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.. _tutorial-ekf: | ||
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=================================================================== | ||
Writing an Extended Kalman Filter (EKF) with mrpt::bayes | ||
=================================================================== | ||
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.. contents:: :local: | ||
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.. toctree:: | ||
:maxdepth: 2 | ||
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In MRPT, the family of `Kalman Filter algorithms <https://en.wikipedia.org/wiki/Kalman_filter>`_ | ||
such as the Extended KF (EKF) or the Iterative EKF (IEKF) are centralized in | ||
one single virtual class, `mrpt::bayes::CKalmanFilterCapable <class_mrpt_bayes_CKalmanFilterCapable.html>`_. | ||
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This C++ class keeps the **system state vector** and the system **covariance matrix**, as well as a | ||
generic method to execute one complete iteration of the selected algorithm. | ||
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In practice, solving a specific problem requires **deriving a new class** from this virtual class | ||
and implementing a few methods such as transforming the state vector through the transition model, | ||
or computing the Jacobian of the observation model linearized at some given value of the state space. | ||
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This page will teach you the implementation details of the 2D target tracking example | ||
shown in this video (`full source code <page_bayes_tracking_example.html>`_): | ||
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.. raw:: html | ||
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<div style="position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden; max-width: 100%; height: auto;"> | ||
<iframe src="https://www.youtube.com/embed/0_gGXYzjcGE" frameborder="0" allowfullscreen style="position: absolute; top: 0; left: 0; width: 100%; height: 100%;"></iframe> | ||
</div> | ||
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1. Kalman Filters in the MRPT | ||
-------------------------------- | ||
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A set of parameters that are problem-independent can be changed in the member `KF_options <struct_mrpt_bayes_TKF_options.html>`_ | ||
of this class, where the most important parameter is the **selection of the KF algorithm**. | ||
Currently it can be chosen from the following ones: | ||
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- Naive EKF: The basic EKF algorithm. | ||
- Iterative Kalman Filter (IKF): This method re-linearizes the Jacobians around increasingly more | ||
accurate values of the state vector. | ||
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.. note:: | ||
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An alternative implementation of Bayesian filtering in MRPT are Particle Filters. | ||
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2. Writing a KF class for a specific problem | ||
---------------------------------------------- | ||
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2.1. Deriving a new class | ||
~~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
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First a new class must be derived from ``mrpt::bayes::CKalmanFilterCapable``. | ||
A public method must be declared as an entry point for the user, which takes the domain-specific input data | ||
(range observations, sonar measurements, temperatures, etc.) and calls the protected method ``runOneKalmanIteration()`` | ||
of the parent class. | ||
There are **two fundamental types of systems** the user can build by deriving a new class: | ||
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- **A regular tracking problem**: The size of the state vector remains unchanged over time. This size must be returned by the virtual method `get_vehicle_size()`. | ||
- **A SLAM-like problem**: The size of the state vector grows as new map elements are incorporated over time. | ||
In this case the first ``get_vehicle_size()`` scalar elements of the state vector correspond to the state of | ||
the vehicle/robot/camera/... and the rest of the state vector is a whole number of ``get_feature_size()`` sub-vectors, | ||
each describing one map element (landmark, feature,...). | ||
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2.2. The internal flow of the algorithm | ||
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
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The method ``mrpt::bayes::CKalmanFilterCapable::runOneKalmanIteration()`` will sequentially call each of the virtual methods, | ||
according to the following order: | ||
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- **During the KF prediction stage**: | ||
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- ``OnGetAction()`` | ||
- ``OnTransitionModel()`` | ||
- ``OnTransitionJacobian()`` | ||
- ``OnTransitionNoise()`` | ||
- ``OnNormalizeStateVector()``: This can be optionally implemented if required for the concrete problem. | ||
- ``OnGetObservations()``: This is the ideal place for generating observations in those applications where it | ||
requires an estimate of the current state (e.g. in visual SLAM, to predict where each landmark will be found in the image). | ||
At this point the internal state vector and covariance contain the "prior distribution", i.e. updated through the transition model. | ||
This is also the place where data association must be solved (in mapping problems). | ||
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- **During the KF update stage**: | ||
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- ``OnObservationModelAndJacobians()`` | ||
- ``OnNormalizeStateVector()`` | ||
- If the system implements a mapping problem, and the data association returned by ``OnGetObservations()`` indicates the | ||
existence of unmapped observations, then the next method will be invoked for each of these new features: ``OnInverseObservationModel()``. | ||
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- ``OnPostIteration()``: A placeholder for any code the user wants to execute after each iteration (e.g. logging, visualization,...). | ||
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3. An example | ||
------------------- | ||
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An example of a KF implementation can be found under `samples/bayesianTracking <page_bayes_tracking_example.html>`_ | ||
for the problem of **tracking a vehicle** from noisy **range and bearing** data. | ||
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Here we will derive the required equations to be implemented, as well as how they are actually implemented in C++. | ||
Note that this problem is also implemented as a Particle Filter in the same example in order to visualize side-to-side | ||
the performance of both approaches. | ||
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3.1. Problem Statement | ||
~~~~~~~~~~~~~~~~~~~~~~~~~~~ | ||
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The problem of **range-bearing tracking** is that of estimating the vehicle state: | ||
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.. math:: | ||
\mathbf{x}=(x ~ y ~ v_x ~ v_y) | ||
where x and y are the Cartesian coordinates of the vehicle, and vx and vy are the linear velocities. | ||
Thus, we will use a simple constant velocity model, where the transition function is: | ||
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.. math:: | ||
\mathbf{x_k} = f(\mathbf{x_{k-1}}, \Delta t) = \left\{ \begin{array}{l} x_{k-1} + v_{x_k} \Delta t \\ y_{k-1} + v_{y_k} \Delta t \end{array} \right. | ||
We will consider the time interval between steps :math:`Delta t` as the action $u$ of the system. | ||
The observation vector :math:`z=(z_b ~ z_r)` consists of the bearing and range of the vehicle from some point (arbitrarily set to the origin): | ||
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.. math:: | ||
z_b = atan2(y,x) | ||
.. math:: | ||
z_r = \sqrt{ x^2 + y^2 } | ||
Then, it is straightforward to obtain the required Jacobian of the transition function: | ||
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.. math:: | ||
\frac{\partial f}{\partial x} = \left( \begin{array}{cccc} 1 ~ 0 ~ \Delta t ~ 0 \\ 0 ~ 1 ~ 0 ~ \Delta t \\ 0 ~ 0 ~ 1 ~ 0 \\ 0 ~ 0 ~ 0 ~ 1 \end{array} \right) | ||
and the Jacobian of the observation model: | ||
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.. math:: | ||
\frac{\partial h}{\partial x} = \left( \begin{array}{cccc} \frac{-y}{x^2+y^2} ~ \frac{1}{x\left(1+\left( \frac{y}{x} \right)^2\right)} ~ 0 ~ 0 \\ \frac{x}{\sqrt{x^2+y^2}} ~ \frac{y}{\sqrt{x^2+y^2}} ~ 0 ~ 0 \end{array} \right) | ||
3.2 Implementation | ||
~~~~~~~~~~~~~~~~~~~~ | ||
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The most important implemented methods are detailed below. For further details refer to the complete sources of the example. | ||
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3.2.1 The transition model | ||
^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
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The constant-velocity model is implemented simply as: | ||
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.. code-block:: cpp | ||
/** Implements the transition model $latex \hat{x}_{k|k-1} = f( \hat{x}_{k-1|k-1}, u_k ) $ | ||
* \param in_u The vector returned by OnGetAction. | ||
* \param inout_x At input has $latex \hat{x}_{k-1|k-1} $, at output must have $latex \hat{x}_{k|k-1} $. | ||
* \param out_skip Set this to true if for some reason you want to skip the prediction step (to do not modify either the vector or the covariance). Default:false | ||
*/ | ||
void CRangeBearing::OnTransitionModel( | ||
const KFArray_ACT &in_u, | ||
KFArray_VEH &inout_x, | ||
bool &out_skipPrediction | ||
) const | ||
{ | ||
// in_u[0] : Delta time | ||
// in_out_x: [0]:x [1]:y [2]:vx [3]: vy | ||
inout_x[0] += in_u[0] * inout_x[2]; | ||
inout_x[1] += in_u[0] * inout_x[3]; | ||
} | ||
3.2.2 The transition model Jacobian | ||
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
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This is just the Jacobian of the state propagation equation above: | ||
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.. code-block:: cpp | ||
/** Implements the transition Jacobian $latex \frac{\partial f}{\partial x} $ | ||
* \param out_F Must return the Jacobian. | ||
* The returned matrix must be $N \times N$latex with N being either the size of the whole state vector or get_vehicle_size(). | ||
*/ | ||
void CRangeBearing::OnTransitionJacobian(KFMatrix_VxV &F) const | ||
{ | ||
F.unit(); | ||
F(0,2) = m_deltaTime; | ||
F(1,3) = m_deltaTime; | ||
} | ||
3.2.3 The observations and the observation model | ||
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
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.. code-block:: cpp | ||
void CRangeBearing::OnGetObservationsAndDataAssociation( | ||
std::vector<KFArray_OBS> &out_z, | ||
vector_int &out_data_association, | ||
const vector<KFArray_OBS> &in_all_predictions, | ||
const KFMatrix &in_S, | ||
const vector_size_t &in_lm_indices_in_S, | ||
const KFMatrix_OxO &in_R | ||
) | ||
{ | ||
out_z.resize(1); | ||
out_z[0][0] = m_obsBearing; | ||
out_z[0][1] = m_obsRange; | ||
out_data_association.clear(); // Not used | ||
} | ||
/** Implements the observation prediction $latex h_i(x) $. | ||
* \param idx_landmark_to_predict The indices of the landmarks in the map whose predictions are expected as output. For non SLAM-like problems, this input value is undefined and the application should just generate one observation for the given problem. | ||
* \param out_predictions The predicted observations. | ||
*/ | ||
void CRangeBearing::OnObservationModel( | ||
const vector_size_t &idx_landmarks_to_predict, | ||
std::vector<KFArray_OBS> &out_predictions | ||
) const | ||
{ | ||
// predicted bearing: | ||
kftype x = m_xkk[0]; | ||
kftype y = m_xkk[1]; | ||
kftype h_bear = atan2(y,x); | ||
kftype h_range = sqrt(square(x)+square(y)); | ||
// idx_landmarks_to_predict is ignored in NON-SLAM problems | ||
out_predictions.resize(1); | ||
out_predictions[0][0] = h_bear; | ||
out_predictions[0][1] = h_range; | ||
} | ||
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<package format="3"> | ||
<name>mrpt2</name> | ||
<!-- Before updating version number, read [MRPT_ROOT]/version_prefix.txt first --> | ||
<version>2.11.0</version> | ||
<version>2.11.1</version> | ||
<description>Mobile Robot Programming Toolkit (MRPT) version 2.x</description> | ||
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<author email="[email protected]">Jose-Luis Blanco-Claraco</author> | ||
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This example illustrates how to implement an Extended Kalman Filter (EKF) | ||
and a particle filter (PF) using mrpt::bayes classes, for the problem of | ||
tracking a 2D mobile target with state space being its location and its | ||
velocity vector. | ||
tracking a 2D mobile target with **state space** being its **location** and its | ||
**velocity vector**. | ||
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Demo video: [https://www.youtube.com/watch?v=0_gGXYzjcGE](https://www.youtube.com/watch?v=0_gGXYzjcGE) | ||
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The equations of this example and the theory behind them are explained in [this tutorial](tutorial-ekf.html). | ||
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