testImuFactor.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
 
 * -------------------------------------------------------------------------- */
 
// #define ENABLE_TIMING // uncomment for timing results
 
#include <gtsam/navigation/ImuFactor.h>
#include <gtsam/navigation/ScenarioRunner.h>
#include <gtsam/geometry/Pose3.h>
#include <gtsam/nonlinear/Values.h>
#include <gtsam/nonlinear/factorTesting.h>
#include <gtsam/linear/Sampler.h>
#include <gtsam/base/TestableAssertions.h>
#include <gtsam/base/numericalDerivative.h>
 
#include <CppUnitLite/TestHarness.h>
#include <list>
 
#include "imuFactorTesting.h"
 
namespace testing {
// Create default parameters with Z-down and above noise parameters
static std::shared_ptr<PreintegrationParams> Params() {
  auto p = PreintegrationParams::MakeSharedD(kGravity);
  p->gyroscopeCovariance = kGyroSigma * kGyroSigma * I_3x3;
  p->accelerometerCovariance = kAccelSigma * kAccelSigma * I_3x3;
  p->integrationCovariance = 0.0001 * I_3x3;
  return p;
}
}
 
/* ************************************************************************* */
namespace {
// Auxiliary functions to test evaluate error in ImuFactor
/* ************************************************************************* */
Rot3 evaluateRotationError(const ImuFactor& factor, const Pose3& pose_i,
    const Vector3& vel_i, const Pose3& pose_j, const Vector3& vel_j,
    const Bias& bias) {
  return Rot3::Expmap(
      factor.evaluateError(pose_i, vel_i, pose_j, vel_j, bias).head(3));
}
 
} // namespace
 
/* ************************************************************************* */
TEST(ImuFactor, PreintegratedMeasurementsConstruction) {
  // Actual pre-integrated values
  PreintegratedImuMeasurements actual(testing::Params());
  EXPECT(assert_equal(Rot3(), actual.deltaRij()));
  EXPECT(assert_equal(kZero, actual.deltaPij()));
  EXPECT(assert_equal(kZero, actual.deltaVij()));
  DOUBLES_EQUAL(0.0, actual.deltaTij(), 1e-9);
}
 
/* ************************************************************************* */
TEST(ImuFactor, PreintegratedMeasurementsReset) {
 
        auto p = testing::Params();
        // Create a preintegrated measurement struct and integrate
        PreintegratedImuMeasurements pimActual(p);
        Vector3 measuredAcc(0.5, 1.0, 0.5);
        Vector3 measuredOmega(0.1, 0.3, 0.1);
        double deltaT = 1.0;
        pimActual.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
 
        // reset and make sure that it is the same as a fresh one
        pimActual.resetIntegration();
        CHECK(assert_equal(pimActual, PreintegratedImuMeasurements(p)));
 
        // Now create one with a different bias ..
        Bias nonZeroBias(Vector3(0.2, 0, 0), Vector3(0.1, 0, 0.3));
        PreintegratedImuMeasurements pimExpected(p, nonZeroBias);
 
        // integrate again, then reset to a new bias
        pimActual.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
        pimActual.resetIntegrationAndSetBias(nonZeroBias);
        CHECK(assert_equal(pimActual, pimExpected));
}
 
/* ************************************************************************* */
TEST(ImuFactor, Accelerating) {
  const double a = 0.2, v = 50;
 
  // Set up body pointing towards y axis, and start at 10,20,0 with velocity going in X
  // The body itself has Z axis pointing down
  const Rot3 nRb(Point3(0, 1, 0), Point3(1, 0, 0), Point3(0, 0, -1));
  const Point3 initial_position(10, 20, 0);
  const Vector3 initial_velocity(v, 0, 0);
 
  const AcceleratingScenario scenario(nRb, initial_position, initial_velocity,
      Vector3(a, 0, 0));
 
  const double T = 3.0; // seconds
  ScenarioRunner runner(scenario, testing::Params(), T / 10);
 
  PreintegratedImuMeasurements pim = runner.integrate(T);
  EXPECT(assert_equal(scenario.pose(T), runner.predict(pim).pose(), 1e-9));
 
  Matrix9 estimatedCov = runner.estimateCovariance(T, 100);
  EXPECT(assert_equal(estimatedCov, pim.preintMeasCov(), 0.1));
}
 
/* ************************************************************************* */
TEST(ImuFactor, PreintegratedMeasurements) {
  // Measurements
  const double a = 0.1, w = M_PI / 100.0;
  Vector3 measuredAcc(a, 0.0, 0.0);
  Vector3 measuredOmega(w, 0.0, 0.0);
  double deltaT = 0.5;
 
  // Expected pre-integrated values
  Vector3 expectedDeltaR1(w * deltaT, 0.0, 0.0);
  Vector3 expectedDeltaP1(0.5 * a * deltaT*deltaT, 0, 0);
  Vector3 expectedDeltaV1(0.05, 0.0, 0.0);
 
  // Actual pre-integrated values
  PreintegratedImuMeasurements actual(testing::Params());
  actual.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
  EXPECT(assert_equal(Rot3::Expmap(expectedDeltaR1), actual.deltaRij()));
  EXPECT(assert_equal(expectedDeltaP1, actual.deltaPij()));
  EXPECT(assert_equal(expectedDeltaV1, actual.deltaVij()));
  DOUBLES_EQUAL(0.5, actual.deltaTij(), 1e-9);
 
  // Check derivatives of computeError
  Bias bias(Vector3(0.2, 0, 0), Vector3(0.1, 0, 0.3)); // Biases (acc, rot)
  NavState x1, x2 = actual.predict(x1, bias);
 
  {
  Matrix9 aH1, aH2;
  Matrix96 aH3;
  actual.computeError(x1, x2, bias, aH1, aH2, aH3);
  std::function<Vector9(const NavState&, const NavState&, const Bias&)> f =
      std::bind(&PreintegrationBase::computeError, actual,
                  std::placeholders::_1, std::placeholders::_2, std::placeholders::_3,
                  nullptr, nullptr, nullptr);
  EXPECT(assert_equal(numericalDerivative31(f, x1, x2, bias), aH1, 1e-9));
  EXPECT(assert_equal(numericalDerivative32(f, x1, x2, bias), aH2, 1e-9));
  EXPECT(assert_equal(numericalDerivative33(f, x1, x2, bias), aH3, 1e-9));
  }
 
  // Integrate again
  Vector3 expectedDeltaR2(2.0 * 0.5 * M_PI / 100.0, 0.0, 0.0);
  Vector3 expectedDeltaP2(0.025 + expectedDeltaP1(0) + 0.5 * 0.1 * 0.5 * 0.5, 0, 0);
  Vector3 expectedDeltaV2 = Vector3(0.05, 0.0, 0.0) +
                            Rot3::Expmap(expectedDeltaR1) * measuredAcc * 0.5;
 
  // Actual pre-integrated values
  actual.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
  EXPECT(assert_equal(Rot3::Expmap(expectedDeltaR2), actual.deltaRij()));
  EXPECT(assert_equal(expectedDeltaP2, actual.deltaPij()));
  EXPECT(assert_equal(expectedDeltaV2, actual.deltaVij()));
  DOUBLES_EQUAL(1.0, actual.deltaTij(), 1e-9);
}
/* ************************************************************************* */
// Common linearization point and measurements for tests
namespace common {
static const Pose3 x1(Rot3::RzRyRx(M_PI / 12.0, M_PI / 6.0, M_PI / 4.0),
    Point3(5.0, 1.0, 0));
static const Vector3 v1(Vector3(0.5, 0.0, 0.0));
static const NavState state1(x1, v1);
 
// Measurements
static const double w = M_PI / 100;
static const Vector3 measuredOmega(w, 0, 0);
static const Vector3 measuredAcc = x1.rotation().unrotate(
    -kGravityAlongNavZDown);
static const double deltaT = 1.0;
 
static const Pose3 x2(Rot3::RzRyRx(M_PI / 12.0 + w, M_PI / 6.0, M_PI / 4.0),
    Point3(5.5, 1.0, 0));
static const Vector3 v2(Vector3(0.5, 0.0, 0.0));
static const NavState state2(x2, v2);
} // namespace common
 
/* ************************************************************************* */
TEST(ImuFactor, PreintegrationBaseMethods) {
  using namespace common;
  auto p = testing::Params();
  p->omegaCoriolis = Vector3(0.02, 0.03, 0.04);
  p->use2ndOrderCoriolis = true;
 
  PreintegratedImuMeasurements pim(p, kZeroBiasHat);
  pim.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
  pim.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
 
  // biasCorrectedDelta
  Matrix96 actualH;
  pim.biasCorrectedDelta(kZeroBias, actualH);
  Matrix expectedH = numericalDerivative11<Vector9, Bias>(
      std::bind(&PreintegrationBase::biasCorrectedDelta, pim,
          std::placeholders::_1, nullptr), kZeroBias);
  EXPECT(assert_equal(expectedH, actualH));
 
  Matrix9 aH1;
  Matrix96 aH2;
  NavState predictedState = pim.predict(state1, kZeroBias, aH1, aH2);
  Matrix eH1 = numericalDerivative11<NavState, NavState>(
      std::bind(&PreintegrationBase::predict, pim, std::placeholders::_1,
          kZeroBias, nullptr, nullptr), state1);
  EXPECT(assert_equal(eH1, aH1));
  Matrix eH2 = numericalDerivative11<NavState, Bias>(
      std::bind(&PreintegrationBase::predict, pim, state1,
          std::placeholders::_1, nullptr, nullptr), kZeroBias);
  EXPECT(assert_equal(eH2, aH2));
}
 
/* ************************************************************************* */
TEST(ImuFactor, MultipleMeasurements) {
  using namespace common;
 
  PreintegratedImuMeasurements expected(testing::Params(), kZeroBiasHat);
  expected.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
  expected.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
 
  Matrix32 acc,gyro;
  Matrix12 dts;
  acc << measuredAcc, measuredAcc;
  gyro << measuredOmega, measuredOmega;
  dts << deltaT, deltaT;
  PreintegratedImuMeasurements actual(testing::Params(), kZeroBiasHat);
  actual.integrateMeasurements(acc,gyro,dts);
 
  EXPECT(assert_equal(expected,actual));
}
 
/* ************************************************************************* */
TEST(ImuFactor, ErrorAndJacobians) {
  using namespace common;
  PreintegratedImuMeasurements pim(testing::Params());
 
  pim.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
  EXPECT(assert_equal(state2, pim.predict(state1, kZeroBias)));
 
  // Create factor
  ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pim);
 
  // Expected error
  Vector expectedError(9);
  expectedError << 0, 0, 0, 0, 0, 0, 0, 0, 0;
  EXPECT(
      assert_equal(expectedError,
          factor.evaluateError(x1, v1, x2, v2, kZeroBias)));
 
  Values values;
  values.insert(X(1), x1);
  values.insert(V(1), v1);
  values.insert(X(2), x2);
  values.insert(V(2), v2);
  values.insert(B(1), kZeroBias);
  EXPECT(assert_equal(expectedError, factor.unwhitenedError(values)));
 
  // Make sure linearization is correct
  double diffDelta = 1e-7;
  EXPECT_CORRECT_FACTOR_JACOBIANS(factor, values, diffDelta, 1e-3);
 
  // Actual Jacobians
  Matrix H1a, H2a, H3a, H4a, H5a;
  (void) factor.evaluateError(x1, v1, x2, v2, kZeroBias, H1a, H2a, H3a, H4a,
      H5a);
 
  // Make sure rotation part is correct when error is interpreted as axis-angle
  // Jacobians are around zero, so the rotation part is the same as:
  Matrix H1Rot3 = numericalDerivative11<Rot3, Pose3>(
      std::bind(&evaluateRotationError, factor, std::placeholders::_1, v1, x2, v2, kZeroBias),
      x1);
  EXPECT(assert_equal(H1Rot3, H1a.topRows(3)));
 
  Matrix H3Rot3 = numericalDerivative11<Rot3, Pose3>(
      std::bind(&evaluateRotationError, factor, x1, v1, std::placeholders::_1, v2, kZeroBias),
      x2);
  EXPECT(assert_equal(H3Rot3, H3a.topRows(3)));
 
  // Evaluate error with wrong values
  Vector3 v2_wrong = v2 + Vector3(0.1, 0.1, 0.1);
  values.update(V(2), v2_wrong);
  expectedError << 0, 0, 0, 0, 0, 0, -0.0724744871, -0.040715657, -0.151952901;
  EXPECT(
      assert_equal(expectedError,
          factor.evaluateError(x1, v1, x2, v2_wrong, kZeroBias), 1e-2));
  EXPECT(assert_equal(expectedError, factor.unwhitenedError(values), 1e-2));
 
  // Make sure the whitening is done correctly
  Matrix cov = pim.preintMeasCov();
  Matrix R = RtR(cov.inverse());
  Vector whitened = R * expectedError;
  EXPECT(assert_equal(0.5 * whitened.squaredNorm(), factor.error(values), 1e-4));
 
  // Make sure linearization is correct
  EXPECT_CORRECT_FACTOR_JACOBIANS(factor, values, diffDelta, 1e-3);
}
 
/* ************************************************************************* */
TEST(ImuFactor, ErrorAndJacobianWithBiases) {
  using common::x1;
  using common::v1;
  using common::v2;
  Bias bias(Vector3(0.2, 0, 0), Vector3(0.1, 0, 0.3)); // Biases (acc, rot)
  Pose3 x2(Rot3::Expmap(Vector3(0, 0, M_PI / 10.0 + M_PI / 10.0)),
      Point3(5.5, 1.0, -50.0));
 
  // Measurements
  Vector3 measuredOmega;
  measuredOmega << 0, 0, M_PI / 10.0 + 0.3;
  Vector3 measuredAcc = x1.rotation().unrotate(-kGravityAlongNavZDown)
      + Vector3(0.2, 0.0, 0.0);
  double deltaT = 1.0;
 
  auto p = testing::Params();
  p->omegaCoriolis = kNonZeroOmegaCoriolis;
 
  Bias biasHat(Vector3(0.2, 0.0, 0.0), Vector3(0.0, 0.0, 0.1));
  PreintegratedImuMeasurements pim(p, biasHat);
  pim.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
 
  // Make sure of biasCorrectedDelta
  Matrix96 actualH;
  pim.biasCorrectedDelta(bias, actualH);
  Matrix expectedH = numericalDerivative11<Vector9, Bias>(
      std::bind(&PreintegrationBase::biasCorrectedDelta, pim,
          std::placeholders::_1, nullptr), bias);
  EXPECT(assert_equal(expectedH, actualH));
 
  // Create factor
  ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pim);
 
  Values values;
  values.insert(X(1), x1);
  values.insert(V(1), v1);
  values.insert(X(2), x2);
  values.insert(V(2), v2);
  values.insert(B(1), bias);
 
  // Make sure linearization is correct
  double diffDelta = 1e-7;
  EXPECT_CORRECT_FACTOR_JACOBIANS(factor, values, diffDelta, 1e-3);
}
 
/* ************************************************************************* */
TEST(ImuFactor, ErrorAndJacobianWith2ndOrderCoriolis) {
  using common::x1;
  using common::v1;
  using common::v2;
  Bias bias(Vector3(0.2, 0, 0), Vector3(0.1, 0, 0.3)); // Biases (acc, rot)
  Pose3 x2(Rot3::Expmap(Vector3(0, 0, M_PI / 10.0 + M_PI / 10.0)),
      Point3(5.5, 1.0, -50.0));
 
  // Measurements
  Vector3 measuredOmega;
  measuredOmega << 0, 0, M_PI / 10.0 + 0.3;
  Vector3 measuredAcc = x1.rotation().unrotate(-kGravityAlongNavZDown)
      + Vector3(0.2, 0.0, 0.0);
  double deltaT = 1.0;
 
  auto p = testing::Params();
  p->omegaCoriolis = kNonZeroOmegaCoriolis;
  p->use2ndOrderCoriolis = true;
 
  PreintegratedImuMeasurements pim(p,
      Bias(Vector3(0.2, 0.0, 0.0), Vector3(0.0, 0.0, 0.1)));
  pim.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
 
  // Create factor
  ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pim);
 
  Values values;
  values.insert(X(1), x1);
  values.insert(V(1), v1);
  values.insert(X(2), x2);
  values.insert(V(2), v2);
  values.insert(B(1), bias);
 
  // Make sure linearization is correct
  double diffDelta = 1e-7;
  EXPECT_CORRECT_FACTOR_JACOBIANS(factor, values, diffDelta, 1e-3);
}
 
/* ************************************************************************* */
TEST(ImuFactor, PartialDerivative_wrt_Bias) {
  // Linearization point
  Vector3 biasOmega(0, 0, 0); // Current estimate of rotation rate bias
 
  // Measurements
  Vector3 measuredOmega(0.1, 0, 0);
  double deltaT = 0.5;
 
  auto evaluateRotation = [=](const Vector3 biasOmega) {
    return Rot3::Expmap((measuredOmega - biasOmega) * deltaT);
  };
 
  // Compute numerical derivatives
  Matrix expectedDelRdelBiasOmega =
      numericalDerivative11<Rot3, Vector3>(evaluateRotation, 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-9));
}
 
/* ************************************************************************* */
TEST(ImuFactor, PartialDerivativeLogmap) {
  // Linearization point
  Vector3 thetaHat(0.1, 0.1, 0); // Current estimate of rotation rate bias
 
  // Measurements
  Vector3 deltaTheta(0, 0, 0);
 
  auto evaluateLogRotation = [=](const Vector3 delta) {
    return Rot3::Logmap(
        Rot3::Expmap(thetaHat).compose(Rot3::Expmap(delta)));
  };
 
  // Compute numerical derivatives
  Matrix expectedDelFdelTheta =
      numericalDerivative11<Vector, Vector3>(evaluateLogRotation, deltaTheta);
 
  Matrix3 actualDelFdelTheta = Rot3::LogmapDerivative(thetaHat);
 
  // Compare Jacobians
  EXPECT(assert_equal(expectedDelFdelTheta, actualDelFdelTheta));
}
 
/* ************************************************************************* */
TEST(ImuFactor, fistOrderExponential) {
  // Linearization point
  Vector3 biasOmega(0, 0, 0); // Current estimate of rotation rate bias
 
  // Measurements
  Vector3 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();
 
  // This is a first order expansion so the equality is only an approximation
  EXPECT(assert_equal(expectedRot, actualRot));
}
 
/* ************************************************************************* */
Vector3 correctedAcc(const PreintegratedImuMeasurements& pim,
    const Vector3& measuredAcc, const Vector3& measuredOmega) {
  Vector3 correctedAcc = pim.biasHat().correctAccelerometer(measuredAcc);
  Vector3 correctedOmega = pim.biasHat().correctGyroscope(measuredOmega);
  return pim.correctMeasurementsBySensorPose(correctedAcc, correctedOmega).first;
}
 
TEST(ImuFactor, ErrorWithBiasesAndSensorBodyDisplacement) {
  const Rot3 nRb = Rot3::Expmap(Vector3(0, 0, M_PI / 4.0));
  const Point3 p1(5.0, 1.0, -50.0);
  const Vector3 v1(0.5, 0.0, 0.0);
 
  const Vector3 a = nRb * Vector3(0.2, 0.0, 0.0);
  const AcceleratingScenario scenario(nRb, p1, v1, a,
      Vector3(0, 0, M_PI / 10.0 + 0.3));
 
  auto p = testing::Params();
  p->body_P_sensor = Pose3(Rot3::Expmap(Vector3(0, M_PI / 2, 0)),
      Point3(0.1, 0.05, 0.01));
  p->omegaCoriolis = kNonZeroOmegaCoriolis;
 
  Bias biasHat(Vector3(0.2, 0.0, 0.0), Vector3(0.0, 0.0, 0.0));
 
  const double T = 3.0; // seconds
  ScenarioRunner runner(scenario, p, T / 10);
 
  //  PreintegratedImuMeasurements pim = runner.integrate(T);
  //  EXPECT(assert_equal(scenario.pose(T), runner.predict(pim).pose, 1e-9));
  //
  //  Matrix6 estimatedCov = runner.estimatePoseCovariance(T);
  //  EXPECT(assert_equal(estimatedCov, runner.poseCovariance(pim), 0.1));
  //
  Pose3 x1(nRb, p1);
 
  // Measurements
  Vector3 measuredOmega = runner.actualAngularVelocity(0);
  Vector3 measuredAcc = runner.actualSpecificForce(0);
 
  // Get mean prediction from "ground truth" measurements
  const Vector3 accNoiseVar2(0.01, 0.02, 0.03);
  const Vector3 omegaNoiseVar2(0.03, 0.01, 0.02);
  PreintegratedImuMeasurements pim(p, biasHat);
 
  // Check updatedDeltaXij derivatives
  Matrix3 D_correctedAcc_measuredOmega = Z_3x3;
  pim.correctMeasurementsBySensorPose(measuredAcc, measuredOmega,
      nullptr, D_correctedAcc_measuredOmega, nullptr);
  Matrix3 expectedD = numericalDerivative11<Vector3, Vector3>(
      std::bind(correctedAcc, pim, measuredAcc, std::placeholders::_1),
      measuredOmega, 1e-6);
  EXPECT(assert_equal(expectedD, D_correctedAcc_measuredOmega, 1e-5));
 
  double dt = 0.1;
 
// TODO(frank): revive derivative tests
//  Matrix93 G1, G2;
//  Vector9 preint =
//      pim.updatedDeltaXij(measuredAcc, measuredOmega, dt, {}, G1, G2);
//
//  Matrix93 expectedG1 = numericalDerivative21<NavState, Vector3, Vector3>(
//      std::bind(&PreintegratedImuMeasurements::updatedDeltaXij, pim,
//          std::placeholders::_1, std::placeholders::_2,
//          dt, {}, {}, {}), measuredAcc,
//      measuredOmega, 1e-6);
//  EXPECT(assert_equal(expectedG1, G1, 1e-5));
//
//  Matrix93 expectedG2 = numericalDerivative22<NavState, Vector3, Vector3>(
//      std::bind(&PreintegratedImuMeasurements::updatedDeltaXij, pim,
//          std::placeholders::_1, std::placeholders::_2,
//          dt, {}, {}, {}), measuredAcc,
//      measuredOmega, 1e-6);
//  EXPECT(assert_equal(expectedG2, G2, 1e-5));
 
  Bias bias(Vector3(0.2, 0, 0), Vector3(0, 0, 0.3)); // Biases (acc, rot)
 
  // integrate at least twice to get position information
  // otherwise factor cov noise from preint_cov is not positive definite
  pim.integrateMeasurement(measuredAcc, measuredOmega, dt);
  pim.integrateMeasurement(measuredAcc, measuredOmega, dt);
 
  // Create factor
  ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pim);
 
  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));
 
  Values values;
  values.insert(X(1), x1);
  values.insert(V(1), v1);
  values.insert(X(2), x2);
  values.insert(V(2), v2);
  values.insert(B(1), bias);
 
  // Make sure linearization is correct
  double diffDelta = 1e-8;
  EXPECT_CORRECT_FACTOR_JACOBIANS(factor, values, diffDelta, 1e-3);
}
 
/* ************************************************************************* */
TEST(ImuFactor, PredictPositionAndVelocity) {
  gttic(PredictPositionAndVelocity);
  Bias bias(Vector3(0, 0, 0), Vector3(0, 0, 0)); // Biases (acc, rot)
 
  // Measurements
  Vector3 measuredOmega;
  measuredOmega << 0, 0, 0; // M_PI/10.0+0.3;
  Vector3 measuredAcc;
  measuredAcc << 0, 1, -kGravity;
  double deltaT = 0.001;
 
  PreintegratedImuMeasurements pim(testing::Params(),
      Bias(Vector3(0.2, 0.0, 0.0), Vector3(0.0, 0.0, 0.0)));
 
  for (int i = 0; i < 1000; ++i)
    pim.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
 
  // Create factor
  ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pim);
 
  // Predict
  Pose3 x1;
  Vector3 v1(0, 0.0, 0.0);
  NavState actual = pim.predict(NavState(x1, v1), bias);
  NavState expected(Rot3(), Point3(0, 0.5, 0), Vector3(0, 1, 0));
  EXPECT(assert_equal(expected, actual));
}
 
/* ************************************************************************* */
TEST(ImuFactor, PredictRotation) {
  gttic(PredictRotation);
  Bias bias(Vector3(0, 0, 0), Vector3(0, 0, 0)); // Biases (acc, rot)
 
  // Measurements
  Vector3 measuredOmega;
  measuredOmega << 0, 0, M_PI / 10; // M_PI/10.0+0.3;
  Vector3 measuredAcc;
  measuredAcc << 0, 0, -kGravity;
  double deltaT = 0.001;
 
  PreintegratedImuMeasurements pim(testing::Params(),
      Bias(Vector3(0.2, 0.0, 0.0), Vector3(0.0, 0.0, 0.0)));
 
  for (int i = 0; i < 1000; ++i)
    pim.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
 
  // Create factor
  ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pim);
 
  // Predict
  NavState actual = pim.predict(NavState(), bias);
  NavState expected(Rot3::Ypr(M_PI / 10, 0, 0), Point3(0,0,0), Z_3x1);
  EXPECT(assert_equal(expected, actual));
}
 
/* ************************************************************************* */
TEST(ImuFactor, PredictArbitrary) {
  gttic(PredictArbitrary);
  Pose3 x1;
  const Vector3 v1(0, 0, 0);
 
  const AcceleratingScenario scenario(x1.rotation(), x1.translation(), v1,
      Vector3(0.1, 0.2, 0), Vector3(M_PI / 10, M_PI / 10, M_PI / 10));
 
  const double T = 3.0; // seconds
  ScenarioRunner runner(scenario, testing::Params(), T / 10);
  //
  //  PreintegratedImuMeasurements pim = runner.integrate(T);
  //  EXPECT(assert_equal(scenario.pose(T), runner.predict(pim).pose, 1e-9));
  //
  //  Matrix6 estimatedCov = runner.estimatePoseCovariance(T);
  //  EXPECT(assert_equal(estimatedCov, runner.poseCovariance(pim), 0.1));
 
  Bias biasHat(Vector3(0.2, 0.0, 0.0), Vector3(0.0, 0.0, 0.0));
 
  // Measurements
  Vector3 measuredOmega = runner.actualAngularVelocity(0);
  Vector3 measuredAcc = runner.actualSpecificForce(0);
 
  auto p = testing::Params();
  p->integrationCovariance = Z_3x3; // MonteCarlo does not sample integration noise
  PreintegratedImuMeasurements pim(p, biasHat);
  Bias bias(Vector3(0, 0, 0), Vector3(0, 0, 0));
//  EXPECT(MonteCarlo(pim, NavState(x1, v1), bias, 0.1, {}, measuredAcc, measuredOmega,
//                    Vector3::Constant(accNoiseVar), Vector3::Constant(omegaNoiseVar), 100000));
 
  double dt = 0.001;
  for (int i = 0; i < 1000; ++i)
    pim.integrateMeasurement(measuredAcc, measuredOmega, dt);
 
  // Create factor
  ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pim);
 
  // Predict
  NavState actual = pim.predict(NavState(x1, v1), bias);
 
  // Regression test for Imu Refactor
  Rot3 expectedR( //
      +0.903715275, -0.250741668, 0.347026393, //
      +0.347026393, 0.903715275, -0.250741668, //
      -0.250741668, 0.347026393, 0.903715275);
  Point3 expectedT(-0.516077031, 0.57842919, 0.0876478403);
  Vector3 expectedV(-1.62337767, 1.57954409, 0.343833571);
  NavState expected(expectedR, expectedT, expectedV);
  EXPECT(assert_equal(expected, actual, 1e-7));
}
 
/* ************************************************************************* */
TEST(ImuFactor, bodyPSensorNoBias) {
  gttic(bodyPSensorNoBias);
  Bias bias(Vector3(0, 0, 0), Vector3(0, 0.1, 0)); // Biases (acc, rot)
 
  // Rotate sensor (z-down) to body (same as navigation) i.e. z-up
  auto p = testing::Params();
  p->n_gravity = Vector3(0, 0, -kGravity); // z-up nav frame
  p->body_P_sensor = Pose3(Rot3::Ypr(0, 0, M_PI), Point3(0, 0, 0));
 
  // Measurements
  // Gyroscope measurement is the angular velocity of sensor w.r.t nav frame in sensor frame
  Vector3 s_omegaMeas_ns(0, 0.1, M_PI / 10);
  // Acc measurement is acceleration of sensor in the sensor frame, when stationary,
  // table exerts an equal and opposite force w.r.t gravity
  Vector3 s_accMeas(0, 0, -kGravity);
  double dt = 0.001;
 
  PreintegratedImuMeasurements pim(p, bias);
 
  for (int i = 0; i < 1000; ++i)
    pim.integrateMeasurement(s_accMeas, s_omegaMeas_ns, dt);
 
  // Create factor
  ImuFactor factor(X(1), V(1), X(2), V(2), B(1), pim);
 
  // Predict
  NavState actual = pim.predict(NavState(), bias);
 
  Pose3 expectedPose(Rot3::Ypr(-M_PI / 10, 0, 0), Point3(0, 0, 0));
  EXPECT(assert_equal(expectedPose, actual.pose()));
 
  Vector3 expectedVelocity(0, 0, 0);
  EXPECT(assert_equal(Vector(expectedVelocity), Vector(actual.velocity())));
}
 
/* ************************************************************************* */
#include <gtsam/nonlinear/NonlinearFactorGraph.h>
#include <gtsam/slam/BetweenFactor.h>
#include <gtsam/nonlinear/LevenbergMarquardtOptimizer.h>
#include <gtsam/nonlinear/Marginals.h>
 
TEST(ImuFactor, bodyPSensorWithBias) {
  gttic(bodyPSensorWithBias);
  using noiseModel::Diagonal;
  typedef Bias Bias;
 
  int numPoses = 10;
  Vector6 noiseBetweenBiasSigma;
  noiseBetweenBiasSigma << Vector3(2.0e-5, 2.0e-5, 2.0e-5), Vector3(3.0e-6,
      3.0e-6, 3.0e-6);
  SharedDiagonal biasNoiseModel = Diagonal::Sigmas(noiseBetweenBiasSigma);
 
  // Measurements
  // Sensor frame is z-down
  // Gyroscope measurement is the angular velocity of sensor w.r.t nav frame in sensor frame
  Vector3 measuredOmega(0, 0.01, 0);
  // Acc measurement is acceleration of sensor in the sensor frame, when stationary,
  // table exerts an equal and opposite force w.r.t gravity
  Vector3 measuredAcc(0, 0, -kGravity);
 
  auto p = testing::Params();
  p->n_gravity = Vector3(0, 0, -kGravity);
  p->body_P_sensor = Pose3(Rot3::Ypr(0, 0, M_PI), Point3(0,0,0));
  p->accelerometerCovariance = 1e-7 * I_3x3;
  p->gyroscopeCovariance = 1e-8 * I_3x3;
  p->integrationCovariance = 1e-9 * I_3x3;
  double deltaT = 0.005;
 
  //   Specify noise values on priors
  Vector6 priorNoisePoseSigmas(
      (Vector(6) << 0.001, 0.001, 0.001, 0.01, 0.01, 0.01).finished());
  Vector3 priorNoiseVelSigmas((Vector(3) << 0.1, 0.1, 0.1).finished());
  Vector6 priorNoiseBiasSigmas(
      (Vector(6) << 0.1, 0.1, 0.1, 0.5e-1, 0.5e-1, 0.5e-1).finished());
  SharedDiagonal priorNoisePose = Diagonal::Sigmas(priorNoisePoseSigmas);
  SharedDiagonal priorNoiseVel = Diagonal::Sigmas(priorNoiseVelSigmas);
  SharedDiagonal priorNoiseBias = Diagonal::Sigmas(priorNoiseBiasSigmas);
  Vector3 zeroVel(0, 0, 0);
 
  // Create a factor graph with priors on initial pose, velocity and bias
  NonlinearFactorGraph graph;
  Values values;
 
  graph.addPrior(X(0), Pose3(), priorNoisePose);
  values.insert(X(0), Pose3());
 
  graph.addPrior(V(0), zeroVel, priorNoiseVel);
  values.insert(V(0), zeroVel);
 
  // The key to this test is that we specify the bias, in the sensor frame, as known a priori
  // We also create factors below that encode our assumption that this bias is constant over time
  // In theory, after optimization, we should recover that same bias estimate
  Bias priorBias(Vector3(0, 0, 0), Vector3(0, 0.01, 0)); // Biases (acc, rot)
  graph.addPrior(B(0), priorBias, priorNoiseBias);
  values.insert(B(0), priorBias);
 
  // Now add IMU factors and bias noise models
  Bias zeroBias(Vector3(0, 0, 0), Vector3(0, 0, 0));
  for (int i = 1; i < numPoses; i++) {
    PreintegratedImuMeasurements pim(p, priorBias);
    for (int j = 0; j < 200; ++j)
      pim.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
 
    // Create factors
    graph.emplace_shared<ImuFactor>(X(i - 1), V(i - 1), X(i), V(i), B(i - 1), pim);
    graph.emplace_shared<BetweenFactor<Bias> >(B(i - 1), B(i), zeroBias, biasNoiseModel);
 
    values.insert(X(i), Pose3());
    values.insert(V(i), zeroVel);
    values.insert(B(i), priorBias);
  }
 
  // Finally, optimize, and get bias at last time step
  Values result = LevenbergMarquardtOptimizer(graph, values).optimize();
  Bias biasActual = result.at<Bias>(B(numPoses - 1));
 
  // And compare it with expected value (our prior)
  Bias biasExpected(Vector3(0, 0, 0), Vector3(0, 0.01, 0));
  EXPECT(assert_equal(biasExpected, biasActual, 1e-3));
}
 
/* ************************************************************************* */
#ifdef GTSAM_TANGENT_PREINTEGRATION
static const double kVelocity = 2.0, kAngularVelocity = M_PI / 6;
 
struct ImuFactorMergeTest {
  std::shared_ptr<PreintegrationParams> p_;
  const ConstantTwistScenario forward_, loop_;
 
  ImuFactorMergeTest()
      : p_(PreintegrationParams::MakeSharedU(kGravity)),
        forward_(kZero, Vector3(kVelocity, 0, 0)),
        loop_(Vector3(0, -kAngularVelocity, 0), Vector3(kVelocity, 0, 0)) {
    // arbitrary noise values
    p_->gyroscopeCovariance = I_3x3 * 0.01;
    p_->accelerometerCovariance = I_3x3 * 0.03;
  }
 
  int TestScenario(TestResult& result_, const std::string& name_,
                           const Scenario& scenario,
                           const Bias& bias01,
                           const Bias& bias12, double tol) {
    // Test merge by creating a 01, 12, and 02 PreintegratedRotation,
    // then checking the merge of 01-12 matches 02.
    PreintegratedImuMeasurements pim01(p_, bias01);
    PreintegratedImuMeasurements pim12(p_, bias12);
    PreintegratedImuMeasurements pim02_expected(p_, bias01);
 
    double deltaT = 0.01;
    ScenarioRunner runner(scenario, p_, deltaT);
    // TODO(frank) can this loop just go into runner ?
    for (int i = 0; i < 100; i++) {
      double t = i * deltaT;
      // integrate the measurements appropriately
      Vector3 accel_meas = runner.actualSpecificForce(t);
      Vector3 omega_meas = runner.actualAngularVelocity(t);
      pim02_expected.integrateMeasurement(accel_meas, omega_meas, deltaT);
      if (i < 50) {
        pim01.integrateMeasurement(accel_meas, omega_meas, deltaT);
      } else {
        pim12.integrateMeasurement(accel_meas, omega_meas, deltaT);
      }
    }
    auto actual_pim02 = ImuFactor::Merge(pim01, pim12);
    EXPECT(assert_equal(pim02_expected.preintegrated(),
                        actual_pim02.preintegrated(), tol));
    EXPECT(assert_equal(pim02_expected, actual_pim02, tol));
 
    auto factor01 =
        std::make_shared<ImuFactor>(X(0), V(0), X(1), V(1), B(0), pim01);
    auto factor12 =
        std::make_shared<ImuFactor>(X(1), V(1), X(2), V(2), B(0), pim12);
    auto factor02_expected = std::make_shared<ImuFactor>(
        X(0), V(0), X(2), V(2), B(0), pim02_expected);
 
    ImuFactor::shared_ptr factor02_merged = ImuFactor::Merge(factor01, factor12);
    EXPECT(assert_equal(*factor02_expected, *factor02_merged, tol));
    return result_.getFailureCount();
  }
 
  void TestScenarios(TestResult& result_, const std::string& name_,
                     const Bias& bias01,
                     const Bias& bias12, double tol) {
    for (auto scenario : {forward_, loop_})
      TestScenario(result_, name_, scenario, bias01, bias12, tol);
  }
};
 
/* ************************************************************************* */
// Test case with zero biases
TEST(ImuFactor, MergeZeroBias) {
  ImuFactorMergeTest mergeTest;
  mergeTest.TestScenarios(result_, name_, kZeroBias, kZeroBias, 1e-4);
}
 
// Test case with identical biases: we expect an exact answer.
TEST(ImuFactor, MergeConstantBias) {
  ImuFactorMergeTest mergeTest;
  Bias bias(Vector3(0.03, -0.02, 0.01), Vector3(-0.01, 0.02, -0.03));
  mergeTest.TestScenarios(result_, name_, bias, bias, 1e-4);
}
 
// Test case with different biases where we expect there to be some variation.
TEST(ImuFactor, MergeChangingBias) {
  ImuFactorMergeTest mergeTest;
  mergeTest.TestScenarios(result_, name_,
      Bias(Vector3(0.03, -0.02, 0.01), Vector3(-0.01, 0.02, -0.03)),
      Bias(Vector3(0.01, 0.02, 0.03), Vector3(0.03, -0.02, 0.01)), 1e-1);
}
 
// Test case with non-zero coriolis
TEST(ImuFactor, MergeWithCoriolis) {
  ImuFactorMergeTest mergeTest;
  mergeTest.p_->omegaCoriolis = Vector3(0.1, 0.2, -0.1);
  mergeTest.TestScenarios(result_, name_, kZeroBias, kZeroBias, 1e-4);
}
#endif
 
/* ************************************************************************* */
// Same values as pre-integration test but now testing covariance
TEST(ImuFactor, CheckCovariance) {
  gttic(CheckCovariance);
  // Measurements
  Vector3 measuredAcc(0.1, 0.0, 0.0);
  Vector3 measuredOmega(M_PI / 100.0, 0.0, 0.0);
  double deltaT = 0.5;
 
  PreintegratedImuMeasurements actual(testing::Params());
  actual.integrateMeasurement(measuredAcc, measuredOmega, deltaT);
  Matrix9 expected;
  expected << 1.0577e-08, 0, 0, 0, 0, 0, 0, 0, 0,     //
      0, 1.0577e-08, 0, 0, 0, 0, 0, 0, 0,             //
      0, 0, 1.0577e-08, 0, 0, 0, 0, 0, 0,             //
      0, 0, 0, 5.00868e-05, 0, 0, 3.47222e-07, 0, 0,  //
      0, 0, 0, 0, 5.00868e-05, 0, 0, 3.47222e-07, 0,  //
      0, 0, 0, 0, 0, 5.00868e-05, 0, 0, 3.47222e-07,  //
      0, 0, 0, 3.47222e-07, 0, 0, 1.38889e-06, 0, 0,  //
      0, 0, 0, 0, 3.47222e-07, 0, 0, 1.38889e-06, 0,  //
      0, 0, 0, 0, 0, 3.47222e-07, 0, 0, 1.38889e-06;
  EXPECT(assert_equal(expected, actual.preintMeasCov()));
}
 
/* ************************************************************************* */
int main() {
  TestResult tr;
  auto result = TestRegistry::runAllTests(tr);
#ifdef ENABLE_TIMING
  tictoc_print_();
#endif
  return result;
}
/* ************************************************************************* */