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.

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

Obtain Measurement Jacobian Matrix of Singer Model

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

state — Input state

real-valued 3N-by-1 vector

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 Degrees	State 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 — Frame to report measurements

'rectangular' (default) | 'spherical'

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

sensorpos — Sensor position

[0;0;0] (default) | real-valued 3-by-1 column vector

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

sensorvel — Sensor velocity

[0;0;0] (default) | real-valued 3-by-1 column vector

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

laxes — Local sensor axes coordinates

[1,0,0;0,1,0;0,0,1] (default) | 3-by-3 orthogonal matrix

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

measurementParameters — Measurement parameters

structure | array of structure

Measurement parameters, specified as a structure or an array of structures. For more details, see Measurement Parameters.

Data Types: struct

Output Arguments

measurementjac — Jacobian of measurement function

real-valued M-by-N matrix

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.

More About

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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2020b