# cameas

Measurement function for constant-acceleration motion

## Syntax

``measurement = cameas(state)``
``measurement = cameas(state,frame)``
``measurement = cameas(state,frame,sensorpos)``
``measurement = cameas(state,frame,sensorpos,sensorvel)``
``measurement = cameas(state,frame,sensorpos,sensorvel,laxes)``
``measurement = cameas(state,measurementParameters)``
``[measurement,bounds] = cameas(___)``

## Description

example

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

example

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

example

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

example

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

example

````[measurement,bounds] = cameas(___)` returns the measurement bounds, used by a tracking filter (`trackingEKF` or `trackingUKF`) in residual calculations. See the `HasMeasurementWrapping` of the filter object for more details.```

## Examples

collapse all

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

```state = [1,10,3,2,20,0.5].'; measurement = cameas(state)```
```measurement = 3×1 1 2 0 ```

The measurement is returned in three-dimensions with the z-component set to zero.

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

```state = [1,10,3,2,20,5].'; measurement = cameas(state,'spherical')```
```measurement = 4×1 63.4349 0 2.2361 22.3607 ```

The elevation of the measurement is zero and the range rate is positive. These results indicate that the object is moving away from the sensor.

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

```state = [1,10,3,2,20,5].'; measurement = cameas(state,'spherical',[20;40;0])```
```measurement = 4×1 -116.5651 0 42.4853 -22.3607 ```

The elevation of the measurement is zero and the range rate is negative indicating that the object is moving toward the sensor.

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

`state2d = [1,10,3,2,20,5].';`

The elevation of the measurement is zero and the range rate is negative indicating that the object is moving toward the sensor.

```frame = 'spherical'; sensorpos = [20;40;0]; sensorvel = [0;5;0]; laxes = eye(3); measurement = cameas(state2d,'spherical',sensorpos,sensorvel,laxes)```
```measurement = 4×1 -116.5651 0 42.4853 -17.8885 ```

The elevation of the measurement is zero and the range rate is negative. These results indicate that the object is moving toward the sensor.

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

```measparm = struct('Frame',frame,'OriginPosition',sensorpos,'OriginVelocity',sensorvel, ... 'Orientation',laxes); measurement = cameas(state2d,measparm)```
```measurement = 4×1 -116.5651 0 42.4853 -17.8885 ```

Specify a 2-D state and specify a measurement structure such that the function outputs azimuth, range, and range-rate measurements.

```state = [10 1 0.1 10 1 0.1]'; % [x vx ax y vy ay]' mp = struct("Frame","Spherical", ... "HasAzimuth",true, ... "HasElevation",false, ... "HasRange",true, ... "HasVelocity",false);```

Output the measurement and wrapping bounds using the `cameas` function.

`[measure,bounds] = cameas(state,mp)`
```measure = 2×1 45.0000 14.1421 ```
```bounds = 2×2 -180 180 -Inf Inf ```

## Input Arguments

collapse all

Kalman filter state for constant-acceleration motion, specified as a real-valued 3D-byN matrix. D is the number of spatial degrees of freedom of motion and N is the number states. For each spatial degree of motion, the state vector, as a column of the `state` matrix, 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.`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`

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

collapse all

Measurement vector, returned as an M-by-N matrix. M is the dimension of the measurement and N, the number of measurement, is the same as the number of states. The form of each measurement depends upon which syntax you use.

• When the syntax does not use the `measurementParameters` argument, the measurement vector is `[x,y,z]` when the `frame` input argument is set to `'rectangular'` and `[az;el;r;rr]` when the `frame` is set to `'spherical'`.

• When the syntax uses the `measurementParameters` argument, the size of the measurement vector depends on the values of the `frame`, `HasVelocity`, and `HasElevation` fields in the `measurementParameters` structure.

framemeasurement
`'spherical'`

Specifies 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

HasElevation
falsetrue
HasVelocityfalse`[az;r]``[az;el;r]`
true`[az;r;rr]``[az;el;r;rr]`

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

`'rectangular'`

Specifies the Cartesian position and velocity coordinates of the tracked object with respect to the ego vehicle coordinate system.

Rectangular measurements

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

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

Data Types: `double`

Measurement residual wrapping bounds, returned as an M-by-2 real-valued matrix, where M is the dimension of the measurement. Each row of the matrix corresponds to the lower and upper bounds for the specific dimension in the `measurement` output.

The function returns different bound values based on the `frame` input.

• If the `frame` input is specified as `'Rectangular'`, each row of the matrix is ```[-Inf Inf]```, indicating the filter does not wrap the measurement residual in the filter.

• If the `frame` input is specified as `'Spherical'`, the returned `bounds` contains the bounds for specific measurement dimension based on the following:

• When `HasAzimuth` = `true`, the matrix includes a row of `[-180 180]`, indicating the filter wraps the azimuth residual in the range of `[-180 180]` in degrees.

• When `HasElevation` = `true`, the matrix includes a row of `[-90 90]`, indicating the filter wraps the elevation residual in the range of `[-90 90]` in degrees.

• When `HasRange` = `true`, the matrix includes a row of `[-Inf Inf]`, indicating the filter does not wrap the range residual.

• When `HasVelocity` = `true`, the matrix includes a row of `[-Inf Inf]`, indicating the filter does not wrap the range rate residual.

If you specify any of the options as `false`, the returned `bounds` does not contain the corresponding row. For example, if `HasAzimuth` = `true`, `HasElevation` = `false`, `HasRange` = `true`, `HasVelocity` = `true`, then `bounds` is returned as

``` -180 180 -Inf Inf -Inf Inf```

The filter wraps the measuring residuals based on this equation:

`${x}_{wrap}=mod\left(x-\frac{a-b}{2},b-a\right)+\frac{a-b}{2}$`

where x is the residual to wrap, a is the lower bound, b is the upper bound, mod is the modules after division function, and xwrap is the wrapped residual.

Data Types: `single` | `double`

collapse all

### 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