constacc

Constant-acceleration motion model

Syntax

updatedstate = constacc(state)

updatedstate = constacc(state,dt)

updatedstate = constacc(state,w,dt)

Description

updatedstate = constacc(state) returns the updated state, state, of a constant acceleration Kalman filter motion model for a step time of one second.

updatedstate = constacc(state,dt) specifies the time step, dt.

updatedstate = constacc(state,w,dt) also specifies the state noise, w.

Examples

Predict State for Constant-Acceleration Motion

Define an initial state for 2-D constant-acceleration motion.

state = [1;1;1;2;1;0];

Predict the state 1 second later.

state = constacc(state)

state = 6×1

    2.5000
    2.0000
    1.0000
    3.0000
    1.0000
         0

Predict State for Constant-Acceleration Motion With Specified Time Step

Define an initial state for 2-D constant-acceleration motion.

state = [1;1;1;2;1;0];

Predict the state 0.5 s later.

state = constacc(state,0.5)

state = 6×1

    1.6250
    1.5000
    1.0000
    2.5000
    1.0000
         0

Input Arguments

state — Kalman filter state vector

real-valued 3N-element vector

Kalman filter 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: double

dt — Time step interval of filter

1.0 (default) | positive scalar

Time step interval of filter, specified as a positive scalar. Time units are in seconds.

Example: 0.5

Data Types: single | double

w — State noise

scalar | real-valued D-by-N matrix

State noise, specified as a scalar or real-valued D-by-N matrix. D is the number of motion dimensions and N is the number of state vectors. If specified as a scalar, the scalar value is expanded to a D-by-N matrix.

Data Types: single | double

Output Arguments

updatedstate — Updated state vector

real-valued column or row vector | real-valued matrix

Updated state vector, returned as a real-valued vector or real-valued matrix with same number of elements and dimensions as the input state vector.

Algorithms

For a two-dimensional constant-acceleration process, the state transition matrix after a time step, T, is block diagonal:

$$\left[\begin{array}{c}{x}_{k+1}\\ v{x}_{k+1}\\ a{x}_{k+1}\\ y_{k+1}\\ v{y}_{k+1}\\ a{y}_{k+1}\end{array}\right]=\left[\begin{array}{cccccc}1& T& \frac{1}{2}{T}^2& 0& 0& 0\\ 0& 1& T& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& T& \frac{1}{2}{T}^2\\ 0& 0& 0& 0& 1& T\\ 0& 0& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_k\\ v{x}_k\\ a{x}_k\\ y_k\\ v{y}_k\\ a{y}_k\end{array}\right]$$

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.

Extended Capabilities

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