# constaccjac

Jacobian for constant-acceleration motion

## Syntax

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

## Description

example

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

example

````jacobian = constaccjac(state,dt) ` also specifies the time step, `dt`.```
````[jacobian,noisejacobian] = constaccjac(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 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 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

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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 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

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Constant-acceleration motion Jacobian, returned as a real-valued 3N-by-3N matrix.

Constant acceleration motion noise Jacobian, 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 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-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:

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