# cameasjac

Jacobian of measurement function for constant-acceleration motion

## Syntax

``measurementjac = cameasjac(state)``
``measurementjac = cameasjac(state,frame)``
``measurementjac = cameasjac(state,frame,sensorpos)``
``measurementjac = cameasjac(state,frame,sensorpos,sensorvel)``
``measurementjac = cameasjac(state,frame,sensorpos,sensorvel,laxes)``
``measurementjac = cameasjac(state,measurementParameters)``

## Description

example

````measurementjac = cameasjac(state)` returns the measurement Jacobian, for constant-acceleration Kalman filter motion model in rectangular coordinates. The `state` argument specifies the current state of the filter.```

example

````measurementjac = cameasjac(state,frame)` also specifies the measurement coordinate system, `frame`.```

example

````measurementjac = cameasjac(state,frame,sensorpos)` also specifies the sensor position, `sensorpos`.```
````measurementjac = cameasjac(state,frame,sensorpos,sensorvel)` also specifies the sensor velocity, `sensorvel`.```
````measurementjac = cameasjac(state,frame,sensorpos,sensorvel,laxes)` also specifies the local sensor axes orientation, `laxes`.```

example

````measurementjac = cameasjac(state,measurementParameters)` specifies the measurement parameters, `measurementParameters`.```

## Examples

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Define the state of an object in 2-D constant-acceleration motion. The state is the position, velocity, and acceleration in both dimensions. Construct the measurement Jacobian in rectangular coordinates.

```state = [1,10,3,2,20,5].'; jacobian = cameasjac(state)```
```jacobian = 3×6 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ```

Define the state of an object in 2-D constant-acceleration motion. The state is the position, velocity, and acceleration in both dimensions. Compute the measurement Jacobian in spherical coordinates.

```state = [1;10;3;2;20;5]; measurementjac = cameasjac(state,'spherical')```
```measurementjac = 4×6 -22.9183 0 0 11.4592 0 0 0 0 0 0 0 0 0.4472 0 0 0.8944 0 0 0.0000 0.4472 0 0.0000 0.8944 0 ```

Define the state of an object in 2-D constant-acceleration motion. The state is the position, velocity, and acceleration in both dimensions. Compute the measurement Jacobian in spherical coordinates with respect to an origin at (5;-20;0) meters.

```state = [1,10,3,2,20,5].'; sensorpos = [5,-20,0].'; measurementjac = cameasjac(state,'spherical',sensorpos)```
```measurementjac = 4×6 -2.5210 0 0 -0.4584 0 0 0 0 0 0 0 0 -0.1789 0 0 0.9839 0 0 0.5903 -0.1789 0 0.1073 0.9839 0 ```

Define the state of an object in 2-D constant-acceleration motion. The state is the position, velocity, and acceleration in both dimensions. Compute the measurement Jacobian in spherical coordinates with respect to an origin at (5;-20;0) meters.

```state2d = [1,10,3,2,20,5].'; sensorpos = [5,-20,0].'; frame = 'spherical'; sensorvel = [0;8;0]; laxes = eye(3); measurementjac = cameasjac(state2d,frame,sensorpos,sensorvel,laxes)```
```measurementjac = 4×6 -2.5210 0 0 -0.4584 0 0 0 0 0 0 0 0 -0.1789 0 0 0.9839 0 0 0.5274 -0.1789 0 0.0959 0.9839 0 ```

Put the measurement parameters in a structure and use the alternative syntax.

```measparm = struct('Frame',frame,'OriginPosition',sensorpos,'OriginVelocity',sensorvel, ... 'Orientation',laxes); measurementjac = cameasjac(state2d,measparm)```
```measurementjac = 4×6 -2.5210 0 0 -0.4584 0 0 0 0 0 0 0 0 -0.1789 0 0 0.9839 0 0 0.5274 -0.1789 0 0.0959 0.9839 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`

Measurement output frame, specified as `'rectangular'` or `'spherical'`. When the frame is `'rectangular'`, a measurement consists of x, y, and z Cartesian coordinates. When specified as `'spherical'`, a measurement consists of azimuth, elevation, range, and range rate.

Data Types: `char`

Sensor position with respect to the navigation frame, specified as a real-valued 3-by-1 column vector. Units are in meters.

Data Types: `double`

Sensor velocity with respect to the navigation frame, specified as a real-valued 3-by-1 column vector. Units are in m/s.

Data Types: `double`

Local sensor coordinate axes, specified as a 3-by-3 orthogonal matrix. Each column specifies the direction of the local x-, y-, and z-axes, respectively, with respect to the navigation frame. That is, the matrix is the rotation matrix from the global frame to the sensor frame.

Data Types: `double`

Measurement parameters, specified as a structure or an array of structures. The fields of the structure are:

FieldDescriptionExample
`Frame`

Frame used to report measurements, specified as one of these values:

• `'Rectangular'` — Detections are reported in rectangular coordinates.

• `'Spherical'` — Detections are reported in spherical coordinates.

`'spherical'`
`OriginPosition`Position offset of the origin of the frame relative to the parent frame, specified as an `[x y z]` real-valued vector.`[0 0 0]`
`OriginVelocity`Velocity offset of the origin of the frame relative to the parent frame, specified as a `[vx vy vz]` real-valued vector.`[0 0 0]`
`Orientation`Frame rotation matrix, specified as a 3-by-3 real-valued orthonormal matrix.`[1 0 0; 0 1 0; 0 0 1]`
`HasAzimuth`

Logical scalar indicating if azimuth is included in the measurement.

This field is not relevant when the `Frame` field is `'Spherical'`.

`1`
`HasElevation`Logical scalar indicating if elevation information is included in the measurement. For measurements reported in a rectangular frame, and if `HasElevation` is false, the reported measurements assume 0 degrees of elevation.`1`
`HasRange`

Logical scalar indicating if range is included in the measurement.

This field is not relevant when the `Frame` is `'Spherical'`.

`1`
`HasVelocity`Logical scalar indicating if the reported detections include velocity measurements. For a measurement reported in the rectangular frame, if `HasVelocity` is `false`, the measurements are reported as ```[x y z]```. If `HasVelocity` is `true`, the measurement is reported as `[x y z vx vy vz]`. For a measurement reported in the spherical frame, if `HasVelocity` is `true`, the measurement contains range-rate information.`1`
`IsParentToChild`Logical scalar indicating if `Orientation` performs a frame rotation from the parent coordinate frame to the child coordinate frame. When `IsParentToChild` is `false`, then `Orientation` performs a frame rotation from the child coordinate frame to the parent coordinate frame.`0`

If you only want to perform one coordinate transformation, such as a transformation from the body frame to the sensor frame, you only need to specify a measurement parameter structure. If you want to perform multiple coordinate transformations, you need to specify an array of measurement parameter structures. To learn how to perform multiple transformations, see the Convert Detections to objectDetection Format (Sensor Fusion and Tracking Toolbox) example.

Data Types: `struct`

## Output Arguments

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Measurement Jacobian, specified as a real-valued 3-by-N or 4-by-N matrix. N is the dimension of the state vector. The interpretation of the rows and columns depends on the `frame` argument, as described in this table.

FrameMeasurement Jacobian
`'rectangular'`Jacobian of the measurements `[x;y;z]` with respect to the state vector. The measurement vector is with respect to the local coordinate system. Coordinates are in meters.
`'spherical'`Jacobian of the measurement vector `[az;el;r;rr]` with respect to the state vector. Measurement vector components specify the azimuth angle, elevation angle, range, and range rate of the object with respect to the local sensor coordinate system. Angle units are in degrees. Range units are in meters and range rate units are in meters/second.

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### Azimuth and Elevation Angle Definitions

Define the azimuth and elevation angles used in the toolbox.

The azimuth angle of a vector is the angle between the x-axis and its orthogonal projection onto the xy plane. The angle is positive in going from the x axis toward the y axis. Azimuth angles lie between –180 and 180 degrees. The elevation angle is the angle between the vector and its orthogonal projection onto the xy-plane. The angle is positive when going toward the positive z-axis from the xy plane. ## Version History

Introduced in R2017a