# cvmeasjac

Jacobian of measurement function for constant velocity motion

## Syntax

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

## Description

example

````measurementjac = cvmeasjac(state)` returns the measurement Jacobian for constant-velocity Kalman filter motion model in rectangular coordinates. `state` specifies the current state of the tracking filter.```

example

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

example

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

example

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

## Examples

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Define the state of an object in 2-D constant-velocity motion. The state is the position and velocity in each spatial dimension. Construct the measurement Jacobian in rectangular coordinates.

```state = [1;10;2;20]; jacobian = cvmeasjac(state)```
```jacobian = 3×4 1 0 0 0 0 0 1 0 0 0 0 0 ```

Define the state of an object in 2-D constant-velocity motion. The state is the position and velocity in each dimension. Compute the measurement Jacobian with respect to spherical coordinates.

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

Define the state of an object in 2-D constant-velocity motion. The state is the position and velocity in each spatial dimension. Compute the measurement Jacobian with respect to spherical coordinates centered at (5;-20;0) meters.

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

Define the state of an object in 2-D constant-velocity motion. The state consists of position and velocity in each spatial dimension. The measurements are in spherical coordinates with respect to a frame located at (20;40;0) meters.

```state2d = [1;10;2;20]; frame = 'spherical'; sensorpos = [20;40;0]; sensorvel = [0;5;0]; laxes = eye(3); measurementjac = cvmeasjac(state2d,frame,sensorpos,sensorvel,laxes)```
```measurementjac = 4×4 1.2062 0 -0.6031 0 0 0 0 0 -0.4472 0 -0.8944 0 0.0471 -0.4472 -0.0235 -0.8944 ```

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

```measparm = struct('Frame',frame,'OriginPosition',sensorpos,'OriginVelocity',sensorvel, ... 'Orientation',laxes); measurementjac = cvmeasjac(state2d,measparm)```
```measurementjac = 4×4 1.2062 0 -0.6031 0 0 0 0 0 -0.4472 0 -0.8944 0 0.0471 -0.4472 -0.0235 -0.8944 ```

## 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`

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

Data Types: `char`

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

Data Types: `double`

Sensor velocity with respect to the global coordinate system, specified as a real-valued 3-by-1 column vector. Units are in meters/second.

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 global coordinate system.

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.`1`
`HasElevation`Logical scalar indicating if elevation 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.`1`
`HasVelocity`Logical scalar indicating if the reported detections include velocity measurements. For measurements reported in the rectangular frame, if `HasVelocity` is false, the measurements are reported as `[x y z]`. If `HasVelocity` is `true`, measurements are reported as `[x y z vx vy vz]`.`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`

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 first dimension and meaning depend on value of the `frame` argument.

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 Automated Driving 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.