# singermeasjac

Jacobian of measurement function for Singer acceleration motion model

Since R2020b

## Syntax

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

## Description

````measurementjac = singermeasjac(state)` returns the Jacobian of the measurement function, `measurementjac`, for a state based on the Singer acceleration motion model, which assumes the target acceleration decays over time. `state` specifies the current state of the track.```

example

````measurementjac = singermeasjac(state,frame)` specifies the measurement Jacobian output coordinate system, `frame`.```
````measurementjac = singermeasjac(state,frame,sensorpos,sensorvel)` specifies the sensor position, `sensorpos`, and the sensor velocity, `sensorvel`.```
````measurementjac = singermeasjac(state,frame,sensorpos,sensorvel,laxes)` specifies the local sensor axes orientation, `laxes`.```
````measurementjac = singermeasjac(state,measurementParameters)` specifies the measurement parameters, `measurementParameters`.```

## Examples

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Define a state for 2-D Singer acceleration motion.

`state = [1;10;0;2;20;1];`

Obtain the measurement Jacobian in a rectangular frame.

`jacobian = singermeasjac(state)`
```jacobian = 3×6 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ```

Obtain the measurement Jacobian in a spherical frame.

`jacobian = singermeasjac(state, 'spherical')`
```jacobian = 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 ```

Obtain the measurement Jacobian in a spherical frame relative to a stationary sensor located at [1;-2;0].

`jacobian = singermeasjac(state, 'spherical', [1;-2;0], [0;0;0])`
```jacobian = 4×6 -14.3239 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 2.5000 0 0 0 1.0000 0 ```

Obtain the measurement Jacobian in a spherical frame relative to a stationary sensor located at [1;-2;0] that is rotated by 90 degrees around the z axis relative to the global frame.

```laxes = [0 -1 0; 1 0 0; 0 0 1]; jacobian = singermeasjac(state, 'spherical', [1;-2;0], [0;0;0], laxes)```
```jacobian = 4×6 -14.3239 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 2.5000 0 0 0 1.0000 0 ```

## Input Arguments

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Input state, specified as a real-valued 3N-by-1 vector. N is the spatial degree of the state. The state vector takes the different forms based on its dimensions.

Spatial DegreesState 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 m/s. Acceleration coordinates are in m/s2.

Example: `[5;0.1;0.01;0;-0.2;-0.01;-3;0.05;0]`

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. For more details, see Measurement Parameters.

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.

### Measurement Parameters

The `MeasurementParameters` property consists of an array of structures that describe a sequence of coordinate transformations from a child frame to a parent frame or the inverse transformations (see Frame Rotation). If `MeasurementParameters` only contains one structure, then it represents the rotation from one frame to the other. If `MeasurementParameters` contains an array of structures, then it represents rotations between multiple frames.

The fields of `MeasurementParameters` are shown here. Not all fields have to be present in the structure.

 Field Description `Frame` Enumerated type indicating the frame used to report measurements. When detections are reported using a rectangular coordinate system, `Frame` is set to `'rectangular'`. When detections are reported in spherical coordinates, `Frame` is set to `'spherical'` for the first `struct`. `OriginPosition` Position offset of the origin of the child frame relative to the parent frame, represented as a 3-by-1 vector. `OriginVelocity` Velocity offset of the origin of the child frame relative to the parent frame, represented as a 3-by-1 vector. `Orientation` 3-by-3 real-valued orthonormal frame rotation matrix. The direction of the rotation depends on the `IsParentTochild` field. `IsParentToChild` A logical scalar indicating if `Orientation` performs a frame rotation from the parent coordinate frame to the child coordinate frame. If `false`, `Orientation` performs a frame rotation from the child coordinate frame to the parent coordinate frame. `HasElevation` A logical scalar indicating if elevation is included in the measurement. For measurements reported in a rectangular frame, and if `HasElevation` is `false`, the measurements are reported assuming 0 degrees of elevation. `HasAzimuth` A logical scalar indicating if azimuth is included in the measurement. `HasRange` A logical scalar indicating if range is included in the measurement. `HasVelocity` A 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]```.

## References

[1] Singer, Robert A. "Estimating optimal tracking filter performance for manned maneuvering targets." IEEE Transactions on Aerospace and Electronic Systems 4 (1970): 473-483.

[2] Blackman, Samuel S., and Robert Popoli. "Design and analysis of modern tracking systems." (1999).

[3] Li, X. Rong, and Vesselin P. Jilkov. "Survey of maneuvering target tracking: dynamic models." Signal and Data Processing of Small Targets 2000, vol. 4048, pp. 212-235. International Society for Optics and Photonics, 2000.

## Version History

Introduced in R2020b