Class sym::ImuFactor#

template<typename Scalar>
class ImuFactor#

A factor for using on-manifold IMU preintegration in a SymForce optimization problem.

By on-manifold, it is meant that the angular velocity measurements are composed as rotations rather than tangent-space approximations.

Assumes IMU bias is constant over the preintegration window. Does not recompute the preintegrated measurements when the IMU bias estimate changes during optimization, but rather uses a first order approximation linearized at the IMU biases given during preintegration.

The gravity argument should be [0, 0, -g] (where g is 9.8, assuming your world frame is gravity aligned so that the -z direction points towards the Earth) unless your IMU reads 0 acceleration while stationary, in which case it should be [0, 0, 0].

More generally, the gravity argument should yield the true acceleration when added to the accelerometer measurement (after the measurement has been converted to the world frame).

Is a functor so as to store the preintegrated IMU measurements between two times. The residual computed is the local-coordinate difference between the final state (pose and velocity) and the state you would expect given the initial state and the IMU measurements.

Based heavily on the on manifold ImuFactor found in GTSAM and on the paper:

Christian Forster, Luca Carlone, Frank Dellaert, and Davide Scaramuzza,
“IMU Preintegration on Manifold for Efficient Visual-Inertial Maximum-a-Posteriori
Estimation”, Robotics: Science and Systems (RSS), 2015.

Example Usage:

enum Var : char {
  POSE = 'p',        // Pose3d
  VELOCITY = 'v',    // Vector3d
  ACCEL_BIAS = 'A',  // Vector3d
  GYRO_BIAS = 'G',   // Vector3d
  GRAVITY = 'g',     // Vector3d
  EPSILON = 'e'      // Scalar

struct ImuMeasurement {
  Eigen::Vector3d acceleration;
  Eigen::Vector3d angular_velocity;
  double duration;

int main() {
  // Dummy function declarations
  std::vector<ImuMeasurement> GetMeasurementsBetween(double start_time, double end_time);
  std::vector<double> GetKeyFrameTimes();
  Eigen::Vector3d GetAccelBiasEstimate(double time);
  Eigen::Vector3d GetGyroBiasEstimate(double time);
  void AppendOtherFactors(std::vector<sym::Factord> & factors);

  // Example accelerometer and gyroscope covariances
  const Eigen::Vector3d accel_cov = Eigen::Vector3d::Constant(1e-5);
  const Eigen::Vector3d gyro_cov = Eigen::Vector3d::Constant(1e-5);

  std::vector<sym::Factord> factors;

  // GetKeyFrameTimes assumed to return at least 2 times
  std::vector<double> key_frame_times = GetKeyFrameTimes();

  // Integrating Imu measurements between keyframes, creating an ImuFactor for each
  for (int i = 0; i < static_cast<int>(key_frame_times.size()) - 1; i++) {
    const double start_time = key_frame_times[i];
    const double end_time = key_frame_times[i + 1];

    const std::vector<ImuMeasurement> measurements = GetMeasurementsBetween(start_time,

    // ImuPreintegrator should be initialized with the best guesses for the IMU biases
    sym::ImuPreintegrator<double> integrator(GetAccelBiasEstimate(start_time),
    for (const ImuMeasurement& meas : measurements) {
      integrator.IntegrateMeasurement(meas.acceleration, meas.angular_velocity, accel_cov,
                                      gyro_cov, meas.duration);

                          .Factor({{Var::POSE, i},
                                   {Var::VELOCITY, i},
                                   {Var::POSE, i + 1},
                                   {Var::VELOCITY, i + 1},
                                   {Var::ACCEL_BIAS, i},
                                   {Var::GYRO_BIAS, i},

  // Adding any other factors there might be for the problem

  sym::Optimizerd optimizer(sym::DefaultOptimizerParams(), factors);

  // Build Values
  sym::Valuesd values;
  for (int i = 0; i < key_frame_times.size(); i++) {
    values.Set({Var::POSE, i}, sym::Pose3d());
    values.Set({Var::VELOCITY, i}, Eigen::Vector3d::Zero());
  for (int i = 0; i < key_frame_times.size() - 1; i++) {
    values.Set({Var::ACCEL_BIAS, i}, GetAccelBiasEstimate(key_frame_times[i]));
    values.Set({Var::GYRO_BIAS, i}, GetGyroBiasEstimate(key_frame_times[i]));
  // gravity should point towards the direction of acceleration
  values.Set({Var::GRAVITY}, Eigen::Vector3d(0.0, 0.0, -9.8));
  values.Set({Var::EPSILON}, sym::kDefaultEpsilond);
  // Initialize any other items in values ...


Public Types

using Pose3 = sym::Pose3<Scalar>#
using Vector3 = sym::Vector3<Scalar>#
using Preintegrator = sym::ImuPreintegrator<Scalar>#
using Measurement = sym::PreintegratedImuMeasurements<Scalar>#
using SqrtInformation = sym::Matrix99<Scalar>#

Public Functions

explicit ImuFactor(const Preintegrator &preintegrator)#

Construct an ImuFactor from a preintegrator.

ImuFactor(const Measurement &measurement, const SqrtInformation &sqrt_information)#

Construct an ImuFactor from a (preintegrated) measurement and its corresponding sqrt information.

sym::Factor<Scalar> Factor(const std::vector<Key> &keys_to_func) const#

Construct a Factor object that can be passed to an Optimizer object given the keys to be passed to the function.

Convenience method to avoid manually specifying which arguments are optimized.

void operator()(const Pose3 &pose_i, const Vector3 &vel_i, const Pose3 &pose_j, const Vector3 &vel_j, const Vector3 &accel_bias_i, const Vector3 &gyro_bias_i, const Vector3 &gravity, Scalar epsilon, Eigen::Matrix<Scalar, 9, 1> *residual = nullptr, Eigen::Matrix<Scalar, 9, 24> *jacobian = nullptr, Eigen::Matrix<Scalar, 24, 24> *hessian = nullptr, Eigen::Matrix<Scalar, 24, 1> *rhs = nullptr) const#

Calculate a between factor on the product manifold of the pose and velocity where the prior is calculated from the preintegrated IMU measurements.

Time step i is the time of the first IMU measurement of the interval. Time step j is the time after the last IMU measurement of the interval.

  • pose_i – Pose at time step i (world_T_body)

  • vel_i – Velocity at time step i (world frame)

  • pose_j – Pose at time step j (world_T_body)

  • vel_j – Velocity at time step j (world frame)

  • accel_bias_i – Bias of the accelerometer measurements between timesteps i and j

  • gyro_bias_i – Bias of the gyroscope measurements between timesteps i and j

  • gravity – Acceleration due to gravity (in the same frame as pose_x and vel_x), i.e., the vector which when added to the accelerometer measurements gives the true acceleration (up to bias and noise) of the IMU.

  • epsilon – epsilon used for numerical stability

  • residual[out] The 9dof whitened local coordinate difference between predicted and estimated state

  • jacobian[out] (9x24) jacobian of res wrt args pose_i (6), vel_i (3), pose_j (6), vel_j (3), accel_bias_i (3), gyro_bias_i (3)

  • hessian[out] (24x24) Gauss-Newton hessian for args pose_i (6), vel_i (3), pose_j (6), vel_j (3), accel_bias_i (3), gyro_bias_i (3)

  • rhs[out] (24x1) Gauss-Newton rhs for args pose_i (6), vel_i (3), pose_j (6), vel_j (3), accel_bias_i (3), gyro_bias_i (3)