modelica stage separation dynamics modeling for end-to-end

Modelica Stage Separation Dynamics Modeling forEnd-to-End Launch Vehicle Trajectory Simulations

Paul Acquatella B., and Matthias J. ReinerGerman Aerospace Center (DLR), Institute of System Dynamics and Control

82234 Wessling, [email protected] · [email protected]

Abstract

Stage separation dynamics modeling is a critical capa-bility of future launchers preparatory studies. The de-velopment of stage separation frameworks integrablein end-to-end launch vehicle trajectory simulationshave been presented in the relevant literature but noneprofiting from the object-oriented and equation-basedacausal modeling properties of MODELICA. The ob-jective of this paper is therefore to present such an ap-proach to this problematic. Based on theConstraintForce Equation (CFE) methodology, two case studiesto evaluate the proposed approach are considered. Re-sults demonstrate that the approach corresponds verywell with the physics behind separation. In addition,we found easiness of implementation of the methodwithin a single environment such as DYMOLA , demon-strating the benefits of an integrated approach.

1 Introduction

Stage separation dynamics modeling is a very chal-lenging task and a critical capability that must beconsidered in the preparatory studies and develop-ment of next generation launchers [14, 16, 17]. Theintegration of such stage separation modeling intoa single environment capable of end-to-end launchvehicle trajectory simulation is also a key technologyto aim for.

The importance of such capability arises from thefact that after separation, the integrity of each stagemust be kept in order to guarantee overall successof the space mission pursued. In this sense, thedevelopment of an integrated framework for analysisand simulation of stage separation is desired.

Early efforts on the subject of multi stage launchvehicle separation from the 60’s and 70’s are mainly

from NASA studies [1, 2, 4] and theirProgramto Optimize Simulated Trajectories (POST) as ageneralized trajectory simulation and optimizationsoftware [5], developed in partnership with the (then)Martin Marietta Corporation. Renewed interest inthe subject in the 2000’s led NASA’s developmentof a stage separation conceptual separation tool,ConSep [11, 12, 13, 14]; which is a MATLAB -basedwrapper to the commercially available ADAMS solver,as its predecessorSepSim. However, beingSepSimand ConSep dependent on the commercial softwareADAMS, they have the disadvantage of not beingeasily integrable in a generic trajectory simulationsoftware. This in turn eludes the capability of per-forming efficient end-to-end launch vehicle trajectorysimulations. As a result, a generalized approachto stage separation problems of launch vehicleswas developed [16]. The approach, coined as theConstraint Force Equation (CFE) methodology, wasimplemented into theProgram to Optimize SimulatedTrajectories II (POST2), the POST follow-up. Sep-aration studies applied to real platforms such as theHyper-X or the Space Shuttle can be found in [18, 10].The thesis [15] studies launcher separation analysiswith OPENMODELICA but results in a tool (OMSep)which is only capable of input-output analysesatseparation time, and not for generic launch vehicletrajectories.

As yet, an object-oriented and equation-based acausalmodeling approach to stage separation dynamicsintegrable in end-to-end launch vehicle trajectorysimulations is still missing. Such approach couldpotentially facilitate the integration of this and othercapabilities within a single multi-physics environmentsuch as DYMOLA .

The objective of this paper is therefore to presentsuch an alternate approach to stage separation dynam-

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ics based on the CFE methodology using MODEL-ICA [7, 8]. We do this by means of the followingsub-objectives: We study first the modeling challengesof multi-stage launcher separation dynamics; then wepresent an approach based on CFE implemented inMODELICA; following, we provide two case studiesfor which we apply the method; and finally we presentsome results and discussion, outlining benefits and dis-advantages.

2 Modeling

For the simulation of launch vehicle stage separationdynamics, it is necessary being able to model two bod-ies connected together according to properly-selectedconstraints prior to their physical separation; and atthe release command of such constraints, their sub-sequent and independent flight motion must continue.This section presents the separation dynamics and theseparation mechanisms modeling aspects.

2.1 Separation dynamics

We refer to separation dynamics in this paper the studyof the effects of forces and torques of a two-bodysystem during their physical separation.

Such separation dynamics modeling clearly exhibitsdiscontinuities similar to those described by otherphenomena such as switching, limiting, friction,etc. Modeling must deal with these problems inspecial ways since this kind of behavior is sensitiveto numerical solution errors, initial condition calcula-tion/propagation, and integration in general.

MODELICA offers the possibility to implement a.o.several methods for such phenomena:

− Stop and restart: The complete system is simu-lated as a single body until separation time. Thenthe system is splitted into two bodies with inde-pendent states, and initial conditions are prop-agated accordingly. This solution however re-quires the split of two (or more) events.

− Regularization: This methodology consists onapplying the constraint between the two bodiesduring their connected motion with a smooth butvery stiff spring-damper system. This avoids theuse of strict discrete or event behaviors. Suchmethodology is commonly used for simulation offriction, stiction, and other similar nonlinear be-havior.

− Hybrid: This methodology consists on treatingthe simulation as a hybrid state machine wherecontinuous and discontinuous behaviors are con-ditioned with data flows and proper transitions.This hybrid state machine framework is howevercomplex to integrate in generic form for launchvehicle trajectory simulations.

− Constraint Force Equation (CFE) Methodology:The CFE methodology [16, 17, 18] consists oncomputing internal constraint forces and mo-ments on two bodies during their connected mo-tion and their application as external forces andtorques to each of them separately. On separationcommand, these internal forces are set to zero,and then each body carries their own flight mo-tion separately.

Of these methods, particular interest due to its appli-cability and easiness of implementation is given to theCFE methodology, which is selected as the primarymethod for the follow up of this study.

2.1.1 Constraint Force Equation Methodology

The Constraint Force Equation (CFE) methodol-ogy [16, 17, 18] is a highly intuitive method consist-ing in the computation of joint loads, namely internalforces and torques, caused by joint constraints; alongwith their application as external forces and torques oneach body independently, see Figure 1.The joint loads which constrain one body’s motionrelative to the other are dependent upon the externalforces acting on each body as well as the type of joint.The net forces and torques on each body are thereforethe sum of the usual external forces and torquesplusthe joint loads applied to each body as additional ex-ternal forces and torques. In consequence, the CFEjoint model simply augments the external loads of thesystem [17]. Quoting step by step [16, 17], the equa-

Figure 1: CFE diagram. Illustration credits: [16].

Modelica Stage Separation Dynamics Modeling for End-to-End Launch Vehicle Trajectory Simulations

590 Proceedings of the 10th International ModelicaConferenceMarch 10-12, 2014, Lund, Sweden

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tions of constrained motion of two rigid bodies (A andB) connected by a single joint (pointA in bodyA andpoint B in bodyB) are as follows:

F(ext)A +F(con)

A = mAxA, (1)

T(ext)A +ρAF(con)

A +T(con)A = IAωA +ωA × IAωA (2)

whereρA is the position vector from the mass centerof A to point A of A at which the constraint force isapplied. Similarly forB:

F(ext)B +F(con)

B = mBxB, (3)

T(ext)B +ρBF(con)

B +T(con)B = IBωB +ωB × IBωB. (4)

There are so far 24 unknowns and 12 equations. An-other set of six equations can be obtained from the lawof action and reaction:

F(con)A +F(con)

B = 0 (5)

T(con)A +T(con)

B +(rB − rA)×F(con)B = 0 (6)

whererA = xA +ρA andrB = xB +ρB.

Six equations are missing. Worth noticing at thispoint, we only consider asingle joint which constrainall six remaining degrees of freedom between the twobodies. This is because our focus is towards trajectorysimulations and having multiple connections is notnecessary unless when considering actuator sizing,sensitivity analyses, etc. In general, the CFE method-ology allows to consider any type of joint whichallows or not any specific relative motion betweenbodies; and redundancy of joints when necessary.

In this sense, for relative translation constraints andebeing unit-vectors of the corresponding (A or B) body-frame, it is required that:

(rB − rA) · eA = 0 (7)

meaning that the distance between the two points ofa particular direction remain fixed. And finally, forrelative rotations constraints, it is required that:

eA · eB = 0 (8)

meaning that three properly selected two-unit-vectorsets must remain perpendicular.Eqs. (7)-(8) would have to be differentiated twice withrespect to time so that the resulting relationships in-volve the unknown accelerations and angular acceler-ations, thus finally being able to couple them with the

equations of motion. In other words, the six missingequations are given by the following generalized con-straint equations of the joint:

g = 0 (9)

whereg represents either of the nondifferentiated con-straints in Eqs. (7) and (8). As it will be demon-strated in the next section, the manual differentiationof Eqs. (7)-(8) and their coupling with the equationsof motion can be avoided altogether by the MODEL-ICA implementation since this is done automatically.The last important aspect of the CFE methodology rel-evant to this work is the accuracy of the joint loadssolution, which is sensitive to computational errorand initial joint missalignment [17]. To handle suchconcern, the CFE algorithm could feature a.o. a sta-bilization technique known as Baumgarte stabiliza-tion [3, 6, 16]. This particular stabilization techniqueconsists on replacing the ODE given by Eq. (9) whichallows perturbations to grow linearly with time, by thefollowing asymptotically stable ODE (η > 0) involv-ing terms of the once differentiated and nondifferenti-ated forms ofg:

g+2η g+η2g = 0 (10)

however at the expense of more computational effort.Many other stabilization techniques [6] could be im-plemented; these other methods, and a guidance forselectingη are however out of the scope of this paper.

2.2 Physical modeling of multi-stage separa-tion mechanisms

Separation mechanism refers in this proposal to amechanical model (or device) that makes separationpossible in simulation (or reality). Physical modelingrefers in this context on the capability to modelseparation behaviour by considering first principles(kinematics, dynamics, mechanics, physics, etc.); andbeing able to get realistic insight from such modelsfor other purposes such as actuator sizing, sensitivityanalyses, control, optimization, etc.

Based on our internalDLR Space Systems Library,separation mechanism physical models of differentcomplexity levels can be studied. Simplified modelsfor preliminary and conceptual studies; and moredetailed ones for engineering validation aspects.These varying degrees of complexity would be helpfulin order to perform separation mechanics analyses and

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to assess the performance of the overall separation.

Configuration details of the separation mechanisms aswell as their physical specifications must be providedto achieve more detailed and realistic models. Con-cerning the simple models, four variants have beenstudied:

− Linear charge (release device): The linear chargemodel performs ideal or benchmark separationbetween two bodies. This mechanism “cuts”the two-body system on command. It simu-lates (ideal) explosive release devices, clamps, di-aphragms, or point-release devices such as explo-sive bolts.

− Bushing (separation impulse device): This modelperforms an impulsive reaction due to the releaseof a smooth but very stiff spring-damper systemwhich keeps the two body system connected untilseparation command.

− Kick-off spring (separation impulse device):Same as before, the impulsive reaction due to therelease of a spring-damper system simulates theproper transmission of forces and moments of thetwo-body system during separation. This modelis implemented with theConstraint Force Equa-tion (CFE) methodology. This element is com-bined with a release device to simulate a realistickick-off spring mechanism.

− Generic (auxiliary devices): Other generic de-vices can be modeled in combination with theprevious models, or with any other physicalmodel from the library.

3 MODELICA implementation

In this section, the MODELICA implementation ofseparation mechanism models is presented. Thechallenges of this implementation strongly dependson the method selected as outlined in Section 2.Since the separation models in this work relies on aproper combination of the CFE methodology withphysically-relevant elements, the implementation isnot a straightforward application of existing MOD-ELICA libraries; other aspects such as proper setupof initial conditions, state selection, modularity, andextendability are also challenging.

Figure 2: DYMOLA simulation layout consisting on aworld model, two instances of rigid bodies, the sepa-ration mechanism model, and a boolean input for theseparation command.

The baseline for the development of separation dy-namics and separation mechanisms is the followingpartial mechanism model:

p a r t i a l model Par t i a lMechan i sm" P a r t i a l s e p a r a t i o n mechanism model "I n t e r f a c e s . F r a m e _ af rame_a

" J o i n t f rame a " ;I n t e r f a c e s . F r a m e _ bframe_b

" J o i n t f rame b " ;I n t e r f a c e s . B o o l e a n I n p u tu ;

end Par t i a lMechan i sm ;

As shown in the code, the partial mechanism in-terface model consists of two frames to connect atwo-body system, and a boolean input for the ignitionor separation command. Such interface allows theuse of several separation models depending on thedesired level of complexity by using repleaceableinstances. The approach here is bottom-up design,where the basis of separation dynamics simulationcomes first from a single instance of a ‘release device’mechanism.

In this work, a release mechanism model is imple-mented to simulate both a linear charge device com-



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monly used in launcher stage separation, where theforces and moments at separation are zero; and as abase model for the next level of complexity. In otherwords, for the implementation of a separation impul-sive device, an instance of a release device providingthe capabilities of joint motion until separation is re-quired on top of another physical model providing thecorresponding impulsive forces or moments at the timeof separation. Therefore, increasing the functionalityto the separation model will consist on adding impul-sive devices or simply improving the physics behindthe device in question.

model Separa t ionMechan ism" S e p a r a t i o n mechanism model "I n t e r f a c e s . F r a m e _ af rame_a

" Mechanism frame a " ;I n t e r f a c e s . F r a m e _ bframe_b

" Mechanism frame b " ;I n t e r f a c e s . B o o l e a n I n p u tu ;r e p l a c e a b l e I n t e r f a c e s . P a r t i a l M e c h a n i s m

end Separa t ionMechan ism ;

The implementation of the CFE procedure in MOD-ELICA is as follows. The generalized constraint equa-tions of the joint (9) have to be differentiated twiceas explained before. Translational and rotational con-straints at the joint are hence implemented as:

equat ion/ / g e n e r a l i z e d c o n s t r a i n t sg_con = f r a m e _ a . r _ 0− f r a m e _ b . r _ 0 ;G_con = F r a m e s . r e l a t i v e R o t a t i o n( f rame_a.R

, f rame_b.R ) ;

/ / g e n e r a l i z e d v e l o c i t y c o n s t r a i n t sg_con_dot = der ( g_con ) ;G_con_dot = F r a m e s . a n g u l a r V e l o c i t y 2( G_con

) ;

/ / g e n e r a l i z e d a c c e l e r a t i o n c o n s t r a i n t sg_con_ddot = der ( g_con_dot ) ;G_con_ddot = der ( G_con_dot ) ;

/ / CFE g e n e r a l i z e d j o i n t c o n s t r a i n t sg_con_ddot = { 0 ,0 ,0} ;G_con_ddot = { 0 ,0 ,0} ;

In short, we present briefly two of the main modelsdeveloped in this work:

− Linear charge (separation release device): A re-lease device is modeled by an instance of theSep-arationMechanism model, called for instancelin-earCharge, which contains the partial interfaceoutlined before, plus a switching mechanism be-tween the CFE methodology and free body mo-

tion.

− Kick-off spring (separation impulse device): Animpulsive device is modeled by an instance of theSeparationMechanism model, called for instancekickOffSpring, which contains alinearChargeinstance, plus a replaceableseparationMecha-nism instance simulating the physics behind theimpulsive device, such as a spring-damper sys-tem.

For a practical scenario to study, consider the trajec-tory phase of a generic launcher where the payload(Body B - the satellite to be placed in orbit) is tobe separated from the remaining launcher upper stage(Body A - assuming a multi stage launcher). In thiscase, the problem consists of two bodies flying to-gether under the effect of gravity in joint motion (thecomposite) up until separation is commanded. Theseparation command is usually given immediately af-ter the shut down of the upper stage main engine. Inthis study however, we provide the separation com-mand at any specified time. Figure 2 shows the DY-MOLA simulation layout while Figure 3 shows a sim-ulation of the physical setup of the case studies.Initial conditions with respect to Earth-Centered-Inertial (ECI) frame of the composite are given toBody A as follows:

xA(t = 0) =

1.1378×107

00

m,

vA(t = 0) =

05.9188×103

0

m/s

and their translational and rotational dynamics are ob-tained from the rigid body model of theModelicaMultibody Library [9]. In the following section, wewill study the separation dynamics implementation inMODELICA by means of two case studies: the first oneconsiders the upper stage and payload (the composite)joint motion, while the second study considers the sep-aration phase. For both cases, the forces due to gravityacceleration are obtained from the EGM96 model im-plemented in our internalDLR Space Systems Library.Both case studies are implemented in DYMOLA andthe solution is computed using the DASSL solver witha tolerance of 1e−7. A smaller tolerance of this solverwould increase significantly the resulting chatteringwhen Baumgarte stabilization is used.


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equat ion. . .

/ / CFE g e n e r a l i z e d j o i n t c o n s t r a i n t s w i th Baumgarte s t a b i li z a t i o ng_con_ddot + 2∗ e t a∗g_con_dot + e t a∗ e t a∗g_con = { 0 ,0 ,0} ;G_con_ddot + 2∗ e t a∗G_con_dot + F r a m e s . O r i e n t a t i o n . e q u a l i t y C o n s t r a i n t( f rame_a.R ,

f rame_b.R ) = { 0 ,0 ,0} ;

Figure 3: Simulation of the physical setup of the casestudies.

Table 1: Mechanical properties of the two-body sys-tem.

Property BodyA Body B Units

Mass 6000 1000 KgI11 23000 800 Kg·m2

I22 23000 800 Kg·m2

I33 18000 600 Kg·m2

I21 = I31 = I32 0 0 Kg·m2

3.1 Case study I: upper stage and payload(composite) joint motion

The joint motion of the composite (BodiesA andB, theupper stage and the payload respectively) is simulatedfor a total time of 2000 s. During such motion, theMODELICA implementation of the CFE methodologyis expected to derive automatically the joint constraintforces and torques such that the two-body system staysproperly connected, with relative zero displacement.This case study therefore accounts for the validity ofsuch implementation.

3.2 Case study II: upper stage payload sepa-ration dynamics

The upper stage payload separation is simulated in apractical scenario setup. It consists of a simulation of20 s, half of which is in connected or joint motion, andthen att = 10 s, the ignition command for separation is

given. At this point, a kick-off spring separation mech-anism model is in charge of the dynamical separationbetween the bodies. The subsequent independent mo-tion of each body is then expected. This case studytherefore accounts for the applicability of the physicalmodels of separation mechanisms implemented.

4 Results and discussion

As outlined in the last section,Case Study I accountsfor the study of internal forces and torques of thecomposite joint motion during a given portion of itstrajectory by means of theConstraint Force Method-ology implemented in MODELICA. During such jointmotion, an important metric to assess the proposedmethod is the relative joint displacement between thetwo bodies when they are supposed to stay connected,as proposed and suggested by [17].

In this respect, Figure 4 presents the resultingconstraint forcesf[i] and torquestau[i] at the jointduring the connected motion, in all ECI directionsi = x,y,z, respectively; while Figure 5 presents theresultingrelative joint positionrrel[i] and therelativejoint velocity vrel[i], in all ECI directionsi = x,y,z,respectively.

Results shows that the corresponding joint constraintforces and torques, obtained automatically by MOD-ELICA in order to satisfy the CFE methodologyconstraints successfully keeps the bodies properlyconnected (hence, the composite) during their con-nected flight motion. Such result is evidenced lookingat the relative joint position and relative joint velocitybetween the two bodies, which are supposed to bezero during the connected flight. A clear disadvantagefor long simulation periods of joint composite motionis the necessity to keep the drift within physicalboundaries, hence requiring a stabilization method.Stability and accuracy of the solution, especially forlarge simulation times, are improved with the additionof the Baumgarte stabilization. Nevertheless at theexpense of chattering as shown in Figures 4-(b), 4-



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f[z] (CFE)f[y] (CFE)f[x] (CFE)

For

cef

[N]

Time [s]0 500 1000 1500 2000

×10−3

−3

−1.5

0

1.5

(a) Constraint forces at joint during connected motion with CFEmethodology, in all ECI directionsi = x,y,z.

f[z] (CFE + Baumgarte)f[y] (CFE + Baumgarte)f[x] (CFE + Baumgarte)

For

cef

[N]

Time [s]0 500 1000 1500 2000

×10−3

−3

−1.5

0

1.5

(b) Constraint forces at joint during connected motion with CFEmethodology plus Baumgarte stabilization withη = 2, in all ECIdirectionsi = x,y,z.

tau[z] (CFE)tau[y] (CFE)tau[x] (CFE)

Torq

ueta

u[N

m]

Time [s]0 500 1000 1500 2000

×10−3

−2

−1

0

0.5

(c) Constraint torques at joint during connected motion withCFE methodology, in all ECI directionsi = x,y,z.

tau[z] (CFE + Baumgarte)tau[y] (CFE + Baumgarte)tau[x] (CFE + Baumgarte)

Torq

ueta

u[N

m]

Time [s]0 500 1000 1500 2000

×10−3

−2

−1

0

0.5

(d) Constraint torques at joint during connected motion withCFE methodology plus Baumgarte stabilization withη = 2, inall ECI directionsi = x,y,z.

Figure 4: Case Study A results: constraint forces and torques at joint during connected motion.

(d), 5-(b), 5-(d), meaning more computational timeand effort.

Case Study II, as outlined in the last section, accountsfor the study of absolute− and relative− position,velocity, and acceleration, respectively, between thetwo bodies from a multi-stage separation dynamicspractical scenario. In here, the ‘release device’ sim-ulated by a linear charge model has been augmentedwith an ‘impulsive device’ in parallel simulated bya kick-off spring model in order to simulate such aseparation mechanism between the two bodies at their

time of release from each other.

In this respect, Figure 6 presents the bodies’relativeposition rrel[i], velocity vrel[i], and accelerationarel[i] along the ECI orbital flight directioni = y(which is valid only for such a very small time frame)during the connected motion (first 10 seconds), andduring their subsequent separation (last 10 seconds).Figure 6 also presents a zoom of the small timewindow just around the separation command.

Results of this separation scenario shows the corre-


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rrel[z] (CFE)rrel[y] (CFE)rrel[x] (CFE)

Rel

ativ

epo

sitio

nrre

l[m

]

Time [s]0 500 1000 1500 2000

×10−7

−5

0

5

(a) Relative joint position during connected motion with CFEmethodology, in all ECI directionsi = x,y,z.

rrel[z] (CFE + Baumgarte)rrel[y] (CFE + Baumgarte)rrel[x] (CFE + Baumgarte)

Rel

ativ

epo

sitio

nrre

l[m

]

Time [s]0 500 1000 1500 2000

×10−7

−5

0

5

(b) Relative joint position during connected motion with CFEmethodology plus Baumgarte stabilization withη = 2, in all ECIdirectionsi = x,y,z.

vrel[z] (CFE)vrel[y] (CFE)vrel[x] (CFE)

Rel

ativ

eve

loci

tyvr

el[m

/s]

Time [s]0 500 1000 1500 2000

×10−9

−1

−0.5

0

0.5

1

(c) Relative joint velocity during connected motion with CFEmethodology, in all ECI directionsi = x,y,z.

vrel[z] (CFE + Baumgarte)vrel[y] (CFE + Baumgarte)vrel[x] (CFE + Baumgarte)

Rel

ativ

eve

loci

tyvr

el[m

/s]

Time [s]0 500 1000 1500 2000

×10−7

−5

0

5

(d) Relative joint velocity during connected motion with CFEmethodology plus Baumgarte stabilization withη = 2, in all ECIdirectionsi = x,y,z.

Figure 5: Case Study A results: relative joint position and velocity during connected motion, in all ECI direc-tionsi = x,y,z.

sponding relative states of the composite up until sep-aration command and then their subsequent indepen-dent flight. Once again, the benefit and ease of use ofthe MODELICA implementation of the CFE method-ology is evidenced during the connected flight of thecomposite, since constraint forces and torques are au-tomatically computed and applied to the system. Atseparation, the relative states suggest an impulsive be-haviour due to the kick-off spring separation mecha-nism model. This model releases a pre-compressedforce stored in a replaceable spring-damper model, ev-

idencing good correspondence with the physics behindseparation. Such devices result in impulsive forces ap-plied to the two-body system. This in turn causes achange in relative velocity and therefore, a successfulphysical separation of the system.

5 Conclusion

The objective of this paper was to present anobject-oriented and equation-based acausal modelingapproach to launch vehicle stage separation dynamics



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with MODELICA. The aim is to develop an integratedapproach for end-to-end launch vehicle trajectorysimulation within a single environment.

Based on theConstraint Force Equation (CFE)methodology, two case studies to evaluate the pro-posed approach were considered. The scenario understudy consisted of two bodies –representing a genericlauncher stage and its payload– prior, during, andafter their separation in orbital flight motion.

Results demonstrated that the approach, mainlythanks to the acausal and equation-based modelingfeatures of the MODELICA language, correspondsvery well with the physics behind separation whileproviding easiness of implementation within a singleenvironment such as DYMOLA . The method computesand applies constraint loads automatically during jointmotion and removes them accordingly at separationtime, all in consistency with the CFE methodology.

A disadvantage for long simulation periods of jointbody motion is the necessity to keep the drift withinphysical boundaries, hence requiring a stabilizationmethod. This in turn increases chattering and com-putational time and effort, thus resulting in a trade-offto consider for the task at hand. Validation studies areleft to future work.

Acknowledgements

This work corresponds to DLR Institute of SystemDynamics and Control activities of the ESA studyUpper Stage Attitude Control Design Framework(USACDF), led by Astrium GmbH as part of Europe’sFuture Launchers Preparatory Program (FLPP).

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[12] Pamadi, B.N., Neirynck, T. A., Covell, P.F., Hotchko, N.J.,and Bose, D.M.Simulation and Analyses of Staging Maneu-vers of Next Generation Reusable Launch Vehicles. AIAAPaper 2004-5185.

[13] Pamadi, B.N., Hotchko, N.J., Jamshid Samareh, Covell,P.F., and Tartabini, P.V.Simulation and Analyses of Multi-Body Separation in Launch Vehicle Staging Environment.AIAA Paper 2006-8033.

[14] Pamadi, B.N., Neirynck, T. A., Hotchko, N.J., ScallionW.I., Murphy, K.J., and Covell, P.F.Simulation and Analy-ses of Stage Separation of Two-Stage Reusable Launch Ve-hicles. Journal of Spacecraft and Rockets, Vol. 44, No. 1,January-February 2007, pp 66-80.

[15] Källdahl M. Separation Analysis with OpenModelica. Stu-dent thesis, Linköping University, Department of ElectricalEngineering, Institutionen fr systemteknik, Sweden, 2007.

[16] Toniolo, M., Tartabini, P.V., and Pamadi, B.N.ConstraintForce Equation Methodology for Modeling Multi-BodyStage Separation Dynamics. 46th AIAA Aerospace Sci-ences Meeting and Exhibit, 2008, 10.2514/6.2008-219.

[17] Tartabini, P.V., Roithmayr, C.M., Toniolo, M.D., Karlgaard,C.D., and Pamadi, B. N.Modeling Multibody Stage Sepa-ration Dynamics Using Constraint Force Equation Method-ology. Journal of Spacecraft and Rockets, Vol. 48, No. 4,July-August 2011, pp 573-583.

[18] Pamadi, B.N., Tartabini, P.V., Toniolo, M.D., Roithmayr,C.M., Karlgaard, C.D., and Samareh, J.A.Application ofConstraint Force Equation Methodology for Launch VehicleStage Separation. Journal of Spacecraft and Rockets, Vol.50, No. 1, January-February 2013, pp 191-205.


DOI10.3384/ECP14096589


597

rrel[y] (along orbital direction)

Rel

ativ

epo

sitio

nrre

l[m

]

Time [s]0 5 10 15 20

5

10

20

30

40

(a) Relative position between bodiesA andB. The initial relativeposition (5.8 m) corresponds to the fixed distance between thebodies center of masses during joint motion.

rrel[y] (along orbital direction)

Rel

ativ

epo

sitio

nrre

l[m

]

Time [s]9.95 10 10.05 10.1

5.8

5.9

6

6.1

(b) Same as (a) with a close view around time of separation.

vrel[y] (along orbital direction)

Rel

ativ

eve

loci

tyvr

el[m

/s]

Time [s]0 5 10 15 20

−0.5

0

1

2

3

4

(c) Relative velocity between bodiesA andB.

vrel[y] (along orbital direction)

Rel

ativ

eve

loci

tyvr

el[m

/s]

Time [s]9.95 10 10.05 10.1

−0.5

0

1

2

3

4

(d) Same as (c) with a close view around time of separation.

arel[y] (along orbital direction)

Rel

ativ

eac

cele

ratio

narel

[m/s

2]

Time [s]0 5 10 15 20

−50

0

100

200

(e) Relative acceleration between bodiesA andB.

arel[y] (along orbital direction)

Rel

ativ

eac

cele

ratio

narel

[m/s

2]

Time [s]9.95 10 10.05 10.1

−50

0

100

200

(f) Same as (e) with a close view around time of separation.

Figure 6: Case Study B results: Relative position, velocity, and acceleration from a kick-off separation scenarioalong orbital flight directioni = 2. Ignition / separation command att = 10 s.



DOI10.3384/ECP14096589

modelica stage separation dynamics modeling for end-to-end

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