# constveljac

Jacobian for constant-velocity motion

## Syntax

``jacobian = constveljac(state)``
``jacobian = constveljac(state,dt) ``
``[jacobian,noisejacobian] = constveljac(state,w,dt)``

## Description

example

````jacobian = constveljac(state)` returns the updated Jacobian , `jacobian`, for a constant-velocity Kalman filter motion model for a step time of one second. The `state` argument specifies the current state of the filter.```

example

````jacobian = constveljac(state,dt) ` specifies the time step, `dt`.```
````[jacobian,noisejacobian] = constveljac(state,w,dt)` specifies the state noise, `w`, and returns the Jacobian, `noisejacobian`, of the state with respect to the noise. ```

## Examples

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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 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

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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. 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 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`

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

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Constant-velocity motion Jacobian, returned as a real-valued 2N-by-2N matrix. N is the number of spatial degrees of motion.

Constant velocity motion noise Jacobian, returned as a real-valued 2N-by-N matrix. N is the number of spatial degrees of motion. The Jacobian is constructed 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:

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