x2, v2, bias, H1a, H2a, H3a, H4a, H5a);

  // Expected Jacobians
  /////////////////// H1 ///////////////////////////
  Matrix H1e = numericalDerivative11<Vector,Pose3>(
      boost::bind(&callEvaluateError, factor, _1, v1, x2, v2, bias), x1);
  // Jacobians are around zero, so the rotation part is the same as:
  Matrix H1Rot3 = numericalDerivative11<Rot3,Pose3>(
      boost::bind(&evaluateRotationError, factor, _1, v1, x2, v2, bias), x1);
  EXPECT(assert_equal(H1Rot3, H1e.bottomRows(3)));
  EXPECT(assert_equal(H1e, H1a));

  /////////////////// H2 ///////////////////////////
  Matrix H2e = numericalDerivative11<Vector,Vector3>(
      boost::bind(&callEvaluateError, factor, x1, _1, x2, v2, bias), v1);
  EXPECT(assert_equal(H2e, H2a));

  /////////////////// H3 ///////////////////////////
  Matrix H3e = numericalDerivative11<Vector,Pose3>(
      boost::bind(&callEvaluateError, factor, x1, v1, _1, v2, bias), x2);
  // Jacobians are around zero, so the rotation part is the same as:
  Matrix H3Rot3 = numericalDerivative11<Rot3,Pose3>(
      boost::bind(&evaluateRotationError, factor, x1, v1, _1, v2, bias), x2);
  EXPECT(assert_equal(H3Rot3, H3e.bottomRows(3)));
  EXPECT(assert_equal(H3e, H3a));

  /////////////////// H4 ///////////////////////////
  Matrix H4e = numericalDerivative11<Vector,Vector3>(
      boost::bind(&callEvaluateError, factor, x1, v1, x2, _1, bias), v2);
  EXPECT(assert_equal(H4e, H4a));

  /////////////////// H5 ///////////////////////////
  Matrix H5e = numericalDerivative11<Vector,imuBias::ConstantBias>(
      boost::bind(&callEvaluateError, factor, x1, v1, x2, v2, _1), bias);
  EXPECT(assert_equal(H5e, H5a));
}

/* ************************************************************************* */
TEST( ImuFactor, ErrorAndJacobianWithBiases )
{
  imuBias::ConstantBias bias(Vector3(0.2, 0, 0), Vector3(0.1, 0, 0.3)); // Biases (acc, rot)
  Pose3 x1(Rot3::RzRyRx(M_PI/12.0, M_PI/6.0, M_PI/10.0), Point3(5.0, 1.0, -50.0));
  Vector3 v1(Vector3(0.5, 0.0, 0.0));
  Pose3 x2(Rot3::Expmap(Vector3(0, 0, M_PI/10.0 + M_PI/10.0)), Point3(5.5, 1.0, -50.0));
  Vector3 v2(Vector3(0.5, 0.0, 0.0));

  // Measurements
  Vector3 gravity; gravity << 0, 0, 9.81;
  Vector3 omegaCoriolis; omegaCoriolis << 0, 0.1, 0.1;
  Vector3 measuredOmega; measuredOmega << 0, 0, M_PI/10.0+0.3;
  Vector3 measuredAcc = x1.rotation().unrotate(-Point3(gravity)).vector() + Vector3(0.2,0.0,0.0);
  double deltaT = 1.0;

  ImuFactor::PreintegratedMeasurements pre_int_data(imuBias::ConstantBias(Vector3(0.2, 0.0, 0.0),
      Vector3(0.0, 0.0, 0.1)), Matrix3::Zero(), Matrix3::Zero(), Matrix3::Zero());
  pre_int_data.integrateMeasurement(measuredAcc, measuredOmega, deltaT);

  // Create factor
  ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pre_int_data, gravity, omegaCoriolis);

  SETDEBUG("ImuFactor evaluateError", false);
  Vector errorActual = factor.evaluateError(x1, v1, x2, v2, bias);
  SETDEBUG("ImuFactor evaluateError", false);

  // Expected error (should not be zero in this test, as we want to evaluate Jacobians
  // at a nontrivial linearization point)
  // Vector errorExpected(9); errorExpected << 0, 0, 0, 0, 0, 0, 0, 0, 0;
  // EXPECT(assert_equal(errorExpected, errorActual, 1e-6));

  // Expected Jacobians
  Matrix H1e = numericalDerivative11<Vector,Pose3>(
      boost::bind(&callEvaluateError, factor, _1, v1, x2, v2, bias), x1);
  Matrix H2e = numericalDerivative11<Vector,Vector3>(
      boost::bind(&callEvaluateError, factor, x1, _1, x2, v2, bias), v1);
  Matrix H3e = numericalDerivative11<Vector,Pose3>(
      boost::bind(&callEvaluateError, factor, x1, v1, _1, v2, bias), x2);
  Matrix H4e = numericalDerivative11<Vector,Vector3>(
      boost::bind(&callEvaluateError, factor, x1, v1, x2, _1, bias), v2);
  Matrix H5e = numericalDerivative11<Vector,imuBias::ConstantBias>(
      boost::bind(&callEvaluateError, factor, x1, v1, x2, v2, _1), bias);

  // Check rotation Jacobians
  Matrix RH1e = numericalDerivative11<Rot3,Pose3>(
      boost::bind(&evaluateRotationError, factor, _1, v1, x2, v2, bias), x1);
  Matrix RH3e = numericalDerivative11<Rot3,Pose3>(
      boost::bind(&evaluateRotationError, factor, x1, v1, _1, v2, bias), x2);
  Matrix RH5e = numericalDerivative11<Rot3,imuBias::ConstantBias>(
      boost::bind(&evaluateRotationError, factor, x1, v1, x2, v2, _1), bias);

  // Actual Jacobians
  Matrix H1a, H2a, H3a, H4a, H5a;
  (void) factor.evaluateError(x1, v1, x2, v2, bias, H1a, H2a, H3a, H4a, H5a);

  EXPECT(assert_equal(H1e, H1a));
  EXPECT(assert_equal(H2e, H2a));
  EXPECT(assert_equal(H3e, H3a));
  EXPECT(assert_equal(H4e, H4a));
  EXPECT(assert_equal(H5e, H5a));
}

/* ************************************************************************* */
TEST( ImuFactor, ErrorAndJacobianWith2ndOrderCoriolis )
{
  imuBias::ConstantBias bias(Vector3(0.2, 0, 0), Vector3(0.1, 0, 0.3)); // Biases (acc, rot)
  Pose3 x1(Rot3::RzRyRx(M_PI/12.0, M_PI/6.0, M_PI/10.0), Point3(5.0, 1.0, -50.0));
  Vector3 v1(Vector3(0.5, 0.0, 0.0));
  Pose3 x2(Rot3::Expmap(Vector3(0, 0, M_PI/10.0 + M_PI/10.0)), Point3(5.5, 1.0, -50.0));
  Vector3 v2(Vector3(0.5, 0.0, 0.0));

  // Measurements
  Vector3 gravity; gravity << 0, 0, 9.81;
  Vector3 omegaCoriolis; omegaCoriolis << 0, 0.1, 0.1;
  Vector3 measuredOmega; measuredOmega << 0, 0, M_PI/10.0+0.3;
  Vector3 measuredAcc = x1.rotation().unrotate(-Point3(gravity)).vector() + Vector3(0.2,0.0,0.0);
  double deltaT = 1.0;

  ImuFactor::PreintegratedMeasurements pre_int_data(imuBias::ConstantBias(Vector3(0.2, 0.0, 0.0),
      Vector3(0.0, 0.0, 0.1)), Matrix3::Zero(), Matrix3::Zero(), Matrix3::Zero());
  pre_int_data.integrateMeasurement(measuredAcc, measuredOmega, deltaT);

  // Create factor
  Pose3 bodyPsensor = Pose3();
  bool use2ndOrderCoriolis = true;
  ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pre_int_data, gravity, omegaCoriolis, bodyPsensor, use2ndOrderCoriolis);

  SETDEBUG("ImuFactor evaluateError", false);
  Vector errorActual = factor.evaluateError(x1, v1, x2, v2, bias);
  SETDEBUG("ImuFactor evaluateError", false);

  // Expected error (should not be zero in this test, as we want to evaluate Jacobians
  // at a nontrivial linearization point)
  // Vector errorExpected(9); errorExpected << 0, 0, 0, 0, 0, 0, 0, 0, 0;
  // EXPECT(assert_equal(errorExpected, errorActual, 1e-6));

  // Expected Jacobians
  Matrix H1e = numericalDerivative11<Vector,Pose3>(
      boost::bind(&callEvaluateError, factor, _1, v1, x2, v2, bias), x1);
  Matrix H2e = numericalDerivative11<Vector,Vector3>(
      boost::bind(&callEvaluateError, factor, x1, _1, x2, v2, bias), v1);
  Matrix H3e = numericalDerivative11<Vector,Pose3>(
      boost::bind(&callEvaluateError, factor, x1, v1, _1, v2, bias), x2);
  Matrix H4e = numericalDerivative11<Vector,Vector3>(
      boost::bind(&callEvaluateError, factor, x1, v1, x2, _1, bias), v2);
  Matrix H5e = numericalDerivative11<Vector,imuBias::ConstantBias>(
      boost::bind(&callEvaluateError, factor, x1, v1, x2, v2, _1), bias);

  // Check rotation Jacobians
  Matrix RH1e = numericalDerivative11<Rot3,Pose3>(
      boost::bind(&evaluateRotationError, factor, _1, v1, x2, v2, bias), x1);
  Matrix RH3e = numericalDerivative11<Rot3,Pose3>(
      boost::bind(&evaluateRotationError, factor, x1, v1, _1, v2, bias), x2);
  Matrix RH5e = numericalDerivative11<Rot3,imuBias::ConstantBias>(
      boost::bind(&evaluateRotationError, factor, x1, v1, x2, v2, _1), bias);

  // Actual Jacobians
  Matrix H1a, H2a, H3a, H4a, H5a;
  (void) factor.evaluateError(x1, v1, x2, v2, bias, H1a, H2a, H3a, H4a, H5a);

  EXPECT(assert_equal(H1e, H1a));
  EXPECT(assert_equal(H2e, H2a));
  EXPECT(assert_equal(H3e, H3a));
  EXPECT(assert_equal(H4e, H4a));
  EXPECT(assert_equal(H5e, H5a));
}

/* ************************************************************************* */
TEST( ImuFactor, PartialDerivative_wrt_Bias )
{
  // Linearization point
  Vector3 biasOmega; biasOmega << 0,0,0; ///< Current estimate of rotation rate bias

  // Measurements
  Vector3 measuredOmega; measuredOmega << 0.1, 0, 0;
  double deltaT = 0.5;

  // Compute numerical derivatives
  Matrix expectedDelRdelBiasOmega = numericalDerivative11<Rot3, Vector3>(boost::bind(
      &evaluateRotation, measuredOmega, _1, deltaT), Vector3(biasOmega));

  const Matrix3 Jr = Rot3::ExpmapDerivative((measuredOmega - biasOmega) * deltaT);

   Matrix3  actualdelRdelBiasOmega = - Jr * deltaT; // the delta bias appears with the minus sign

  // Compare Jacobians
  EXPECT(assert_equal(expectedDelRdelBiasOmega, actualdelRdelBiasOmega, 1e-3));  // 1e-3 needs to be added only when using quaternions for rotations
}

/* ************************************************************************* */
TEST( ImuFactor, PartialDerivativeLogmap )
{
  // Linearization point
  Vector3 thetahat; thetahat << 0.1,0.1,0; ///< Current estimate of rotation rate bias

  // Measurements
  Vector3 deltatheta; deltatheta << 0, 0, 0;

  // Compute numerical derivatives
  Matrix expectedDelFdeltheta = numericalDerivative11<Vector,Vector3>(boost::bind(
      &evaluateLogRotation, thetahat, _1), Vector3(deltatheta));

  Matrix3 actualDelFdeltheta = Rot3::LogmapDerivative(thetahat);

  // Compare Jacobians
  EXPECT(assert_equal(expectedDelFdeltheta, actualDelFdeltheta));
}

/* ************************************************************************* */
TEST( ImuFactor, fistOrderExponential )
{
  // Linearization point
  Vector3 biasOmega; biasOmega << 0,0,0; ///< Current estimate of rotation rate bias

  // Measurements
  Vector3 measuredOmega; measuredOmega << 0.1, 0, 0;
  double deltaT = 1.0;

  // change w.r.t. linearization point
  double alpha = 0.0;
  Vector3 deltabiasOmega; deltabiasOmega << alpha,alpha,alpha;

  const Matrix3 Jr = Rot3::ExpmapDerivative((measuredOmega - biasOmega) * deltaT);

  Matrix3  delRdelBiasOmega = - Jr * deltaT; // the delta bias appears with the minus sign

  const Matrix expectedRot = Rot3::Expmap((measuredOmega - biasOmega - deltabiasOmega) * deltaT).matrix();

  const Matrix3 hatRot = Rot3::Expmap((measuredOmega - biasOmega) * deltaT).matrix();
  const Matrix3 actualRot =
      hatRot * Rot3::Expmap(delRdelBiasOmega * deltabiasOmega).matrix();
  //hatRot * (Matrix3::Identity() + skewSymmetric(delRdelBiasOmega * deltabiasOmega));

  // This is a first order expansion so the equality is only an approximation
  EXPECT(assert_equal(expectedRot, actualRot));
}

/* ************************************************************************* */
TEST( ImuFactor, FirstOrderPreIntegratedMeasurements )
{
  // Linearization point
  imuBias::ConstantBias bias; ///< Current estimate of acceleration and rotation rate biases

  Pose3 body_P_sensor(Rot3::Expmap(Vector3(0,0.1,0.1)), Point3(1, 0, 1));

  // Measurements
  list<Vector3> measuredAccs, measuredOmegas;
  list<double> deltaTs;
  measuredAccs.push_back(Vector3(0.1, 0.0, 0.0));
  measuredOmegas.push_back(Vector3(M_PI/100.0, 0.0, 0.0));
  deltaTs.push_back(0.01);
  measuredAccs.push_back(Vector3(0.1, 0.0, 0.0));
  measuredOmegas.push_back(Vector3(M_PI/100.0, 0.0, 0.0));
  deltaTs.push_back(0.01);
  for(int i=1;i<100;i++)
  {
    measuredAccs.push_back(Vector3(0.05, 0.09, 0.01));
    measuredOmegas.push_back(Vector3(M_PI/100.0, M_PI/300.0, 2*M_PI/100.0));
    deltaTs.push_back(0.01);
  }

  // Actual preintegrated values
  ImuFactor::PreintegratedMeasurements preintegrated =
      evaluatePreintegratedMeasurements(bias, measuredAccs, measuredOmegas, deltaTs);

  // Compute numerical derivatives
  Matrix expectedDelPdelBias = numericalDerivative11<Vector,imuBias::ConstantBias>(
      boost::bind(&evaluatePreintegratedMeasurementsPosition, _1, measuredAccs, measuredOmegas, deltaTs), bias);
  Matrix expectedDelPdelBiasAcc   = expectedDelPdelBias.leftCols(3);
  Matrix expectedDelPdelBiasOmega = expectedDelPdelBias.rightCols(3);

  Matrix expectedDelVdelBias = numericalDerivative11<Vector,imuBias::ConstantBias>(
      boost::bind(&evaluatePreintegratedMeasurementsVelocity, _1, measuredAccs, measuredOmegas, deltaTs), bias);
  Matrix expectedDelVdelBiasAcc   = expectedDelVdelBias.leftCols(3);
  Matrix expectedDelVdelBiasOmega = expectedDelVdelBias.rightCols(3);

  Matrix expectedDelRdelBias = numericalDerivative11<Rot3,imuBias::ConstantBias>(
      boost::bind(&evaluatePreintegratedMeasurementsRotation, _1, measuredAccs, measuredOmegas, deltaTs), bias);
  Matrix expectedDelRdelBiasAcc   = expectedDelRdelBias.leftCols(3);
  Matrix expectedDelRdelBiasOmega = expectedDelRdelBias.rightCols(3);

  // Compare Jacobians
  EXPECT(assert_equal(expectedDelPdelBiasAcc, preintegrated.delPdelBiasAcc()));
  EXPECT(assert_equal(expectedDelPdelBiasOmega, preintegrated.delPdelBiasOmega()));
  EXPECT(assert_equal(expectedDelVdelBiasAcc, preintegrated.delVdelBiasAcc()));
  EXPECT(assert_equal(expectedDelVdelBiasOmega, preintegrated.delVdelBiasOmega()));
  EXPECT(assert_equal(expectedDelRdelBiasAcc, Matrix::Zero(3,3)));
  EXPECT(assert_equal(expectedDelRdelBiasOmega, preintegrated.delRdelBiasOmega(), 1e-3)); // 1e-3 needs to be added only when using quaternions for rotations
}

/* ************************************************************************* */
TEST( ImuFactor, JacobianPreintegratedCovariancePropagation )
{
  // Linearization point
  imuBias::ConstantBias bias; ///< Current estimate of acceleration and rotation rate biases
  Pose3 body_P_sensor = Pose3(); // (Rot3::Expmap(Vector3(0,0.1,0.1)), Point3(1, 0, 1));

  // Measurements
  list<Vector3> measuredAccs, measuredOmegas;
  list<double> deltaTs;
  measuredAccs.push_back(Vector3(0.1, 0.0, 0.0));
  measuredOmegas.push_back(Vector3(M_PI/100.0, 0.0, 0.0));
  deltaTs.push_back(0.01);
  measuredAccs.push_back(Vector3(0.1, 0.0, 0.0));
  measuredOmegas.push_back(Vector3(M_PI/100.0, 0.0, 0.0));
  deltaTs.push_back(0.01);
  for(int i=1;i<100;i++)
  {
    measuredAccs.push_back(Vector3(0.05, 0.09, 0.01));
    measuredOmegas.push_back(Vector3(M_PI/100.0, M_PI/300.0, 2*M_PI/100.0));
    deltaTs.push_back(0.01);
  }
  bool use2ndOrderIntegration = false;
  // Actual preintegrated values
  ImuFactor::PreintegratedMeasurements preintegrated =
      evaluatePreintegratedMeasurements(bias, measuredAccs, measuredOmegas, deltaTs, use2ndOrderIntegration);

  // so far we only created a nontrivial linearization point for the preintegrated measurements
  // Now we add a new measurement and ask for Jacobians
  const Vector3 newMeasuredAcc = Vector3(0.1, 0.0, 0.0);
  const Vector3 newMeasuredOmega = Vector3(M_PI/100.0, 0.0, 0.0);
  const double newDeltaT = 0.01;
  const Rot3    deltaRij_old = preintegrated.deltaRij();    // before adding new measurement
  const Vector3 deltaVij_old = preintegrated.deltaVij();    // before adding new measurement
  const Vector3 deltaPij_old = preintegrated.deltaPij();    // before adding new measurement

  Matrix oldPreintCovariance = preintegrated.preintMeasCov();

  Matrix Factual, Gactual;
  preintegrated.integrateMeasurement(newMeasuredAcc, newMeasuredOmega, newDeltaT,
      body_P_sensor, Factual, Gactual);

  //////////////////////////////////////////////////////////////////////////////////////////////
  // COMPUTE NUMERICAL DERIVATIVES FOR F
  //////////////////////////////////////////////////////////////////////////////////////////////
  // Compute expected f_pos_vel wrt positions
  Matrix dfpv_dpos =
      numericalDerivative11<Vector, Vector3>(boost::bind(&updatePreintegratedPosVel,
          _1, deltaVij_old, deltaRij_old,
          newMeasuredAcc, newMeasuredOmega, newDeltaT, use2ndOrderIntegration), deltaPij_old);

  // Compute expected f_pos_vel wrt velocities
  Matrix dfpv_dvel =
      numericalDerivative11<Vector, Vector3>(boost::bind(&updatePreintegratedPosVel,
          deltaPij_old, _1, deltaRij_old,
          newMeasuredAcc, newMeasuredOmega, newDeltaT, use2ndOrderIntegration), deltaVij_old);

  // Compute expected f_pos_vel wrt angles
  Matrix dfpv_dangle =
      numericalDerivative11<Vector, Rot3>(boost::bind(&updatePreintegratedPosVel,
          deltaPij_old, deltaVij_old, _1,
          newMeasuredAcc, newMeasuredOmega, newDeltaT, use2ndOrderIntegration), deltaRij_old);

  Matrix FexpectedTop6(6,9);   FexpectedTop6 << dfpv_dpos, dfpv_dvel, dfpv_dangle;

  // Compute expected f_rot wrt angles
  Matrix dfr_dangle =
      numericalDerivative11<Rot3, Rot3>(boost::bind(&updatePreintegratedRot,
          _1, newMeasuredOmega, newDeltaT), deltaRij_old);

  Matrix FexpectedBottom3(3,9);
  FexpectedBottom3 << Z_3x3, Z_3x3, dfr_dangle;
  Matrix Fexpected(9,9);  Fexpected << FexpectedTop6, FexpectedBottom3;

  EXPECT(assert_equal(Fexpected, Factual));

  //////////////////////////////////////////////////////////////////////////////////////////////
  // COMPUTE NUMERICAL DERIVATIVES FOR G
  //////////////////////////////////////////////////////////////////////////////////////////////
  // Compute jacobian wrt integration noise
  Matrix dgpv_dintNoise(6,3);
  dgpv_dintNoise << I_3x3 * newDeltaT, Z_3x3;

  // Compute jacobian wrt acc noise
  Matrix dgpv_daccNoise =
      numericalDerivative11<Vector, Vector3>(boost::bind(&updatePreintegratedPosVel,
          deltaPij_old, deltaVij_old, deltaRij_old,
          _1, newMeasuredOmega, newDeltaT, use2ndOrderIntegration), newMeasuredAcc);

  // Compute expected F wrt gyro noise
  Matrix dgpv_domegaNoise =
      numericalDerivative11<Vector, Vector3>(boost::bind(&updatePreintegratedPosVel,
          deltaPij_old, deltaVij_old, deltaRij_old,
          newMeasuredAcc, _1, newDeltaT, use2ndOrderIntegration), newMeasuredOmega);
  Matrix GexpectedTop6(6,9);
  GexpectedTop6 << dgpv_dintNoise, dgpv_daccNoise, dgpv_domegaNoise;

  // Compute expected f_rot wrt gyro noise
  Matrix dgr_dangle =
      numericalDerivative11<Rot3, Vector3>(boost::bind(&updatePreintegratedRot,
          deltaRij_old, _1, newDeltaT), newMeasuredOmega);

  Matrix GexpectedBottom3(3,9);
  GexpectedBottom3 << Z_3x3, Z_3x3, dgr_dangle;
  Matrix Gexpected(9,9);  Gexpected << GexpectedTop6, GexpectedBottom3;

  EXPECT(assert_equal(Gexpected, Gactual));

  // Check covariance propagation
  Matrix9 measurementCovariance;
  measurementCovariance << intNoiseVar*I_3x3, Z_3x3,             Z_3x3,
                           Z_3x3,             accNoiseVar*I_3x3, Z_3x3,
                           Z_3x3,             Z_3x3,             omegaNoiseVar*I_3x3;

  Matrix newPreintCovarianceExpected = Factual * oldPreintCovariance * Factual.transpose() +
      (1/newDeltaT) * Gactual * measurementCovariance * Gactual.transpose();

  Matrix newPreintCovarianceActual = preintegrated.preintMeasCov();
  EXPECT(assert_equal(newPreintCovarianceExpected, newPreintCovarianceActual));
}

/* ************************************************************************* */
TEST( ImuFactor, JacobianPreintegratedCovariancePropagation_2ndOrderInt )
{
  // Linearization point
  imuBias::ConstantBias bias; ///< Current estimate of acceleration and rotation rate biases
  Pose3 body_P_sensor = Pose3(); // (Rot3::Expmap(Vector3(0,0.1,0.1)), Point3(1, 0, 1));

  // Measurements
  list<Vector3> measuredAccs, measuredOmegas;
  list<double> deltaTs;
  measuredAccs.push_back(Vector3(0.1, 0.0, 0.0));
  measuredOmegas.push_back(Vector3(M_PI/100.0, 0.0, 0.0));
  deltaTs.push_back(0.01);
  measuredAccs.push_back(Vector3(0.1, 0.0, 0.0));
  measuredOmegas.push_back(Vector3(M_PI/100.0, 0.0, 0.0));
  deltaTs.push_back(0.01);
  for(int i=1;i<100;i++)
  {
    measuredAccs.push_back(Vector3(0.05, 0.09, 0.01));
    measuredOmegas.push_back(Vector3(M_PI/100.0, M_PI/300.0, 2*M_PI/100.0));
    deltaTs.push_back(0.01);
  }
  bool use2ndOrderIntegration = true;
  // Actual preintegrated values
  ImuFactor::PreintegratedMeasurements preintegrated =
      evaluatePreintegratedMeasurements(bias, measuredAccs, measuredOmegas, deltaTs, use2ndOrderIntegration);

  // so far we only created a nontrivial linearization point for the preintegrated measurements
  // Now we add a new measurement and ask for Jacobians
  const Vector3 newMeasuredAcc = Vector3(0.1, 0.0, 0.0);
  const Vector3 newMeasuredOmega = Vector3(M_PI/100.0, 0.0, 0.0);
  const double newDeltaT = 0.01;
  const Rot3    deltaRij_old = preintegrated.deltaRij();    // before adding new measurement
  const Vector3 deltaVij_old = preintegrated.deltaVij();    // before adding new measurement
  const Vector3 deltaPij_old = preintegrated.deltaPij();    // before adding new measurement

  Matrix oldPreintCovariance = preintegrated.preintMeasCov();

  Matrix Factual, Gactual;
  preintegrated.integrateMeasurement(newMeasuredAcc, newMeasuredOmega, newDeltaT,
      body_P_sensor, Factual, Gactual);

  //////////////////////////////////////////////////////////////////////////////////////////////
  // COMPUTE NUMERICAL DERIVATIVES FOR F
  //////////////////////////////////////////////////////////////////////////////////////////////
  // Compute expected f_pos_vel wrt positions
  Matrix dfpv_dpos =
      numericalDerivative11<Vector, Vector3>(boost::bind(&updatePreintegratedPosVel,
          _1, deltaVij_old, deltaRij_old,
          newMeasuredAcc, newMeasuredOmega, newDeltaT, use2ndOrderIntegration), deltaPij_old);

  // Compute expected f_pos_vel wrt velocities
  Matrix dfpv_dvel =
      numericalDerivative11<Vector, Vector3>(boost::bind(&updatePreintegratedPosVel,
          deltaPij_old, _1, deltaRij_old,
          newMeasuredAcc, newMeasuredOmega, newDeltaT, use2ndOrderIntegration), deltaVij_old);

  // Compute expected f_pos_vel wrt angles
  Matrix dfpv_dangle =
      numericalDerivative11<Vector, Rot3>(boost::bind(&updatePreintegratedPosVel,
          deltaPij_old, deltaVij_old, _1,
          newMeasuredAcc, newMeasuredOmega, newDeltaT, use2ndOrderIntegration), deltaRij_old);

  Matrix FexpectedTop6(6,9);   FexpectedTop6 << dfpv_dpos, dfpv_dvel, dfpv_dangle;

  // Compute expected f_rot wrt angles
  Matrix dfr_dangle =
      numericalDerivative11<Rot3, Rot3>(boost::bind(&updatePreintegratedRot,
          _1, newMeasuredOmega, newDeltaT), deltaRij_old);

  Matrix FexpectedBottom3(3,9);
  FexpectedBottom3 << Z_3x3, Z_3x3, dfr_dangle;
  Matrix Fexpected(9,9);  Fexpected << FexpectedTop6, FexpectedBottom3;

  EXPECT(assert_equal(Fexpected, Factual));

  //////////////////////////////////////////////////////////////////////////////////////////////
  // COMPUTE NUMERICAL DERIVATIVES FOR G
  //////////////////////////////////////////////////////////////////////////////////////////////
  // Compute jacobian wrt integration noise
  Matrix dgpv_dintNoise(6,3);
  dgpv_dintNoise << I_3x3 * newDeltaT, Z_3x3;

  // Compute jacobian wrt acc noise
  Matrix dgpv_daccNoise =
      numericalDerivative11<Vector, Vector3>(boost::bind(&updatePreintegratedPosVel,
          deltaPij_old, deltaVij_old, deltaRij_old,
          _1, newMeasuredOmega, newDeltaT, use2ndOrderIntegration), newMeasuredAcc);

  // Compute expected F wrt gyro noise
  Matrix dgpv_domegaNoise =
      numericalDerivative11<Vector, Vector3>(boost::bind(&updatePreintegratedPosVel,
          deltaPij_old, deltaVij_old, deltaRij_old,
          newMeasuredAcc, _1, newDeltaT, use2ndOrderIntegration), newMeasuredOmega);
  Matrix GexpectedTop6(6,9);
  GexpectedTop6 << dgpv_dintNoise, dgpv_daccNoise, dgpv_domegaNoise;

  // Compute expected f_rot wrt gyro noise
  Matrix dgr_dangle =
      numericalDerivative11<Rot3, Vector3>(boost::bind(&updatePreintegratedRot,
          deltaRij_old, _1, newDeltaT), newMeasuredOmega);

  Matrix GexpectedBottom3(3,9);
  GexpectedBottom3 << Z_3x3, Z_3x3, dgr_dangle;
  Matrix Gexpected(9,9);  Gexpected << GexpectedTop6, GexpectedBottom3;

  EXPECT(assert_equal(Gexpected, Gactual));

  // Check covariance propagation
  Matrix9 measurementCovariance;
  measurementCovariance << intNoiseVar*I_3x3, Z_3x3,             Z_3x3,
                           Z_3x3,             accNoiseVar*I_3x3, Z_3x3,
                           Z_3x3,             Z_3x3,             omegaNoiseVar*I_3x3;

  Matrix newPreintCovarianceExpected = Factual * oldPreintCovariance * Factual.transpose() +
      (1/newDeltaT) * Gactual * measurementCovariance * Gactual.transpose();

  Matrix newPreintCovarianceActual = preintegrated.preintMeasCov();
  EXPECT(assert_equal(newPreintCovarianceExpected, newPreintCovarianceActual));
}

//#include <gtsam/linear/GaussianFactorGraph.h>
///* ************************************************************************* */
//TEST( ImuFactor, LinearizeTiming)
//{
//  // Linearization point
//  Pose3 x1(Rot3::RzRyRx(M_PI/12.0, M_PI/6.0, M_PI/4.0), Point3(5.0, 1.0, -50.0));
//  Vector3 v1(Vector3(0.5, 0.0, 0.0));
//  Pose3 x2(Rot3::RzRyRx(M_PI/12.0 + M_PI/100.0, M_PI/6.0, M_PI/4.0), Point3(5.5, 1.0, -50.0));
//  Vector3 v2(Vector3(0.5, 0.0, 0.0));
//  imuBias::ConstantBias bias(Vector3(0.001, 0.002, 0.008), Vector3(0.002, 0.004, 0.012));
//
//  // Pre-integrator
//  imuBias::ConstantBias biasHat(Vector3(0, 0, 0.10), Vector3(0, 0, 0.10));
//  Vector3 gravity; gravity << 0, 0, 9.81;
//  Vector3 omegaCoriolis; omegaCoriolis << 0.0001, 0, 0.01;
//  ImuFactor::PreintegratedMeasurements pre_int_data(biasHat, Matrix3::Identity(), Matrix3::Identity(), Matrix3::Identity());
//
//  // Pre-integrate Measurements
//  Vector3 measuredAcc(0.1, 0.0, 0.0);
//  Vector3 measuredOmega(M_PI/100.0, 0.0, 0.0);
//  double deltaT = 0.5;
//  for(size_t i = 0; i < 50; ++i) {
//    pre_int_data.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
//  }
//
//  // Create factor
//  noiseModel::Base::shared_ptr model = noiseModel::Gaussian::Covariance(pre_int_data.MeasurementCovariance());
//  ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pre_int_data, gravity, omegaCoriolis);
//
//  Values values;
//  values.insert(X(1), x1);
//  values.insert(X(2), x2);
//  values.insert(V(1), v1);
//  values.insert(V(2), v2);
//  values.insert(B(1), bias);
//
//  Ordering ordering;
//  ordering.push_back(X(1));
//  ordering.push_back(V(1));
//  ordering.push_back(X(2));
//  ordering.push_back(V(2));
//  ordering.push_back(B(1));
//
//  GaussianFactorGraph graph;
//  gttic_(LinearizeTiming);
//  for(size_t i = 0; i < 100000; ++i) {
//    GaussianFactor::shared_ptr g = factor.linearize(values, ordering);
//    graph.push_back(g);
//  }
//  gttoc_(LinearizeTiming);
//  tictoc_finishedIteration_();
//  std::cout << "Linear Error: " << graph.error(values.zeroVectors(ordering)) << std::endl;
//  tictoc_print_();
//}

/* ************************************************************************* */
TEST( ImuFactor, ErrorWithBiasesAndSensorBodyDisplacement )
{

  imuBias::ConstantBias bias(Vector3(0.2, 0, 0), Vector3(0, 0, 0.3)); // Biases (acc, rot)
  Pose3 x1(Rot3::Expmap(Vector3(0, 0, M_PI/4.0)), Point3(5.0, 1.0, -50.0));
  Vector3 v1(Vector3(0.5, 0.0, 0.0));
  Pose3 x2(Rot3::Expmap(Vector3(0, 0, M_PI/4.0 + M_PI/10.0)), Point3(5.5, 1.0, -50.0));
  Vector3 v2(Vector3(0.5, 0.0, 0.0));

  // Measurements
  Vector3 gravity; gravity << 0, 0, 9.81;
  Vector3 omegaCoriolis; omegaCoriolis << 0, 0.1, 0.1;
  Vector3 measuredOmega; measuredOmega << 0, 0, M_PI/10.0+0.3;
  Vector3 measuredAcc = x1.rotation().unrotate(-Point3(gravity)).vector() + Vector3(0.2,0.0,0.0);
  double deltaT = 1.0;

  const Pose3 body_P_sensor(Rot3::Expmap(Vector3(0,0.10,0.10)), Point3(1,0,0));

  ImuFactor::PreintegratedMeasurements pre_int_data(imuBias::ConstantBias(Vector3(0.2, 0.0, 0.0),
        Vector3(0.0, 0.0, 0.0)), Matrix3::Zero(), Matrix3::Zero(), Matrix3::Zero());

  pre_int_data.integrateMeasurement(measuredAcc, measuredOmega, deltaT);

    // Create factor
    ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pre_int_data, gravity, omegaCoriolis);

    // Expected Jacobians
    Matrix H1e = numericalDerivative11<Vector,Pose3>(
        boost::bind(&callEvaluateError, factor, _1, v1, x2, v2, bias), x1);
    Matrix H2e = numericalDerivative11<Vector,Vector3>(
        boost::bind(&callEvaluateError, factor, x1, _1, x2, v2, bias), v1);
    Matrix H3e = numericalDerivative11<Vector,Pose3>(
        boost::bind(&callEvaluateError, factor, x1, v1, _1, v2, bias), x2);
    Matrix H4e = numericalDerivative11<Vector,Vector3>(
        boost::bind(&callEvaluateError, factor, x1, v1, x2, _1, bias), v2);
    Matrix H5e = numericalDerivative11<Vector,imuBias::ConstantBias>(
        boost::bind(&callEvaluateError, factor, x1, v1, x2, v2, _1), bias);

    // Check rotation Jacobians
    Matrix RH1e = numericalDerivative11<Rot3,Pose3>(
        boost::bind(&evaluateRotationError, factor, _1, v1, x2, v2, bias), x1);
    Matrix RH3e = numericalDerivative11<Rot3,Pose3>(
        boost::bind(&evaluateRotationError, factor, x1, v1, _1, v2, bias), x2);
    Matrix RH5e = numericalDerivative11<Rot3,imuBias::ConstantBias>(
        boost::bind(&evaluateRotationError, factor, x1, v1, x2, v2, _1), bias);

    // Actual Jacobians
    Matrix H1a, H2a, H3a, H4a, H5a;
    (void) factor.evaluateError(x1, v1, x2, v2, bias, H1a, H2a, H3a, H4a, H5a);

    EXPECT(assert_equal(H1e, H1a));
    EXPECT(assert_equal(H2e, H2a));
    EXPECT(assert_equal(H3e, H3a));
    EXPECT(assert_equal(H4e, H4a));
    EXPECT(assert_equal(H5e, H5a));
}

/* ************************************************************************* */
TEST(ImuFactor, PredictPositionAndVelocity){
  imuBias::ConstantBias bias(Vector3(0, 0, 0), Vector3(0, 0, 0)); // Biases (acc, rot)

  // Measurements
  Vector3 gravity; gravity << 0, 0, 9.81;
  Vector3 omegaCoriolis; omegaCoriolis << 0, 0, 0;
  Vector3 measuredOmega; measuredOmega << 0, 0, 0;//M_PI/10.0+0.3;
  Vector3 measuredAcc; measuredAcc << 0,1,-9.81;
  double deltaT = 0.001;

  Matrix I6x6(6,6);
  I6x6 = Matrix::Identity(6,6);

  ImuFactor::PreintegratedMeasurements pre_int_data(imuBias::ConstantBias(Vector3(0.2, 0.0, 0.0),
        Vector3(0.0, 0.0, 0.0)), Matrix3::Zero(), Matrix3::Zero(), Matrix3::Zero(), true);

  for (int i = 0; i<1000; ++i)   pre_int_data.integrateMeasurement(measuredAcc, measuredOmega, deltaT);

    // Create factor
    ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pre_int_data, gravity, omegaCoriolis);

    // Predict
    Pose3 x1;
    Vector3 v1(0, 0.0, 0.0);
    PoseVelocityBias poseVelocity = pre_int_data.Predict(x1, v1, bias, gravity, omegaCoriolis);
    Pose3 expectedPose(Rot3(), Point3(0, 0.5, 0));
    Vector3 expectedVelocity; expectedVelocity<<0,1,0;
    EXPECT(assert_equal(expectedPose, poseVelocity.pose));
    EXPECT(assert_equal(Vector(expectedVelocity), Vector(poseVelocity.velocity)));
}

/* ************************************************************************* */
TEST(ImuFactor, PredictRotation) {
  imuBias::ConstantBias bias(Vector3(0, 0, 0), Vector3(0, 0, 0)); // Biases (acc, rot)

  // Measurements
  Vector3 gravity; gravity << 0, 0, 9.81;
  Vector3 omegaCoriolis; omegaCoriolis << 0, 0, 0;
  Vector3 measuredOmega; measuredOmega << 0, 0, M_PI/10;//M_PI/10.0+0.3;
  Vector3 measuredAcc; measuredAcc << 0,0,-9.81;
  double deltaT = 0.001;

  Matrix I6x6(6,6);
  I6x6 = Matrix::Identity(6,6);

  ImuFactor::PreintegratedMeasurements pre_int_data(imuBias::ConstantBias(Vector3(0.2, 0.0, 0.0),
        Vector3(0.0, 0.0, 0.0)), Matrix3::Zero(), Matrix3::Zero(), Matrix3::Zero(), true);

  for (int i = 0; i<1000; ++i)   pre_int_data.integrateMeasurement(measuredAcc, measuredOmega, deltaT);

    // Create factor
    ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pre_int_data, gravity, omegaCoriolis);

    // Predict
    Pose3 x1, x2;
    Vector3 v1 = Vector3(0, 0.0, 0.0);
    Vector3 v2;
    ImuFactor::Predict(x1, v1, x2, v2, bias, factor.preintegratedMeasurements(), gravity, omegaCoriolis);
    Pose3 expectedPose(Rot3().ypr(M_PI/10, 0, 0), Point3(0, 0, 0));
    Vector3 expectedVelocity; expectedVelocity<<0,0,0;
    EXPECT(assert_equal(expectedPose, x2));
    EXPECT(assert_equal(Vector(expectedVelocity), Vector(v2)));
}

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