# cameasjac

Jacobian of measurement function for constant-acceleration motion

## Syntax

## Description

returns the measurement Jacobian, for constant-acceleration Kalman filter motion
model in rectangular coordinates. The `measurementjac`

= cameasjac(`state`

)`state`

argument specifies
the current state of the filter.

also specifies the measurement coordinate system, `measurementjac`

= cameasjac(`state`

,`frame`

)`frame`

.

also specifies the sensor position, `measurementjac`

= cameasjac(`state`

,`frame`

,`sensorpos`

)`sensorpos`

.

also
specifies the sensor velocity, `measurementjac`

= cameasjac(`state`

,`frame`

,`sensorpos`

,`sensorvel`

)`sensorvel`

.

specifies the measurement parameters,
`measurementjac`

= cameasjac(`state`

,`measurementParameters`

)`measurementParameters`

.

## Examples

### Measurement Jacobian of Accelerating Object in Rectangular Frame

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

state = [1,10,3,2,20,5].'; jacobian = cameasjac(state)

`jacobian = `*3×6*
1 0 0 0 0 0
0 0 0 1 0 0
0 0 0 0 0 0

### Measurement Jacobian of Accelerating Object in Spherical Frame

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

```
state = [1;10;3;2;20;5];
measurementjac = cameasjac(state,'spherical')
```

`measurementjac = `*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

### Measurement Jacobian of Accelerating Object in Translated Spherical Frame

Define the state of an object in 2-D constant-acceleration motion. The state is the position, velocity, and acceleration in both dimensions. Compute the measurement Jacobian in spherical coordinates with respect to an origin at *(5;-20;0)* meters.

```
state = [1,10,3,2,20,5].';
sensorpos = [5,-20,0].';
measurementjac = cameasjac(state,'spherical',sensorpos)
```

`measurementjac = `*4×6*
-2.5210 0 0 -0.4584 0 0
0 0 0 0 0 0
-0.1789 0 0 0.9839 0 0
0.5903 -0.1789 0 0.1073 0.9839 0

### Create Measurement Jacobian of Accelerating Object Using Measurement Parameters

Define the state of an object in 2-D constant-acceleration motion. The state is the position, velocity, and acceleration in both dimensions. Compute the measurement Jacobian in spherical coordinates with respect to an origin at *(5;-20;0)* meters.

```
state2d = [1,10,3,2,20,5].';
sensorpos = [5,-20,0].';
frame = 'spherical';
sensorvel = [0;8;0];
laxes = eye(3);
measurementjac = cameasjac(state2d,frame,sensorpos,sensorvel,laxes)
```

`measurementjac = `*4×6*
-2.5210 0 0 -0.4584 0 0
0 0 0 0 0 0
-0.1789 0 0 0.9839 0 0
0.5274 -0.1789 0 0.0959 0.9839 0

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

measparm = struct('Frame',frame,'OriginPosition',sensorpos,'OriginVelocity',sensorvel, ... 'Orientation',laxes); measurementjac = cameasjac(state2d,measparm)

`measurementjac = `*4×6*
-2.5210 0 0 -0.4584 0 0
0 0 0 0 0 0
-0.1789 0 0 0.9839 0 0
0.5274 -0.1789 0 0.0959 0.9839 0

## Input Arguments

`state`

— Kalman filter state vector

real-valued *3N*-element vector

Kalman filter state vector for constant-acceleration motion,
specified as a real-valued *3N*-element vector. *N* is
the number of spatial degrees of freedom of motion. For each spatial
degree of motion, the state vector takes the form shown in this table.

Spatial Dimensions | 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 meters/second. Acceleration coordinates are in meters/second^{2}.

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

**Data Types: **`double`

`frame`

— Measurement output frame

`'rectangular'`

(default) | `'spherical'`

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`

`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: **`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: **`double`

`laxes`

— Local sensor coordinate axes

`[1,0,0;0,1,0;0,0,1]`

(default) | 3-by-3 orthogonal matrix

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`

`measurementParameters`

— Measurement parameters

structure | array of structure

Measurement parameters, specified as a structure or an array of structures. The fields of the structure are:

Field | Description | Example |
---|---|---|

`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. This
field is not relevant when the | `1` |

`HasElevation` | Logical 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 | `1` |

`HasVelocity` | Logical 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` |

`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

`measurementjac`

— Measurement Jacobian

real-valued 3-by-*N* matrix | real-valued 4-by-*N* matrix

Measurement Jacobian, specified as a real-valued 3-by-*N* or
4-by-*N* matrix. *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.

Frame | Measurement Jacobian |
---|---|

`'rectangular'` | Jacobian of the measurements
`[x;y;z]` with respect to the state
vector. The measurement vector is with respect to the
local coordinate system. Coordinates are in
meters. |

`'spherical'` | Jacobian of the measurement vector
`[az;el;r;rr]` with respect to the
state vector. Measurement vector components specify the
azimuth angle, elevation angle, range, and range rate of
the object with respect to the local sensor coordinate
system. Angle units are in degrees. Range units are in
meters and range rate units are in
meters/second. |

## More About

### 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.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

## Version History

**Introduced in R2017a**

## See Also

### Functions

`constacc`

|`constaccjac`

|`cameas`

|`constturn`

|`constturnjac`

|`ctmeas`

|`ctmeasjac`

|`constvel`

|`constveljac`

|`cvmeas`

|`cvmeasjac`

### Objects

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