Constant velocity (CV) motion model in MSC frame
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 .
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
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];
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,
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;
state— Relative state
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)
vNoise— Target acceleration noise
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
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 Systems.
dt— Time difference
Time difference between current state and the time at which the state is to be calculated, specified as a real finite numeric scalar.
u— Observer input
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
state, the input
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
assumed to be constant acceleration of the observer, specified in the scenario
frame during the time interval,
state— State at next time step
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,
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.