A model and sounding rocket simulation tool with

Mathematica R©

Tiago Miguel Oliveira Pinto

tiagomopinto@tecnico.ulisboa.pt

Instituto Superior Tecnico, Lisboa, Portugal

November 2015

Abstract

This work aims to simulate trajectories from model to sounding rockets taking into account the windand the atmospheric boundary layer. The developed tool is also capable to forecast the most probablelanding region and perform rocket design optimization using Monte Carlo methods. A study of theatmospheric boundary layer is carried out and are considered three wind profiles that take into accountthe atmospheric conditions and the terrain type in the launch site. Presenting the developed tool’sflexibility, a Monte Carlo simulation is performed in order to deduce the dispersion of the landing sitesdue to the wind parameters’ uncertainties. Owing to the Mathematica R©’s potentialities, the developedtool allows to define any number of stages, with external or internal boosters in the first stage, and toeasily modify the required models to compute the trajectory or to implement new ones in the database.

A test simulation using a two-stage model rocket was performed and the results showed a largedependence of the trajectory on the wind profile. Considering the defined uncertainties from the windspeed and direction, as well from the surface type, they presented a strong influence on the dispersionof the landing sites.

Keywords: Trajectory Simulation, Model Rocketry, Atmospheric Boundary Layer, Monte Carlo.

1. IntroductionThis paper focuses on passively controlled rockets,whose trajectories are influenced by the wind thatchanges according the land surface [1] and time ofthe day [2]. Thus, if the wind profile is foreseen un-der given conditions, the simulations’ errors can beminimized and simulations in different conditionscan provide to users important information to helpselecting the site and day of the launch to achievesuccessful flights. Sounding rockets consist of oneor more stages (propelled by solid or liquid fuel)carrying a scientific payload in order to study theatmosphere and carry out microgravity research be-tween 40 km to 2000 km height [3]. From 40 km to200 km, these rockets are specially interesting be-cause neither balloons nor low-earth satellites canreach these altitudes. Hence, allowing the devel-oped tool to simulate trajectories of fin-stabilizedsounding rockets up to these heights, brings addi-tional useful applications to our project.

Although the simulations from the developed toolmay cost a larger processing time due to the inter-preted language of Mathematica R© [4], it presentsvaluable features as: simulation of trajectories frommicro to sounding rockets (and even any type of

missiles); flexibility to change easily a certain inputin a given model and figure out its influence in thetrajectory; freedom to edit or implement new func-tions and values into the simulator’s database; andprocess the resulting data in order to obtain fur-ther conclusions. Besides these, are also embeddedother basic features such as multi-staging, boosters,launch direction and atmospheric, wind and dragcoefficient models. The simulator does not includea modeling interface, however, it allows to definea versatile rocket model that is functional to ev-ery type of rockets. This is accomplished by us-ing a default scheme of lists structured accordingthe number of stages and boosters. Therefore, thisstructure of lists, allied to the simulator code, allowsthe rocket model to have any number of stages, al-though this scheme of lists must always be respectedso that the program runs properly. As this tool isdeveloped under a modular programming philoso-phy, the necessary inputs to the trajectory functionare independent from each other. Hence, all of themcan be easily modified without worrying about theother models. This allows to run several simulationsunder very different conditions and compare theirimpact in the trajectory. Two databases are already

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implemented as default, one for the rocket’s mo-tors and another for the atmospheric models. Thedatabases are defined in a way that simplifies theaccess to the data and their properties. Besides, an-other models can be joined in the databases to ex-pand the options to use during the simulations. Thedeveloped simulator also allows the user to computetrajectories in loop, changing a determined groupof parameters. This procedure employs the MonteCarlo methods enabling the formulation of furtherstudies. These analysis may consist of calculationsto determine a probable landing region giving cer-tain parameters’ uncertainties or a rocket design op-timization evaluating the combination of possiblevalues for the components that achieve the desiredgoal in the trajectory.

2. Theoretical backgroundA rocket is stable if its Center of Pressure (CP) isbehind the Center of Mass (CM) (seeing from thenose to the tail of the rocket) [5]. The CP is thepoint where acts the resultant aerodynamic forceproduced by the air pressure and moves toward thenose with increasing angle of attack (α). This force,when there is an angle of attack, may be decom-posed as an axial and normal force where the lattercreates a moment about the CM of the rocket. Instable conditions, this moment produces a dampedoscillating movement between the air flow directionuntil the rocket flies without angle of attack and thenormal force vanishes [5]. The arm of the momentis the length between the CM and the CP, calledstatic margin. Decreasing the static margin untilthe CP overtakes the CM, makes the rocket unsta-ble because the resulting moment reverses its direc-tion. In this situation, if a perturbation deviatesthe rocket from the air flow direction, the aerody-namic moment will amplify the disturbance causingthe rocket to spin and crash. With a larger staticmargin, however, the rocket may become overstableand reaches a lower apogee because it turns soonerto the wind [5]. This effect, called weathercocking,is due to the contributions of the rocket and wind’svelocities (v and w, respectively) that define thedirection and intensity of the true air speed as

vt = v −w . (1)

The angle of attack is maximum when the rocketleaves the guiding rods and is found by (see Fig.(1))

α = arccosv · vt

‖v‖‖vt‖. (2)

Note that v may describe other angles in respectto the ground, so that Figure 1 no longer depicts aright triangle.

w

vvt α

Figure 1: Air speed components.

2.1. WindThe lowest layer of the troposphere is directly influ-enced by the ground surface characteristics. There-fore, a wind speed profile is worth to be imple-mented since its effect will be noticed, mainly, in thetrajectory of small and medium model rockets. Thislayer is known as the Atmospheric Boundary Layer(ABL) and its thickness may change from about ahundred meters to a few kilometers varying in timeand with geographic region [6]. The bottom 10%of the ABL is named the Surface Layer (SL) with,in average, 100 m height [7] which makes the ABLabout 1 km thick.

Above the ABL stays the free atmosphere, amore stable region where the frictional influencesof the surface can be ignored and the wind is nearlygeostrophic [6]. This wind, moving parallel to theisobars, results from the balance between the Cori-olis and the pressure gradient forces [8]. Consider-ing only geostrophic balance, which is the case ofa barotropic atmosphere, the wind is constant withheight and its intensity can be assumed as equal tothe one at the top of the ABL.

Regarding atmospheric stability, the ABL is clas-sified in three different classes: neutral, stable andunstable. A neutral atmosphere implies an adia-batic lapse rate and no convection which is the caseof a partially or highly cloudy atmosphere that mayreduce the insolation at the surface [6]. Stable con-ditions occur mostly at night but can also appearwhen the ground surface is colder than the sur-rounding air. An unstable atmosphere is formed inclear weather during the day when there is high ra-diation from the sun causing ascending heat trans-fer. These conditions influence the wind profile andmust be taken into account due to convective ef-fects.

From measurements of the wind speed, the litera-ture [9] presents the following wind profiles modeledconsidering the entire ABL and its stability:

u(z) =u∗0κ

[ln

(z

z0

)+

4.7z

L

(1− z

2h

)+

+z

LMBL− z

h

(z

2LMBL

)], (3)

for stable conditions,

u(z) =u∗0

κ

[ln(zz0

)+ z

LMBL− z

h

(z

2LMBL

)], (4)

2

for neutral conditions, and

u(z) =u∗0κ

[ln

(z

z0

)− ψ

(z

L

)+

z

LMBL−

− zh

(z

2LMBL

)], (5)

for unstable conditions, where h is the ABL depth,z0 is the surface roughness length, κ is the vonKarman constant, LMBL is the length scale in themiddle of the ABL for its respective conditions, u∗0is the local surface friction velocity near the ground,L is the Obukhov length and ψ is a stability correc-tion for the SL.

The surface type influences z0 which is quantifiedand can be obtained from Table 1.

Surface type z0 (cm)

Sand 0.01...0.1

Cut grass (∼ 0.01 m) 0.1...1

Small grass, steppe 1...4

Uncultivated land 2...3

High grass 4...10

Coniferous forest 90...100

Uptown, suburbs 20...40

Downtown 35...45

Big towns 60...80

Table 1: Surface roughness values for different landsurfaces [1].

The Obukhov length (L) represents the thicknessnear the surface in which the shear stress dominatesover the buoyancy effects in generating turbulence.This parameter is hard to estimate since we get con-siderable errors even for measurements taken hun-dreds of meter above the surface. Nevertheless, ifthe atmospheric stability is deduced qualitativelyfrom the weather conditions and time of day (asdescribed before in the section), we get to knowwhich wind profile to use and a value for L can beestimated from Table 2.

2.2. TrajectoryThe launch site is a non-inertial topocentric-horizoncoordinate frame with the x axis pointing south-ward and the y eastward. Thus, as this is a right-handed coordinate frame, the z axis points up-ward collinear with the Earth’s radius direction(or zenith). The total acceleration, in the Earth-centered inertial coordinate frame, is given by [10]

a = ao + ∂2r∂t2 + dΩ

dt × r + 2Ω× ∂r∂t + Ω× (Ω× r) , (6)

where Ω is the Earth’s angular velocity, r is the po-sition of the rocket relative to the local frame and

Obukhov length interval (m) Stability class

10 ≤ L ≤ 50 Very stable

50 ≤ L ≤ 200 Stable

200 ≤ L ≤ 500 Near stable

|L| ≥ 500 Neutral

−500 ≤ L ≤ −200 Near unstable

−200 ≤ L ≤ −100 Unstable

−100 ≤ L ≤ −50 Very unstable

Table 2: Atmospheric stability classes according tointervals of Obukhov length, L [9].

ao is the linear acceleration of this frame given asΩ × (Ω ×Ro), where Ro is the position of the lo-cal frame’s origin relative to the Earth’s center. Therocket’s velocity seen from an observer at the launch

site is v = ∂2r∂t , where the partial derivative stands

for the time derivative of r with reference to thelocal frame. The term dΩ

dt × r is the angular accel-

eration of the moving frame, 2Ω× ∂r∂t is the Coriolis

acceleration and Ω× (Ω× r) is the centripetal ac-celeration. In order to get the trajectory seen byan observer at the launch site, we must write thesevectors in the local frame coordinates.

After using the momentum conservation (tak-ing into account the loss of mass and the externalforces) and applying (6), the trajectory equationresults in

m∂2r

∂t2= −mao −m

dΩ

dt× r− 2mΩ× ∂r

∂t−

−mΩ× (Ω× r) + T + D + W , (7)

where m is the rocket’s mass, T is the thrust, Dis the drag and W is the weight. Solving (7), theposition of the rocket seen from the launch site canbe obtained.

The thrust is always aligned with the rocket’s lon-gitudinal axis as the rocket is not steerable. Sincethe rocket may fly with angle of attack until it sta-bilizes and the thrust becomes collinear with thetrue speed vector, the effective propulsive force maybe lower in this direction. Therefore, the adoptedmodel to the thrust force is

T = Tp cosαvt

‖vt‖(8)

where Tp is the propulsion from the motors’ thrustprofile. The aerodynamic force is assumed to bemerely the drag since the rocket only flies with an-gle of attack during a short interval of time after thelaunch (or after any perturbation) and it describesa low amplitude oscillation. The lift that is pro-duced during these moments contributes essentiallyto stabilize the rocket, its average is considered to

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be zero, and its effect can be implemented as de-scribed in Section 2.3. Thus, the aerodynamic forcecomes

D =1

2ρ‖vt‖SCDvt , (9)

where ρ is the air density, S is the reference area(usually chosen as the maximum sectional area ofthe rocket or, after the apogee, the reference area ofthe recovery device) and CD is the drag coefficient.The weight points to the Earth’s center and is foundby

W = − µ⊕m

(R0 + z)2z , (10)

where µ⊕ = 3.986× 105 km3 s−2 is the Earth’sgravitational parameter and z is the unit vector ofthe vertical coordinate.

During the launch, when the rocket is constrainedby the guiding rods, the drag and weight normalto the launch direction are counterbalanced by therod’s reaction and only the tangential forces to therods contribute to the launch. Thus, the last threeterms in (7) become

T + D + W = − 12ρ‖v‖SCDv + (Tp + W sinEl)el , (11)

where el is the unit vector of the rods’ direction de-termined by the launch elevation (El) and azimuth(Az) as

el = (cosEl cosAz, cosEl sinAz, sinEl) . (12)

2.3. Dynamic stabilityAfter leaving the platform, and when a distur-bance occurs, the rocket acquires an angle of attackthat will gradually decrease and force it to oscil-late about the true air speed direction. To generatethis damped oscillating movement, we conclude therocket is subjected to two moments [11]: a stabiliz-ing (or restoring) moment and a damping moment.

The stabilizing moment comes from the normalaerodynamic force acting on the CP which causesthe rocket to rotate about the CM since these pointsmust distance from each other by the static margin.Therefore, assuming small angles of attack, the sta-bilizing moment is found by

Ms =1

2ρV 2SCNαα(xcp − xcm) , (13)

where xcm and xcp are, respectively, the rocket’sCM and CP distances from the reference point (of-ten considered as the nose tip, which we also con-sider throughout this work), V is the flow veloc-ity and CNα is the rocket’s normal force coefficientderivative .

One source of the damping moment results fromthe aerodynamic resistance of the air while the

rocket is rotating [11]. During the rotation, the an-gle of attack of each part of the rocket changes dueto the tangential velocity of this motion. Like thestabilizing moment, only the normal force of thisadditional resistance contributes to dampen the ro-tation. Thus, since the aerodynamic damping mo-ment (Mda) is determined by summing all the ele-mental moments along the rocket [11], we get [12]

Mda =1

2ρV Sα

n∑i=1

CNαi (xcpi − xcm)2 , (14)

where n is the total number of components of therocket that contribute to this moment.

The other contribution to the damping momentcomes from the Coriolis acceleration due to thechange of the gas flow through the nozzle, alsocalled jet damping, given by [11]

Mdj = αm(xnozzle − xcm)2 , (15)

where xnozzle is the nozzle distance from the refer-ence point and m is the rocket’s burning rate.

The governing equation for the motion of the an-gle of attack, assuming the rotation is two dimen-sional (in the plane formed by the rocket’s velocityand wind vectors), is found by the angular momen-tum knowing that the stabilizing and damping mo-ments counteract the rotation. Thus, we get

−Ms −Mda −Mdj = Iα+ Iα , (16)

where I is the transversal moment of inertia relativeto the CM. Applying (13), (14) and (15) in (16), weget a second-order differential equation that repre-sents a damped harmonic oscillator (considering therocket is stable) giving the angle of attack.

3. Rocket trajectory simulatorThe simulator’s main function, which integrates thetrajectory equations, receives all the required in-puts and calls the function that computes every pa-rameter of the rocket over time. These propertiesare determined by combining the data of the sev-eral stages regarding the developed structure thatgathers the data of the rocket and considering theinstants of the ignitions and ejections. The mostrelevant inputs are the rocket, density, wind andCD models that, since they are external parame-ters, the user is allowed to define or change themaccording to his needs. The outputs from the sim-ulation are three functions giving the position (onefor each cartesian coordinate) plus a function givingthe angle of attack. These are computed in threesteps: one for the constrained trajectory due to theguiding rod, another for the climbing phase and thelast one for the recovery trajectory. The angle ofattack starts to be computed only in the secondphase, since it is when the rocket suffers the first

4

perturbation from the wind. After this process, thethree parts of the trajectory are merged togetherand, having these time dependent functions, it ispossible to determine its derivatives and get otherflight data such as total velocity, total acceleration,Mach and flight path angle.

3.1. Rocket assemblyThe proposed structure for a general multi-stagerocket developed in the present thesis is shown inFigure 2. Although the figure represents a three-

3rd Stage 2nd Stage 1st Stage

Figure 2: Multi-stage rocket design.

stage rocket, this structure can be extended to anynumber of stages and was developed with the goal ofmodeling several types of rockets, specially the mostcommon. For each stage the fundamental compo-nents are: a connector (or nose, in the case of thelast stage), a body tube and fins. Internally, it isonly considered the motors since data from addi-tional components can be joined with the body tubeproperties. In the first stage the rocket can also besupplied with external boosters in order to achieveparallel staging.

Rocket

Assembly List

Delays List

Miscellany List

Motor Cluster List

Number Motors

Motor List

Recovery List

Reference Area

Drag Coefficient

Body List

Fins List

Connector/Nose List

Motor List

Structure Data List

Delay Booster

Stage i List

Stage 2 List

Boosters List

Stages List

Number

Position

Stage 1 List

Thrust Data List

Figure 3: Rocket assembly list structure.

The defined structure of lists enables to accesseasily the rocket’s data and to be quickly processedby the simulator. This structure has three majorlists concerning, respectively, the external boosters,

the stages and the recovery system plus the motorcluster in the first stage (see Figure 3). The stageslist contains as many lists as the number of stages.Each one of these lists holds other lists respectingthe stages components plus a list to indicate thetime intervals between the discharges. All of thisdata (dimensions, mass, CM, inertia, CP and CNα)can be defined manually by the user or, in somecases, estimated by default functions implementedin the simulator.

3.2. Developed modelsThe atmospheric model stored in the database isthe CIRA-86 which gives pressure and temperaturedata up to 120 km (the latter is also dependent onthe latitude and the month of the year). This orother atmospheric models can take into accountthe conditions at the launch site using developedfunctions that match the measured pressure andtemperature at the corresponding altitude with thevalues from the model at higher altitudes (see Fig-ure 4a). One of this functions also determines thedensity from the equation of state.

The developed wind model uses the ABL pro-files (taking into account the atmospheric stability)that approximate the wind data also provided in theCIRA-86 model at higher altitudes. If desired, thewind may be constant above the ABL. This modelrequires local wind speed, its respective measure-ment altitude, the terrain’s specification and thedownwind direction. Wind gusts can also be imple-mented providing their intensity and the instants ofthe respective perturbations as another input of thetrajectory function. These disturbances are simu-lated considering they change instantly the angle ofattack to another value determined by the rocketand wind gust’s velocities at the perturbation’s in-stant. After that moment, the simulator computesnew oscillations for the angle of attack (see examplein Figure 4b).

CIRA-86

CIRA-86 with local conditions

5 10 15 20 25Altitude (km)

220

240

260

280

300

Temperature (K)

(a) Temperature profilesfor the CIRA-86 model.

1 2 3 4 5 6 7t (s)

-10

-5

5

10

α (º)

(b) Angle of attack untilthe apogee with a windgust perturbation.

Figure 4: Temperature and angle of attack models.

The simulator also provides a drag coefficientmodel that takes the structure of lists respectingthe rocket’s data and determines the drag coeffi-cients for each one of the external components de-pending on Mach and Rex. Then it gathers all the

5

coefficients to compute the rocket’s CD in respectto its reference area. Using this drag coefficientmodel, we are considering that all the componentscontribute to the skin friction drag but the pressuredrag comes only from the nose, connectors and fins(due to the normal surface facing the flow). In thismodel the rocket’s nose is the only wave drag sourcewhose effect is implemented by a fitted function inthe transonic regime under M = 1 and by an em-pirical function above this Mach. The final dragcoefficient function for the conical nose, includingthe wave drag, is represented in Fig.(5).

0.0 0.5 1.0 1.5 2.0 2.5 3.0Mach

0.32

0.33

0.34

0.35

0.36

0.37

0.38CD

Figure 5: Conical nose drag coefficient with Mach.

3.3. Validation of the toolThe developed tool requires validation in order toassure the simulator meets the specifications andcorrectly computes the expected output. This maybe performed by comparing our results with aknown analytic solution from a specific launch.

One of developed tests consists in determine theerror presented between the altitude from a verti-cal sounding rocket launch and the tool’s outputunder the same conditions. The analytical solutionrequires that these conditions correspond to a casewithout atmosphere and the effects of the Earth’srotation, constant thrust and constant accelerationof gravity. The obtained relative error over time(until the burnout) is represented in Fig.(6). It can

5 10 15 20 25 30t (s)

1.× 10-6

2.× 10-6

3.× 10-6

Relative Error (%)

Figure 6: Altitude’s relative error between analyti-cal and numerical solution until the burnout.

be observed that the error presents values in theorder of magnitude of 10−6%. Although this errorgives reasonable results under the mentioned con-ditions, more tests should be performed in orderto increase the confidence in the solutions from thedeveloped simulator.

To check the drag effect, we can make a terminalvelocity test for a given rocket in free fall after theapogee. From an equilibrium between the gravita-tional force and the drag force, we can find the ter-minal velocity and conclude that if the rocket losesmass, it must reach a slower velocity. In Fig.(7)it is represented this quasi-static process computedby the simulator. As we can see, the rocket tendsasymptotically to a new terminal velocity, as ex-pected.

15 20 25 30 35 40t (s)

-50

-40

-30

-20

-10

Velocity (m/s)

Figure 7: Descent velocity from a vertical launch inwhich the rocket loses mass at 25 s.

4. Test resultsIn Fig.(8) are presented the results from the as-sembling of a two-stage model rocket. Since modelrockets reach lower velocities and apogees, they are

1 2 3 4 5t (s)

5

10

15

20

25

30

Thrust (N)

(a) Thrust.

1 2 3 4 5t (s)

50

60

70

80

90

100

110

Mass (g)

(b) Mass.

0 1 2 3 4 5t (s)

25

30

35

40CM (cm)

(c) CM relative to the nosetip.

0 1 2 3 4 5t (s)

25

30

35

40CP (cm)

(d) CP relative to the nosetip.

1 2 3 4 5t (s)

3.0

3.5

4.0

Mom. Inertia (g.m3)

(e) Transversal moment ofinertia relative to the CM.

1 2 3 4 5t (s)

6

8

10

12

Ref. Area (cm2)

(f) Reference area.

Figure 8: Model rocket properties’ functions.

suitable to observe the effects of the wind and the

6

ABL on the trajectory simulation. Due to the firstdiscontinuity shown in Fig.(8b), Fig.(8b), Fig.(8c),Fig.(8d) and Fig.(8e), we can see that the firststage’s structure (not considering the connector) isdiscarded at the first burnout. Also in these fig-ures, during the coasting phase (lasting 1 s), we seethe connector’s discharge represented by the seconddiscontinuity. Except from these sharp changes,the rocket maintains its properties during coasting,leading the mentioned figures to match with Fig-ure 8a when there is no thrust between the burn-ings. The reference area is defined as the largestcross section of the rocket (without consideringboosters, for other cases) at each instant. This def-inition agrees with Figure 8f, where it shows therocket’s reference area changing nearly 2 s after thelaunch, which is the instant when the connector isdiscarded and the rocket loses its largest sectionalarea. Comparing Figures 8c and 8d, we concludethis rocket is always stable since the CM, insteadof the CP, is closer to the nose all the time. Hence,the rocket is well assembled and able to be used ina trajectory simulation.

4.1. Trajectory simulationFrom the previous rocket, we get the trajectory de-picted in Fig.(9). This rocket is launched vertically(in order to be easily observed the deviation fromthe launch direction) under a neutral atmospherewith a constant wind speed above the ABL, blow-ing northeastward in a high grass terrain.

(a) 3D trajectory.

-400 -200 0 200

100

200

300

400

500

Up (m)

Ground Track (m)

(b) Trajectory profile.

Figure 9: Model rocket trajectory with a verticallaunch and a neutral ABL; Parachute ejection atthe apogee.

In this trajectory the weathercock effect is ob-servable. Since the wind blows northeastward, therocket climbs in the opposite direction (southwest-ward) to follow the true air speed. Although thiseffect increases smoothly with altitude due to theABL, we can see a slightly discontinuity in the up-ward flight path around 200 m of altitude. Thishappens because the weathercock is enhanced dur-ing the coasting time, between the first burnout andthe second ignition, as the rocket decreases its ve-locity. During the recovery phase, the rocket slowly

descends toward the downwind direction. Hence, itsurpasses the launch site and lands 380 m north and382 m east away from that place (coordinates takenfrom the simulator). The influence of the ABL isalso observable during the recovery phase in whichthe trajectory describes a smooth curvature sincethe wind speed is decreasing as the rocket falls. Ifthe wind was constant with altitude, the recoverytrajectory would describe a linear path.

In Fig.(10) the flight data from this launch is rep-

20 40 60 80 100 120t (s)

100

200

300

400

500

Altitude (m)

(a) Profile of altitude vstime.

2 4 6 8 10 12 14t (s)

20

40

60

80

100

120Velocity (m/s)

(b) Profile of vertical veloc-ity vs time.

2 4 6 8 10 12t (s)

-20

-10

10

20

Accel.(/g)

(c) Longitudinal accelera-tion vs time.

2 4 6 8 10 12 14t (s)

0.1

0.2

0.3

0.4

Mach

(d) Mach vs time.

0.5 1.0 1.5 2.0 2.5 3.0t (s)

-5

5

10

15

α (º)

(e) Angle of attack vs time.

20 40 60 80 100 120t (s)

-50

50

100Flight Path (º)

(f) Flight path angle vstime until the landing.

0 2 4 6 8t (s)

0.5

0.6

0.7

0.8

0.9CD

(g) Drag coefficient vs timeuntil the apogee.

Figure 10: Vertical model rocket launch flight data.

resented. From Fig.(10a) we deduce that the rocketreaches the apogee at about 560 m of altitude andlands 120 s after the lift off. Observing when thevertical velocity turns negative, in Fig.(10b), weconclude that the apogee is reached at 9.5 s, ap-proximately. As already mentioned, we also seethat the rocket decreases velocity due to the coast-ing between stages. This is also observable from theMach number, in Fig.(10d), which remains subsonic

7

throughout the entire flight.During the burnings, in Fig.(10c), the accelera-

tion describes a profile similar to the thrust (seeFig.(8a)). However, it is decreasing over time asthe rocket gains velocity, which increases the dragopposing the thrust force. In contrast, when therocket is not burning, the acceleration is negativeand increases over time (decreases in absolute value)because the velocity is decreasing. The peak accel-eration at 9.5 s results from the parachute ejection.

The angle of attack, in Fig.(10e), shows therocket rapidly damps its oscillations and follows thetrue air speed during the majority of the ascendingflight. During the first instants the angle of attackis not computed because its model receives the im-pulse to start the oscillations only when the rocketleaves the guiding rod.

From Fig.(10f), we can see the flight path angleas 90 at the ignition since the launch is vertical.This angle decreases until the apogee due to theweathercock effect that causes a curvature in thetrajectory. Reaching the apogee, the flight pathangle becomes negative because the rocket starts todescend. There is a peak value of about −90 sincethe rocket reverses its direction and stabilizes in a30 slope, approximately, toward the ground dueto the drag force on the parachute. As the rocketdescends, the flight path decreases, which meansthe velocity is becoming steeper. This fact agreeswith the curvature described by the trajectory dueto the ABL, observed in Fig.(9).

The drag coefficient, in Fig.(10g), is dependenton the Reynolds and the Mach numbers. There-fore, when the velocity is increasing, the drag co-efficient decreases its value due to the skin frictiondrag. Inversely, as the velocity decreases, the dragcoefficient starts to increase. When the first stageis discarded, the CD decreases instantaneously be-cause the rocket loses a great source of drag. Onthe other hand, when the connector is discarded,although it is lost another source of drag, the CDincreases significantly. This happens because this isthe instant when the reference area becomes smallerand the CD has to change in order to be given inrespect to another section.

5. Stochastic methodsMany of the parameters required to determine thetrajectory cannot be known (or well defined, atleast) and the measured data may vary randomly,presenting some uncertainties. Therefore, in orderto analyze the outcome of several uncertain scenar-ios, it lead us to perform simulations that rely on re-peated random sampling and statistical analysis tocompute the results, known as Monte Carlo simula-tions [14]. To get the outcome due to the uncertain-ties, we must identify a suitable probability distri-

bution for each input parameter from which a set ofdata is randomly generated. Then, all the possiblecombinations between the created data correspondto the inputs given in each trajectory simulation,which highly increases the number of simulationsif we are studying many parameters with a largesample. Therefore, since we do not own such greatcomputational resources to minimize the simulationtime, a better approach to perform what if simula-tions in which it is analyzed the outcome from theuncertainties of only two or three parameters withsmaller samples.

In the following simulations we use a single-stagehigh power rocket in order to present results froma more powerful rocket.

5.1. Landing site uncertaintiesAfter studying the uncertainties presented in thelanding site due to the parameters required to de-termine the atmospheric and CD models, we con-cluded that only the wind speed and direction mea-sured at the launch site and the surface roughnesslength (terrain type) presented a large uncertaintyin the results under the defined conditions. There-fore, these are the three parameters to taking intoaccount in the Monte Carlo simulation. The windspeed and direction data are randomly generatedfrom a normal distribution, creating a sample of twohundred points for each parameter. On the otherhand, only the worst case scenarios for the rough-ness length are considered (for a high grass terrain).Also, in order to reduce the simulation time, andsince it is not expected to impact the results much,a mean CD is determined from a single simulationunder the same conditions. The resulting landingsites dispersion is represented in Fig.(11). Since

Figure 11: Landing positions and respective confi-dence ellipses from the Monte Carlo simulation.

the rocket follows the wind speed direction, it isexpected the uncertainties from the local wind ve-locity and the surface roughness length (both onlyinfluencing the wind intensity) change the landingsite along this direction, describing a straight line.

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Analogously, the changes in the wind direction, dueto its uncertainties, make the landing points to de-scribe an arc of a circumference that gets larger asthe distance to the launch site increases.

The dispersion of the landing points, in Fig.(11),clearly shows the influence of the normal distribu-tions used to generate the local wind direction andspeed applied in each iteration. Since a mean of45 was defined for the wind direction’s normal dis-tribution, the orange dots are heavily concentratedalong the northeast direction. In the same way, therocket lands less often apart from the northeast di-rection as there are fewer inputs that deviate morefrom the mean. Also due to this principle, the windspeed uncertainties contribute to the lower densityof points near the closest and furthest distancesfrom the launch site. This pattern can also be seenin Fig.(12) where the probability density function

0

0.02

0.04

0.06

Figure 12: Probability density function of the land-ing coordinates resulting from the Monte Carlo sim-ulation.

of the landing sites is represented. Furthermore,this figure enables us to distinguish better the mostprobable landing area since Fig.(11) gets saturatedin this region due to the great amount of trajecto-ries simulated.

The confidence ellipses in Fig.(11) are rather ec-centric. Their major axes (aligned northeastward)measure 10.4 km, for a confidence of 95%, and9.1 km, for 90% of confidence. As the furthest land-ing site, collinear with the major axes, is 14.6 kmaway from the launch location, the major axis fromthe lowest confidence level represents almost twothirds of the furthest distance. Thus, we concludethe measured wind speed, allied with the surfaceroughness length range, brings a strong uncertaintyto the landing site estimation under the specifiedconditions. Regarding the minor axes compared totheir distances from the launch site, we see that thewind direction also induces a considerable uncer-tainty.

5.2. Rocket design optimizationTaking the previous simulation from a different ap-proach, we get another tool that allows us to per-form optimization. Thus, after defining an opti-mization criteria (maximization or minimization ofan output) given an universe of input variables, wecan deduce the best alternative that matches ourgoal.

Changing the rocket parameters in each MonteCarlo iteration (opposing to the previous simula-tion, where the rocket properties are kept constant),we can optimize the rocket characteristics. For ex-ample, we are aiming to determine the best bodytube’s length and diameter combination so that therocket reaches the highest apogee keeping constantthe other properties. To this example a windlessatmosphere is defined and three options for eachtube’s dimension, which are selected randomly froman uniform distribution. The minimum and maxi-mum length limits (constraints) are 1 m and 2.5 m,respectively, and for the diameter are 7.5 cm and13 cm, respectively.

In Fig.(13) the altitude over time is depicted (un-til the highest apogee’s instant) of these nine con-figurations. . As shown, the configuration with the

5 10 15 20 25t (s)

1

2

3

4

5

Altitude (km) l=1.686m d=10.1cm

l=1.686m d=8.4cm

l=1.686m d=12.2cm

l=1.215m d=10.1cm

l=1.215m d=8.4cm

l=1.215m d=12.2cm

l=2.266m d=10.1cm

l=2.266m d=8.4cm

l=2.266m d=12.2cm

Figure 13: Altitude over time from nine launcheswith different rocket’s length and diameter config-urations.

smaller dimensions (1.215 m and 8.4 cm) reaches thehighest apogee, as opposed to the rocket with thelargest dimensions that reach the lowest apogee.These facts are expected due to the major influ-ence of the rocket’s dimensions on the drag coeffi-cient and the reference area. The longer and largerthe rocket is, the greater is the drag that decreasesthe velocity and forces the rocket to reach soonera lower apogee. From this result, we may say thismethod works properly but a more extensive opti-mization should be performed in the future sincethe geometric parameters of the rocket also influ-ence the mass of the components.

Even though this is a simple and expected exam-ple, it gives a glimpse of the many possibilities atour disposal after developing a Monte Carlo simula-tion to our tool. From this starting point, and hav-ing enough computational resources, more complex

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optimizations may be processed. Besides, other effi-cient optimization methods (not considered in thiswork) may be developed and applied to the pre-sented tool so that the expected result can be fastand directly found, saving resources finding the bestalternative from the outputs.

6. ConclusionsThe present work focused on the development ofa trajectory simulator tool for fin-stabilized modeland sounding rockets addressing the effects of anABL profile. This tool was developed under Math-ematica R© and supports from model to soundingrockets with any number of stages. This work pre-sented the tool’s flexibility to customize the de-fault models and to compute the outcome from theinputs’ uncertainties (or perform an optimization)from a Monte Carlo simulation.

The developed structure to describe the rocketwas presented and is adaptable to any type of com-mon rockets. The developed atmospheric modeltakes into account the local conditions at the launchsite which tend to the CIRA-86 profile at a speci-fied rate. Likewise, above the ABL, the wind modelapproximates the values given in the CIRA-86 ta-bles. Within the ABL, as the wind plays an impor-tant role in the model rockets trajectory, three pro-files were implemented considering the atmosphericstability, which can be deduced from the Obukhovlength. As this parameter is hard to determine withlow errors using common measuring instruments, aqualitative alternative was presented to figure outthe atmospheric stability from the weather condi-tions. Besides the local atmospheric data, the ABLprofiles also take into account the type of the sur-face. In order to improve the simulations consider-ing the rocket’s attitude, as our tool possesses threedegrees of freedom, we developed a model to com-pute the angle of attack over time that can be in-fluenced by wind gusts appearing along the winddirection. A drag coefficient model was also imple-mented depending on the Mach, the Reynolds andthe configuration of the rocket.

A simulation from a model rocket launch was per-formed in order to present examples from the tool’soutputs. The results revealed a large impact ofthe wind in the trajectories (due to the weather-cock effect), which leads us to reassure the impor-tance to forecast the atmospheric stability and tomodel a suitable wind profile. Exploring the poten-tialities from the tool, from a Monte Carlo simula-tion, we also conclude that the wind profiles mustalways concern the local conditions at the launchsince, for the defined uncertainties, they impose astrong influence in estimating the landing site. Us-ing the same simulation process, we also presented amethod to optimize the rocket characteristics which

showed expected results.

References[1] Degeratu, M., Georgescu, A. M., Bandoc, G.,

Alboiu, N. I., Cosoiu, C. I., and Golumbeanu,M. Atmospheric Boundary Layer Modelling asMean Velocity Profile Used for Wind TunnelTests on Contaminant Dispersion in the At-mosphere. Journal of Environmental Protec-tion and Ecology, 14(1):22–28, 2013.

[2] Thuillier, R. H. and Lappe, U. Wind and Tem-perature Profile Characteristics from Observa-tions on a 1400 ft Tower. Journal of AppliedMeteorology, 3:299–306, June 1964.

[3] Seibert, G. The History of Sounding Rocketsand Their Contribution to European Space Re-search. ESA Publications Division, November2006.

[4] Wolfram Research, Inc., Mathematica, Version10.2, Champaign, IL (2015).

[5] Barrowman, J. Stability of a Model Rocketin Flight. Technical Information Report 30,Centuri Engineering Company, 1970.

[6] Stull, R. An Introduction to Boundary LayerMeteorology. Kluwer Academic Publishers,1988.

[7] Tennekes, H. The Logarithmic Wind Pro-file. Journal of the Atmospheric Sciences,30(2):234–238, March 1973.

[8] Arya, S. P. Introduction to Micrometeorology.Academic Press, Inc., 1988.

[9] Gryning, S. E., Batchvarova, E., Brummer,B., Jørgensen, H., and Larsen, S. On the ex-tension of the wind profile over homogeneousterrain beyond the surface boundary layer.Boundary-Layer Meteorology, 124(2):251–268,August 2007.

[10] Tewari, A. Atmospheric and Space Flight Dy-namics. Birkhauser, 2007.

[11] Feodosiev, V. I. and Siniarev, G. B. Intro-duction to Rocket Technology. Academic PressInc., 1959.

[12] Milligan, T. Basics of Dynamic Flight Analysis(Part 3) - The Damping Moment Coefficient.Peak of Flight Newsletter, (195), October 2007.

[13] Curtis, H. Orbital Mechanics for EngineeringStudents. Elsevier, 2nd edition, 2010.

[14] Raychaudhuri, S. Introduction to Monte Carlosimulation. In Proceedings of the 2008 WinterSimulation Conference, pages 91–100, 2008.

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