singer

State transition function for Singer acceleration motion model

Syntax

predictedState = singer(state)

predictedState = singer(state,dt)

predictedState = singer(state,dt,tau)

Description

predictedState = singer(state) returns the predicted state, predictedState, obtained from the current state, state, based on the Singer acceleration motion model, which assumes the target acceleration decays over time. The default time step is 1 second.

predictedState = singer(state,dt) specifies the time step, dt, in seconds.

predictedState = singer(state,dt,tau) specifies the target maneuver time constant, tau, in seconds. The default target maneuver time constant is 20 seconds.

Examples

Predict Multiple Singer Acceleration States

Define a state matrix for a 2-D Singer acceleration motion.

states = [1 2 2.5;1 2.5 3;0 -1 2;2 3 -1;5 0 3;-2 4 2];

Predict the states by using a default time step interval dt = 1 second.

states = singer(states)

states = 6×3

    2.0000    4.0082    6.4835
    1.0000    1.5246    4.9508
         0   -0.9512    1.9025
    6.0165    4.9671    2.9835
    3.0492    3.9016    4.9508
   -1.9025    3.8049    1.9025

Predict the state by using dt = 0.1 second.

states = singer(states,0.1)

states = 6×3

    2.1000    4.1559    6.9881
    1.0000    1.4297    5.1406
         0   -0.9465    1.8930
    6.3119    5.3762    3.4881
    2.8594    4.2812    5.1406
   -1.8930    3.7859    1.8930

Predict and Measure Position Using Singer Model

Define a state vector for a 2-D Singer acceleration motion.

state = [10;-10;3;0;10;-3];
dt = 0.2; % time step in seconds
tau = 10; % maneuver time in seconds

Use the singer function to create a trajectory and measure the positions using the singermeas function.

positions = zeros(2,100); % Pre-allocate memory
measurements = zeros(3,100); % Pre-allocate memory
for i = 1:1:100
    state = singer(state, dt, tau);
    positions(:,i) = [state(1); state(4)];
    measurements(:,i) = singermeas(state);
end

Visualize the results.

plot(positions(1,:), positions(2,:))
hold on
plot(measurements(1,:), measurements(2,:), '.')
title('Singer Acceleration Model'); 
xlabel('X[m]'); ylabel('Y[m]');
legend('Trajectory', 'Measurements'); 

Input Arguments

state — Current state

real-valued 3N-by-1 vector | real-valued 3N-by-M matrix

Current state for Singer acceleration motion model, specified as a real-valued 3N-by-1 vector or a real-valued 3N-by-M matrix. N is the spatial degree of the state, and M is the number of states.

The state vector in each column takes different forms based on its spatial dimensions.

Spatial Degrees	State Vector Structure
1-D	[x;vx;ax]
2-D	[x;vx;ax;y;vy;ay]
3-D	[x;vx;ax;y;vy;ay;z;vz;az]

For example, x represents the x-coordinate, vx represents the velocity in the x-direction, and ax represents the acceleration in the x-direction. If the motion model is in one-dimensional space, the y- and z-axes are assumed to be zero. If the motion model is in two-dimensional space, values along the z-axis are assumed to be zero. Position coordinates are in meters. Velocity coordinates are in meters/second. Acceleration coordinates are in m/s².

Example: [5;0.1;0.01;0;-0.2;-0.01;-3;0.05;0]

dt — Time step

1.0 (default) | positive scalar

Time step, specified as a positive scalar in seconds.

Example: 0.5

tau — Target maneuver time constant

20 (default) | positive scalar | N-element vector of scalar

Target maneuver time constant, specified as a positive scalar or an N-element vector of scalars in seconds. N is the spatial degree of the state. When specified as a vector, each element applies to the corresponding spatial dimension.

Example: 30

Output Arguments

predictedState — Predicted state at next time step

real-valued 3N-by-1 vector | real-valued 3N-by-M matrix

Predicted state, returned as a real-valued 3N-by-1 vector or a real-valued 3N-by-M matrix. N is the spatial degree of the state, and M is the number of states. The predictedStates output has the exactly same form as the states input.

Algorithms

The Singer acceleration model assumes the acceleration at time step k+1, which depends on the acceleration at time step k with exponential decay as:

$$a(k+1)=a(k)*\exp (-T/\tau )$$

where a(k) is the acceleration at time step k, T is the time step, and τ is the target maneuver time constant.

For a 1-D singer model state p = [x, vx, ax]^T, the state propagation is:

$$p(k)=\left[\begin{array}{ccc}1& T& (\alpha T-1-e^{-\alpha T})/{\alpha }^2\\ 0& 1& (1-e^{-\alpha T})/\alpha \\ 0& 0& e^{-\alpha T}\end{array}\right]p(k)+w(k)$$

where α = 1/τ is the reciprocal of the target maneuver time constant and w(k) is the Singer model process noise at time step k. See singerProcessNoise for more details on the process noise.

References

[1] Singer, Robert A. "Estimating optimal tracking filter performance for manned maneuvering targets." IEEE Transactions on Aerospace and Electronic Systems 4 (1970): 473-483.

[2] Blackman, Samuel S., and Robert Popoli. "Design and analysis of modern tracking systems." (1999).

[3] Li, X. Rong, and Vesselin P. Jilkov. "Survey of maneuvering target tracking: dynamic models." Signal and Data Processing of Small Targets 2000, vol. 4048, pp. 212-235. International Society for Optics and Photonics, 2000.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2020b