constaccjac

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

Syntax

Jx = constaccjac(state)

Jx = constaccjac(state,dt)

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

Description

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

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

[Jx,Jw] = constaccjac(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-Acceleration Motion

Compute the state Jacobian for two-dimensional constant-acceleration motion.

Define an initial state and compute the state Jacobian for a one second update time.

state = [1,1,1,2,1,0];
jacobian = constaccjac(state)

jacobian = 6×6

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

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

Compute the state Jacobian for two-dimensional constant-acceleration motion. Set the step time to 0.5 seconds.

state = [1,1,1,2,1,0].';
jacobian = constaccjac(state,0.5)

jacobian = 6×6

    1.0000    0.5000    0.1250         0         0         0
         0    1.0000    0.5000         0         0         0
         0         0    1.0000         0         0         0
         0         0         0    1.0000    0.5000    0.1250
         0         0         0         0    1.0000    0.5000
         0         0         0         0         0    1.0000

Input Arguments

state — State vector

real-valued 3N-element vector

State vector for constant-acceleration motion, specified as a real-valued 3N-element vector. N is the number of spatial degrees of freedom of motion. 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;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 meters/second².

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

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 a N-by-1 vector.

Data Types: single | double

Output Arguments

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

real-valued 3N-by-3N matrix

Jacobian of the state transition function with respect to the input state, returned as a real-valued 3N-by-3N matrix. 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 3N-by-N matrix

Jacobian of the state transition function with respect to the noise, returned as a real-valued 3N-by-N matrix. N is the number of spatial degrees of motion. For example, N = 2 for the 2-D 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-acceleration process, the Jacobian matrix after a time step, T, is block diagonal:

$$\left[\begin{array}{cccccc}1& T& \frac{1}{2}{T}^2& 0& 0& 0\\ 0& 1& T& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& T& \frac{1}{2}{T}^2\\ 0& 0& 0& 0& 1& T\\ 0& 0& 0& 0& 0& 1\end{array}\right]$$

The block for each spatial dimension has this form:

$$\left[\begin{array}{ccc}1& T& \frac{1}{2}{T}^2\\ 0& 1& T\\ 0& 0& 1\end{array}\right]$$

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 R2021a