Documentation

# constvel

Constant velocity state update

## Syntax

``updatedstate = constvel(state)``
``updatedstate = constvel(state,dt) ``

## Description

example

````updatedstate = constvel(state)` returns the updated state, `state`, of a constant-velocity Kalman filter motion model after a one-second time step. ```

example

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

## Examples

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Update the state of two-dimensional constant-velocity motion for a time interval of one second.

```state = [1;1;2;1]; state = constvel(state)```
```state = 4×1 2 1 3 1 ```

Update the state of two-dimensional constant-velocity motion for a time interval of 1.5 seconds.

```state = [1;1;2;1]; state = constvel(state,1.5)```
```state = 4×1 2.5000 1.0000 3.5000 1.0000 ```

## Input Arguments

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Kalman filter 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. For each spatial degree of motion, the state vector takes the form shown in this table.

Spatial DimensionsState 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`

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

Example: `0.5`

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-velocity process, the state transition matrix after a time step, T, is block diagonal as shown here.

`$\left[\begin{array}{c}{x}_{k+1}\\ {v}_{x,k+1}\\ {y}_{k+1}\\ {v}_{y,k+1}\end{array}\right]=\left[\begin{array}{cccc}1& T& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& T\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_{k}\\ v{x}_{k}\\ {y}_{k}\\ v{y}_{k}\end{array}\right]$`

The block for each spatial dimension is:

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