constveljac

Jacobian of state transition function based on constant-velocity motion model

Syntax

Jx = constveljac(state)

Jx = constveljac(state,dt)

[Jx,Jw] = constveljac(state,w,dt)

Description

Jx = constveljac(state) returns the Jacobian of the state transition function based on the constant-velocity motion model. The default time step is 1 second. By default, constveljac returns the Jacobian Jx with respect to the input state, state.

Jx = constveljac(state,dt) specifies the time step, dt.

[Jx,Jw] = constveljac(state,w,dt) specifies the state noise, w, and returns the Jacobian, Jw, of the state with respect to the noise.

Examples

Compute State Jacobian for Constant-Velocity Motion

Compute the state Jacobian for a two-dimensional constant-velocity motion model for a one second update time.

state = [1,1,2,1].';
jacobian = constveljac(state)

jacobian = 4×4

     1     1     0     0
     0     1     0     0
     0     0     1     1
     0     0     0     1

Compute State Jacobian for Constant-Velocity Motion with Specified Time Step

Compute the state Jacobian for a two-dimensional constant-velocity motion model for a half-second update time.

state = [1;1;2;1];

Compute the state update Jacobian for 0.5 second.

jacobian = constveljac(state,0.5)

jacobian = 4×4

    1.0000    0.5000         0         0
         0    1.0000         0         0
         0         0    1.0000    0.5000
         0         0         0    1.0000

Input Arguments

state — State vector

real-valued 2N-element vector

State vector for constant-velocity motion, specified as a real-valued 2N-element column vector where N is the number of spatial degrees of freedom of motion. The state is expected to be Cartesian state. For each spatial degree of motion, the state vector takes the form shown in this table.

Spatial Dimensions	State Vector Structure
1-D	[x;vx]
2-D	[x;vx;y;vy]
3-D	[x;vx;y;vy;z;vz]

For example, x represents the x-coordinate and vx represents the velocity in the x-direction. If the motion model is 1-D, values along the y and z axes are assumed to be zero. If the motion model is 2-D, values along the z axis are assumed to be zero. Position coordinates are in meters and velocity coordinates are in meters/sec.

Example: [5;.1;0;-.2;-3;.05]

Data Types: single | double

dt — Time step interval

1.0 (default) | scalar

Time step interval, specified as a scalar. Time units are in seconds. If dt is negative, this function performs retrodiction and propagates the state estimate and covariance backward in time.

Example: 0.5

Data Types: single | double

w — State noise

scalar | real-valued N-by-1 vector

State noise, specified as a scalar or real-valued real valued N-by-1 vector. N is the number of motion dimensions. For example, N = 2 for the 2-D motion. If specified as a scalar, the scalar value is expanded to an N-by-1 vector.

Data Types: single | double

Output Arguments

Jx — Jacobian of state transition function with respect to input state

real-valued 2N-by-2N matrix

Jacobian of the state transition function with respect to the input state, returned as a real-valued 2N-by-2N matrix. N is the number of spatial degrees of motion. The function constructs the Jacobian from the partial derivatives of the state at the updated time step with respect to the state at the input time step.

Jw — Jacobian of state transition function with respect to noise

real-valued 2N-by-N matrix

Jacobian of the state transition function with respect to the noise, returned as a real-valued 2N-by-N matrix. N is the number of spatial degrees of motion. The function constructs the Jacobian from the partial derivatives of the state at the updated time step with respect to the noise components.

Algorithms

For a two-dimensional constant-velocity motion, the Jacobian matrix for a time step, T, is block diagonal:

$$\left[\begin{array}{cccc}1& T& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& T\\ 0& 0& 0& 1\end{array}\right]$$

The block for each spatial dimension has this form:

$$\left[\begin{array}{cc}1& T\\ 0& 1\end{array}\right]$$

For each additional spatial dimension, add an identical block.

References

[1] 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 R2018b