# Parachute Simulation Study with Monte Carlo Analysis

This example shows the simulation of a package with a parachute being dropped from an airplane. The parachute opens out of the package and induces a drag force, slowing the descent. The example uses Monte Carlo analysis to perform the sensitivity study of the parachute simulation with uncertainty in wind disturbance, airdrop altitude of the package from the aircraft, and opening altitude of the parachute.

### Simulink Model Description

To understand the parachute simulation architecture, open the Simulink™ model.

`open_system('ParachuteSim.slx');`

The top model consists of two subsystems:

Environment defines the environment in which the parachute operates. It consists of the following blocks:

Flat Earth to LLA block converts Earth position to latitude, longitude, and altitude. The initial heading angle and the initial latitude and longitude of the package drop are specified as parameters in the block. The variable

*headingInitial*corresponds to the initial heading angle in degrees. The variables*latInitial*and*longInitial*correspond to the geodetic latitude and longitude in degrees.

headingInitial = 45; latInitial = 43.5319; longInitial = 1.2438;

COESA Atmosphere Model calculates the density value corresponding to the altitude.

WGS84 Gravity Model computes the Earth gravity at a specific latitude, longitude, and altitude.

A realistic wind environment is created using the combination of Wind Shear Model to calculate the mean wind speed, Dryden Wind Turbulence Model (Continuous) to add wind turbulence, and Discrete Wind Gust Model to generate gust wind.

Parachute System Model uses the 6DOF (Euler Angles) block to simulate the parachute model based on the initial conditions, weight, and drag force. When a package containing the parachute is dropped from an aircraft, two forces act on it, the weight of the package and the aerodynamic drag. The aerodynamic drag force of the package is negligible before the parachute opens, resulting in a projectile motion of the package. Once the parachute opens, the aerodynamic drag on the parachute slows the descent. The default values of the parachute are based on the EPC (Ensemble de Parachutage du Combattant), one of the French parachutes manufactured by IrvinGQ (formerly Airborne Systems).

As the aerodynamic drag and the weight of the parachute are opposing forces, the net vertical force is given by the difference between the weight $\left(\mathrm{mass}\times \mathit{g}\right)$ and the drag force $\left(\frac{1}{2}\times \mathrm{rho}\times {\mathrm{velocity}}^{2}\times {\mathit{C}}_{\mathit{d}}\times \mathrm{area}\right)$, where:

*mass*represents the mass of the parachute in kg*g*is the acceleration due to gravity approximated to 9.81 $\frac{\mathit{m}}{{\mathit{s}}^{2}}$*rho*is the density of air approximated to 1.229 $\frac{\mathrm{kg}}{{\mathit{m}}^{3}}$*velocity*corresponds to the velocity of parachute in $\frac{\mathit{m}}{\mathit{s}}$${\mathit{C}}_{\mathit{d}}$ is the drag coefficient

*area*represents the area of the parachute in ${\mathit{m}}^{2}$

During the descent, a point is quickly reached when the net force on the parachute is zero and the drag equals weight. This relation is used to compute the drag coefficient of the parachute, as follows:

area = 115; mass = 165; velocity = 6; g = 9.81; rho = 1.229; Cd = 2*mass*g/(rho*(velocity^2)*area);

The gravity and density values corresponding to the altitude and wind velocity computed in the Environment subsystem are used with the drag coefficient, opening altitude, and mass of the parachute to obtain the net force acting on the parachute. The opening altitude, denoted by the variable *hOpening*, corresponds to the altitude (in meters) at which the parachute is deployed from the package. Observe that the opening altitude should be less than the airdrop altitude from an aircraft. The variable *packageMass* corresponds to the mass of the package with the parachute in kg. By default, the opening altitude and the package mass are set to 500 m and 130 kg, respectively.

hOpening = 500; packageMass = 130;

The net moments on the parachute are assumed zero. The net forces and moments are subsequently fed to the 6DOF (Euler Angles) block to simulate the parachute model based on the specified initial conditions in the block. The initial position in the inertial axis is specified below. The variable *ZeInitial* corresponds to the airdrop altitude (in meters) of the package from the aircraft. By default, the airdrop altitude of the package is set to 2000 m.

XeInitial = 0; YeInitial = 0; ZeInitial = 2000;

The initial velocity of the package along the *x*-axis corresponds to the velocity of the aircraft when the package is dropped, and is given a default value of 235 km/h. The initial velocities of the package along *y* and *z*-axes are considered zero (in km/h). The initial Euler angle and Euler rates are considered zero. The initial velocities in the three body axes are converted to m/s using the `convvel`

function as shown below:

UInitialKMPH = 235; UInitialMPS = convvel(UInitialKMPH,'km/h','m/s'); VInitialKMPH = 0; VInitialMPS = convvel(VInitialKMPH,'km/h','m/s'); WInitialKMPH = 0; WInitialMPS = convvel(WInitialKMPH,'km/h','m/s');

### Parachute Simulation

By default, the wind disturbance is introduced in the parachute simulation. The parameter **Gust start time** in Discrete Wind Gust Model corresponds to the time the gust begins. As the time taken for the parachute to reach the ground is around 100 seconds, the variable *windSeed* that corresponds to the **Gust start time** parameter is taken as a random integer within a range of 1 to 100 using the `randi`

function. To simulate the parachute model without wind disturbance, set the w*indDisturbance* variable to `false`

. This switching is carried out in the Simulink model using a Variant Subsystem, Variant Model, Variant Assembly Subsystem with *windDisturbance* as the variant choice variable.

windDisturbance = true; if windDisturbance == true windSeed = randi(100,1); end

After initializing and setting up all the parameters of the parachute simulation model, the `sim`

function simulates it until the parachute touches the ground. The results are stored in the variable *SimResults* as a `Simulink.SimulationOutput`

object.

`simResults = sim('ParachuteSim.slx');`

The simulation results characterize the response of the parachute. An overview is given by the state of the parachute at the touchdown, as follows.

The time taken for the parachute to hit the ground in seconds:

TFinal = simResults.tout(end)

TFinal = 108.3668

The position along the *x*-axis where the parachute hits the ground in meters:

XeFinal = simResults.yout{1}.Values.Data(end,1)

XeFinal = 1.3581e+03

The position along the *y*-axis where the parachute hits the ground in meters:

YeFinal = simResults.yout{1}.Values.Data(end,2)

YeFinal = -33.1586

The velocity along the *x*-axis with which the parachute hits the ground in m/s:

UeFinal = simResults.yout{2}.Values.Data(end,1)

UeFinal = -1.7877

The velocity along the *y*-axis with which the parachute hits the ground in m/s:

VeFinal = simResults.yout{2}.Values.Data(end,2)

VeFinal = -2.5148

The velocity along the *z*-axis with which the parachute hits the ground in m/s:

WeFinal = simResults.yout{2}.Values.Data(end,3)

WeFinal = -5.1849

The variables *YeFinal* and *VeFinal* depend on the wind disturbance. If an ideal scenario with no wind condition is considered, then these variables are zero. This assumption results in only longitudinal motion of the parachute, represented in 2D with *x* and *z* coordinates. However, in a realistic environment, the package exhibits both lateral and longitudinal motion, which is plotted below in 3D. From the plot, observe that the package exhibits a projectile motion until the parachute opens and induces an aerodynamic drag. This drag considerably slows down the descent.

figure; plot3(simResults.yout{1}.Values.Data(:,1),simResults.yout{1}.Values.Data(:,2),simResults.yout{1}.Values.Data(:,3)); zlim([0 max(simResults.yout{1}.Values.Data(:,3))]); grid on; xlabel('Xe (m)'); ylabel('Ye (m)'); zlabel('Ze (m)');

The planar satellite view of the parachute simulation is depicted below. The `flat2lla`

function estimates latitude, longitude, and altitude from the Earth position coordinates. The `geoplot`

function plots the resulting geographic coordinates. The `geobasemap`

function changes the basemap to `satellite`

.

figure; llaPosition = flat2lla([simResults.yout{1}.Values.Data(:,1) simResults.yout{1}.Values.Data(:,2) -1*simResults.yout{1}.Values.Data(:,3)],[latInitial, longInitial], headingInitial,0); geoplot(llaPosition(:,1), llaPosition(:,2),'r','LineWidth',2) geolimits([min(llaPosition(:,1))-0.01 max(llaPosition(:,1))+0.01],[min(llaPosition(:,2))-0.01 max(llaPosition(:,2))+0.01]) geobasemap satellite

### Monte Carlo Analysis Using Parallel Simulation

The Monte Carlo simulations assess the response of the parachute under uncertain conditions. The variable *numSim* provides the number of simulations in the Monte Carlo analysis. In the parachute simulation, the uncertainty in wind disturbance, airdrop altitude of the package, and the opening altitude of the parachute are considered. The randomness in wind disturbance is accommodated by varying the start time of the gust in the Discrete Wind Gust Model block within the range of 1 to 100.

numSim = 100; windDisturbance = true; gustStartTime = randi(100,[1 numSim]);

The uncertainty in the airdrop altitude of the package and the opening altitude of the parachute is assumed to follow a normal distribution curve with the mean set at the initial conditions of *ZeInitial* and *hOpening*, respectively. The standard deviations of the airdrop altitude and the opening altitude are specified below:

ZeDeviation = 5; hOpeningDeviation = 5;

The random values of the airdrop altitude and the opening altitude are then drawn from a normal distribution with the specified mean and standard deviation using the `randn`

function.

airdropAltitude = ZeDeviation.*randn(numSim,1) + ZeInitial; openingAltitude = hOpeningDeviation.*randn(numSim,1) + hOpening;

The Monte Carlo analysis requires simulating the parachute system multiple times, so this example uses the `parsim`

function. The `Simulink.SimulationInput`

object specifies the varying inputs of the simulations. The `parsim`

function is configured to run simulations using fast restart, to monitor progress using Simulation Manager, to track the progress of simulations, and to transfer the variables in the base workspace to the parallel workers using Parallel Computing Toolbox™.

in(1:numSim) = Simulink.SimulationInput('ParachuteSim'); for simNumber = 1:1:numSim in(simNumber) = in(simNumber).setVariable('hOpening',openingAltitude(simNumber)); in(simNumber) = in(simNumber).setVariable('Ze',airdropAltitude(simNumber)); in(simNumber) = in(simNumber).setVariable('windSeed',gustStartTime(simNumber)); end simResults = parsim(in, 'UseFastRestart','on','ShowSimulationManager', 'on','ShowProgress','on','TransferBaseWorkspaceVariables','on');

[07-Nov-2022 11:02:13] Checking for availability of parallel pool... Starting parallel pool (parpool) using the 'Processes' profile ... Connected to parallel pool with 6 workers. [07-Nov-2022 11:03:03] Starting Simulink on parallel workers... [07-Nov-2022 11:03:55] Configuring simulation cache folder on parallel workers... [07-Nov-2022 11:03:56] Transferring base workspace variables used in the model to parallel workers... [07-Nov-2022 11:03:56] Loading model on parallel workers... [07-Nov-2022 11:04:11] Running simulations... [07-Nov-2022 11:04:59] Cleaning up parallel workers...

The results of the multiple simulations are plotted below. From the plot, it is seen that the input uncertainties result in a small deviation of the landing site of the parachute.

figure; for simNumber = 1:1:numSim plot3(simResults(1,simNumber).yout{1}.Values.Data(:,1),simResults(1,simNumber).yout{1}.Values.Data(:,2),simResults(1,simNumber).yout{1}.Values.Data(:,3)); hold on; end grid on; title(["Monte Carlo Simulation Results for Variations in the";"Wind Disturbance, Airdrop and Opening Altitudes"]); xlabel('Xe (m)'); ylabel('Ye (m)'); zlabel('Ze (m)');

The simulation results are also represented as a geoplot in the satellite view.

figure; gx = geoaxes; for simNumber = 1:1:numSim llaPosition = flat2lla([simResults(1,simNumber).yout{1}.Values.Data(:,1) simResults(1,simNumber).yout{1}.Values.Data(:,2) -1*simResults(1,simNumber).yout{1}.Values.Data(:,3)],[latInitial, longInitial], headingInitial,0); geoplot(gx,llaPosition(:,1), llaPosition(:,2),'LineWidth',2) geolimits(gx,[min(llaPosition(:,1,:))-0.01 max(llaPosition(:,1,:))+0.01],[min(llaPosition(:,2,:))-0.01 max(llaPosition(:,2,:))+0.01]) geobasemap(gx,"satellite") hold on end

The plot is zoomed in to reflect the differences in the landing sites corresponding to each of the parachute simulations.

figure; gx = geoaxes; for simNumber = 1:1:numSim llaPosition = flat2lla([simResults(1,simNumber).yout{1}.Values.Data(:,1) simResults(1,simNumber).yout{1}.Values.Data(:,2) -1*simResults(1,simNumber).yout{1}.Values.Data(:,3)],[latInitial, longInitial], headingInitial,0); geoplot(gx,llaPosition(:,1), llaPosition(:,2),'LineWidth',2) geolimits(gx,[43.5375 43.5420],[1.2495 1.2570]) geobasemap(gx,"satellite") hold on end

The example demonstrates Monte Carlo analysis for parachute simulation with variations in the wind disturbance, airdrop and opening altitudes. Observe the simulation results of Monte Carlo analysis by varying the start time of the wind gust and standard deviations of the airdrop and opening altitudes.