# ctmeasjac

Jacobian of measurement function for constant turn-rate and velocity- magnitude motion model

## Syntax

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

## Description

````measurementjac = ctmeasjac(state)` returns the Jacobian of the measurement function, `measurementjac`, for a state based on the constant turn-rate and velocity-magnitude motion model. `state` specifies the current state of the track. Note`ctmeasjac` represents velocity in the xy-plane with its Cartesian components, `Vx` and `Vy`. For the Jacobian of measurement function for constant turn-rate and velocity-magnitude motion model using velocity magnitude and course direction, see `ctrvmeasjac`. ```

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

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````measurementjac = ctmeasjac(state,frame,sensorpos)` also specifies the sensor position, `sensorpos`.```

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````measurementjac = ctmeasjac(state,frame,sensorpos,sensorvel)` also specifies the sensor velocity, `sensorvel`.```
````measurementjac = ctmeasjac(state,frame,sensorpos,sensorvel,laxes)` also specifies the local sensor axes orientation, `laxes`.```
````measurementjac = ctmeasjac(state,measurementParameters)` specifies the measurement parameters, `measurementParameters`.```

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## Examples

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

```state = [1;10;2;20;5]; jacobian = ctmeasjac(state)```
```jacobian = 3×5 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ```

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

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

Define the state of an object in 2-D constant turn-rate motion. The state is the position and velocity in each dimension, and the turn rate. Compute the measurement Jacobian with respect to spherical coordinates centered at `[5;-20;0]`.

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

Define the state of an object in 2-D constant turn-rate motion. The state is the position and velocity in each dimension, and the turn rate. Compute the measurement Jacobian with respect to spherical coordinates centered at `[25;-40;0]`.

```state2d = [1;10;2;20;5]; sensorpos = [25,-40,0].'; frame = 'spherical'; sensorvel = [0;5;0]; laxes = eye(3); measurementjac = ctmeasjac(state2d,frame,sensorpos,sensorvel,laxes)```
```measurementjac = 4×5 -1.0284 0 -0.5876 0 0 0 0 0 0 0 -0.4961 0 0.8682 0 0 0.2894 -0.4961 0.1654 0.8682 0 ```

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

```measparm = struct('Frame',frame,'OriginPosition',sensorpos,'OriginVelocity',sensorvel, ... 'Orientation',laxes); measurementjac = ctmeasjac(state2d,measparm)```
```measurementjac = 4×5 -1.0284 0 -0.5876 0 0 0 0 0 0 0 -0.4961 0 0.8682 0 0 0.2894 -0.4961 0.1654 0.8682 0 ```

## Input Arguments

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State vector for a constant turn-rate motion model in two or three spatial dimensions, specified as a real-valued vector.

• When specified as a 5-element vector, the state vector describes 2-D motion in the xy-plane. You can specify the state vector as a row or column vector. The components of the state vector are `[x;vx;y;vy;omega]` where `x` represents the x-coordinate and `vx` represents the velocity in the x-direction. `y` represents the y-coordinate and `vy` represents the velocity in the y-direction. `omega` represents the turn rate.

• When specified as a 7-element vector, the state vector describes 3-D motion. You can specify the state vector as a row or column vector. The components of the state vector are `[x;vx;y;vy;omega;z;vz]` where `x` represents the x-coordinate and `vx` represents the velocity in the x-direction. `y` represents the y-coordinate and `vy` represents the velocity in the y-direction. `omega` represents the turn rate. `z` represents the z-coordinate and `vz` represents the velocity in the z-direction.

Position coordinates are in meters. Velocity coordinates are in meters/second. Turn rate is in degrees/second.

Example: `[5;0.1;4;-0.2;0.01]`

Data Types: `single` | `double`

Frame to report measurements, specified as `'rectangular'` or `'spherical'`. When you specify frame as `'rectangular'`, a measurement consists of x, y, and z Cartesian coordinates. When you specify frame as `'spherical'`, a measurement consists of azimuth, elevation, range, and range rate.

Data Types: `char` | `string`

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

Data Types: `single` | `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: `single` | `double`

Local sensor axes coordinates, 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. The matrix is the rotation matrix from the global frame to the sensor frame.

Data Types: `single` | `double`

Measurement parameters, specified as a structure or an array of structures. This table lists the fields in the structure.

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.

Tip

In Simulink, when you create an object detection Bus, specify `Frame` as an enumeration object of `fusionCoordinateFrameType.Rectangular` or `fusionCoordinateFrameType.Spherical` because Simulink does not support variables such as a character vector that can vary in size.

`'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 `'Rectangular'`.

`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 `'Rectangular'`.

`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 want to perform only one coordinate transformation, such as a transformation from the body frame to the sensor frame, you must specify a measurement parameter structure. If you want to perform multiple coordinate transformations, you must specify an array of measurement parameter structures. To learn how to perform multiple transformations, see the Convert Detections to objectDetection Format example.

Data Types: `struct`

## Output Arguments

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Jacobian of the measurement function, returned as a real-valued M-by-N matrix. The function constructs the Jacobian from the partial derivatives of the measurement vector with respect to the input state. The form of the measurement vector depends on the syntax.

• When you do not specify the `measurementParameters` argument and set the `frame` argument to `'rectangular'`, the function outputs measurement vectors in the format of `[x;y;z]`.

• When you do not specify the `measurementParameters` argument and set the `frame` argument to `'spherical'`, the function outputs measurement vectors in the format of `[az;el;r;rr]`.

• When you specify the `measurementParameters` argument and set the `frame` field to `'rectangular'`, the size of the measurement vector depends on the value of the `HasVelocity` field in the `measurementParameters` structure. The measurement vector includes the Cartesian position and velocity coordinates of the tracked object with respect to the ego vehicle coordinate system.

Rectangular Measurements

 `HasVelocity` = `'false'` `[x;y;z]` `HasVelocity` = `'true'` `[x;y;z;vx;vy;vz]`

Position units are in meters and velocity units are in m/s.

• When you specify the `measurementParameters` argument and set the `frame` field to `'spherical'`, the size of the measurement vector depends on the value of the `HasVelocity`, `HasRange`, and `HasElevation` fields in the `measurementParameters` structure. The measurement vector includes the azimuth angle, az, elevation angle, el, range, r, and range rate, rr, of the object with respect to the local ego vehicle coordinate system. Positive values for range rate indicate that an object is moving away from the sensor.

Spherical Measurements

`HasRange` = `'true'``HasRange` = `'false'`
`HasElevation` = `'false'``HasElevation` = `'true'``HasElevation` = `'false'``HasElevation` = `'true'`
`HasVelocity` = `'false'``[az;r]``[az;el;r]``[az]``[az;el]`
`HasVelocity` = `'true'``[az;r;rr]``[az;el;r;rr]``[az]``[az;el]`

Angle units are in degrees, range units are in meters, and range rate units are in m/s.

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### Azimuth and Elevation Angle Definitions

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 when 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 R2018b