#include <visp3/core/vpLinearKalmanFilterInstantiation.h>

Inheritance diagram for vpLinearKalmanFilterInstantiation:

Public Types
enum	vpStateModel { stateConstVel_MeasurePos , stateConstVelWithColoredNoise_MeasureVel , stateConstAccWithColoredNoise_MeasureVel , unknown }

Public Member Functions
	vpLinearKalmanFilterInstantiation ()

vpStateModel	getStateModel ()

void	filter (vpColVector &z)

void	setNumberOfSignal (unsigned int n_signal)

void	init (unsigned int size_state, unsigned int size_measure, unsigned int n_signal)

void	prediction ()

void	filtering (const vpColVector &z)

unsigned int	getStateSize ()

unsigned int	getMeasureSize ()

unsigned int	getNumberOfSignal ()

long	getIteration ()

void	verbose (bool on)

Generic linear filter initializer
void	setStateModel (vpStateModel model)

void	initFilter (unsigned int nsignal, vpColVector &sigma_state, vpColVector &sigma_measure, double rho, double dt)

Linear filter initializer with constant velocity models
void	initStateConstVel_MeasurePos (unsigned int nsignal, vpColVector &sigma_state, vpColVector &sigma_measure, double dt)

void	initStateConstVelWithColoredNoise_MeasureVel (unsigned int nsignal, vpColVector &sigma_state, vpColVector &sigma_measure, double rho)

Public Attributes
vpColVector	Xest

vpColVector	Xpre

vpMatrix	F

vpMatrix	H

vpMatrix	R

vpMatrix	Q

double	dt

vpMatrix	Ppre

vpMatrix	Pest

Protected Attributes
long	iter

unsigned int	size_state

unsigned int	size_measure

unsigned int	nsignal

bool	verbose_mode

vpMatrix	W

vpMatrix	I

Linear filter initializer with constant acceleration models
vpStateModel	model

void	initStateConstAccWithColoredNoise_MeasureVel (unsigned int nsignal, vpColVector &sigma_state, vpColVector &sigma_measure, double rho, double dt)

Detailed Description

This class provides an implementation of some specific linear Kalman filters.

Examples: servoAfma4Point2DCamVelocityKalman.cpp, servoViper850Point2DCamVelocityKalman.cpp, testKalmanAcceleration.cpp, and testKalmanVelocity.cpp.

Definition at line 53 of file vpLinearKalmanFilterInstantiation.h.

Member Enumeration Documentation

◆ vpStateModel

enum vpLinearKalmanFilterInstantiation::vpStateModel

Selector used to set the Kalman filter state model.

Enumerator
stateConstVel_MeasurePos	Consider the state as a constant velocity model with white noise. Measures available are the successive positions of the target. To know more about this state model, see initStateConstVel_MeasurePos().
stateConstVelWithColoredNoise_MeasureVel	Consider the state as a constant velocity model with colored noise measurements as acceleration terms. Measured available are the velocities of the target. To know more about this state model, see initStateConstVelWithColoredNoise_MeasureVel().
stateConstAccWithColoredNoise_MeasureVel	Consider the state as a constant acceleration model with colored noise measurements as acceleration terms. Measured available are the velocities of the target. To know more about this state model, see initStateConstAccWithColoredNoise_MeasureVel().
unknown	Used to indicate that the state model is not initialized.

Definition at line 59 of file vpLinearKalmanFilterInstantiation.h.

Constructor & Destructor Documentation

◆ vpLinearKalmanFilterInstantiation()

vpLinearKalmanFilterInstantiation::vpLinearKalmanFilterInstantiation ( )

inline

Default linear Kalman filter.

By default the state model is unknown and set to vpLinearKalmanFilterInstantiation::unknown.

Definition at line 86 of file vpLinearKalmanFilterInstantiation.h.

Member Function Documentation

◆ filter()

void vpLinearKalmanFilterInstantiation::filter ( vpColVector & z )

Do the filtering and prediction of the measure signal.

Parameters

z	: Measures ${\bf z}_k$ used to initialise the filter. The dimension of this vector is equal to the number of signal to filter (given by getNumberOfSignal()) multiplied by the size of the measure vector (given by getMeasureSize()) .

Exceptions

vpException::notInitialized : If the filter is not initialized. To initialize the filter see initFilter().

Examples: servoAfma4Point2DCamVelocityKalman.cpp, servoViper850Point2DCamVelocityKalman.cpp, testKalmanAcceleration.cpp, and testKalmanVelocity.cpp.

Definition at line 803 of file vpLinearKalmanFilterInstantiation.cpp.

References vpKalmanFilter::dt, vpKalmanFilter::filtering(), vpKalmanFilter::iter, model, vpException::notInitialized, vpKalmanFilter::nsignal, vpKalmanFilter::prediction(), vpKalmanFilter::size_measure, vpKalmanFilter::size_state, stateConstAccWithColoredNoise_MeasureVel, stateConstVel_MeasurePos, stateConstVelWithColoredNoise_MeasureVel, unknown, and vpKalmanFilter::Xest.

◆ filtering()

void vpKalmanFilter::filtering ( const vpColVector & z )

inherited

Update the Kalman filter by applying the filtering equations and increment the filter iteration (vpKalmanFilter::iter).

Parameters

z	: Measure (or observation) ${\bf z}_k$ provided at iteration .

The filtering equation is given by:

${\bf x}_{k \mid k} = {\bf x}_{k \mid k-1} + {\bf W}_k \left[ {\bf z}_k - {\bf H} {\bf x}_{k \mid k-1} \right]$

where ${\bf W_k}$ is the filter gain computed using the formula:

${\bf W_k} = {\bf P}_{k \mid k-1} {\bf H}^T \left[ {\bf H P}_{k \mid k-1} {\bf H}^T + {\bf R}_k \right]^{-1}$

and where the updated covariance of the state is given by

${\bf P}_{k \mid k} = \left({\bf I} - {\bf W}_k {\bf H} \right) {\bf P}_{k \mid k-1}$

or in a symmetric form

${\bf P}_{k \mid k} = {\bf P}_{k \mid k-1} - {\bf W}_k {\bf S}_k {\bf W}^T_k$

with

${\bf S}_k = {\bf H P}_{k \mid k-1} {\bf H}^T + {\bf R}_k$

Definition at line 199 of file vpKalmanFilter.cpp.

References vpKalmanFilter::H, vpMatrix::inverseByLU(), vpKalmanFilter::iter, vpKalmanFilter::Pest, vpKalmanFilter::Ppre, vpKalmanFilter::R, vpMatrix::t(), vpKalmanFilter::verbose_mode, vpKalmanFilter::W, vpKalmanFilter::Xest, and vpKalmanFilter::Xpre.

Referenced by filter().

◆ getIteration()

long vpKalmanFilter::getIteration ( )

inlineinherited

Return the iteration number.

Definition at line 154 of file vpKalmanFilter.h.

◆ getMeasureSize()

unsigned int vpKalmanFilter::getMeasureSize ( )

inlineinherited

Return the size of the measure vector ${\bf z}_{(k)}$ for one signal.

Examples: testKalmanAcceleration.cpp.

Definition at line 146 of file vpKalmanFilter.h.

◆ getNumberOfSignal()

unsigned int vpKalmanFilter::getNumberOfSignal ( )

inlineinherited

Return the number of signal to filter.

Definition at line 150 of file vpKalmanFilter.h.

◆ getStateModel()

vpStateModel vpLinearKalmanFilterInstantiation::getStateModel ( )

inline

Return the current state model.

Definition at line 91 of file vpLinearKalmanFilterInstantiation.h.

◆ getStateSize()

unsigned int vpKalmanFilter::getStateSize ( )

inlineinherited

Return the size of the state vector ${\bf x}_{(k)}$ for one signal.

Examples: servoAfma4Point2DCamVelocityKalman.cpp, servoViper850Point2DCamVelocityKalman.cpp, and testKalmanAcceleration.cpp.

Definition at line 142 of file vpKalmanFilter.h.

◆ init()

BEGIN_VISP_NAMESPACE void vpKalmanFilter::init	(	unsigned int	size_state_vector,
		unsigned int	size_measure_vector,
		unsigned int	n_signal
	)

inherited

Initialize the Kalman filter.

Parameters

size_state_vector	: Size of the state vector ${\bf x}_{k}$ for one signal.
size_measure_vector	: Size of the measure vector ${\bf z}_{k}$ for one signal.
n_signal	: Number of signal to filter.

Definition at line 58 of file vpKalmanFilter.cpp.

References vpKalmanFilter::dt, vpKalmanFilter::F, vpKalmanFilter::H, vpKalmanFilter::I, vpKalmanFilter::iter, vpKalmanFilter::nsignal, vpKalmanFilter::Pest, vpKalmanFilter::Q, vpKalmanFilter::R, vpColVector::resize(), vpArray2D< Type >::resize(), vpKalmanFilter::size_measure, vpKalmanFilter::size_state, vpKalmanFilter::Xest, and vpKalmanFilter::Xpre.

Referenced by initStateConstAccWithColoredNoise_MeasureVel(), initStateConstVel_MeasurePos(), initStateConstVelWithColoredNoise_MeasureVel(), and vpKalmanFilter::vpKalmanFilter().

◆ initFilter()

BEGIN_VISP_NAMESPACE void vpLinearKalmanFilterInstantiation::initFilter	(	unsigned int	n_signal,
		vpColVector &	sigma_state,
		vpColVector &	sigma_measure,
		double	rho,
		double	delta_t
	)

Initialize the Kalman filter material depending on the selected state model set with setStateModel(). This function is provided as a wrapper over all the other initializer functions like initStateConstVel_MeasurePos(), initStateConstVelWithColoredNoise_MeasureVel(), initStateConstAccWithColoredNoise_MeasureVel().

Warning: It is required to set the state model before using this method.

Parameters

n_signal	: Number of signal to filter.
sigma_state	: Vector that contains the variance of the state noise. The dimension of this vector is equal to the state vector ${\bf x}$ size multiplied by the number of signal to filter. Values are used to initialize the ${\bf Q}$ state covariance matrix.
sigma_measure	: Vector that contains the variance of the measurement noise. The dimension of this vector is equal to the measure vector ${\bf x}$ size multiplied by the number of signal to filter. Values are used to initialize the ${\bf R}$ measure covariance matrix.
rho	: Degree of correlation between successive accelerations. Values are in [0:1[.
delta_t	: Sampling time $\Delta t$ expressed is second. Depending on the filter modelization, this value may not be used. This is for example the case for the vpLinearKalmanFilterInstantiation::stateConstVelWithColoredNoise_MeasureVel model implemented in initStateConstVelWithColoredNoise_MeasureVel().

Exceptions

vpException::badValue	: Bad rho value which is not in [0:1[.
vpException::notInitialized	: If the state model is not initialized. To initialize it you need to call setStateModel().

The example below shows how to initialize the filter for a one dimensional signal.

#include <visp3/core/vpLinearKalmanFilterInstantiation.h>
 
#ifdef ENABLE_VISP_NAMESPACE
using namespace VISP_NAMESPACE_NAME;
#endif
 
int main()
{
  vpLinearKalmanFilterInstantiation kalman;
  // Select a constant velocity state model with colored noise
  // Measures are velocities
  kalman.setStateModel(vpLinearKalmanFilterInstantiation::stateConstVelWithColoredNoise_MeasureVel);
 
  // Initialise the filter for a one dimension signal
  int signal = 1;
  vpColVector sigma_state(2);   // State vector size is 2
  vpColVector sigma_measure(1); // Measure vector size is 1
  double rho = 0.9;
  double dummy = 0; // non used parameter for the selected state model
 
  kalman.initFilter(signal, sigma_state, sigma_measure, rho, dummy);
}

The example below shows a more complete example to filter a two dimensional target trajectory with an estimation of the target velocities from velocity measures.

#include <visp3/core/vpLinearKalmanFilterInstantiation.h>
 
#ifdef ENABLE_VISP_NAMESPACE
using namespace VISP_NAMESPACE_NAME;
#endif
 
int main()
{
  vpLinearKalmanFilterInstantiation kalman;
 
  // Set the constant velocity state model used for the filtering
  vpLinearKalmanFilterInstantiation::vpStateModel model;
  model = vpLinearKalmanFilterInstantiation::stateConstVelWithColoredNoise_MeasureVel;
  kalman.setStateModel(model);
 
  // We are now able to retrieve the size of the state vector and measure vector
  int state_size = kalman.getStateSize();
 
  // Filter the x and y velocities of a target (2 signals are to consider)
  int nsignal = 2;
 
  // Initialize the filter parameters:
  // - Firstly, the state variance
  int size = state_size*nsignal;
  vpColVector sigma_state(size);
  sigma_state[1] = 0.001; // Variance on the acceleration for the 1st signal (x)
  sigma_state[3] = 0.002; // Variance on the acceleration for the 2nd signal (y)
  // - Secondly, the measures variance
  vpColVector sigma_measure(nsignal); // 2 velocity measures available
  sigma_measure[0] = 0.03; // Variance on the x velocity measure
  sigma_measure[1] = 0.06; // Variance on the y velocity measure
  // - Thirdly, the correlation between successive accelerations
  double rho = 0.9;
 
  double dummy = 0; // non used parameter for the selected state model
 
  // Initialize the filter
  // The state model was set before
  kalman.initFilter(nsignal, sigma_state, sigma_measure, rho, dummy);
 
  // Does the filtering
  vpColVector vm(2); // Measured velocities
  for ( ; ; ) {
    // Get the two dimensional velocity measures
    // vm[0] = ...;
    // vm[1] = ...;
 
    // Compute the filtering and the prediction
    kalman.filter(vm);
    // Print the estimation of the velocities (1st value of the state vector)
    std::cout << "Estimated x velocity: kalman.Xest[0]" << std::endl;
    std::cout << "Estimated y velocity: kalman.Xest[kalman.getStateSize()]"
              << std::endl;
    // The one step prediction is available in kalman.Xpre variable
  }
}

Examples: servoAfma4Point2DCamVelocityKalman.cpp, servoViper850Point2DCamVelocityKalman.cpp, testKalmanAcceleration.cpp, and testKalmanVelocity.cpp.

Definition at line 179 of file vpLinearKalmanFilterInstantiation.cpp.

References initStateConstAccWithColoredNoise_MeasureVel(), initStateConstVel_MeasurePos(), initStateConstVelWithColoredNoise_MeasureVel(), model, vpException::notInitialized, stateConstAccWithColoredNoise_MeasureVel, stateConstVel_MeasurePos, stateConstVelWithColoredNoise_MeasureVel, and unknown.

◆ initStateConstAccWithColoredNoise_MeasureVel()

void vpLinearKalmanFilterInstantiation::initStateConstAccWithColoredNoise_MeasureVel	(	unsigned int	n_signal,
		vpColVector &	sigma_state,
		vpColVector &	sigma_measure,
		double	rho,
		double	delta_t
	)

Modelization of a constant acceleration state model with colored noise. The measure is assumed to be the velocity of the target.

This state model assume that there is some memory associated with noise measurements as acceleration terms. They can be represented as remaining correlated (or colored) over successive time intervals, leading to the following state model:

$\left\{ \begin{array}{rll} x_{(k+1)} & = x_{(k)} + \Delta t \; \dot{x}_{(k)} + \nu_{(k)} &\\ \nu_{(k+1)}& = \rho \nu_{(k)} &+w_{1(k)} \\ \dot{x}_{(k+1)} & = \dot{x}_{(k)} &+w_{2(k)}\\ \end{array} \right.$

The terms $w_{1(k)}$ and $w_{2(k)}$ account for deviations from the assumed constant acceleration trajectory. They are assumed zero-mean, white, mutually uncorrelated, stationary random variable with variance $\sigma^2_{Q_1}$ and $\sigma^2_{Q_2}$ . The term $\rho$ is the degree of correlation between successive accelerations. Values can range from 0 to 1.

We recall that the recursive state evolution equation is given by

${\bf x}_k= {\bf F}_{k-1} {\bf x}_{k-1} + {\bf w}_{k-1} \\$

From this state model, the transition matrix ${\bf F}$ and the state covariance matrix ${\bf Q}$ are given by:

${\bf F} = \left[ \begin{array}{ccc} 1 & 1 & \Delta t\\ 0 & \rho & 0 \\ 0 & 0 & 1 \end{array} \right]$

and

${\bf Q} = \left[ \begin{array}{ccc} 0 & 0 & 0\\ 0 & \sigma^2_{Q_1} & 0\\ 0 & 0& \sigma^2_{Q_2} \\ \end{array} \right]$

The measurement model is given by:

$z_{(k)} = {\bf H} {\bf x}_{(k)} + r_{(k)}$

where ${\bf H} = [1 \; 0 \; 0]$ , $z_{(k)}$ is the measure of the velocity and $r_{(k)}$ is the measurement noise, assumed zero-mean, white mutually uncorrelated stationary random variables with variance $\sigma^2_R$ , giving the covariance matrix:

${\bf R} = \left[\sigma^2_R\right]$

The initial value of the state vector is set to:

${\bf x_{(0)}} = \left[ \begin{array}{c} z_{(0)}\\ 0 \\ 0 \end{array} \right]$

The initial value $P_{(0|0)}$ of the prediction covariance matrix is given by:

${\bf P_{(0|0)}} = \left[ \begin{array}{ccc} \sigma^2_R & 0 & \sigma^2_R / \Delta t\\ 0 & \sigma^2_{Q_1}/(1-\rho^2) & -\rho \sigma^2_{Q_1} / (1-\rho^2)\Delta t \\ \sigma^2_R / \Delta t & -\rho \sigma^2_{Q_1} / (1-\rho^2)\Delta t & (2\sigma^2_R +\sigma^2_{Q_1}/(1-\rho^2) )/\Delta t^2 \end{array} \right]$

Parameters

n_signal	: Number of signal to filter.
sigma_state	: Vector that fix the variance of the state covariance matrix $[0 \; \sigma^2_{Q_1} \; \sigma^2_{Q_2}]^T$ . The dimension of this vector is 3 multiplied by the number of signal to filter.
sigma_measure	: Variance $\sigma^2_R$ of the measurement noise. The dimension of this vector is equal to the number of signal to filter.
rho	: Degree of correlation between successive accelerations. Values are in [0:1[.
delta_t	: Sampling time $\Delta t$ expressed is second.

Exceptions

vpException::badValue : Bad rho value which is not in [0:1[.

The example below shows how to filter a two dimensional target trajectory with an estimation of the target velocity from velocity measures.

#include <visp3/core/vpLinearKalmanFilterInstantiation.h>
 
#ifdef ENABLE_VISP_NAMESPACE
using namespace VISP_NAMESPACE_NAME;
#endif
 
int main()
{
  vpLinearKalmanFilterInstantiation kalman;
  // Filter the x and y velocities of a target (2 signals are to consider)
  int nsignal = 2;
 
  // Initialize the filter parameters:
  // - Firstly, the state variance
  vpColVector sigma_state(6); // 6 = 3 for the state size x 2 signal
  sigma_state[1] = 0.001; // Variance on the acceleration for the 1st signal (x)
  sigma_state[2] = 0.001; // Variance on the acceleration for the 1st signal (x)
  sigma_state[4] = 0.002; // Variance on the acceleration for the 2nd signal (y)
  sigma_state[5] = 0.002; // Variance on the acceleration for the 2nd signal (y)
  // - Secondly, the measures variance
  vpColVector sigma_measure(nsignal); // 2 velocity measures available
  sigma_measure[0] = 0.03; // Variance on the x velocity measure
  sigma_measure[1] = 0.06; // Variance on the y velocity measure
  // - Thirdly, the correlation between successive accelerations
  double rho = 0.9;
  // - Lastly, the sampling time
  double dt = 0.020; // 20 ms
 
  // Initialize the filter
  kalman.initStateConstAccWithColoredNoise_MeasureVel(nsignal, sigma_state, sigma_measure, rho, dt);
 
  // Does the filtering
  vpColVector vm(2); // Measured velocities
  for ( ; ; ) {
    // Get the two dimensional velocity measures
    // vm[0] = ...;
    // vm[1] = ...;
 
    // Compute the filtering and the prediction
    kalman.filter(vm);
    // Print the estimation of the velocities (1st value of the state vector)
    std::cout << "Estimated x velocity: kalman.Xest[0]" << std::endl;
    std::cout << "Estimated y velocity: kalman.Xest[kalman.getStateSize()]"
              << std::endl;
    // The one step prediction is available in kalman.Xpre variable
  }
}

Definition at line 727 of file vpLinearKalmanFilterInstantiation.cpp.

References vpException::badValue, vpKalmanFilter::dt, vpKalmanFilter::F, vpKalmanFilter::H, vpKalmanFilter::init(), vpKalmanFilter::iter, vpKalmanFilter::nsignal, vpKalmanFilter::Pest, vpKalmanFilter::Q, vpKalmanFilter::R, setStateModel(), vpKalmanFilter::size_measure, vpKalmanFilter::size_state, stateConstAccWithColoredNoise_MeasureVel, and vpKalmanFilter::Xest.

Referenced by initFilter().

◆ initStateConstVel_MeasurePos()

void vpLinearKalmanFilterInstantiation::initStateConstVel_MeasurePos	(	unsigned int	n_signal,
		vpColVector &	sigma_state,
		vpColVector &	sigma_measure,
		double	delta_t
	)

Modelization of a constant speed state model with white noise. The measure is assumed to be the position of the target.

The considered state model is the following

$\left\{ \begin{array}{rlrl} x_{(k+1)} & = x_{(k)} & + \Delta t \; {\dot{x}}_{(k)} & + {w_1}_{(k)} \\ {\dot{x}}_{(k+1)} & = & {\dot{x}}_{(k)} &+{w_2}_{(k)} \end{array} \right.$

The terms ${w_1}_{(k)}$ and ${w_2}_{(k)}$ account for deviations from the assumed constant velocity trajectory. They are assumed zero-mean, white, mutually uncorrelated, stationary random variable with variance $\sigma^2_{Q_1}$ and $\sigma^2_{Q_2}$ .

We recall that the recursive state evolution equation is given by

${\bf x}_k= {\bf F}_{k-1} {\bf x}_{k-1} + {\bf w}_{k-1} \\$

From this state model, the transition matrix ${\bf F}$ and the state covariance matrix ${\bf Q}$ are given by:

${\bf F} = \left[ \begin{array}{cc} 1 & \Delta t\\ 0 & 1 \end{array} \right]$

and

${\bf Q} = \sigma^2_Q \left[ \begin{array}{cc} \frac{1}{3}\Delta t^3 & \frac{1}{2}\Delta t^2\\ \frac{1}{2}\Delta t^2 & \Delta t \end{array} \right]$

The initial value of the state vector at iteration 0 is set to:

${\bf x_{(0)}} = \left[ \begin{array}{c} z_{(0)}\\ 0 \end{array} \right]$

The value at iteration 1 is set to:

${\bf x_{(1)}} = \left[ \begin{array}{c} z_{(1)}\\ (z_{(1)} - z_{(0)})/ \Delta t \end{array} \right]$

The initial value $P_{(0|0)}$ of the prediction covariance matrix is given by:

${\bf P_{(0|0)}} = \left[ \begin{array}{cc} \sigma^2_R & \frac{\sigma^2_R}{2 \Delta t}\\ \frac{\sigma^2_R}{2 \Delta t} & \frac{2}{3}\sigma^2_Q \Delta t + \frac{\sigma^2_R}{2 \Delta t^2} \end{array} \right]$

Parameters

n_signal	: Number of signal to filter.
sigma_state	: Vector that fix the variance of the state covariance matrix $[\sigma^2_Q \; 0]^T$ . The dimension of this vector is 2 multiplied by the number of signal to filter.
sigma_measure	: Variance $\sigma^2_R$ of the measurement noise. The dimension of this vector is equal to the number of signal to filter.
delta_t	: Sampling time $\Delta t$ expressed is second.

Definition at line 293 of file vpLinearKalmanFilterInstantiation.cpp.

References vpKalmanFilter::dt, vpKalmanFilter::F, vpKalmanFilter::H, vpKalmanFilter::init(), vpKalmanFilter::iter, vpKalmanFilter::Pest, vpKalmanFilter::Q, vpKalmanFilter::R, setStateModel(), vpKalmanFilter::size_measure, vpKalmanFilter::size_state, stateConstVel_MeasurePos, and vpKalmanFilter::Xest.

Referenced by initFilter().

◆ initStateConstVelWithColoredNoise_MeasureVel()

void vpLinearKalmanFilterInstantiation::initStateConstVelWithColoredNoise_MeasureVel	(	unsigned int	n_signal,
		vpColVector &	sigma_state,
		vpColVector &	sigma_measure,
		double	rho
	)

Modelization of a constant speed state model with colored noise. The measure is assumed to be the velocity of the target.

This state model assume that there is some memory associated with noise measurements as acceleration terms. They can be represented as remaining correlated (or colored) over successive time intervals, leading to the following state model:

$\left\{ \begin{array}{rll} x_{(k+1)} & = x_{(k)} + \nu_{(k)} &\\ \nu_{(k+1)}& = \rho \nu_{(k)} &+w_{(k)} \end{array} \right.$

The term $w_{(k)}$ account for deviations from the assumed constant velocity trajectory. It is assumed zero-mean, white, mutually uncorrelated, stationary random variable with variance $\sigma^2_Q$ . The term $\rho$ is the degree of correlation between successive accelerations. Values can range from 0 to 1.

We recall that the recursive state evolution equation is given by

${\bf x}_k= {\bf F}_{k-1} {\bf x}_{k-1} + {\bf w}_{k-1} \\$

From this state model, the transition matrix ${\bf F}$ and the state covariance matrix ${\bf Q}$ are given by:

${\bf F} = \left[ \begin{array}{cc} 1 & 1\\ 0 & \rho \end{array} \right]$

and

${\bf Q} = \left[ \begin{array}{cc} 0 & 0\\ 0 & \sigma^2_Q \end{array} \right]$

The measurement model is given by:

$z_{(k)} = {\bf H} {\bf x}_{(k)} + r_{(k)}$

where ${\bf H} = [1 \; 0 \; 0]$ , $z_{(k)}$ is the measure of the velocity and $r_{(k)}$ is the measurement noise, assumed zero-mean, white mutually uncorrelated stationary random variables with variance $\sigma^2_R$ , giving the covariance matrix:

${\bf R} = \left[\sigma^2_R\right]$

The initial value of the state vector is set to:

${\bf x_{(0)}} = \left[ \begin{array}{c} z_{(0)}\\ 0 \end{array} \right]$

The initial value $P_{(0|0)}$ of the prediction covariance matrix is given by:

${\bf P_{(0|0)}} = \left[ \begin{array}{cc} \sigma^2_R & 0\\ 0 & \sigma^2_Q/(1-\rho^2) \end{array} \right]$

Parameters

n_signal	: Number of signal to filter.
sigma_state	: Vector that fix the variance of the state covariance matrix $[0 \; \sigma^2_Q]^T$ . The dimension of this vector is 2 multiplied by the number of signal to filter.
sigma_measure	: Variance $\sigma^2_R$ of the measurement noise. The dimension of this vector is equal to the number of signal to filter.
rho	: Degree of correlation between successive accelerations. Values are in [0:1[.

Exceptions

vpException::badValue : Bad rho value which is not in [0:1[.

The example below shows how to filter a two dimensional target trajectory with an estimation of the target velocity from velocity measures.

#include <visp3/core/vpLinearKalmanFilterInstantiation.h>
 
#ifdef ENABLE_VISP_NAMESPACE
using namespace VISP_NAMESPACE_NAME;
#endif
 
int main()
{
  vpLinearKalmanFilterInstantiation kalman;
  // Filter the x and y velocities of a target (2 signals are to consider)
  int nsignal = 2;
 
  // Initialize the filter parameters:
  // - Firstly, the state variance
  vpColVector sigma_state(4); // 4 = 2 for the state size x 2 signal
  sigma_state[1] = 0.001; // Variance on the acceleration for the 1st signal (x)
  sigma_state[3] = 0.002; // Variance on the acceleration for the 2nd signal (y)
  // - Secondly, the measures variance
  vpColVector sigma_measure(nsignal); // 2 velocity measures available
  sigma_measure[0] = 0.03; // Variance on the x velocity measure
  sigma_measure[1] = 0.06; // Variance on the y velocity measure
  // - Thirdly, the correlation between successive accelerations
  double rho = 0.9;
 
  // Initialize the filter
  kalman.initStateConstVelWithColoredNoise_MeasureVel(nsignal, sigma_state, sigma_measure, rho);
 
  // Does the filtering
  vpColVector vm(2); // Measured velocities
  for ( ; ; ) {
    // Get the two dimensional velocity measures
    // vm[0] = ...;
    // vm[1] = ...;
 
    // Compute the filtering and the prediction
    kalman.filter(vm);
    // Print the estimation of the velocities (1st value of the state vector)
    std::cout << "Estimated x velocity: kalman.Xest[0]" << std::endl;
    std::cout << "Estimated y velocity: kalman.Xest[kalman.getStateSize()]"
              << std::endl;
    // The one step prediction is available in kalman.Xpre variable
  }
}

Definition at line 503 of file vpLinearKalmanFilterInstantiation.cpp.

References vpException::badValue, vpKalmanFilter::F, vpKalmanFilter::H, vpKalmanFilter::init(), vpKalmanFilter::iter, vpKalmanFilter::Pest, vpKalmanFilter::Q, vpKalmanFilter::R, setStateModel(), vpKalmanFilter::size_measure, vpKalmanFilter::size_state, stateConstVelWithColoredNoise_MeasureVel, and vpKalmanFilter::Xest.

Referenced by initFilter().

◆ prediction()

void vpKalmanFilter::prediction ( )

inherited

Update the Kalman filter by applying the prediction equations.

The predicted state is given by

${{\bf x}}_{k|k-1} = {\bf F}_{k-1} {\bf x}_{k-1\mid k-1}$

and the state prediction covariance by

${\bf P}_{k \mid k-1} = {\bf F}_{k-1} {\bf P}_{k-1 \mid k-1} {\bf F}^T_{k-1} + {\bf Q}_k$

Definition at line 142 of file vpKalmanFilter.cpp.

References vpKalmanFilter::F, vpException::fatalError, vpArray2D< Type >::getRows(), vpKalmanFilter::nsignal, vpKalmanFilter::Pest, vpKalmanFilter::Ppre, vpKalmanFilter::Q, vpKalmanFilter::size_state, vpMatrix::t(), vpKalmanFilter::verbose_mode, vpKalmanFilter::Xest, and vpKalmanFilter::Xpre.

Referenced by filter().

◆ setNumberOfSignal()

void vpKalmanFilter::setNumberOfSignal ( unsigned int n_signal )

inlineinherited

Set the number of signal to filter.

Definition at line 133 of file vpKalmanFilter.h.

◆ setStateModel()

void vpLinearKalmanFilterInstantiation::setStateModel ( vpStateModel mdl )

inline

Set the Kalman state model. Depending on the state model, we set the state vector size and the measure vector size.

The example below shows how to use this method and then to get the size of the state and measure vectors.

#include <visp3/core/vpLinearKalmanFilterInstantiation.h>
 
#ifdef ENABLE_VISP_NAMESPACE
using namespace VISP_NAMESPACE_NAME;
#endif
 
int main()
{
  vpLinearKalmanFilterInstantiation kalman;
 
  kalman.setStateModel(vpLinearKalmanFilterInstantiation::stateConstVelWithColoredNoise_MeasureVel);
  // Value is 2
  std::cout << "State vector size: " << kalman.getStateSize() << std::endl;
  // Value is 1
  std::cout << "Measure vector size: " << kalman.getMeasureSize() << std::endl;
}

Examples: servoAfma4Point2DCamVelocityKalman.cpp, servoViper850Point2DCamVelocityKalman.cpp, testKalmanAcceleration.cpp, and testKalmanVelocity.cpp.

Definition at line 145 of file vpLinearKalmanFilterInstantiation.h.

References model, vpKalmanFilter::size_measure, vpKalmanFilter::size_state, stateConstAccWithColoredNoise_MeasureVel, stateConstVel_MeasurePos, stateConstVelWithColoredNoise_MeasureVel, and unknown.

Referenced by initStateConstAccWithColoredNoise_MeasureVel(), initStateConstVel_MeasurePos(), and initStateConstVelWithColoredNoise_MeasureVel().

◆ verbose()

void vpKalmanFilter::verbose ( bool on )

inlineinherited

Sets the verbose mode.

Parameters

on	: If true, activates the verbose mode which consists in printing the Kalman filter internal values.

Examples: testKalmanAcceleration.cpp, and testKalmanVelocity.cpp.

Definition at line 160 of file vpKalmanFilter.h.

Member Data Documentation

◆ dt

double vpKalmanFilter::dt

inherited

Sampling time $\Delta t$ in second between two successive iterations. Only used in some specific state models implemented in vpLinearKalmanFilterInstantiation.

Definition at line 188 of file vpKalmanFilter.h.

Referenced by filter(), vpKalmanFilter::init(), initStateConstAccWithColoredNoise_MeasureVel(), and initStateConstVel_MeasurePos().

◆ F

vpMatrix vpKalmanFilter::F

inherited

Transition matrix ${\bf F}$ that describes the evolution of the state.

Definition at line 176 of file vpKalmanFilter.h.

Referenced by vpKalmanFilter::init(), initStateConstAccWithColoredNoise_MeasureVel(), initStateConstVel_MeasurePos(), initStateConstVelWithColoredNoise_MeasureVel(), and vpKalmanFilter::prediction().

◆ H

vpMatrix vpKalmanFilter::H

inherited

Matrix ${\bf H}$ that describes the evolution of the measurements.

Definition at line 179 of file vpKalmanFilter.h.

Referenced by vpKalmanFilter::filtering(), vpKalmanFilter::init(), initStateConstAccWithColoredNoise_MeasureVel(), initStateConstVel_MeasurePos(), and initStateConstVelWithColoredNoise_MeasureVel().

◆ I

vpMatrix vpKalmanFilter::I

protectedinherited

Identity matrix $\bf I$ .

Definition at line 213 of file vpKalmanFilter.h.

Referenced by vpKalmanFilter::init().

◆ iter

long vpKalmanFilter::iter

protectedinherited

Filter step or iteration. When set to zero, initialize the filter.

Definition at line 111 of file vpKalmanFilter.h.

Referenced by filter(), vpKalmanFilter::filtering(), vpKalmanFilter::init(), initStateConstAccWithColoredNoise_MeasureVel(), initStateConstVel_MeasurePos(), and initStateConstVelWithColoredNoise_MeasureVel().

◆ model

vpStateModel vpLinearKalmanFilterInstantiation::model

protected

Definition at line 116 of file vpLinearKalmanFilterInstantiation.h.

Referenced by filter(), initFilter(), and setStateModel().

◆ nsignal

unsigned int vpKalmanFilter::nsignal

protectedinherited

Number of signal to filter.

Definition at line 118 of file vpKalmanFilter.h.

Referenced by filter(), vpKalmanFilter::init(), initStateConstAccWithColoredNoise_MeasureVel(), and vpKalmanFilter::prediction().

◆ Pest

vpMatrix vpKalmanFilter::Pest

inherited

The updated covariance of the state ${\bf P}_{k \mid k}$ where ${\bf P}_{k \mid k} = \left({\bf I - W}_k {\bf H} \right) {\bf P}_{k \mid k-1}$ .

Definition at line 202 of file vpKalmanFilter.h.

Referenced by vpKalmanFilter::filtering(), vpKalmanFilter::init(), initStateConstAccWithColoredNoise_MeasureVel(), initStateConstVel_MeasurePos(), initStateConstVelWithColoredNoise_MeasureVel(), and vpKalmanFilter::prediction().

◆ Ppre

vpMatrix vpKalmanFilter::Ppre

inherited

The state prediction covariance ${\bf P}_{k \mid k-1}$ where ${\bf P}_{k \mid k-1} = {\bf F}_{k-1} {\bf P}_{k-1 \mid k-1} {\bf F}^T_{k-1} + {\bf Q}_k$ .

Definition at line 195 of file vpKalmanFilter.h.

Referenced by vpKalmanFilter::filtering(), and vpKalmanFilter::prediction().

◆ Q

vpMatrix vpKalmanFilter::Q

inherited

Process noise covariance matrix ${\bf Q}$ .

Definition at line 184 of file vpKalmanFilter.h.

Referenced by vpKalmanFilter::init(), initStateConstAccWithColoredNoise_MeasureVel(), initStateConstVel_MeasurePos(), initStateConstVelWithColoredNoise_MeasureVel(), and vpKalmanFilter::prediction().

◆ R

vpMatrix vpKalmanFilter::R

inherited

Measurement noise covariance matrix ${\bf R}$ .

Definition at line 182 of file vpKalmanFilter.h.

Referenced by vpKalmanFilter::filtering(), vpKalmanFilter::init(), initStateConstAccWithColoredNoise_MeasureVel(), initStateConstVel_MeasurePos(), and initStateConstVelWithColoredNoise_MeasureVel().

◆ size_measure

unsigned int vpKalmanFilter::size_measure

protectedinherited

Size of the measure vector ${\bf z}_k$ .

Definition at line 116 of file vpKalmanFilter.h.

Referenced by filter(), vpKalmanFilter::init(), initStateConstAccWithColoredNoise_MeasureVel(), initStateConstVel_MeasurePos(), initStateConstVelWithColoredNoise_MeasureVel(), and setStateModel().

◆ size_state

unsigned int vpKalmanFilter::size_state

protectedinherited

Size of the state vector ${\bf x}_k$ .

Definition at line 114 of file vpKalmanFilter.h.

Referenced by filter(), vpKalmanFilter::init(), initStateConstAccWithColoredNoise_MeasureVel(), initStateConstVel_MeasurePos(), initStateConstVelWithColoredNoise_MeasureVel(), vpKalmanFilter::prediction(), and setStateModel().

◆ verbose_mode

bool vpKalmanFilter::verbose_mode

protectedinherited

When set to true, print the content of internal variables during filtering() and prediction().

Definition at line 122 of file vpKalmanFilter.h.

Referenced by vpKalmanFilter::filtering(), and vpKalmanFilter::prediction().

◆ W

vpMatrix vpKalmanFilter::W

protectedinherited

Filter gain ${\bf W}_k$ where ${\bf W}_k = {\bf P}_{k \mid k-1} {\bf H}^T \left[ {\bf H P}_{k \mid k-1} {\bf H}^T + {\bf R}_k \right]^{-1}$ .

Definition at line 210 of file vpKalmanFilter.h.

Referenced by vpKalmanFilter::filtering().

◆ Xest

vpColVector vpKalmanFilter::Xest

inherited

The updated state estimate ${\bf x}_{k \mid k}$ where ${\bf x}_{k \mid k} = {\bf x}_{k \mid k-1} + {\bf W}_k \left[ {\bf z}_k - {\bf H x}_{k \mid k-1} \right]$ .

Examples: servoAfma4Point2DCamVelocityKalman.cpp, servoViper850Point2DCamVelocityKalman.cpp, testKalmanAcceleration.cpp, and testKalmanVelocity.cpp.

Definition at line 168 of file vpKalmanFilter.h.

Referenced by filter(), vpKalmanFilter::filtering(), vpKalmanFilter::init(), initStateConstAccWithColoredNoise_MeasureVel(), initStateConstVel_MeasurePos(), initStateConstVelWithColoredNoise_MeasureVel(), and vpKalmanFilter::prediction().

◆ Xpre

vpColVector vpKalmanFilter::Xpre

inherited

The predicted state ${\bf x}_{k \mid k-1}$ where ${\bf x}_{k|k-1} = {\bf F}_{k-1} {\bf x}_{k-1\mid k-1}$ .

Examples: testKalmanAcceleration.cpp.

Definition at line 173 of file vpKalmanFilter.h.

Referenced by vpKalmanFilter::filtering(), vpKalmanFilter::init(), and vpKalmanFilter::prediction().

Public Types

Public Member Functions

Public Attributes

Protected Attributes

Linear filter initializer with constant acceleration models

Detailed Description

Member Enumeration Documentation

◆ vpStateModel

Constructor & Destructor Documentation

◆ vpLinearKalmanFilterInstantiation()

Member Function Documentation

◆ filter()

◆ filtering()

◆ getIteration()

◆ getMeasureSize()

◆ getNumberOfSignal()

◆ getStateModel()

◆ getStateSize()

◆ init()

◆ initFilter()

◆ initStateConstAccWithColoredNoise_MeasureVel()

◆ initStateConstVel_MeasurePos()

◆ initStateConstVelWithColoredNoise_MeasureVel()

◆ prediction()

◆ setNumberOfSignal()

◆ setStateModel()

◆ verbose()

Member Data Documentation

◆ dt

◆ F

◆ H

◆ I

◆ iter

◆ model

◆ nsignal

◆ Pest

◆ Ppre

◆ Q

◆ R

◆ size_measure

◆ size_state

◆ verbose_mode

◆ W

◆ Xest

◆ Xpre