# constacc

Constant-acceleration motion model

## Description

example

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

example

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

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

## Examples

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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

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

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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 DimensionsState 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

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

Example: 0.5

Data Types: single | double

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

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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:

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

## Extended Capabilities

### Objects

Introduced in R2017a