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  • Only dependency is Eigen 3 (3.3+ should work)

  • Requires C++ 11 (but not 14) support.



  • Include UKF.h
  • Define a UKFModel class
    • typedefs StateVec, MeasureVec to specify vector formats
    • functions dF, H: to specify process, measurement models
    • function init: for initializing the UKF
  • Create a UKF<UKFModel[, InputType]> object
  • Call ukf.update(time_since_last, measurement, input) on each measurement
  • Read off ukf.state and ukf.stateRootCov

The UKF Class

This provides the core SR-UKF functionality. All template arguments possible: UKF<UKFModel, InputType = Eigen::MatrixXd, UKFParams = DefaultUKFParams<UKFModel>, Integrator = kalman::util::integrator::RK4>. For typical uses you would specify the UKFModel and the input type.

The UKF class has the following fields:

  • ukf.state: type StateVec. The state vector.
  • ukf.stateRootCov: type MatrixXd. Square root of state covariance matrix. Square with size StateVec::COV_SIZE.
  • ukf.processNoiseRootCov: type MatrixXd. Square root of process covariance matrix. Square with size StateVec::COV_SIZE.
  • ukf.measureNoiseRootCov: type MatrixXd. Square root of measurement covariance matrix. Square with size MeasureVec::COV_SIZE.

After updates, you may retrieve the state and root covariance of the filter by accessing ukf.state and ukf.stateRootCov.

The Vector class

State and measurement vectors are implemented in the Vector<NUM_SCALARS, NUM_3D_VECS, NUM_QUATERNIONS> class. The template arguments specify the number of scalars, 3-vectors and quaternions in the state vector. Note that 3D vecs are unnecessary (could add three scalars) and are included for cleanness. Vector is actually a subclass of the Eigen::Matrix class and thus all computations with Vector are internally handled by Eigen.

Model Definition

The UKF model definition is a class with only static members. This helps to improve efficiency. You should create this class according to your requirements and pass it as the first template argument for the UKF class.

State and Measurement Vector Definitions Simply typedef Vector<x, y, z> StateVec and typedef Vector<x, y, z> MeasureVec in the model definition, where x,y,z are set appropriately to the numbers of scalars, 3-vectors, and quaternions.

Description of Process and Measurement Models The model definition class should have two static functions specifying the process and measurement models.

  • The function StateVec dF(state, input) should specify the derivative (over time) of the process model given a current state and an input. The use of the derivative allows us to have non-uniform time steps. Below are some functions that may be helpful:
    • kalman::util::diffQuaternion(state, omega_start) helper to compute process model derivatives for all quaternions. state is the state vector, while omega_start is the index in the state vector where angular velocities begin.
    • kalman::util::diffPosition(state, p_start, p_end, v_start) similar to above, but for positions. Positions are between p_start, p_end and velocities corresponding are between v_start and v_start + (p_end - p_start)
    • The full signature is static StateVec dF(const StateVec & state, const TheInputType & u)
  • MeasureVec H(state, input) should compute the measurement vector corresponding to a state and input. Note that the input is not really necessary here but is included for convenience.
    • The full signature is static MeasureVec H(const StateVec & state, const TheInputType & u)

UKF Initialization The UKF model definition class should also contain a static function called init: static void init(kalman::UKF<TheUKFModelClass> & ukf)

This is ran on each UKF instance upon creation. The init function is expected to initialize the state, stateRootCov, processNoiseRootCov, measureNoiseRootCov fields.

You may use ukf.defaultInitialize(state, proc, measure) to set the three matrices to diagonal matrices with diagonal coefficients equal to state, proc, and measure respectively.

Optional: Input

You may pass an input type as the second template parameter to UKF. The input is provided by the user in the ukf.update() function and passed to the process and measurement model functions. Modifying this allows you to make custom classes available to the measurement and process models.

Optional: UKF Parameters

Optionally, you may also define a class to specify UKF parameters, and pass it as the third template argument to UKF. For example:

class MyUKFParams {
        // parameters
        static constexpr double alphaSquared = 1.0;
        static constexpr double beta = 0.0;
        static constexpr double lambda = alphaSquared * (UKFModel::StateVec::COV_SIZE + 3.0)
                                         - UKFModel::StateVec::COV_SIZE;

        Definitions for parameters used to calculated MRP vectors.
        See the Markley paper for further details.

        Note: By default, we use Gibbs vectors, which have a singularity at 180
        degrees. This is to avoid numerical issues calculating the covariance
        matrix when the quaternion covariance vector is large and the SUT scaling
        parameter alpha is set very small.

        The singularity being 180 degrees instead of 360 is not a problem unless
        the rotation is expected to change by more than 180 degrees in a single
        filter iteration; if it is, setting the MRP_A parameter to 1.0 moves the
        singularity to 360 degrees.
        static constexpr double MRP_A = 0.0;
        static constexpr double MRP_F = 2.0 * (MRP_A + 1.0);

        Definitions for sigma point weights. The naming convention follows that used
        in in Van der Merwe 2001.
        static constexpr double sigma_WM0 = lambda / (UKFModel::StateVec::COV_SIZE + lambda);
        static constexpr double sigma_WC0 = sigma_WM0 + (1.0 - alphaSquared + beta);
        static constexpr double sigma_WMI = 1.0 / (2.0 * (UKFModel::StateVec::COV_SIZE + lambda));
        static constexpr double sigma_WCI = sigma_WMI;

Then pass MyUKFParams<MyUKFModel> as the third template argument. Typically the default parameters should work well.

Optional: Numerical Integrators

You may also use a different numerical integrator for the process model by passing the integrator class as the fourth argument to the UKF.

  • kalman::util::integrator::RK4 4th order integrator. Default and usually the best choice.
  • kalman::util::integrator::Heun 2nd order integrator.
  • kalman::util::integrator::Euler 1st order integrator. Recommended if the process model is very simple, e.g. for parameter estimation.

It is also possible, though unlikely to be useful, to define your own integrator, following the following format:

 class RK4 {
        template <class Derivative, class InputType>
        static typename Derivative::StateVec integrate(double delta, const typename Derivative::StateVec & state,
                                                       const InputType & u) {
            typename Derivative::StateVec a = Derivative::dF(state, u);
            typename Derivative::StateVec b = Derivative::dF(typename Derivative::StateVec(state + 0.5 * delta * a), u);
            typename Derivative::StateVec c = Derivative::dF(typename Derivative::StateVec(state + 0.5 * delta * b), u);
            typename Derivative::StateVec d = Derivative::dF(typename Derivative::StateVec(state + delta * c), u);
            return state + (delta / 6.0) * (a + (b * 2.0) + (c * 2.0) + d);

Usage Example

Please see example.cpp for usage example.

To Compile Example:

  • Install CMake
  • mkdir build && cd build
  • cmake ..
    • if this doesn't find Eigen automatically, you may need cmake .. -DEIGEN_INCLUDE_DIRS="path/to/eigen/includes"
  • cmake --build . --config Release


Significant parts of this implementation are based on sfwa/ukf. Publications referenced include:

  • Kraft, "A Quaternion-based Unscented Kalman Filter for Orientation Tracking", 2003
  • Van der Merwe and Wan, "The Square-root Unscented Kalman Filter for state and Parameter-estimation", 2001
  • Wan and Van der Merwe, "The Unscented Kalman Filter for Nonlinear Estimation", 2000
  • Markley, "Attitude Error Representations for Kalman Filtering", 2002


apache 2.0


a minimal header-only C++ square root UKF library, with support for quaternion state vectors







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