AAS 03-532

DYNAMICS AND CONTROL OF TETHERED SATELLITE SYSTEMSFOR NASA’S SPECS MISSION

Mischa Kim∗ and Christopher D. Hall†

The control problems of two different configurations of Tethered SatelliteSystems (TSS) for NASA’s Submillimeter Probe of the Evolution of CosmicStructure (SPECS) mission are studied and their control performances arecompared. The configuration of main focus is the TetraStar model com-prised of three controlled spacecraft and three uncontrolled counterweights.This system is compared to a triangular TSS consisting of three controlledspacecraft. The equations of motion are derived using Lagrange’s equa-tions. Several mission scenarios for the SPECS mission considering theoperation of an infrared telescope are introduced and asymptotic trackinglaws based on Lyapunov control are developed.

INTRODUCTION

Over the past three decades, a variety of concepts have been proposed for space exploration usingTethered Satellite Systems (TSS). These include scientific experiments in the microgravity environ-ment, upper atmospheric research, cargo transfer between orbiting bodies, generation of electricity,and deep space observation.1–3 Numerous missions have already been launched to verify the teth-ered system concept for space application. Important milestones include retrieval of a tether inspace (TSS-1, 1992), successful deployment of a 20 km tether in space (SEDS-1, 1993), closed-loopcontrol of a tether deployment (SEDS-2, 1994), and operation of an electrodynamic tether used inboth power and thrust mode (PMG, 1993).4,5 The idea of interconnecting spacecraft by meansof lightweight deployable tethers has been proven to be particularly attractive also for space ob-servations for various reasons. Variable-baseline interferometric observations can be achieved by acarefully controlled deployment and retrieval procedure of the tethers. In addition, the observationalplane can be densely covered by spinning the tethered system. Lastly, the high levels of propellantconsumption demanded by separated spacecraft in formation can be significantly reduced by tensionor length control of the interconnecting tethers.

The paper is organized as follows. We begin with a discussion of some relevant papers andarticles on the dynamics of TSS and adaptive control of mechanical systems. In the followingsection we present a brief description of the science mission which motivated this study and introducetwo particular satellite formations. Subsequently, the system model is introduced and the motionequations are derived. Relative equilibria are identified and mission scenarios are presented followedby a thorough discussion on control law development. Finally, the performance of the controllersare validated considering the different mission scenarios.

∗Graduate Assistant, Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and StateUniversity, Blacksburg, Virginia. mischa@vt.edu. Student Member AIAA, Student Member AAS.

†Professor, Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University,Blacksburg, Virginia. cdhall@vt.edu. Associate Fellow AIAA, Member AAS.

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LITERATURE REVIEW

The problem of dynamics and control of tethered satellite formations has attracted considerableattention over the past decade. Of particular interest in the context of this paper are publicationsby DeCou,6,7 Keshmiri and Misra,8 Misra et al.9 and Farquhar.10 For a thorough literature reviewwe refer to Ref. 11.

Nonlinear adaptive control of dynamical systems has emerged as an increasingly important ap-proach to nonlinear controller design in recent years. Major breakthroughs in adaptive output–feedback control were achieved amongst others by Marino and Tomei12,13 whose work triggereda remarkable development in the following years. In Ref. 12 the authors managed to remove thestructural restriction that the nonlinearities in the output were not allowed to precede the input. Amore general class of systems was considered in their companion paper13 where the system was notrequired to be linear with respect to the unknown parameter vector. The latter results were onlyobtained, however, for set–point regulation problems. In a more recent paper Marino and Tomeiextended the results obtained in Ref. 12 for a class of systems with time–varying parameters.14 Thepaper by Krstic and Kokotovic15 refined the control approaches developed by Marino and Tomeiusing such concepts as “tuning functions” and “swapping–based” schemes to allow for incorporationof any standard gradient update law and to deal with overparameterization. Khalil16 considered theproblem of adaptive output–feedback control of single–input–single–output systems which can benonlinearly dependent on the control input. However, only semiglobal stability could be achieved.Furthermore, a–priori knowledge of bounds on parameters and on initial conditions was requiredand, more importantly, persistence of system excitation was sufficient for both parameter and track-ing error convergence. Loria presented one of the first papers on global output–feedback controlfor one–degree–of–freedom Euler–Lagrange systems.17 The control design exploited the propertiesof hyperbolic functions to define a “nonlinear approximate differentiation filter” to automaticallyenlarge the domain of attraction. Ref. 18 uses a similar approach to achieve global stability forgeneral nth order uncertain systems and is the major reference for the control approach used here.

THE SPECS MISSION CONCEPT

NASA’s future Earth and Space science missions involve formation flying of multiple coordinatedspacecraft. Several space science missions include distributed instruments, large phased arrays oflightweight reflectors, and long variable baseline space interferometers. An array of collectors andcombiner spacecraft will form variable–baseline space interferometers for a variety of science missionssuch as the Submillimeter Probe of the Evolution of Cosmic Structure (SPECS).19 This particularmission concept was initiated by a NASA science team and proposes a 1 km baseline submillimeterinterferometer (λ ≈ 40 − 500 µm). It comprises possibly as few as three 3 − 4 meter diametermirrors and rotates about the primary optical axis collecting (3 − 0.25 eV) photons which arethen preprocessed by a central beam collector. Since operating such systems in any kind of Earthorbit is not feasible due to extensive fuel consumption and unsatisfactory photon yield, the secondLagrangian point in the Sun–Earth system L2 was chosen as the operational environment. TheSPECS spacecraft formation is intended to be placed in a Halo orbit about this libration point.20

Possible satellite formations for SPECS

The two tethered satellite configuration we compare in this paper are a triangular system and amore complex formation called TetraStar.21 The latter system consists of a total of three con-trolled spacecraft and three uncontrolled counterweights. Figures 1(a–c) show a schematic layout ofTetraStar during tether deployment. The three inner point masses denote the controlled spacecraft

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and form what we refer to as the “inner” system, the outer point masses (“outer” system) markthe uncontrolled counterweights. Additionally, the three tethers interconnecting the controllablespacecraft are extensible whereas the remaining six tethers are assumed to be of fixed length. Notethat the triangular configuration can be obtained from the TetraStar model by removing the outerpoint masses and the corresponding tethers.

eu

ev

ew(a) (b) (c)

Accelerations dueto tether forces

centripetal–Coriolisacceleration spin

direction

l(t)

L(t)

=co

nst.

ρ(t)

Figure 1: Deployment of the TetraStar system.

From a controls point of view TetraStar offers significant benefits over the triangular TSS. As weshowed in an earlier work,11 the overall control effort for a triangular TSS can be divided into twoparts. The first and major contribution to the total control impulse is required in angular directionand regulates the angular momentum of the TSS according to the desired trajectory. The secondcontrol part is a radial control component which significantly contributes to the overall controleffort only during times when the TSS is in a transitional phase between steady–spin and tetherdeployment or retrieval and vice versa. The TetraStar model can improve the control performanceon both counts. Firstly, the three counterweights in combination with the corresponding tethersprovide additional control authority in the radial direction for ρ(t) < π/3. In Figure 1(a) theseadditional force components are identified by full arrows, the broken arrow denotes a centripetal–Coriolis acceleration component. Secondly and more importantly, if dimensioned correctly thesecounterweights can act as a buffer to balance the increase and decrease in angular momentumduring tether deployment and retrieval which is due to the controlled spacecraft.

SYSTEM MODEL AND EQUATIONS OF MOTION

In this section equations of motions (EOM) of the system are formulated using the system descriptiondeveloped in Ref. 22. The mechanical system considered is shown in Figure 2 with TetraStar.It is comprised of a system of n point masses interconnected arbitrarily by m idealized tethers.The tethers are assumed to be massless and extensible capable of exerting force only along thestraight–line connecting the respective masses. Also, the tethers do not support compression orany components of shear forces or bending moments and are therefore assumed to be perfectlyflexible. The constitutive character for the tethers is taken as visco–elastic, allowing intrinsic energydissipation. Since the system is ultimately being operated at the second Lagrangian point L2 in theSun–Earth system, gravitational and other environmental forces are assumed to be negligible, butare readily incorporated into the model if necessary.

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ex

ey

ez

mj

mi, qi = (ri, θi, zi)T

lij , Eij , Aij , aij

Target direction

FP

FI

Figure 2: System model in cylindrical coordinates.

The EOM are derived using Lagrange’s equations defined in a “prescribed motion” referenceframe FP rather than in an inertial reference frame FI . Cylindrical coordinates relative to FPare chosen to describe the position of the point masses. As illustrated in Figure 2, ri, θi, and zi

are the radial distance in the exey plane, the angle between the ex–axis and the projection of theposition vector qi onto the exey plane, and the axial distance in ez direction of the point mass mi,respectively. Note that during an observation the TSS rotates in the so–called uv plane which isdefined as the plane perpendicular to the target direction. The motion equations for the controlledsystem are

Miqi = Q(e)i + Ni + Si + Ui , i = 1, 2, . . . , n (1)

where

Mi = mi

1 0 00 r2

i 00 0 1

, qi =

ri

θi

zi

, Q

(e)i =

Fri

riFti

Fzi

, Si = Q

(i)i (2)

In equations (1) Mi, qi, Q(e)i , Ni, Si, and Ui are the mass matrix, the vector of generalized

coordinates, and the vectors of external, coupling, spring (or internal), and control forces for the ithpoint mass, respectively. The spring forces in equations (1) are obtained assuming a Kelvin–Voigtconstitutive law; that is, the tensile stress, τij , and the longitudinal strain, εij , are related by

τij = Eij(εij + aij εij) i, j = 1, 2, . . . n and i 6= j (3)

where Eij is the effective Young’s modulus and aij is a constant dissipation parameter for the tetherconnecting point masses mi and mj . The longitudinal strain in the tether is defined as

εij =lij − dij

dij(4)

4

where lij and dij are the actual and the unstrained tether length, respectively. Hence, if Aij is theeffective cross sectional area of the tether then the tension is given by

fij =

Aijτij if εij ≥ 0 and τij ≥ 00 otherwise (5)

which introduces discontinuities since the tethers do not support compression. Defining the vectorof generalized coordinate vectors for the system as

q =(qT

1 , qT2 , . . . , qT

n−1, qTn

)T(6)

equations (1) can be combined to yield

Mq = Q(e) + N + S + U (7)

where

M =

M1 03 . . . 03

03 M2 . . . 03

......

. . ....

03 03 . . . Mn

, Q(e) =

Q(e)1

Q(e)2...

Q(e)n

, N =

N1

N2

...Nn

, S =

S1

S2

...Sn

, U =

U1

U2

...Un

(8)

where 03 is the 3× 3 null matrix. Notice that equations (7) are readily integrated after beingtransformed to a set of 2× 3n first order ODEs; that is, defining x = (q, q)T , equations (7) can bewritten in compact form as

x = f(x) , where f(x) =(

qM−1(Q(e) + N + S + U)

)(9)

For the subsequent analysis we consider two different TSS configurations. The triangular systemcomprises n = 3 point masses and m = 3 tethers whereas the TetraStar system consists of n = 6point masses and m = 9 tethers. In any case, we are only interested in symmetrical TSS. That is,for the triangular system all the point masses are equal (mi = m). The same applies to all of thetether parameters. For TetraStar all point masses and tethers belonging to the “outer” system havesimilar characteristics and so do the point masses and tethers of the “inner” system. For the sakeof convenience we introduce the following notation: subscripts and • are used to denote physicalquantities such as states and parameters corresponding to the inner and outer system, respectively.For example for TetraStar:

mi =

m for i = 1, 2, 3m• for i = 4, 5, 6 (10)

Figure 3 shows a schematic layout of the labelling of the tethers and point masses for the two satelliteconfigurations. The “inner” subsystems are marked by a hatched border.

In the following section we identify relative equilibria for the triangular system and for TetraStar.

RELATIVE EQUILIBRIA

Reviewing Gates’ system description we note that the spring forces appearing in equations (1) canbe written as

Si =n=3∑

j 6=ij=1

fij

lij

σij

τij

ζij

(11)

5

m1 = m

m2 = m

m3 = m

m1 = m

m2 = m

m4 = m•

m5 = m•

m6 = m•

Figure 3: Labelling of different system configurations.

where we have introduced the quantities

σij = rj cos(θj − θi)− ri

τij = rirj sin(θj − θi)ζij = zj − zi

(12)

Additionally, in the rotating reference frame FP the relative equilibrium motion satisfies

xe = f (xe) = 0 (13)

where the superscript denotes the relative equilibrium state. Assuming perfectly symmetrical TSSconfigurations in the case of the triangular system it is sufficient to analyze either of the subsystemsdescribed by equations (1). For TetraStar two subsystems have to be solved simultaneously, one ofwhich corresponds to an inner point mass. The second subsystem corresponds to one of the twocorresponding outer point masses.

Relative equilibria for a triangular system

Solving equation (13) for xe we find three conditions for each of the three point masses of the form

re = const. (14)

θe (re

) =

√√3Æ(

√3re − d)

mre d,

(note: le =

√3re)

(15)

ze = const. (16)

where we have introduced the variable Æ = AE. Therefore, the relative equilibrium angularvelocity for the TSS is

θe (le ) =

√3Æ(le − d)

mle d(17)

which shows that the rotating TSS has to be under tension (le > d).

Relative equilibria for TetraStar

In the case of TetraStar conditions for a relative equilibrium are obtained effortlessly using d’Alem-bert’s principle or the principle of virtual work.

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These conditions are

re = const. (18)re• = const. (19)

m(θe)2re =

√3Æ(le − de

)de

− (re• − 2re

)Æ•(le• − d•)

le• d•, where le• =

√re 2 + re• 2 − re re• (20)

m•(θe)2re• = (2re

• − re )

Æ•(le• − d•)le• d•

(21)

ze = ze

• = const. (22)

Note that for the symmetrical TetraStar system θe = θe

• ≡ θe. Equation (21) can be rewritten toyield a quartic equation in re

• of the following form

(re• )4 + (re

• )3re

(2β

α− 1

)+ (re

• )2

re

2

(β

α− 1

)2

−(

2γ

α

)2

+

re•

(re )3

β

α

(2− β

α

)+ re

(2γ

α

)2

+ (re )2

(re )2

(β

α

)2

−(γ

α

)2

= 0

(23)

whereα = m•(θe)2d• − 2Æ• , β = Æ• , and γ = Æ•d• (24)

For known tether parameters and a given fixed length of the outer tethers d• equation (23) can besolved for re

• in terms of re using a Newton–Raphson method. Equations (20) and (21) can then be

used to solve for the only remaining unknown – the unstretched tether length of the inner tethersde – which yields

de = re

3Æ

(θe)2(

mre + m•re•re• − 2re

2re• − re

)+√

3Æ

(25)

Note that in the limit as Æ →∞ (extremely stiff tethers, rigid TSS) de → le =

√3re , as expected.

Figures 4 and 5 show uncontrolled initial conditions perturbation responses for the triangularsystem and for TetraStar. Unperturbed initial conditions result in relative equilibrium motion.Additionally, in Figure 5 we show the effect of tether damping for aij = 0.5 s. Comparing the twosystem responses we point out that coordinate errors measured relative to the unperturbed systemmotions are notably greater for TetraStar, especially in angular direction. The system responsesignals can be thought of as a superposition of three signal components which are best explainedwith the upper two plots of Figure 5. The high–frequency oscillations reflect tether vibrations whichcan be significantly reduced by increasing the dissipation parameters aij . Notice, that the radialerror signal for the counterweight carries an additional high–frequency signal component comparedto the inner point mass which results from increased tether stiffness of the outer tethers (see Table 1).The signal component with a time period of approximately 95 s is due to the rotation of the TSS. Thesignal amplitudes are increasing since the choice of initial conditions results in a non–zero systemlinear momentum. The set of initial conditions perturbations chosen for TetraStar yields a relativelyincreased system angular momentum; hence the TSS rotates notably faster than the unperturbedsatellite formation.

MISSION SCENARIOS FOR SPECS

This section briefly discusses two mission scenarios to validate the control law developed in thefollowing section. For a detailed analysis we refer to Ref. 11. The first mission scenario considers the

7

-0.10

-0.05

0

0.05

0.10Coordinate error e

r1 = r

1opt - r

1

Err

or e

r1 in

m

0 50 100 150 200 250 300 350 400 450 500-0.01

0

0.01

0.02

0.03Coordinate error eθ1

= θ1opt - θ

1

Err

or e

θ1 in

rad

Time t in seconds

Figure 4: Triangle: Uncontrolled initial conditions perturbation response. The output error is mea-sured relative to a relative equilibrium motion.

stabilization of a particular relative equilibrium motion of the TSS. This case is of importance forthose phases during the sequence of observations when the satellite formation needs to be reorientedto point at specific targets of interest. The second mission scenario represents a SPECS relevanttrajectory including a tether deployment and retrieval procedure which allows the system to denselycover the observational plane in an optimal fashion. For this particular scenario an additionalscientifically motivated constraint is formulated which requires that the instantaneous tangentialvelocity of the controlled point masses never exceeds a maximum value.

In Ref. 11 we commented on the importance of implementing “smooth” mission trajectories tominimize the required control input and we introduced smoothing functions for those critical periodsof time when the TSS in a transitional phase between a relative equilibrium motion and a tetherdeployment or retrieval process.

Sizing of outer point masses for TetraStar

Up to this point the outer point masses have not entered the analysis. As mentioned before, theirpurpose is to provide additional control authority in radial direction and serve as a buffer for thesystem angular momentum during tether deployment and retrieval. To use the interferometer ef-ficiently we require the instantaneous tangential velocity of the point masses to be equal to themaximum allowable velocity. However, we also require the outer tethers to be inextensible whichleaves the outer point masses as the only free parameters to work with in order to obtain increasedcontrol performance compared to the triangular TSS. Note, that these constraints do not allow anoptimal solution to the problem of deploying and retrieving the TSS such that the system angularmomentum h(t) = const. However, we can size the outer point masses such that h(ti) = h(tf ), whereti and tf are the initial and final times for tether deployment or retrieval. It is straightforward toshow that,

h(ti) = h(tf ) ⇐⇒ m(ri − rf

)

+ m•

((ri•)

2

ri− (rf

• )2

rf

)= 0 (26)

Note that equation (26) contains three unknowns, namely m•, ri• , and rf

• . Evaluating equation (23)at t = ti and t = tf adds two more equations and the resulting system of equations can be solved

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-0.10

-0.05

0

0.05

0.10Coordinate error e

r1 = r

1opt - r

1

Err

or e

r1 in

m

-0.01

0

0.01

0.02

0.03Coordinate error eθ1

= θ1opt - θ

1

Err

or e

θ1 in

rad

noisy data: aij = 0.0 s

smooth data: aij = 0.5 s

-0.10

-0.05

0

0.05

0.10Coordinate error e

r6 = r

6opt - r

4

Err

or e

r6 in

m

0 50 100 150 200 250 300 350 400 450 500-0.01

0

0.01

0.02

0.03Coordinate error eθ6

= θ6opt - θ

6

Err

or e

θ6 in

rad

Time t in seconds

noisy data: aij = 0.0 s

smooth data: aij = 0.5 s

Figure 5: TetraStar: Uncontrolled initial conditions perturbation response. The output error ismeasured relative to a relative equilibrium motion. Damping (aij > 0) reduces high–frequencyoscillations.

using a Newton–Raphson method. For the system parameters listed in Table 1 and an unstretchedand constant tether length of the outer tethers of d• = 65.0 m we obtain an outer point mass ofm• = 5.64 kg.

Applying the outlined methodology to a realistic 1 km baselength TSS with inner system pointmasses of m = 500 kg, and an unstretched outer tether length of d• = 1100 m results counterweightsof m• = 3.56 kg for a full aperture scan.

In the next section the tools necessary to develop a nonlinear adaptive output feedback controllerare introduced.

CONTROL LAW DEVELOPMENT

A drawback of the observer–based control approach as used by the authors in an earlier paper is thatan accurate physical model of the dynamical system is required to guarantee asymptotic stability.Although the mathematical structure of the system is usually well defined, the physical parametersare often not precisely known. Additionally, the cost of implementing a controller based on input–

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state feedback typically includes the cost of motion sensors which in turn results in elevated sensorcount. Adaptive output–feedback controllers utilizing a velocity–generated filter and an estimatorfor system parameters offer an elegant solution to the problem.

For this analysis the positions of the inner point masses mi are chosen as outputs; that is, yi ≡qi = (ri, θi, zi)T , i = 1, 2, 3. Also, due to the particular choice of control inputs in equations (1) onecan show that the triangular system is feedback linearizable and therefore completely controllable.11

TetraStar, on the other hand is clearly uncontrollable due to a lack of control authority (outersystem). However, we will show that for the mission scenarios described in the previous section thecontrol of the inner system is sufficient. With these results we derive an adaptive output–feedbackcontrol law based on the work developed in Ref. 18.

Problem statement

Motivated by the subsequent analysis we rewrite the motion equations (7) in the following form

Ω(q, q, q)Ψ , M(q)q + Vm(q, q)q + G(q) + Fdq = U (27)

to collect and reorder corresponding terms in q, q, and q. We have used the fact that the dynamicsequations are linearly parameterizable to define the product Ω(q, q, q)Ψ where Ω(q, q, q) is thesystem regression matrix, which contains known functions of q, q, and q and where Ψ is the constantsystem parameter vector. In equations (27) Vm(q, q) represents the centripetal–Coriolis matrix,G(q) denotes gravity effects, and Fd is the viscous friction coefficient matrix.

The primary control objective is to design the control input U such that the tracking error vectore = yd − y → 0 as t → ∞, while the state remains bounded and with the constraint that onlyposition measurements are available. Note that for both TSS under consideration only the innerpoint masses are controlled, thus we define

e = (e1, e2,e3)T , where ei = ydi − yi , i = 1, 2, 3 (28)

Filter and controller design

To aid the control design and analysis of this section, we define the vector function tanh ξ ,(tanh ξ1, . . . , tanh ξn)T and the matrix function Cosh(ξ) , diagcosh ξ1, . . . , cosh ξn for ξ ∈ Rn.The filtered tracking error–like variable η is then defined as

η = e + tanh e + tanh ef (29)

where we introduce the filter output ef which is defined to have the following dynamics

ef = − tanh ef + tanh e− k Cosh2(ef )η , with ef (0) = 0 (30)

and where k is a positive control gain. To obtain the open–loop dynamics in terms of the filteredtracking error–like variable we differentiate equations (29) with respect to time and premultiply theresulting expression by M(q) to obtain

M(q)η = M(q)qd −M(q)q + M(q)[Cosh2(e)

]−1 (η − tanh e− tanh ef

)

+ M(q)[Cosh2(ef )

]−1 (tanh e− tanh ef − k Cosh2(ef )η

) (31)

10

where we have used equations (29) and (30). Next we use equation (27) to substitute for the term−M(q)q which contains time derivatives of the unmeasurable velocity vector. The centripetal–Coriolis term can be expanded into

Vm(q, q)q = Vm(q, q)(qd + tanh e + tanh ef − e− tanh e− tanh ef

)(32)

= − Vm(q, q)(e + tanh e + tanh ef

)(33)

+ Vm(q, qd + tanh e + tanh ef )(q − qd) (34)+ Vm(q, qd)

(qd + tanh e + tanh ef

)(35)

= − Vm(q, q)η + Υ(q, qd, η, tanh e, tanh ef

)+ Vm(q, qd)qd (36)

where we have used the symmetry property Vm(q, ξ)ζ = Vm(q, ζ)ξ,23 repeatedly. The open–loopdynamics (31) then yields

M(q)η = − Vm(q, q)η − kM(q)η + ΩdΨ + χ + Y −U (37)

where χ(q, qd,η, tanh e, tanh ef

)and Y

(q, qd, qd, qd

)are introduced as

χ = M(q)[

Cosh2(e)]−1 (

η − tanh e− tanh ef

)+

[Cosh2(ef )

]−1 (tanh e− tanh ef

)

+Υ(q, qd, η, tanh e, tanh ef

) (38)

andY = M(q)qd + Vm(q, qd)qd + G(q) + Fdq −ΩdΨ (39)

In equations (37,39) we have introduced the desired regression matrix Ωd which is obtained from Ωby exchanging q(i) ←→ q

(i)d , i = 0, 1, 2. Note that the desired regression matrix plays the role of the

desired trajectory when using input–state feedback linearization, for instance. For the open–loopdynamics described by equation (37) it is proven in Ref. 18 that the following control input yieldsasymptotic stability:

U = ΩdΨ− k Cosh2(ef ) tanh ef + tanh e (40)

where the parameter estimate vector Ψ is generated according to a gradient update law

˙Ψ = ΓΩTd η, with Γ = ΓT > 0 (41)

Note that the control input in equations (40) requires the computation of ef (or rather η) andtherefore e. To obtain a control law with only position measurements define υ = υi , tanh ef andnotice that

υi =efi

cosh2(efi)=

(υ2

i − 1)(υi − tanh ei)− k(ei + tanh ei + υi) , with υi(0) = 0 (42)

which can be rewritten as follows

φi =(φi − kei)2 − 1

(φi − kei − tanh ei)− k(tanh ei + φi − kei) , υi = φi − kei (43)

where φi(0) = kei(0). With the help of equation (43) υ and therefore tanh ef can be calculatedwith position measurements only. Using filter (43) the parameter update law can be integrated withrespect to time to yield

Ψ = ΓΩTd e + Γ

∫ t

0

ΩT

d

(tanh e + υ

)− ΩTd e

dτ (44)

11

where we have used definition (29) and integrated by parts to replace the term containing timederivatives of the error vector, namely e. Finally, the control law (40) can be written componentwise

Ui =(ΩdΨ

)i− kυi

1− υ2i

+ tanh ei (45)

Equations (43–45) represent the velocity filter, the parameter estimator, and the control law, re-spectively. Note that the controller (45) has the same basic structure as a conventional controllerbased on input–state feedback linearization. The first term is a feedforward term which containsthe desired system dynamics and for e = e = 0 exactly cancels the right–hand–side of the motionequations. The second and third term regulate the velocity–like and the position error vectors,respectively.

EXAMPLES

We developed and implemented a generic simulation code in C++ to validate the performance of thecontrol laws presented in the previous sections. In particular, the mission scenarios described earlierwere used to define desired trajectories. System and simulation parameters are listed in Table 1.

For the perturbed initial conditions response simulation the perturbation vectors were generatedrandomly with an upper bound of δx ≤ 10−3 for each element of the vectors. The initial and finalradial distance of the point masses for mission scenarios 2 were chosen to be R0 = r0+ = 15m andRf = r0− = 35 m. Note that we chose the outer tether length d• such that the angle between twocorresponding outer tethers is ρ(t) < π/3 (Figure 1) at all times; hence the additional radial forcecomponent due to the outer tethers can assist in the deployment and control process even when thesystem is fully deployed.

Control parameters for all simulations were tuned to yield comparable control performance interms of control effort and output error for the triangular system and TetraStar. In particular, thecontrol parameters were chosen as k = 40.0 and Γ = diag1, 1, 1, 5, . . . , 5, where Γ is of properdimension.

Table 1: System and simulation parameters.

Point masses m in kg 50.0 Damping parameters a• in s 0.0Point masses m• in kg 5.644 Mirror diameters Di in m 3.0Young’s moduli E in N/mm2 10, 000 Unstretched tether length d• in m 65.0Young’s moduli E• in N/mm2 15, 000 Initial radial distance R0 in m 15.0Tether cross sections A in mm2 1.0 Final radial distance Rf in m 35.0Tether cross sections A• in mm2 1.0 Integration time step dt in s 0.01Damping parameters a in s 0.0 IC disturbances δxi,j in [δxi,j ] ≤ 10−3

Figures 6–13 show simulation results for the mission scenarios introduced in a previous section.The initial conditions perturbation responses are illustrated in Figures 6 and 7 for the triangularsystem and in Figure 8 for TetraStar. Comparing Figures 6 and 8 the triangular system convergesconsiderably faster towards “steady–state” conditions than TetraStar. The controllability of theformer system on the one hand and the increased number of parameters to be estimated for the morecomplex TetraStar configuration on the other hand are the crucial factors resulting in an increasein overall required control input. Note that the parameter estimates for mi and Æi, i = 1, 2, 3in Figure 7 reach steady states which are, however, notably different from the actual parameter

12

values. According to Ref. 18 the difference in estimated and actual parameters is not uncommon formechanical systems and lies in the fact that the primary goal of the controller is to track a desiredoutput and not to accurately estimate the unknown parameter vector.

The results for the SPECS relevant mission scenarios are shown in Figures 9–13, Figures 9 and 11illustrate the corresponding trajectories. Similar to the previous simulation results the triangularsystems converges rapidly towards an “optimal” control state lacking high–frequency oscillations.Both control input and signal error are comparable in magnitude for both satellite formations,however, we observe a significant decrease in required control effort in radial and angular directionin the case of TetraStar. The bumps in the radial control signal shown in Figure 10 have disappearedin Figure 12 due to the additional radial force component of the outer tethers. More importantly,the control effort in angular direction is significantly reduced for TetraStar as shown in Figure 12.The corresponding (negative) control effort for the triangular system is shown here for comparison(dashed line). For this particular simulation we obtain a decrease in overall required control effort forTetraStar of about 75%, about 68% in angular direction. We point out, however, that the particularstrategy of choosing the outer point masses following the approach developed in a previous sectionmight not yield an optimal solution to the problem. Especially for large final baselengths witha ratio of rf

/ri ≈ 40.0 TetraStar shows significant benefits when the outer tethers are allowed

to be extensible, as preliminary simulation results show. Figure 13 shows the parameter estimatehistory for the SPECS relevant mission scenario for TetraStar. Other than for mission scenario 1the parameters are not approaching final constant values but show a cyclic symmetry owing to thesymmetry of the TetraStar configuration. The time–dependence of the parameter vector comes tono surprise since for the SPECS relevant mission scenario we introduced explicit time dependencein the motion equations during tether deployment and retrieval.

SUMMARY AND CONCLUSIONS

Techniques to control the motion of TSS comprised of n point masses and interconnected by midealized tethers are presented. Specifically, the control problems of a triangular TSS and a satelliteconfiguration called TetraStar are discussed. Mission scenarios for NASA’s SPECS mission areintroduced and asymptotic tracking laws based on output–feedback control are developed. Thesimulation results show that triangular system is superior in terms of parameter estimation comparedto the more complex TetraStar formation which is mainly due to the fact that we require the outerspacecraft of TetraStar to be uncontrolled which renders the satellite formation uncontrollable andunstabalizable in general. From a controls point of view TetraStar shows significant benefits. Thecontrol effort in radial direction for TetraStar becomes negligible, in angular direction we obtain asignificant decrease in overall control impulse by requiring that the system angular momentum at t =ti and t = tf are equal. Even though, this approach proofs to be a reasonable technique of choosingthe mass of the outer spacecraft we come to the conclusion that the a better way of dramaticallydecreasing the overall control effort is to allow the outer tethers to be extensible. As a final notice,we point out that the lack of control authority for the TetraStar configuration with uncontrollablecounterweights turns out to be immaterial for in–plane maneuvers. The uncontrollability of thesystem greatly effects, however, plane changes such as re–targeting maneuvers, which are certainlyinfeasible.

ACKNOWLEDGEMENTS

This work was supported by David Quinn of NASA Goddard Space Flight Center and by ArjeNachman of the Air Force Office of Scientific Research.

13

-2

-1

0

1x 10

-3 Coordinate error er1

= r1d - r

1

Err

or e

r1 in

m

-1

0

1x 10

-4 Coordinate error eθ1 = θ

1d - θ

1

Err

or e

θ1 in

rad

-0.2

0

0.2Control effort in r-direction for m

1

Con

trol

forc

e u r1

in N

0 10 20 30 40 50 60-0.5

0

0.5Control effort in θ-direction for m

1

Con

trol

torq

ue u

θ1 in

Ns

Time t in seconds

Figure 6: Triangle: Output error and control vector history for an initial conditions perturbationresponse. The controlled system performs a relative equilibrium motion.

-1

0

1x 10

-4 Parameter estimates for masses m1, m

2, and m

3

Est

imat

es fo

r m

i

m1

m2

m3

0 10 20 30 40 50 60-2

0

2x 10

-4 Parameter estimates for EA1, EA

2, and EA

3

Est

imat

es fo

r E

Ai

Time t in seconds

EA1

EA2

EA3

Figure 7: Triangle: Parameter estimate history for a relative equilibrium motion.

14

-5

0

5x 10

-3 Coordinate error er1

= r1d - r

1

Err

or e

r1 in

m

-5

0

5x 10

-4 Coordinate error eθ1 = θ

1d - θ

1

Err

or e

θ1 in

rad

-1

0

1Control effort in r-direction for m

1

Con

trol

forc

e u r1

in N

0 20 40 60 80 100 120 140 160 180 200-5

0

5Control effort in θ-direction for m

1

Con

trol

torq

ue u

θ1 in

Ns

Time t in seconds

Figure 8: TetraStar: Output error and control vector history for an initial conditions perturbationresponse. The controlled system performs a relative equilibrium motion.

15

0 150 300 450 600 750 90010

15

20

25

30

35

40Desired radial distance r

1(t) for SPECS mission

r 1(t)

in m

Time t in seconds-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

40 Trace of desired trajectory for mass m

1

x(t) in m

y(t)

in m

Figure 9: Triangle: Trajectory of point mass m1 for SPECS relevant mission scenario.

-0.02

0

0.02Coordinate error e

r1 = r

1d - r

1

Err

or e

r1 in

m

-2

0

2x 10

-3 Coordinate error eθ1 = θ

1d - θ

1

Err

or e

θ1 in

rad

-1

0

1Control effort in r-direction for m

1

Con

trol

forc

e u r1

in N

0 100 200 300 400 500 600 700 800 900-5

0

5Control effort in θ-direction for m

1

Con

trol

torq

ue u

θ1 in

Ns

Time t in seconds

Figure 10: Triangle: Output error and control vector history for SPECS relevant mission scenario.

16

0 150 300 450 600 750 90010

20

30

40

50

60

70

80Radial distances r

1 and r

6

r 1(t)

and

r 6(t)

in m

Time t in seconds-80 -60 -40 -20 0 20 40 60 80

-80

-60

-40

-20

0

20

40

60

80 Trace of trajectory for masses m

1 and m

6

x(t) in m

y(t)

in m

Figure 11: TetraStar: Trajectories of point masses m1 and m6 for SPECS relevant mission scenario.

0 100 200 300 400 500 600 700 800 900-5

0

5Control effort in θ-direction for m

1

Con

trol

torq

ue u

θ1 in

Ns

Time t in seconds

-5

0

5x 10

-3 Coordinate error er1

= r1d - r

1

Err

or e

r1 in

m

-1

0

1x 10

-3 Coordinate error eθ1 = θ

1d - θ

1

Err

or e

θ1 in

rad

-1

0

1Control effort in r-direction for m

1

Con

trol

forc

e u r1

in N

Figure 12: TetraStar: Output error and control vector history for SPECS relevant mission scenario.Angular control effort for the triangular system (dashed line) and savings for TetraStar (dotted line)are shown for comparison.

17

-5

0

5x 10

-4 Parameter estimates for m1, m

2, and m

3

Est

imat

es fo

r m

i

-5

0

5x 10

-4 Parameter estimates for EA1, EA

4, and EA

6

Est

imat

es fo

r E

Ai

EA1

EA4

EA6

-5

0

5x 10

-4 Parameter estimates for EA5, EA

7, and EA

9

Est

imat

es fo

r E

Ai

EA5

EA7

EA9

0 100 200 300 400 500 600 700 800 900-5

0

5x 10

-4 Parameter estimates for EA2, EA

3, and EA

8

Est

imat

es fo

r E

Ai

Time t in seconds

EA2

EA3

EA8

Figure 13: TetraStar: Parameter estimate history for SPECS relevant mission scenario.

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