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Aerospace Science and Technology 8 (2004) 73–81www.elsevier.com/locate/aesc

Adaptive neural control of the deployment procedure fortether-assisted re-entry

Holger Gläßela,∗, Frank Zimmermannb, Steffen Brücknera, Ulrich M. Schöttlec,Stephan Rudolpha

a Institut für Statik und Dynamik der Luft- und Raumfahrtkonstruktionen, Universität Stuttgart, Pfaffenwaldring 27, D-70550 Stuttgart, Germab VEGA Informations-Technologien GmbH, Julius-Reiber Straße 19, D-64293 Darmstadt, Germany

c Institut für Raumfahrtsysteme, Universität Stuttgart, Pfaffenwaldring 31, D-70550 Stuttgart, Germany

Received 5 June 2002; received in revised form 16 April 2003; accepted 7 August 2003

Abstract

An adaptive neural control concept for the deployment of a tethered re-entry capsule is presented. The control concept appliesneural controller that combines two neural networks, a controller network and a plant model network. While the controller neinitialized by means of multiple conventional linear quadratic regulator designs, the plant model network is trained to predict fututhus deviations from an optimized reference path.

System inputs are found by means of an online optimization process, which minimizes a user defined cost function, that influperformance of the neural controller. Due to the special structure of the controller network, stability investigations of the closed conbecome possible.

After introducing the tether deployment scenario, assumptions and simplifications are applied to the mathematical system mnumerical simulations focus on the effects of perturbations concerning the initial states and the plant model. The simulation resuperformance comparison of the linear quadratic regulator and the neural control concept. 2003 Elsevier SAS. All rights reserved.

Keywords:Adaptive control; Neural networks; Predictive control; Optimal control; Tether-assisted deorbit

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1. Introduction

Among the many possible space applications of tesystems identified in the past [12], the highlight of tethexperimentation is still the Tethered Satellite ExperimTSS-1 and its reflight TSS-1R, where the investigatifocused on the electrodynamic interaction of a conductether with the magnetic field of the Earth as wellon the controlled deployment and retrieval of the tethesatellite [3]. More recently the application of tethersre-entry systems has received much attention [9,11Efficient utilization of the International Space Station (ISrequires the frequent return of small payloads or mateprocessed in space to ensure quick access by the

* Corresponding author.E-mail addresses:glaessel@isd.uni-stuttgart.de (H. Gläßel),

frank.zimmermann@vega.de (F. Zimmermann),schoettle@irs.uni-stuttgart.de (U.M. Schöttle).

1270-9638/$ – see front matter 2003 Elsevier SAS. All rights reserved.doi:10.1016/j.ast.2003.08.007

s

on Earth. Such return capabilities could be providedmeans of small unmanned re-entry vehicles, preferacontrollable capsules with medium lift-to-drag ratios. Whinitiating the deorbiting manoeuvre by deploying a lothin tether from the space station, no deorbit propulssystem is necessary. Therefore, the tether-assisted deorrepresents an important application for the ISS and becoan alternative to conventional return manoeuvres thatfuel consuming propulsion systems [21]. Such missirequire the deployment and time of spacecraft release texecuted very accurately in order to attain narrow re-ewindows even under non-nominal conditions. Assumthat it is difficult to identify a complete model of thtethered capsule in the real space environment a mtask is to handle the unexpected behaviour resulting fan insufficient plant model. Hence, adaptive control oftether deployment is mandatory. The objectives of this stare to investigate an adaptive neural control concept

74 H. Gläßel et al. / Aerospace Science and Technology 8 (2004) 73–81

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this task and to provide a performance comparison wiconventional linear quadratic regulator approach.

2. Tether deployment scenario

In a tether deployment scenario the re-entry capsuattached at the end of a tether which is deployed fromspace station. The capsule will orbit at a smaller velothan required to stay at its lower altitude. Thus, if the tetis cut, the capsule will enter into a re-entry trajectory.

Two deployment procedures may be distinguished,static and the dynamic release. The dynamic release ore-entry capsule is preferable, since an additional velodecrease is achieved by a dynamic swing back, which ala reduction of the necessary tether length [19]. The mproperties of the dynamic deorbit manoeuvre are depicteFig. 1.

During deployment, a forward swing due to orbit mchanical coupling effects is invoked. Active braking pvided by a reusable braking mechanism on board the sstation is used to control this deployment phase. At a prefined time when the required tether length is achieved,deployment is stopped and the swing back is initiated. Cto the local vertical the tether is cut, resulting in a reductof the inertial velocity of the capsule such that it enters ia predefined return trajectory.

This deployment procedure can be further divided iseveral phases. The initial ejection by means of a spmechanism leads to a separation velocity of the cap(0.5–1 m s−1) in a downward direction. After that, an initiadeployment phase follows, where the tether is deplo

Fig. 1. Tether-assisted deorbiting manoeuvre (dynamic release). Thetrolled main deployment phase is indicated by a thick line between thedeployment to the tether lengthl0 and the uncontrolled swing back.

close to the local vertical and the capsule is stabili1–2 km below the station. This early deployment phaccounts for the high sensitivity with respect to disturbanwhile the stabilizing gravity gradient forces are still low [1From this stable position the main deployment is initiatThis strategy results in significantly reduced deviations frthe nominal landing site [11].

During the subsequent main deployment phase a larggle in flight direction is achieved through fast deploymentwards the maximum tether length. The deployment folloa reference trajectory which is determined minimizingcontrol input [20]. The deorbiting task requires an accurexecution of the tether deployment and the capsule releaattain the designated recovery area [6]. Therefore a cloloop controller is mandatory to compensate disturbancesatmospheric drag, gravitational disturbances, and electrnamic forces in case of conductive tethers. In additionthis, model uncertainties related to e.g. the tether dynamthe environment and to friction in the deployment mecnism have to be accounted for. To cover a representativof disturbances, the current analyses focus on initial detions from the nominal tether states at the beginning ofmain deployment phase and on a deviation from the nonal braking force applied along the deployment. Both hasignificant effect on the deployment path. The differencetween a controlled and uncontrolled main deployment phfor an initial deviation is exemplified in Fig. 2. The apprpriate coordinate system is defined below.

In case of unexpected system behaviour, thus to accfor model uncertainties, there is a demand for adaptiveof the control system. Subsequent deviations fromoptimal trajectory must be compensated even during the

Fig. 2. Controlled (closed loop, LQR) and uncontrolled (open loop) patthe capsule during deployment for an initial deviation of�θ(t = 0) = +2◦versus reference path.

H. Gläßel et al. / Aerospace Science and Technology 8 (2004) 73–81 75

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deployment procedure, because important model paramsuch as the friction in the deployment mechanism subjea microgravity environment are difficult to estimate.

For the development of the neural adaptive controonly the main deployment phase of the tether deploymprocedure is considered, beginning at an initial tether lenof 1000 m and ending when active braking is terminatethe maximum tether length and the swing back starts.

3. Modelling the tether system

Various methods for the dynamic modelling of a tethsystem exist, where the tether is described either as a couum, as connected point masses, or massless [19]. To mtain the advantage of low computational effort requiredoptimization and controller design, the following assumtions are made:

• Because typical tether masses for this type of applicaamount to 0.3–0.7 kg km−1, the system is idealized atwo point masses, representing the space stationthe return capsule, connected by a massless inextentether.

• The center of orbit, center of gravity, and center of mare not distinguished and considered to be situated adeployer position. This assumption is justified sincespace station mass (≈ 415 t) exceeds by far the massthe capsule (170 kg) [19].

• The space station moves in a circular orbit at a distaof r0 = 6771 km from the center of Earth, thusapproximately 400 km altitude.

• The atmospheric influence in terms of a drag comnent, which could lead to a curved tether, is neglecte

The main acting forces in the co-orbiting frame with its ogin at the centre of mass are the centrifugal and the grtational force resulting in a net force called gravity gradiforce. This force vector can be divided into a stabilizing nmal component and a radial component stretching the te

The corresponding coordinate system is shown in Figwhere θ indicates the in-plane angle,ϕ the out-of-planeangle, andl the tether length. With the center of orbitr0situated at the space station position, the following equatof motion for the tether system are obtained [15]:

θ + (θ − ω)

(2l

l− 2ϕ tanϕ

)+ 3ω2 sinθ cosθ = 0, (1)

ϕ + 2l

lϕ + [

(θ − ω)2 + 3ω2 cos2 θ]sinϕ cosϕ = 0, (2)

l − l[ϕ2 + (θ − ω)2 cos2 ϕ + ω2(3 cos2 ϕ cos2 θ − 1

)]+ T

m= 0, (3)

wherem denotes the capsule mass andT the tether tensionThe orbit angular velocityω is calculated byω2 = µ/r3

0 ,

s

--

e

.

Fig. 3. Coordinate system of the tether. Thex andz coordinate are withinthe orbital plane, they coordinate is perpendicular to it. Thex coordinatepoints in flight direction of the space station, thez coordinate to the centrof Earth.

where µ = 3.986005× 1014m3 s−2 is the gravitationaparameter of the Earth.

An important property of the above equations is ta pure in-plane motion will not excite any out-of-plamotion, whereas the opposite is not true. Furthermthere is no controllability of the out-of-plane motion viamodulation of the tether tension forϕ = ϕ = 0. Therefore,the controller design is limited to the in-plane motion. Tcorresponding equations are rewritten as a set of first oequations

x1 = x3, (4)

x2 = x4, (5)

x3 = −(x3 − ω)2x4

x2− 3ω2 sinx1 cosx1, (6)

x4 = x2[(x3 − ω)2 + ω2(3 cos2 x1 − 1

)] − u

m, (7)

that can be solved numerically and wherex1 to x4 denote theelements of the state vector�x = [θ l θ l]T andu = T withT � 0 represents the braking force as the control vector.the massless tether, the braking force is equal to the tetension.

The optimal tether deployment path is derived fromoptimization according to [20]. It is based on the aboequations of motion and applies the performance index

J =tf∫

t0

T 2 dt, (8)

where t0 and tf denote the initial and the final time othe main deployment phase. Following this trajectory wminimize the overall control input during deployment, suthat an effort and thereby energy optimized controlaccomplished.

4. Neural adaptive control

Neural networks show several properties which mthem useful for control tasks. Their behaviour can be

76 H. Gläßel et al. / Aerospace Science and Technology 8 (2004) 73–81

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fected by the means of different network topologies, sas feedforward or recurrent networks. Where recurrentworks are dynamic systems and may become unstable,feedforward networks can be proven to be unconditionstable. Basically, neural networks consist of small comtational units called neurons which form a network usweighted connections. By selecting an appropriate action function within the neurons, neural networks can pform linear as well as non-linear mappings [13].

The input-output behaviour of these networks is train(i.e. adapted) using representative data samples. Diffelearning algorithms are available in standard software pages, such as MATLAB [5], which has been used in twork. The training procedure adapts the weights of the cnections to perform the mapping underlying the presentraining data.

In control theory, the non-linearity and adaptabilityneural networks makes them an alternative techniquconventional control designs based on linearized sysmodels, despite the fact that neural control concepts areas well-elaborated as classical control theory yet.

4.1. Neural control strategy

For non-linear and adaptive control, different neural ctrol strategies have been proposed [2,16–18]. One proing approach is given by an indirect control concept, whtwo components, a neural plant model network and a necontroller network are used (see Fig. 4). The plant netwusually serves to back-propagate output errors in ordeidentify better system input values and to adjust the freerameters of the controller network within a learning (opmization) process. Therefore, many training samples fnumerous deployments would be necessary and a longoptimization over many deorbit manoeuvres could be rized. An immediate reaction already during the first deplment is no more possible. For this reason the classical

c

t

t

of neural controller adaption was not investigated here,is of course still possible.

In this study the controller network is structured ainitialized according to the design of a conventional linquadratic regulator (LQR) based on a linearized temodel (see also section Neural Controller Network). Toffline trained controller as well as the plant networknot adapted during operation in the closed control loDuring control computation only the optimizer is active aresponsible for the adaptability of the control input. Enoutime remains to find an improved control modification onlbetween two updates, such that the requirement to proadaptiveness already during the first deployment proceis fulfilled. Therefore, this approach seems to be well sufor the tether deployment problem.

The structure of the closed control loop is shown in FigThe deviations of the current state vector from the referetrajectory��x = �x − �xref are fed back. Based on these, tcontroller network calculates a control modification�u. Itis assumed that all state vector elements can be measThis could be provided by a cooperative capsule whicequipped with a satellite navigation system like GPS.pending on the current and former state vectors (�x(t) and�x(t − �t)), the optimizer identifies an improved contrmodification�u′ = �uopt near�u according to the dynamics represented by the plant network such that the weigcost function

C =4∑

j=1

(xj,pred(t + �t) − xj,ref(t + �t)

gj

)2

+(

�u(t)′

g5

)2

(9)

is minimized. Thegj weight the specific state deviation�xj,pred in comparison to the control deviation�u′. Thiscost function is similar to the cost function used byLQR design according to Eq. (10) but minimizes only tpredicted future state deviations��xpred(t + �t) from the

Fig. 4. Closed neural control loop. Main components are the neural plant network, the neural controller network and the optimizer.

H. Gläßel et al. / Aerospace Science and Technology 8 (2004) 73–81 77

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plant network and the suggested control deviation�u(t)′.The optimization finds the best control modification for tcurrent situation and is fast enough to be done online duthe simple plant network.

Adding the optimized control modification�u′ = �uoptto the reference control inputuref(t) results in the commanded input valueuc(t), which is applied to the system acontrolling input until the next control update. Whereascontrol modification�u is constant during one time step tcommanded input remains independent from the controsample rate, because the reference input is defined conously.

4.2. System identification using a neural plant model

The neural plant network predicts the future state ve�xpred(t +�t), which will be reached one time step�t = 10 slater subject to the meanwhile applied system input (FigThis system input is composed of the reference inputuref andthe control modification�u. To benefit from the stability oa static feedforward network but enable a dynamic mappthe current state vector input�x(t) and one delayed vecto�x(t − �t) are used [4]. The linear plant network consistsan input layer with 9 inputs ([x1(t) x2(t) x3(t) x4(t) x1(t −�t) x2(t − �t) x3(t − �t) x4(t − �t) �u(t)] i.e. the stateand delayed state vector each with 4 components pluscontrol modification input), no hidden layer and an outlayer with 4 linear neurons. Additionally, the input aoutput data sets are normalized such that they showmean and unity standard deviation.

Solving the above stated equations of motion (Eqs. ((7)) by a Runge–Kutta integration with different initial statdistributed along the reference trajectory and simulatingeffect of high positive and negative control modificatio(Fig. 5) the training data set is obtained. The validatof the plant network confirms high performance overwhole state space. In particular the ability to back-propacontrol modifications according to existing state deviatiwas verified [8].

Fig. 5. Schematic example of three training data samples for diffecontrol modifications.

-

4.3. Linear quadratic regulator as initial solution

The initial neural controller network is based on a lear quadratic controller design (LQ-Regulator) to provreasonable behaviour already from the beginning. Forapproach the non-linear problem is transferred into muple linear problems at certain operating points alongreference trajectory. Linearizing the equations of mot(Eqs. (4)–(7)) at each of these operating points in timesults in corresponding time-independent linear systemtricesA(t) andB(t), which are only valid in vicinity of thesepoints. The LQR design minimizes the quadratic cost fution

J =∞∫

0

(��x(t)TQ��x(t) + �u(t)TR�u(t)

)dt (10)

by solving the Riccatti equation [10] and leads to a timdependent linear control law of the form�u = −�k(t) ·��x, where�k(t) denotes the gain vector. Simulations shthat by using this control law initial disturbances are wcompensated over the deployment procedure (see FigLow computational effort, a short design process,evident stable behaviour are the main features of the Lapproach.

4.4. Neural controller network

The structure of the controller network is derived frothe time-dependent LQR design outlined above andillustrated in Fig. 6. Splitting the mappingF :��x �→ �u

into two steps,F1 : t �→ �k(t) and F2 : {��x, �k(t)} �→ �u,the time-dependent correlations can be learned easily.four subnetworks evaluating the gain vector componenki

are radial basis networks with one hidden layer. Theywell suited for this function approximation purpose, duetheir arbitrary mapping precision and their simple desprocess, that allows to select the network size meetingdesired precision and computational effectiveness [5,7].controller network represents a control law similar toLQR and so features the corresponding design advant(especially stability), but allows adaption through a lalearning process as well (as already mentioned in sec“Neural Control Strategy” by back-propagating predicstate deviations through the plant network).

In using the outlined controller network structure coclusions on the stability of the adapted closed loop syswith the new control law could be derived from the modificontroller network, as long as the optimization is not actThus, this approach enables the application of conventistability criteria on neurally controlled systems.

5. Results from closed loop simulation

Both the LQ-Regulator and the neural controller habeen investigated and compared in numerous simula

78 H. Gläßel et al. / Aerospace Science and Technology 8 (2004) 73–81

Fig. 6. Structure of the controller network. MappingF1 : t �→ �k(t) is provided by the four radial basis subnetworksk1 to k4, mappingF2 : {��x, �k(t)} �→ �u isdone by one linear neuron in the output layer.

st,

nt

asitiale

lledller

isg to-fo-

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LQ-

ons

pt.

t

ares theromafter

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cess

othbe

ly, a

focusing on the effect of two kinds of perturbations. Firdeviations from the initial reference state�xref(t = 0) =[0 1000m 0 0]T after termination of the early deploymephase are considered.

As Fig. 7 exemplifies, the LQ-Regulator as wellthe neural control concept compensate the applied indisturbance of�θ(t = 0) = +2◦ and return to the referenctrajectory quite well. Compared to an open loop controdeployment, the final deviations are significantly sma(see Fig. 2).

In addition to this, an imperfect braking mechanismsupposed conveying a disturbed system input accordinuc,real= kf uc with kf = 1.2 over the whole deployment procedure neglecting initial perturbations. These simulationscus on the applicability of both control concepts with regto assumed model uncertainties. None of the controllersigns is tuned towards the new situation.

Fig. 8 shows, that in contrast to the LQ-Regulatorneural control concept adapts to the unexpected behavemphasizing that there is no learning process during thetrolled deployment procedure. The reason for this predinance can be ascribed to two facts. On the one handneural plant network mapping is rather general. Thus,the disturbed system behaviour can be reproduced wethe plant network. On the other hand, any new control inresults from an optimization observing the current sysstatus and searching for better input values taking intocount also future states (see Eq. (9)), in contrast to the

,-

Regulator, which optimizes all control and state deviatiover an infinite time horizon (see Eq. (10)).

Fig. 9 points out the adaptability of the neural conceDue to improper braking the commanded inputuc,real is toolarge by a factor of aboutkf = 1.2. In fact the controllerhas to command a varied inputuc, which should be abou1/kf = 0.833 smaller than the reference inputuref to stayon the reference trajectory if no other perturbationsconsidered. The main advantage of the neural system iuse of future state predictions, such that deviations fthe optimized reference trajectory are not compensatedtheir occurrence only, but already before they arise.

6. Conclusions

An efficient neural adaptive control concept based olinear quadratic regulator was successfully developed.demand to respond to unexpected system behaviourperturbations already during the first deployment leadthe adaptive concept, that identifies a better input througonline optimization process. Thereby, the performancebe tailored by the composition of the weighted cost functespecially by varying individual weights.

The required computational effort is rather low, sinthe plant network included in the optimization proceis compact enough. Additionally, an adaptation of bnetworks after several deployment procedures wouldpossible, but was not investigated here. Consequent

H. Gläßel et al. / Aerospace Science and Technology 8 (2004) 73–81 79

Fig. 7. State histories for an initial deviation of�θ(t = 0) = +2◦, both neural and LQR controlled. Deviations from the reference trajectory are plotted againsttime.

Fig. 8. State histories for an imperfect braking mechanism, both neural and LQR controlled. Deviations from the reference trajectory are plotted against time.

80 H. Gläßel et al. / Aerospace Science and Technology 8 (2004) 73–81

e. W

Fig. 9. Commanded inputs of both controllers applied to the imperfect braking mechanism in relation to the reference input are plotted against timithoutexternal perturbations the input ratio ofuc

uref= 1

kfwould not lead to state deviations.

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long term optimization considering data from several tetdeployments is conceivable. The structure of the contronetwork allows investigations concerning the stability ofneural control concept as soon as the controller netwomodified and differs from the original LQR initialization.

Acknowledgements

This work was performed at the Institute of Space Stems (IRS) in cooperation with the Institute of Statics aDynamics of Aerospace Structures (ISD), UniversityStuttgart, as a thesis work, while the second author still ha position at the IRS. The financial support in part by“Deutsche Forschungsgemeinschaft (DFG)” within the Claborative Research Project SFB 409 “Adaptive Struktuim Flugzeugbau und Leichtbau” is acknowledged.

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