Unscented Kalman filter for object tracking
trackingUKF object is a
discrete-time unscented Kalman filter used to track the positions and velocities of
target platforms. An unscented Kalman filter is a recursive algorithm for estimating the
evolving state of a process when measurements are made on the process. The unscented
Kalman filter can model the evolution of a state that obeys a nonlinear motion model.
The measurements can also be nonlinear functions of the state, and the process and
measurements can have noise. Use an unscented Kalman filter when the current state is a
nonlinear function of the previous state, when the measurements are nonlinear functions
of the state, or when both conditions apply. The unscented Kalman filter estimates the
uncertainty about the state, and its propagation through the nonlinear state and
measurement equations, by using a fixed number of sigma points. Sigma points are chosen
by using the unscented transformation, as parameterized by the
filter = trackingUKF creates an unscented Kalman filter object for a
discrete-time system by using default values for the
The process and measurement noises are assumed to be additive.
the state transition function,
filter = trackingUKF(
the measurement function,
the initial state of the system,
configures the properties of the unscented Kalman filter object using one or
filter = trackingUKF(___,
Name,Value pair arguments and any of the previous
syntaxes. Any unspecified properties have default values.
Alpha— Sigma point spread around state
1.0e-3(default) | positive scalar greater than 0 and less than or equal to 1
Sigma point spread around state, specified as a positive scalar greater than 0 and less than or equal to 1.
Beta— Distribution of sigma points
2(default) | nonnegative scalar
Distribution of sigma points, specified as a nonnegative scalar. This
parameter incorporates knowledge of the noise distribution of states for
generating sigma points. For Gaussian distributions, setting
Beta to 2 is optimal.
Kappa— Secondary scaling factor for generating sigma points
0(default) | scalar from 0 to 3
Secondary scaling factor for generation of sigma points, specified as a scalar from 0 to 3. This parameter helps specify the generation of sigma points.
|Predict state and state estimation error covariance of tracking filter|
|Correct state and state estimation error covariance using tracking filter|
|Correct state and state estimation error covariance using tracking filter and JPDA|
|Distances between current and predicted measurements of tracking filter|
|Likelihood of measurement from tracking filter|
|Create duplicate tracking filter|
|Measurement residual and residual noise from tracking filter|
|Initialize state and covariance of tracking filter|
trackingUKF object using the predefined constant-velocity motion model,
constvel, and the associated measurement model,
cvmeas. These models assume that the state vector has the form [x;vx;y;vy] and that the position measurement is in Cartesian coordinates, [x;y;z]. Set the sigma point spread property to 1e-2.
filter = trackingUKF(@constvel,@cvmeas,[0;0;0;0],'Alpha',1e-2);
Run the filter. Use the
correct functions to propagate the state. You can call
correct in any order and as many times as you want.
meas = [1;1;0]; [xpred, Ppred] = predict(filter); [xcorr, Pcorr] = correct(filter,meas); [xpred, Ppred] = predict(filter); [xpred, Ppred] = predict(filter)
xpred = 4×1 1.2500 0.2500 1.2500 0.2500
Ppred = 4×4 11.7500 4.7500 -0.0000 0.0000 4.7500 3.7500 -0.0000 0.0000 -0.0000 -0.0000 11.7500 4.7500 0.0000 0.0000 4.7500 3.7500
This table relates the filter model parameters to the object properties. M is the size of the state vector. N is the size of the measurement vector.
|Model Parameter||Description||Filter Property||Size|
|f||State transition function that specifies the equations of motion
of the object. This function determines the state at time ||Function returns M-element vector|
|h||Measurement function that specifies how the measurements are functions of the state and measurement noise.||Function returns N-element vector|
|xk||Estimate of the object state.||M|
|Pk||State error covariance matrix representing the uncertainty in the values of the state||M-by-M|
|Qk||Estimate of the process noise covariance matrix at step || ||M-by-M when |
|Rk||Estimate of the measurement noise covariance at step k. Measurement noise reflects the uncertainty of the measurement and is assumed to be zero-mean white Gaussian noise.||N-by-N when |
|α||Determines spread of sigma points.||scalar|
|β||A priori knowledge of sigma point distribution.||scalar|
|κ||Secondary scaling parameter.||scalar|
The unscented Kalman filter estimates the state of a process governed by a nonlinear stochastic equation
where xk is the state at step k. f() is the state transition function, uk are the controls on the process. The motion may be affected by random noise perturbations, wk. The filter also supports a simplified form,
To use the simplified
In the unscented Kalman filter, the measurements are also general functions of the state,
where 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 these. The measurements can include noise as well, represented by vk. Again the class offers a simpler formulation
To use the simplified form, set
These equations represent the actual motion of the object and the actual measurements. However, the noise contribution at each step is unknown and cannot be modeled exactly. Only statistical properties of the noise are known.
 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.
 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.
 Wan, Eric A. and R. van der Merwe. “The Unscented Kalman Filter for Nonlinear Estimation”. Adaptive Systems for Signal Processing, Communications, and Control. AS-SPCC, IEEE, 2000, pp.153–158.
 Wan, Merle. “The Unscented Kalman Filter.” In Kalman Filtering and Neural Networks. Edited by Simon Haykin. John Wiley & Sons, Inc., 2001.
 Sarkka S. “Recursive Bayesian Inference on Stochastic Differential Equations.” Doctoral Dissertation. Helsinki University of Technology, Finland. 2006.
 Blackman, Samuel. Multiple-Target Tracking with Radar Applications. Artech House, 1986.