Documentation

# trackingEKF

Extended Kalman filter for object tracking

## Description

A trackingEKF object is a discrete-time extended Kalman filter used to track the positions and velocities of objects that can be encountered in an automated driving scenario. Such objects include automobiles, pedestrians, bicycles, and stationary structures or obstacles. A Kalman filter is a recursive algorithm for estimating the evolving state of a process when measurements are made on the process. The extended Kalman filter can model the evolution of a state when the state follows a nonlinear motion model, when the measurements are nonlinear functions of the state, or when both conditions apply. The extended Kalman filter is based on the linearization of the nonlinear equations. This approach leads to a filter formulation similar to the linear Kalman filter, trackingKF.

The process and measurements can have Gaussian noise, which you can include in these ways:

• Add noise to both the process and the measurements. In this case, the sizes of the process noise and measurement noise must match the sizes of the state vector and measurement vector, respectively.

• Add noise in the state transition function, the measurement model function, or in both functions. In these cases, the corresponding noise sizes are not restricted.

## Creation

### Description

filter = trackingEKF creates an extended Kalman filter object for a discrete-time system by using default values for the StateTransitionFcn, MeasurementFcn, and State properties. The process and measurement noises are assumed to be additive.

filter = trackingEKF(transitionfcn,measurementfcn,state) specifies the state transition function, transitionfcn, the measurement function, measurementfcn, and the initial state of the system, state.

example

filter = trackingEKF(___,Name,Value) configures the properties of the extended Kalman filter object by using one or more Name,Value pair arguments and any of the previous syntaxes. Any unspecified properties have default values.

## Properties

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Kalman filter state, specified as a real-valued M-element vector, where M is the size of the filter state.

Example: [200; 0.2]

Data Types: double

State error covariance, specified as a positive-definite real-valued M-by-M matrix where M is the size of the filter state. The covariance matrix represents the uncertainty in the filter state.

Example: [20 0.1; 0.1 1]

State transition function, specified as a function handle. This function calculates the state vector at time step k from the state vector at time step k – 1. The function can take additional input parameters, such as control inputs or time step size. The function can also include noise values.

The valid syntaxes for the state transition function depend on whether the filter has additive process noise. The table shows the valid syntaxes based on the value of the HasAdditiveProcessNoise property.

x(k) = statetransitionfcn(x(k-1))
x(k) = statetransitionfcn(x(k-1),parameters)
• x(k) is the state at time k.

• parameters stands for all additional arguments required by the state transition function.

x(k) = statetransitionfcn(x(k-1),w(k-1))
x(k) = statetransitionfcn(x(k-1),w(k-1),dt)
x(k) = statetransitionfcn(__,parameters)
• x(k) is the state at time k.

• w(k) is a value for the process noise at time k.

• dt is the time step of the trackingEKF filter, filter, specified in the most recent call to the predict function. The dt argument applies when you use the filter within a tracker and call the predict function with the filter to predict the state of the tracker at the next time step. For the nonadditive process noise case, the tracker assumes that you explicitly specify the time step by using this syntax: predict(filter,dt).

• parameters stands for all additional arguments required by the state transition function.

Example: @constacc

Data Types: function_handle

Jacobian of the state transition function, specified as a function handle. This function has the same input arguments as the state transition function.

The valid syntaxes for the Jacobian of the state transition function depend on whether the filter has additive process noise. The table shows the valid syntaxes based on the value of the HasAdditiveProcessNoise property.

Jx(k) = statejacobianfcn(x(k))
Jx(k) = statejacobianfcn(x(k),parameters)
• x(k) is the state at time k.

• Jx(k) denotes the Jacobian of the predicted state with respect to the previous state. This Jacobian is an M-by-M matrix at time k. The Jacobian function can take additional input parameters, such as control inputs or time-step size.

• parameters stands for all additional arguments required by the Jacobian function, such as control inputs or time-step size.

[Jx(k),Jw(k)] = statejacobianfcn(x(k),w(k))
[Jx(k),Jw(k)] = statejacobianfcn(x(k),w(k),dt)
[Jx(k),Jw(k)] = statejacobianfcn(__,parameters)
• x(k) is the state at time k

• w(k) is a sample Q-element vector of the process noise at time k. Q is the size of the process noise covariance. The process noise vector in the nonadditive case does not need to have the same dimensions as the state vector.

• Jx(k) denotes the Jacobian of the predicted state with respect to the previous state. This Jacobian is an M-by-M matrix at time k. The Jacobian function can take additional input parameters, such as control inputs or time-step size.

• Jw(k) denotes the M-by-Q Jacobian of the predicted state with respect to the process noise elements.

• dt is the time step of the trackingEKF filter, filter, specified in the most recent call to the predict function. The dt argument applies when you use the filter within a tracker and call the predict function with the filter to predict the state of the tracker at the next time step. For the nonadditive process noise case, the tracker assumes that you explicitly specify the time step by using this syntax: predict(filter,dt).

• parameters stands for all additional arguments required by the Jacobian function, such as control inputs or time-step size.

If this property is not specified, the Jacobians are computed by numeric differencing at each call of the predict function. This computation can increase the processing time and numeric inaccuracy.

Example: @constaccjac

Data Types: function_handle

Process noise covariance, specified as a scalar or matrix.

• When HasAdditiveProcessNoise is true, specify the process noise covariance as a positive real scalar or a positive-definite real-valued M-by-M matrix. M is the dimension of the state vector. When specified as a scalar, the matrix is a multiple of the M-by-M identity matrix.

• When HasAdditiveProcessNoise is false, specify the process noise covariance as a Q-by-Q matrix. Q is the size of the process noise vector.

You must specify ProcessNoise before any call to the predict function. In later calls to predict, you can optionally specify the process noise as a scalar. In this case, the process noise matrix is a multiple of the Q-by-Q identity matrix.

Example: [1.0 0.05; 0.05 2]

Option to model process noise as additive, specified as true or false. When this property is true, process noise is added to the state vector. Otherwise, noise is incorporated into the state transition function.

Measurement model function, specified as a function handle. This function can be a nonlinear function that models measurements from the predicted state. Input to the function is the M-element state vector. The output is the N-element measurement vector. The function can take additional input arguments, such as sensor position and orientation.

• If HasAdditiveMeasurementNoise is true, specify the function using one of these syntaxes:

z(k) = measurementfcn(x(k))

z(k) = measurementfcn(x(k),parameters)
x(k) is the state at time k and z(k) is the predicted measurement at time k. The parameters argument stands for all additional arguments required by the measurement function.

• If HasAdditiveMeasurementNoise is false, specify the function using one of these syntaxes:

z(k) = measurementfcn(x(k),v(k))

z(k) = measurementfcn(x(k),v(k),parameters)
x(k) is the state at time k and v(k) is the measurement noise at time k. The parameters argument stands for all additional arguments required by the measurement function.

Example: @cameas

Data Types: function_handle

Jacobian of the measurement function, specified as a function handle. The function has the same input arguments as the measurement function. The function can take additional input parameters, such sensor position and orientation.

• If HasAdditiveMeasurmentNoise is true, specify the Jacobian function using one of these syntaxes:

Jmx(k) = measjacobianfcn(x(k))

Jmx(k) = measjacobianfcn(x(k),parameters)
x(k) is the state at time k. Jx(k) denotes the N-by-M Jacobian of the measurement function with respect to the state. The parameters argument stands for all arguments required by the measurement function.

• If HasAdditiveMeasurmentNoise is false, specify the Jacobian function using one of these syntaxes:

[Jmx(k),Jmv(k)] = measjacobianfcn(x(k),v(k))

[Jmx(k),Jmv(k)] = measjacobianfcn(x(k),v(k),parameters)
x(k) is the state at time k and v(k) is an R-dimensional sample noise vector. Jmx(k) denotes the N-by-M Jacobian of the measurement function with respect to the state. Jmv(k) denotes the Jacobian of the N-by-R measurement function with respect to the measurement noise. The parameters argument stands for all arguments required by the measurement function.

If not specified, measurement Jacobians are computed using numerical differencing at each call to the correct function. This computation can increase processing time and numerical inaccuracy.

Example: @cameasjac

Data Types: function_handle

Measurement noise covariance, specified as a positive scalar or positive-definite real-valued matrix.

• When HasAdditiveMeasurementNoise is true, specify the measurement noise covariance as a scalar or an N-by-N matrix. N is the size of the measurement vector. When specified as a scalar, the matrix is a multiple of the N-by-N identity matrix.

• When HasAdditiveMeasurementNoise is false, specify the measurement noise covariance as an R-by-R matrix. R is the size of the measurement noise vector.

You must specify MeasurementNoise before any call to the correct function. After the first call to correct, you can optionally specify the measurement noise as a scalar. In this case, the measurement noise matrix is a multiple of the R-by-R identity matrix.

Example: 0.2

Option to enable additive measurement noise, specified as true or false. When this property is true, noise is added to the measurement. Otherwise, noise is incorporated into the measurement function.

## Object Functions

 predict Predict state and state estimation error covariance of tracking filter correct Correct state and state estimation error covariance using tracking filter correctjpda Correct state and state estimation error covariance using tracking filter and JPDA distance Distances between current and predicted measurements of tracking filter likelihood Likelihood of measurement from tracking filter clone Create duplicate tracking filter residual Measurement residual and residual noise from tracking filter initialize Initialize state and covariance of tracking filter

## Examples

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Create a two-dimensional trackingEKF object and use name-value pairs to define the StateTransitionJacobianFcn and MeasurementJacobianFcn properties. Use the predefined constant-velocity motion and measurement models and their Jacobians.

EKF = trackingEKF(@constvel,@cvmeas,[0;0;0;0], ...
'StateTransitionJacobianFcn',@constveljac, ...
'MeasurementJacobianFcn',@cvmeasjac);

Run the filter. Use the predict and correct functions to propagate the state. You may call predict and correct in any order and as many times you want. Specify the measurement in Cartesian coordinates.

measurement = [1;1;0];
[xpred, Ppred] = predict(EKF);
[xcorr, Pcorr] = correct(EKF,measurement);
[xpred, Ppred] = predict(EKF);
[xpred, Ppred] = predict(EKF)
xpred = 4×1

1.2500
0.2500
1.2500
0.2500

Ppred = 4×4

11.7500    4.7500         0         0
4.7500    3.7500         0         0
0         0   11.7500    4.7500
0         0    4.7500    3.7500

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## Algorithms

The extended Kalman filter estimates the state of a process governed by this nonlinear stochastic equation:

${x}_{k+1}=f\left({x}_{k},{u}_{k},{w}_{k},t\right)$

xk is the state at step k. f() is the state transition function. Random noise perturbations, wk, can affect the object motion. The filter also supports a simplified form,

${x}_{k+1}=f\left({x}_{k},{u}_{k},t\right)+{w}_{k}$

To use the simplified form, set HasAdditiveProcessNoise to true.

In the extended Kalman filter, the measurements are also general functions of the state:

${z}_{k}=h\left({x}_{k},{v}_{k},t\right)$

h(xk,vk,t) is the measurement function that determines the measurements as functions of the state. Typical measurements are position and velocity or some function of position and velocity. The measurements can also include noise, represented by vk. Again, the filter offers a simpler formulation.

${z}_{k}=h\left({x}_{k},t\right)+{v}_{k}$

To use the simplified form, set HasAdditiveMeasurmentNoise to true.

These equations represent the actual motion and the actual measurements of the object. However, the noise contribution at each step is unknown and cannot be modeled deterministically. Only the statistical properties of the noise are known.

## References

[1] Brown, R.G. and P.Y.C. Wang. Introduction to Random Signal Analysis and Applied Kalman Filtering. 3rd Edition. New York: John Wiley & Sons, 1997.

[2] Kalman, R. E. “A New Approach to Linear Filtering and Prediction Problems.” Transactions of the ASME–Journal of Basic Engineering. Vol. 82, Series D, March 1960, pp. 35–45.

[3] Blackman, Samuel and R. Popoli. Design and Analysis of Modern Tracking Systems. Artech House.1999.

[4] Blackman, Samuel. Multiple-Target Tracking with Radar Applications. Artech House. 1986.