# cameas

Measurement function for constant-acceleration motion

## 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, trackingUKF, trackingCKF, trackingIMM, trackingMSCEKF, or trackingGSF) in residual calculations. See the HasMeasurementWrapping of the filter object for more details.

## 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. 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

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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.

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 variable-size variables such as a character vector.

'spherical'
OriginPositionPosition offset of the origin of the frame relative to the parent frame, specified as an [x y z] real-valued vector.[0 0 0]
OriginVelocityVelocity offset of the origin of the frame relative to the parent frame, specified as a [vx vy vz] real-valued vector.[0 0 0]
OrientationFrame 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
HasElevationLogical 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
HasVelocityLogical 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
IsParentToChildLogical 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 example.

Data Types: struct

## Output Arguments

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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

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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 R2018b