constvelmscjac

Jacobian of constant velocity (CV) motion model in MSC frame

Syntax

[jacobianState,jacobianNoise] = constvelmscjac(state,vNoise)
[jacobianState,jacobianNoise] = constvelmscjac(state,vNoise,dt)
[jacobianState,jacobianNoise] = constvelmscjac(state,vNoise,dt,u)

Description

[jacobianState,jacobianNoise] = constvelmscjac(state,vNoise) calculates the Jacobian matrix of the motion model with respect to the state vector and the noise. The input state defines the current state, and vNoise defines the target acceleration noise in the observer's Cartesian frame. The function assumes a time interval, dt, of one second, and zero observer acceleration in all dimensions.

The trackingEKF object allows you to specify the StateTransitionJacobianFcn property. The function can be used as a StateTransitionJacobianFcn when the HasAdditiveProcessNoise is set to false.

[jacobianState,jacobianNoise] = constvelmscjac(state,vNoise,dt) specifies the time interval, dt. The function assumes zero observer acceleration in all dimensions.

example

[jacobianState,jacobianNoise] = constvelmscjac(state,vNoise,dt,u) specifies the observer input, u, during the time interval, dt.

Examples

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Define a state vector for 2-D MSC.

state = [0.5;0.01;0.001;0.01];

Calculate the Jacobian matrix assuming dt = 1 second, no observer maneuver, and zero target acceleration noise.

[jacobianState,jacobianNoise] = constvelmscjac(state,zeros(2,1)) %#ok
jacobianState = 4×4

    1.0000    0.9900   -0.0000   -0.0098
   -0.0000    0.9800         0   -0.0194
    0.0000   -0.0000    0.9901   -0.0010
   -0.0000    0.0194   -0.0000    0.9800

jacobianNoise = 4×2
10-3 ×

   -0.2416    0.4321
   -0.4851    0.8574
   -0.0004   -0.0002
    0.8574    0.4851

Calculate the Jacobian matrix, given dt = 0.1 seconds, no observer maneuver, and a unit standard deviation target acceleration noise.

[jacobianState,jacobianNoise] = constvelmscjac(state,randn(2,1),0.1) %#ok
jacobianState = 4×4

    1.0000    0.0999    0.0067   -0.0001
   -0.0001    0.9980    0.1348   -0.0020
   -0.0000   -0.0000    0.9990   -0.0001
    0.0001    0.0020    0.1351    0.9980

jacobianNoise = 4×2
10-4 ×

   -0.0240    0.0438
   -0.4800    0.8755
   -0.0000   -0.0000
    0.8755    0.4800

Calculate the Jacobian matrix, given dt = 0.1 seconds and observer acceleration = [0.1 0.3] in the 2-D observer's Cartesian coordinates.

[jacobianState,jacobianNoise] = constvelmscjac(state,randn(2,1),0.1,[0.1;0.3])
jacobianState = 4×4

    1.0000    0.0999    0.0081   -0.0001
    0.0002    0.9980    0.1625   -0.0020
   -0.0000   -0.0000    0.9990   -0.0001
    0.0002    0.0020   -0.1795    0.9980

jacobianNoise = 4×2
10-4 ×

   -0.0240    0.0438
   -0.4800    0.8756
   -0.0000   -0.0000
    0.8756    0.4800

Input Arguments

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State that is defined relative to an observer in modified spherical coordinates, specified as a vector. For example, if there is a constant velocity target state, xT, and a constant velocity observer state, xO, then the state is defined as xT - xO transformed in modified spherical coordinates.

The two-dimensional version of modified spherical coordinates (MSC) is also referred to as the modified polar coordinates (MPC).

In case the motion is in:

  • 2-D space –– State is equal to [az azRate 1/r vr/r]

  • 3-D space –– State is equal to [az omega el elRate 1/r vr/r]

The variables used in the convention are:

  • az –– Azimuth angle (rad)

  • el –– Elevation angle (rad)

  • azRate –– Azimuth rate (rad/s)

  • elRate –– Elevation rate (rad/s)

  • omega –– azRate × cos(el) (rad/s)

  • 1/r –– 1/range (1/m)

  • vr/r –– range-rate/range or inverse time-to-go (1/s)

Data Types: single | double

Target acceleration noise in scenario, specified as a vector of 2 or 3 elements.

Data Types: double

Time difference between the current state and the time at which the state is to be calculated, specified as a real finite numeric scalar.

Data Types: single | double

Observer input, specified as a vector or a matrix. The observer input can have the following impact on state-prediction based on its dimensions:

  • When the number of elements in u equals the number of elements in state, the input u is assumed to be the maneuver performed by the observer during the time interval, dt. A maneuver is defined as motion of the observer higher than first order (or constant velocity).

  • When the number of elements in u equals half the number of elements in state, the input u is assumed to be constant acceleration of the observer, specified in the scenario frame during the time interval, dt.

Data Types: double

Output Arguments

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Jacobian of the predicted state with respect to the previous state, returned as an n-by-n matrix, where n is the number of states in the state vector.

Data Types: double

Jacobian of the predicted state with respect to the noise elements, returned as an n-by-m matrix. The variable n is the number of states in the state vector, and the variable m is the number of process noise terms. That is, m = 2 for state in 2-D space, and m = 3 for state in 3-D space.

For example, if the state vector is a 4-by-1 vector in a 2-D space, vNoise must be a 2-by-1 vector, and jacobianNoise is a 4-by-2 matrix.

If the state vector is a 6-by-1 vector in 3-D space, vNoise must be a 3-by-1 vector, and jacobianNoise is a 6-by-3 matrix.

Data Types: double

Extended Capabilities

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

See Also

Classes

Functions

Introduced in R2018b