singermeasjac

Jacobian of measurement function for Singer acceleration motion model

Description

example

jacobian = singermeasjac(state) returns the measurement Jacobian in rectangular coordinates with respect to the state for the Singer acceleration motion model.

jacobian = singermeasjac(state,frame) specifies the measurement Jacobian output coordinate system, frame.

jacobian = singermeasjac(state,frame,sensorpos,sensorvel) specifies the sensor position, sensorpos, and the sensor velocity, sensorvel.

jacobian = singermeasjac(state,frame,sensorpos,sensorvel,laxes) specifies the local sensor axes orientation, laxes.

jacobian = 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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Current 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]

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. For more details, see Measurement Parameters.

Data Types: struct

Output Arguments

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The measurement Jacobian for a Singer model, returned as a real-valued 3-by-N for a rectangular frame or 4-by-N matrix for a spherical frame. N is the dimension of the state vector. The interpretation of the rows and columns depends on the frame argument, as described in this table.

FrameMeasurement Jacobian
'rectangular'Jacobian of the measurements [x;y;z] with respect to the state vector. Coordinates are in meters.
'spherical'Jacobian of the measurement vector [az;el;r;rr] with respect to the state vector. Measurement vector is the azimuth angle, elevation angle, range, and range rate of the object in the local sensor coordinate system. Angle units are in degrees. Range units are in meters and range rate units are in ms/s.

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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. 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].

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

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

 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.