# constvelmsc

Constant velocity (CV) motion model in MSC frame

## Syntax

## Description

## Examples

### Predict Constant Velocity MSC State with Different Inputs

Define a state vector for a 3-D MSC state.

mscState = [0.1;0.01;0.1;0.01;0.001;1]; dt = 0.1;

Predict the state with zero observer acceleration.

mscState = constvelmsc(mscState,zeros(3,1),dt)

`mscState = `*6×1*
0.1009
0.0083
0.1009
0.0083
0.0009
0.9091

Predict the state with [5;3;1] observer acceleration in scenario.

mscState = constvelmsc(mscState,zeros(3,1),dt,[5;3;1])

`mscState = `*6×1*
0.1017
0.0067
0.1017
0.0069
0.0008
0.8329

Predict the state with observer maneuver and unit standard deviation random noise in target acceleration. Let observer acceleration in the time interval be $\left[\mathrm{sin}\left(\mathit{t}\right)\text{\hspace{0.17em}}\mathrm{cos}\left(\mathit{t}\right)\right]$.

velManeuver = [1 - cos(dt);sin(dt);0]; posManeuver = [-sin(dt);cos(dt) - 1;0]; u = zeros(6,1); u(1:2:end) = posManeuver; u(2:2:end) = velManeuver; mscState = constvelmsc(mscState,randn(3,1),dt,u)

`mscState = `*6×1*
0.1023
0.0058
0.1023
0.0057
0.0008
0.7689

### Predict and Measure State of Constant Velocity Target in Modified Spherical Coordinates

Define a state vector for a motion model in 2-D. The time interval is 2 seconds.

mscState = [0.5;0.02;1/1000;-10/1000]; dt = 2;

As modified spherical coordinates (MSC) state is relative, let the observer state be defined by a constant acceleration model in 2-D.

observerState = [100;10;0.5;20;-5;0.1];

Pre-allocate memory. `rPlot`

is the range for plotting bearing measurements.

observerPositions = zeros(2,10); targetPositions = zeros(2,10); azimuthMeasurement = zeros(1,10); bearingHistory = zeros(2,30); rPlot = 2000;

Use a loop to predict the state multiple times. Use `constvelmsc`

to create a trajectory with constant velocity target and measure the angles using the measurement function, `cvmeasmsc`

.

for i = 1:10 obsAcceleration = observerState(3:3:end); % Use zeros(2,1) as process noise to get true predictions mscState = constvelmsc(mscState,zeros(2,1),dt,obsAcceleration); % Update observer state using constant acceleration model observerState = constacc(observerState,dt); observerPositions(:,i) = observerState(1:3:end); % Update bearing history with current measurement. az = cvmeasmsc(mscState); bearingHistory(:,3*i-2) = observerState(1:3:end); bearingHistory(:,3*i-1) = observerState(1:3:end) + [rPlot*cosd(az);rPlot*sind(az)]; bearingHistory(:,3*i) = [NaN;NaN]; % Use the 'rectangular' frame to get relative positions of the % target using cvmeasmsc function. relativePosition = cvmeasmsc(mscState,'rectangular'); relativePosition2D = relativePosition(1:2); targetPositions(:,i) = relativePosition2D + observerPositions(:,i); end

plot(observerPositions(1,:),observerPositions(2,:)); hold on; plot(targetPositions(1,:),targetPositions(2,:)); plot(bearingHistory(1,:),bearingHistory(2,:),'-.'); title('Constant velocity model in modified spherical coordinates');xlabel('X[m]'); ylabel('Y[m]') legend('Observer Positions', 'Target Positions', 'Bearings Measurements'); hold off;

## Input Arguments

`state`

— Relative state

vector | 2-D matrix

State that is defined relative to an observer in modified spherical coordinates,
specified as a vector or a 2-D matrix. For example, if there is a constant velocity
target state, *xT*, and a constant velocity observer state,
*xO*, then the `state`

is defined as *xT -
xO* transformed in modified spherical coordinates.

The two-dimensional version of modified spherical coordinates (MSC) is also referred to as the modified polar coordinates (MPC). In the case of:

2-D space –– State is equal to [

*az**azRate*1/*r**vr*/*r*]3-D space –– State is equal to [

*az**omega**el**elRate*1/*r**vr*/*r*]

If specified as a matrix, states must be concatenated along columns, where each column represents a state following the convention specified above.

The variables used in the convention are:

*az*–– Azimuth angle (rad)*el*–– Elevation angle (rad)*azRate*–– Azimuth rate (rad/s)*elRate*–– Elevation rate (rad/s)*omega*––*azRate*× cos(*el*) (rad/s)1/

*r*–– 1/range (1/m)*vr*/*r*–– range-rate/range or inverse time-to-go (1/s)

**Data Types: **`single`

| `double`

`vNoise`

— Target acceleration noise

vector | matrix

Target acceleration noise in the scenario, specified as a vector of 2 or 3 elements
or a matrix with dimensions corresponding to `state`

. That is, if the
dimensions of the `state`

matrix is 6-by-10, then the acceptable
dimensions for `vNoise`

is 3-by-10. If the dimensions of the
`state`

matrix is 4-by-10, then the acceptable dimensions for
`vNoise`

is 2-by-10. For more details, see Orientation, Position, and Coordinate Convention.

**Data Types: **`double`

`dt`

— Time difference

scalar

Time difference between current state and the time at which the state is to be calculated, specified as a real finite numeric scalar.

**Data Types: **`single`

| `double`

`u`

— Observer input

vector

Observer input, specified as a vector. The observer input can have the following impact on state-prediction based on its dimensions:

When the number of elements in

`u`

equals the number of elements in`state`

, the input`u`

is assumed to be the maneuver performed by the observer during the time interval,`dt`

. A maneuver is defined as motion of the observer higher than first order (or constant velocity).When the number of elements in

`u`

equals half the number of elements in`state`

, the input`u`

is assumed to be constant acceleration of the observer, specified in the scenario frame during the time interval,`dt`

.

**Data Types: **`double`

## Output Arguments

`state`

— State at next time step

vector | 2-D matrix | 3-D matrix

State at the next time step, returned as a vector and a matrix of two or three
dimensions. The state at the next time step is calculated based on the current state and
the target acceleration noise, `vNoise`

.

**Data Types: **`double`

## Algorithms

The function provides a constant velocity transition function in modified spherical coordinates (MSC) using a non-additive noise structure. The MSC frame assumes a single observer and the state is defined relative to it.

## Extended Capabilities

### C/C++ Code Generation

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

## Version History

**Introduced in R2018b**

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