Optimal Current Switching Control of ElectrodynamicTethers for Fast Deorbit

R. Zhong∗

Beijing University of Aeronautics and Astronautics, 100191 Beijing, People’s Republic of China

and

Z. H. Zhu†

York University, Toronto, Ontario M3J 1P3, Canada

DOI: 10.2514/1.G000385

Electrodynamic tethers have the ability to remove space debris from Earth’s orbit without the use of propellant.

Unfortunately, the periodic variations of electrodynamic forcewill lead to the tumbling of tethers, and attitude control

is needed to achieve a successful deorbit of spacedebris at the cost of deorbit efficiency.This paperdevelops anoptimal

current switching control scheme to enable a fast and stable spacecraft deorbit simultaneously by electrodynamic

tethers. In addition, the computational effort of the proposed optimal control is significantly reduced by using a

piecewise treatment that discretizes the deorbit process into consecutive time intervals. Within each interval, the

systemdynamicmodel is simplified based on different timescales of state variables and the current on–off switching is

optimizedby solvinga constrainedminimization problemof a control index representing the deorbit efficiency.Direct

Hermite–Simpson discretization is adopted to convert the optimal control problem into a standard nonlinear

programming problem. The validity and efficiency of the proposed control strategy is shown by numerical simu-

lations. The deorbit rate increases significantly with the proposed optimal current switching control compared with

the existing simple current on–off switch, whereas the computational efforts are reduced.

Nomenclature

Aα = maximum allowed libration amplitudein pitch direction, rad

Aβ = maximum allowed libration amplitudein roll direction, rad

a = semimajor axis, mB = local geomagnetic field strength vector, TEm = projection of motional electric field along

tether length, V · m−1

e = orbital eccentricityeoxoy;oz = unit vectors along the x y; z axis

of orbital coordinate systemexy = orbital elements used to replace e and ω to

avoid singularity in near circular orbitFe = electrodynamic force vector, N

Ia = average current across electrodynamic tether, A

Is = current across electrodynamic tether at thelocation with coordinate s

Isc = short-circuit current across electrodynamictether, A

i = orbital inclination angle, radKp = penalty parameter

ksc = efficiency parameter of current

L = length of electrodynamic tether, ml = unit vector along tether lengthM = mean anomaly, radmt = mass of tether, kgm12 = mass of main satellite (subsatellite), kgp = semilatus rectum of an orbit, m

Qα, Qβ = perturbative torques in pitch and rolldirections, respectively

R = resistance of electrodynamic tether

r = geocentric radius of mass center ofelectrodynamic tether, m

s = local coordinate along electrodynamic tether, m

u = sum of true anomaly and argument of perigee, rad

V = orbital velocity of electrodynamic tether, m · s−1

Vx = orbital velocity of electrodynamic tetherin eox direction, m · s−1

v = orbital velocity of electrodynamic tether, m · s−1

vr = orbital velocity with respect to environmentplasma, m · s−1

x = state vector of tether libration in state spacexini = initial value of state vector

xmax,xmin = upper and lower limits of state vector, respectively

α = in-plane libration angle of electrodynamictether, rad

αamp = libration amplitude of roll motion, rad

α

B = right ascension angle of geomagnetic dipoleaxis, rad

αmax, αmin = upper and lower limits of libration amplitudeof roll motion, respectively, rad

β = out-of-plane libration angle of electrodynamictether, rad

βamp = libration amplitude of pitch motion, rad

β

B = tilted angle of geomagnetic dipole axis, rad

βmax,βmin = upper and lower limits of libration amplitudeof pitch motion, respectively, rad

ε = current distribution parameterϑj = orbital elementsμ = Earth’s gravitational constant, m3 · s−2

μm = magnetic moment of the Earth’s dipole, T · m3

ν = true anomaly, radρ = linear mass density of electrodynamic tetherσxy;z = components of perturbative accelerations along

xy; z axis in orbital coordinate system, m · s−2

τf, τ0 = dimensionless final and initial time of a timeinterval

υ = control input

Received 15 November 2013; revision received 22 March 2014; acceptedfor publication 22 April 2014; published online 15 July 2014. Copyright ©2014 by the American Institute of Aeronautics and Astronautics, Inc. Allrights reserved. Copies of this paper may be made for personal or internal use,on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 1533-3884/14 and $10.00 in correspondence with the CCC.

*Lecturer, School of Astronautics, 37 XueYuan Road, Haidian.†Professor, Department of Earth and Space Science and Engineering, 4700

Keele Street; gzhu@yorku.ca. Associate Fellow AIAA (CorrespondingAuthor).

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JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS

Vol. 37, No. 5, September–October 2014

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υmax = upper limit of control input, turned on currentφ = sum of the mean anomaly and argument of

perigee, radΩ = longitude of ascending node of orbit, radΩG = longitude of prime median with respect to

vernal equinox direction, radω = argument of perigee, rad

I. Introduction

E LECTRODYNAMIC tethers (EDTs) have received increasingattention since they were proposed [1], due to their ability in

power generation and orbital maneuvering without the need forpropellant. Over the past four decades, 12 EDT missions have beentested in space to validate its physical principle [2]. Recently, therehas been renewed interest in the EDT-propulsion technology fromNASA [3], Japan Aerospace Exploration Agency [4], ESA [5], andother leading space agencies due to its potential application in spacedebris mitigation and removal. The continuously growing numbersof nonoperational satellites, spent rockets, and other debris producedobjects, and the relatively concentrated distribution of these objects inorbits [6] makes the space around the Earth extremely crowded,especially in the low Earth orbit (LEO). Compared with other deorbitconcepts or technologies such as the rocket or thruster, the EDT-propulsion technology has the advantages of lowmass, compact size,no propellant, system simplicity, and no requirement for spacecraft tobe operational during the deorbit process [7]. These merits make itmost appealing for the fast-growing micro/nanosatellites [8] becausetheir small mass/size ratio limits the use of the techniques commonlyemployed by large spacecraft to deorbit at the end of their missions.One of the key issues in the EDT deorbit is the stability of tether

libration because the EDT is intrinsically unstable under periodicexcitations of the electrodynamic force. Extensive studies [9–22]have been conducted to address the issue from different aspects.Although Pelaez and Lorenzini [9] suggested avoiding the tethertumbling by applying control forces on end bodies, the mostcommonly adopted control method is to limit the tether librationenergy through current regulation to control the electrodynamicforce. This is especially suitable for the tethered micro/nanosatellitesbecause it is practically prohibitive to install extra thrusters on endbodies. However, the current in an EDTwill be the only controllablevariable if the tether length is not allowed to change, making the EDTsystem underactuated and hard to control. Many effective controlmethods have been proposed to stabilize the underactuated system.For instance, the energy-based error correction schemes could leadthe tether libration to a periodical motion [13] or the librationamplitude could be reduced by stopping the energy pumping fromelectromagnetic induction [15]. A detailed summary of representa-tive studies can be found in the present authors’ recent work [22].Although the EDT libration control is well studied, less attention

has been devoted to the orbital transfer efficiency of the EDTwhilecontrolling libration stability, which is the most important index forspace debris removal by the EDT-propulsion technology. Stevens andBaker [10] and Williams [11] considered the orbital transfer timeoptimization problem using grossly oversimplified environmentalperturbationmodels (e.g., dipole geomagnetic field and the neglect ofother environmental perturbation forces except the electrodynamicforce). Furthermore, neither of them considered the in-plane and out-of-plane tether libration dynamics simultaneously. Thus, the open-loop approaches adopted in their optimal control schemes werevulnerable to the accumulated errors resulting from the simplifiedEDT dynamics and environmental perturbation model in the longterm. Consequently, they are limited to the applications of short EDTdeorbit processes. In terms of current control, there are two kinds ofcurrent regulation approaches in the literature, where one is the on–off current switch and the other is the continuous time-varyingcurrent regulation. Both of them have advantages and disadvantagesin application. The on–off switch is usually derived based on a simplejudgment (i.e., cut off the current when the electrodynamic force doespositive work in the libration direction). The control is activated onlyafter the libration angles of energy exceed the allowable range.

Although it is effective in reducing the libration amplitudes back intothe allowable range quickly, the orbital transfer efficiency is notoptimal. The continuous current regulation is usually derived usingthe optimal control theory, which is capable of includine the orbitalinformation in the control scheme. However, additional hardware isrequired to implement such control schemes, such as an ammeter tomonitor the current and a varistor to follow the current profile. Thisis not favorable for nanosatellites that deorbit by EDTs becausegenerating a continuous time-varying current is far more demandingthan the simple on–off switch in spacemissions; the former approachincreases the system complexity, risk, and cost significantly.The present work proposes an optimal current control scheme by

integrating the easily implementable simple current on–off switchwith the optimal control method to achieve a fast EDT deorbit withbounded librationmotion simultaneously. First, the deorbit process isoptimized for the fastest descending time. To improve the com-putational efficiency of the proposed optimal control for real-timeimplementation, the whole deorbit process is discretized into a seriesof time intervals to decouple the orbital and libration dynamics ofEDT based on their different timescales. Within each interval, theslowly varying orbital elements are assumed constant and only thefast-varying tether librationmotion is considered. Second, the currenton–off switch control within the interval is optimized for fast deorbitand libration stability by minimizing a composite control indexrepresenting the deorbit efficiency subject to system constraints. Thecurrent is restrained to the discretized on and off values by penaltyterms in the control index. However, this open-loop optimal currenton–off switch cannot guarantee the libration stability because of thesimplified EDT dynamics and environmental models adopted in thecontrol optimization. Thus, the proposed optimal current controlscheme will change between the closed-loop simple current on–offcontrol and the open-loop optimal current on–off control, where theformer is based on the feedback of libration amplitudes. Finally, thevalidity and effectiveness of the proposed control scheme forEDT deorbit with different orbital parameters are demonstrated bynumerical simulations. A brief diagram of the control scheme isshown in Fig. 1.

II. Orbital and Libration Dynamicsof Electrodynamic Tether

A. Orbital Dynamics

The EDT system considered is a conductive tether that connectstwo end masses. In the orbital dynamic analysis, the system can besimplified as a lumped mass located at the system’s c.m. [14]. Itsorbital motion is generally described in a geocentric inertial frame ofthe Earth oxyz with the origin o at the Earth’s center, as shown inFig. 2. The x axis directs to the point of vernal equinox, the z axisaligns with the Earth’s rotational axis, and the y axis completes aright-hand coordinate system, respectively. The equation of motioncan bewritten in the form ofGaussian perturbation equations, such as

dϑjdtfjϑ1;ϑ2;ϑ3;ϑ4;ϑ5;ϑ6;σx;σy;σz; j1;2 : : : ;6 (1)

The orbital elements ϑj describes the orbital motion of the EDT’sc.m. and can be obtained by integrating Eq. (1) with the perturbativeaccelerations σx, σy, and σz (see Fig. 2). The commonly adopted

orbital elements are a, e, i, ω, Ω, M, or ν. In the present work, theelements e, ω, and M are replaced by ex, ey, and φ to avoid the

singularity when e approaches zero, such that

ex e cos ω ey e sin ω φ M ω (2)

Furthermore, the following major perturbations are considered in theenvironmental perturbative model, namely, 1) the electrodynamicforce exerting on a current-carrying EDT, 2) the Earth’s atmosphericdrag, 3) the Earth’s nonhomogeneity and oblateness, 4) the lunisolargravitational perturbations, and 5) the solar radiation pressure, re-spectively. It is worth noting that the local strength of the geo-magnetic field should be described by the International Geomagnetic

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Reference Field 2000 model with up to seventh-order terms toeffectively reflect the inhomogeneous variation of geomagneticstrength with an accuracy of over 99% [2]. The present authorsshowed the commonly used tiled dipole model is inadequate toaccount for this significant inhomogeneous variation, which iscritical in EDT dynamics. The detailed expressions of the majorperturbations 1–5 can be found in a previous work [21]. Generally,the environmental perturbations 2–5, especially the last two, arerather small comparedwith the electrodynamic force. Therefore, theyare often ignored for simplicity in the study of EDT stability andcontrol in the literature. However, these perturbations will haveimportant effects on the stability of the EDT deorbit process in thelong term if the EDT is relatively short and the available current islimited [2]. More specifically, the atmosphere and geomagnetic andambient plasma fields are assumed to corotate with the Earth at thesame rate in the present work. To be more accurate, the geodeticheight is used instead of the commonly used geocentric height toevaluate height-dependent parameters such as the air density.

B. Libration Dynamics

The EDT system is simplified as a dumbbell model [9,15], wherethe tether is assumed rigid and the endmasses aremodeled as lumpedmasses. The masses of two end masses are assumed to be in the sameor similar order so that the c.m. is located between the end masses.Furthermore, the tether mass is in the same or similar order of the endmasses and not negligible.The EDT libration is described in a local coordinate system with

the origino at the c.m. of the EDT. The z axis of the coordinate systempoints from the Earth’s center to the c.m. The x axis lies in the orbitalplane and is perpendicular to the z axis with its positive directionpointing to the EDT’s orbital motion. The y axis completes a right-hand coordinate system. The instant attitude of the tether is describedby an in-plane angle α(the pitch angle, rotating around the y axis) andis followed by an out-of-plane angle β (the roll angle, rotating aroundthe x 0 axis), as shown in Fig. 3,where coordinate systemsox 0y 0z 0 andox 0 0y 0 0z 0 0 represent the local coordinate system oxyz after in-plane

and out-of-plane rotation, respectively. Thus, the equations of EDTlibration motion can be derived as [14]

α ν − 2 _α _ν_β tan β 3

2

μ

r3sin 2α Qα

meqL2 cos2 β

(3)

β 1

2

_α _ν2 3

μ

r3cos2 α

sin 2β

Qβ

meqL2

(4)

where meq is the system’s equivalent moment of inertia, such that

meqm1m2

1

3m1m2mt

1

12m2t

∕m1m2mt (5)

The perturbative torquesQα andQβ can be derived by the principleof virtual work in terms of the perturbative accelerationsσx, σy, andσz, defined in Eq. (1), such that

Optimal TrajectoryMax energy dissipation

by the Lorentz force

Open-loop Optimal Control

Optimal On-off Current Profile

Simplified Dynamic ModelKeep slow-varying orbital

parameters constant

Full Dynamic

Model

Stop

Terminate and update the states

for the next interval

No

No

Yes

Yes

Yes

Simple

Current

Yes

OptimalOn-offCurrent

SW=1 SW=0

YesNo

No No

Interval Initial States

System Constraints

Libration amplitude too large?

On-off

Libration amplitude small enough?

SW=0

Target height reached?

Interval end reached?

Fig. 1 Flow chart of the hybrid optimal control algorithm in one interval.

x

z

y

C.M.

OEquatorial

Plane

σ z

σ x

σ y

Perigee

r

νωΩ

i

Apogee

a

o EDT

v

Fig. 2 Illustration of system coordinates for EDT’s orbital motion.

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Qα ZL

0

cos α cos βeox − sin α cos βeozs − λ

· σxeox σyeoy σzeozρ ds (6)

Qβ ZL

0

− sin α sin βeox − cos βeoy − cos α sin βeozs − λ

· σxeox σyeoy σzeozρ ds (7)

where λ is the mass distribution parameter defined as

λ m1 0.5mt∕m1 m2 mt (8)

The integrands in Eqs. (6) and (7) are highly nonlinear along thetether and difficult to be integrated analytically. As an alternative,they are evaluated numerically by the finite element approach wherethe tether is discretized into a series of segments and the integrandswithin each segment are assumed constant. The more segments arediscretized, the more accurate the perturbative torques.

III. Optimal Current On–Off Switching Control

A. Piecewise Treatment for Fast Deorbit of Electrodynamic TetherSystem

The deorbit of an EDT system is generally quite long comparedwith other deorbit technologies, such as rockets. Accordingly, theorbital parameters of an EDT system, except the true anomaly,change slowly in one or a few orbits due to the slow descending rate.In contrast, the libration angles vary relatively much faster. Thissuggests a possibility of treating the orbital and libration motions atdifferent timescales [10,11], because treating the instantaneous sloworbital and fast libration motions with the same timescale is not onlycomputationally unnecessary but also vulnerable to accumulatednumerical errors in slow motion over a long deorbit process.Based on the timescale separation concept, the deorbit control of

an EDT system is solved piecewise by dividing the whole processinto consecutive time intervals. Within each interval, the orbital andlibration dynamics of the EDT system is decoupled by assuming theslow-varying orbital parameters to be constant and only the fast-varying libration dynamics is considered so that the order of theproblem is reduced dramatically. Then, an open-loop optimal currenton–off switching control for fast and stable deorbit of theEDT systemwithin the interval is derived, where the libration dynamics is furthersimplified by considering only the electrodynamic force with asimplified dipole geomagnetic field model. The energy conservationat the beginning and end of the time interval τ0; τf requires

−μ

rτ0 1

2V2τ0

Eopt

m1 m2 mt

p3

μ

s −

μ

rτf 1

2V2τf

(9)

where Eopt is the work done by the electrodynamic force associatedwith the optimal trajectory determined and τ t

μ∕p3

pis the

dimensionless time.Because the orbit eccentricity is small during the long-term deorbit

if the target is initially in a circular or near circular orbit, it isreasonable to assume the position of the EDT’s c.m. is approximatelyequal to the semimajor axis, such that r ≈ a. The orbital heightdeterioration Δh, defined as the decadence of the semimajor axishere, can be derived from Eq. (9), approximately

Δh aτ0 − aτf

aτ0 −μ

μ∕aτ0 − 2Eoptμ∕p−0.5m1 m2 mt−1(10)

The time interval should be reduced if Δh is too large, such that theassumption of constant slow-varying orbital parameters cannot besatisfied sufficiently. Otherwise, the time interval should be increasedif Δh is too small to increase the control efficiency.

B. Open-Loop Optimal Current On–Off Switching Control WithinInterval

The EDTdeorbit is achieved by dissipating the orbital energy of anEDT system through thework done by the electrodynamic force. Theopen-loop optimal current on–off switching control for fast EDTdeorbit is defined intuitively by searching a profile for the state-control pair fx; υg over the time interval τ0; τf to minimize the costfunction

J Z

τf

τ0

gx; υ; τ dτ Z

τf

τ0

Fe · v dτ (11)

subject to the constraints

x 0 fx; υ; τ (12)

xτ0 xini (13)

xmin < x < xmax (14)

jυj 0 or jυj υmax (15)

It is noted that the initial value of the state vector xini for theoptimization is known in advance. Constraints in Eq. (15) are addedto ensure that the resultant current profile is in a binary on or offstatus. Moreover, the minimization of the cost function in Eq. (11)will actually maximize the work done by the electrodynamic forcebecause the work is always negative for passive EDTs in LEO [23].In the present work, the control input of the EDToptimal control is

defined as the averaged current Ia along the tether, such as

υ Ia 1

L

ZL0

Is ds (16)

This approach allows us to handle an insulated tether (a uniformcurrent) or a bare tether (nonuniform current) uniformly. The statevariable vector of the EDT system x includes the libration angles andvelocities (α, β, α 0, β 0) and is defined as

x fx1; x2; x3; x4g fα; β; α 0; β 0g (17)

where the prime notation denotes the derivatives with respect to thedimensionless time τ.The nonlinear state constraint (12) is the dynamic equation of EDT

libration motion. For simplicity in the optimization, all perturbationsexcept electrodynamic force are ignored, and a simplified nontilteddipolemodel of geomagnetic field is used in the electrodynamic forcecalculation. Thus, it is easy to obtain the analytical expressions for the

Z

X

Y

O

α

x,eox

z, eoz

c.m.o

s

Earth

z’

x’

β z’’

α

β

m1

m2

Fig. 3 Local coordinate system for tether libration motion.

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perturbative torquesQα andQβ byEqs. (6) and (7). The scalar formofEq. (12) is finally derived as

x 01 x2x 02 2η3e sin ν 2x2 η2x4 tan x3 − 3η3 sin x1 cos x1

sin i tan x32 sin u cos x1 − cos u sin x1 − cos i

× ε − λIaμmμmeq

η3

x 03 x4x 04 −x2 η22 sin x3 cos x3 − 3η3 cos2 x1 sin x3 cos x3

− sin i2 sin u sin x1 cos u cos x1ε − λIaμmμmeq

η3 (18)

The variables u; η and the current distribution parameter ε aredefined as follows:

u νω; η 1e cos ν; ε 1

IaL2

ZL0

sIsds (19)

It is noted that ε 0.5 for the insulated tether with a uniform current.Next, assume only the constraints on the libration amplitude of

pitch and roll angles are considered in Eq. (14), whereas the librationrates are not constrained in this study, such that

jαj ≤ Aα β ≤ Aβ (20)

The magnitude of the turned-on current υmax is determined by thecurrent available in theEDTsystemand shouldbe less than the system’sshort-circuit current. Here, the υmax is estimated by introducing anefficiency parameter ksc to the short-circuit current, such that

υmax kscIsc kscEmLR

(21)

The projection of themotional electric field on the unit vector along theEDT length direction Em can be expressed as

Em −vr × B · l (22)

The precise expression of Em is difficult to derive. By assuming thetether is aligned with the geocentric direction, the relative orbitalvelocity of the EDT is equal to the orbit velocity (vr ≈ v), and thesimplified nontilted dipole geomagnetic field [18] is used, the ex-pression of Em can be derived from Eq. (22) approximately as

Emμmr3sin i sin β

B sinΩG α

B−Ω−cos i cos β

BVx (23)

The dipole direction angles α

B and β

B vary slowly with time and theyare assumed constant here. Substituting Eq. (23) into Eq. (21) yields

υmaxkscvxμmr3L

Rjsin i sin β

B sinΩG α

B−Ω−cos icos β

Bj (24)

It shows that the turned-on current is the function of both orbital altitudeandposition generally. In the special caseof equatorial orbit, υmax variesonly with respect to the altitude.Finally, it is noted that the discrete constraints in Eq. (15) are not

convenient for mathematic optimization. Therefore, the constraintsare replaced by adding a continuous penalty function to the costfunction in Eq. (11), such that

J Zτfτ0

gx; υ; τ Kpυ − υlow2υ − υtop2 dτ ≜Zτfτ0

gx; υ; τ dτ

(25)

where

Kp > 0;

υlow 0; υtop υmax if Em > 0

υlow −υmax; υtop 0 if Em < 0(26)

With the sufficiently large penalty parameterKp, the minimization ofthe cost function in Eq. (25) will be equivalent to minimize the costfunction in Eq. (11), whereas the constraints in Eq. (15) will besatisfied approximately.The optimal control problem formulated in Eqs. (11–15), (25), and

(26) is solved by the Hermite–Simpson scheme [24,25], whichdiscretizes the continuous optimization problem into a discrete var-iable optimization problem of nonlinear programing. A commonlyused numerical software package, the sparse sequential quadraticprogramming software SNOPT [26] is used to solve the precedingnonlinear programming problem, resulting in discretized optimaltrajectories of state vector and control input in a piecewisemanner. Theresultant optimal control input υoptimalt is not exactly the on–offcurrents due to the approximate treatment in Eq. (25), but very close tothe on–off switch values.

C. Closed-Loop Simple Current On–Off Switch Based on LibrationAmplitudes Monitoring

Direct application of the optimal current on–off trajectoryυoptimalt as an open-loop control does not guarantee the preventionof the EDT system from tumbling. This is due to the approximationson the EDT dynamics and the environment models adopted inSec. III.B. Thus, the optimal on–off current switching control shouldbe used in concert with the existing simple current on–off switchfeedback control [13,21] that monitors the libration energy or anglesto prevent the tether from tumbling. The present authors developed asimple current on–off switch criterion based on the libration energyanalysis [21], such that

13p

αamp2 4β2amp < 3 (27)

In the highly inclined orbits (HIO), the roll destabilizes faster than thepitch if no control is applied. Accordingly, Eq. (27) can be simplifiedby replacing the pitch amplitude αamp with the roll amplitude βamp

conservatively,

3p

βamp 3.5β2amp < 1 (28)

Based on Eq. (27) or (28), a new optimal current on–off switchcontrol scheme is proposed. The optimal current on–off switchingcontrol is applied when the instantaneous libration amplitudes ofEDT libration are within the acceptable ranges, that is, αamp ≤ αmax

and βamp ≤ βmax, or βamp ≤ βmax in HIO, where αmax and βmax are the

upper limits for the libration amplitudes. Otherwise, the currentcontrol is changed to the simple current on–off switching control,where the current is turned on only if the electrodynamic force isdoing negative work, until the libration amplitudes are reduced tosufficiently small values, that is, αamp ≤ αmin and βamp ≤ βmin, or

βamp ≤ βmin only in HIO, where αmin and βmin are the lower limits for

the libration amplitudes.The allowable libration amplitudes of the pitch and roll motion Aα

and Aβ in the open-loop optimization procedure can be chosen

according to Eq. (27) [or Eq. (28) in the HIO]. However, themaximum libration amplitudes αmax and βmax for the switch ofcontrol scheme can be more tolerated. This is because the tumbling-avoiding-constraints for libration amplitudes in Eq. (27) or (28) arederived by assuming a free-libration motion [21], whereas theoptimal control profile resulting from the open-loop control alreadytakes the libration stability into consideration, as shown in Eq. (20).

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IV. Numerical Simulation and ControlStrategy Validation

The performance of the proposed optimal current on–off switchingcontrol strategy has been examined and validated by numericalsimulation. A typical nano-EDT system, which was used in ourprevious simple current on–off switching control [21], is adopted inthe numerical simulation, which is proposed for engineering andscientific applications in a mission concept study. The systemparameters are shown in Table 1. Please note that the parameters arebased on the assumption of a bare tether and a Spindt type of field-effect emitter for electron current collection and emission.Fast deorbit by the EDT was simulated in three different cases,

namely, the equatorial, 57-deg-inclined and polar orbits, respectively.Once the orbit inclination exceeds the 57 deg, the motion-inducedvoltage over the EDTwill start to reverse its polarity in one orbit. TheEDT is assumed initially stabilized in the geocentric direction withzero libration angles and velocities. Table 2 gives the orbitalinformation of the EDT and the constraints for the tether librationmotion. The libration amplitudes for the switching control schemefrom the open-loop optimal control to the closed-loop simple on–offswitch are set as 1.5 times the maximum allowable amplitudesdefined by Eq. (27) or (28). The libration amplitudes for switchingcontrol scheme back to the open-loop optimal control from theclosed-loop simple on–off switch are set to 25% of the maximumallowable amplitudes.It should be noted that the electric current efficiency factor ksc is

determined based on the available current of the system. In thesimulation, the factor ksc is set to 0.25, indicating that the turned-oncurrent is assumed as one-quarter of the short-circuit current. Thisvalue is set by trial and error as a compromise between deorbitefficiency and libration stability. The simple on–off current controlcan reduce the libration amplitude effectively but does not ensure thefast deorbit. If ksc is larger than 25%, the libration stability will bereduced because the libration angles would quickly approach themaximum allowable amplitudes under the open-loop optimal on–offcurrent. Otherwise, if ksc is smaller than 25%, the time period for thesimple on–off switch control needs to increase and the deorbitefficiency is reduced. The penalty parameter Kp is set to 10,considering that a too large Kp is not favorable for minimizing thework done by the electrodynamic force, whereas a too small Kp willresult in large errors. The initial interval length for the piecewiseoptimal control is set to two orbits. The numerical simulations wereperformed by MATLAB/Simulink with a time step of 10 s, which is

chosen as a balance between accuracy and speed of the numericalcomputation.

A. Fast Deorbit Under Optimal Control in the Equatorial Orbit

First, Fig. 4 shows the current on–off profiles in different stages ofthe control strategy. The peak or turned-on value of the currentincreases steadily as the orbital height decreases due to the increase ofgeomagnetic field strength and plasma density. Figure 4b gives theoptimal current on–off trajectory solvedby the nonlinear programmingprocess. It shows that the current values are not exactly the turned-onand turned-off (zero) values of the current, because the discrete on–offconstraint is replaced by the continuous penalty function in the costfunction. However, the difference is not significant compared with theadjusted optimal current profile, as shown in Fig. 4c, where themagnitudeof the turned-on current is adjusted to as perEq. (21) and themagnitudeof the turned-off current is adjusted to zero. Figure 4a showsthe final current control input to the EDT system, which combines theoptimal and simple current on–off switching control. The closed-loopcurrent control turns on and off more frequently than the open-loopoptimal current profile shown in Fig. 4c due to the requirement ofreducing the libration amplitudes immediately when they exceed theacceptable ranges. The relationship of the current profiles in differentstages can be found more clearly in the detailed comparison shownin Fig. 5.Next, the simulation results of the tether libration during the

deorbit process are shown in Figs. 6–9, including both the optimaltrajectory (dotted line) resulting from the open-loop control optimi-zation and the “real” EDT trajectories (solid line, obtained bydynamic simulation using non-simplified EDT dynamics and spaceenvironmental perturbation models subject to the control input). Thereal trajectories of the libration show that the libration angles are wellbounded. The pitch amplitudes are larger than the roll amplitudes dueto the larger perturbative toques in the pitch direction in the equatorialorbit. Most of the time, the peak pitch amplitudes are about 30 deg,whereas the maximum roll amplitude is less than 20 deg, whichshows the effectiveness of the proposed optimal current switchingcontrol in keeping the libration stable. In spite of the instant states ofreal libration trajectory at the end of the previous interval being usedas the initial conditions for the control optimization in the nextinterval (see the discontinuity in the optimal trajectory as shown inFig. 7), the optimal trajectory obtained from the simplified EDTdynamics and environmental perturbation models is still verydifferent from the real libration trajectory. Thus, the closed-loopcurrent on–off switching control, such as the simple current on–offregulation, is necessary to prevent the tether from tumbling.Finally, the deorbit efficiency of the proposed optimal current

switching control is compared with the existing simple current on–offregulation. The deorbit by the simple current on–off regulation alone isused as a baseline, where all of the parameters such as the maximumallowable libration angles and turned-on current are the same as thosein the optimal control. Figure 10 shows that the EDT deorbit from 800to 500 km under the proposed optimal current switching control takesabout eight days, whereas the simple current on–off switching controltakes over 55 days, a significant improvement in deorbit time byroughly 6 times. Simulations of a nonelectrodynamic tether with thesame parameter and an electrodynamic tether without control are also

Table 1 System parameters of nanosatellitesconnected by a short EDT

Parameters Values

Mass of primary satellite 5.0 kgMass of secondary satellite 1.75 kgMass of tether 0.25 kgDimensions of primary satellite 0.2 × 0.2 × 0.2 mDimensions of secondary satellite 0.1 × 0.17 × 0.1 mTether length 500 mTether diameter 0.0005 mTether conductivity, aluminum 3.4014 × 107 Ω−1 m−1

Table 2 Orbital parameters and libration constraints of the EDT system

Parameters Values

Initial orbital altitude 800 kmTarget orbital altitude 500 kmInitial orbital eccentricity 0Initial orbital inclination 0, 57, 90 degMaximum amplitude for pitch motion αmax (limit for open-loop optimal control) 30 degMaximum amplitude for roll motion βmax (limit for open-loop optimal control) 30 degMinimum amplitude for pitch motion αmin (limit for closed-loop simple control) 5 degMinimum amplitude for roll motion βmin (limit for closed-loop simple control) 5 degMaximum amplitude for pitch motion Aα (open-loop optimization) 20 degMaximum amplitude for roll motion Aβ (Open-loop optimization) 20 deg

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provided. It is found that the orbit height of the non-EDT does notchange much due to its small size and the tether tumbling happens inless than one day for the EDT without control, which validates theimprovement of the control method proposed in this paper for thedeorbit application.

B. Fast Deorbit Under Optimal Control in the Fifty-Seven-DegreeInclined Orbit

The deorbit in the 57-deg-inclined orbit can be achieved in about 24days (longer than the equatorial orbit), due to the smaller currentavailable in the EDT system. Detailed results are given in Figs. 11–17.The variation of the current control profiles is shown in Figs. 11

and 12. It should be noted that the maximum current value is reduced

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compared with the equatorial orbit case. The EDT current in theinclined orbit shows not only an increasing tendency as the orbitalheight decreases but also a periodical variation with the orbital rate.As shown in Fig. 12, the comparison of the control current (solid linerepresenting optimal current on–off control, thick line representingadjusted optimal current on–off control, and dotted line representingcombined optimal and simple current on–off switching control) indifferent stage leads to the similar conclusion as in the equatorialorbit: The penalty function in the performance index effectivelygenerates an optimal control current close to its on–off value withminor differences (see the difference between the solid curve andthick curve). The dotted curve shows that the current on–offswitching control is applied more frequently to avoid the tetherfrom tumbling due to the errors of the open-loop optimal control.However, by comparing with Fig. 7, it is found that the switchingfrequency increased in the optimal control is much less than in theequatorial case.Figures 13 and14give thevariation of the libration angles during the

deorbit process. In this case of HIO, only the roll motion is monitored,as shown in Eq. (28). Although the real libration trajectory is differentfrom the optimal trajectory, the resulted libration angles are bounded.Unlike in the equatorial case, the amplitudes of the roll motion arelarger than the pitch motion, indicating a larger out-of-plane electro-dynamic perturbation in HIO.

The comparison of the deorbit efficiency between the proposedoptimal control and the simple current on–off regulation withoutswitch optimization is shown in Fig. 15. Under the same conditions,the deorbit process under the optimal current on–off switchingcontrol is faster than the simple current on–off regulation. Theadvantage of the proposed control in this inclined orbit case is not asobvious as in the equatorial orbit, but the deorbit time is still reducedby more than 10 days or 35%. Just like in the equatorial orbit, thecontrol method here shows a large advantage compared with the caseof non-EDT and the case of EDTwithout control (which fails due tothe tether tumbling within five days).

C. Fast Deorbit Under Optimal Control in the Polar Orbit

Finally, the simulation results of the EDT deorbit in the polar orbitare given in Figs. 16–20. The EDT deorbit in the polar orbit is leasteffective because the angles between the directions of the geo-magnetic field and the orbit velocity are small and the availablecurrent is limited. The deorbit time is sharply increased. Fig. 16shows that the maximum current generated in this case is about 0.05A and the deorbit time is less than 350 days, which is still acceptable.It is also shown in Fig. 17 that the current (solid line representingoptimal current on–off control, thick line representing adjusted opti-mal current on–off control, and dotted line representing combinedoptimal and simple current on–off switching control) changes thedirection in one orbit, which is different from the equatorial and57-deg-inclined orbits. Moreover, the switch frequency in the finaloptimal current control is almost the same as the open-loop optimalcontrol profile (see the difference between the dashed curve and thickcurve in Fig. 17). It indicates that there is less need to apply the

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Fig. 10 Comparison of EDT deorbit efficiency in the equatorial orbit using different control schemes (zoomed view on right-hand side).

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Fig. 11 Time history of the average current in the 57-deg-inclined orbit.

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closed-loop current on–off switch regulation to avoid the tethertumbling in the polar orbit case. This is because the longer deorbitprocess makes the assumptions in the open-loop control design,where the slow-varying orbit parameters are assumed constant, moreaccurate in comparison with the real situation. Thus, the open-loopoptimal current profile is less likely to induce tether tumbling.

The librations shown in Figs. 18 and 19 validate the effectivenessof the proposed optimal control in keeping libration stability in thepolar orbit. It shows clearly the roll angle reaches the maximumallowed value much more frequently than the pitch angle, and theperturbation in the roll motion is the major source that induces thetether tumbling. This proves again that the roll angle monitoring issufficient to keep the libration stable in HIO.

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Fig. 13 Time history of the pitch motion in the 57-deg-inclined orbitduring the deorbit process.

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Fig. 14 Time history of the roll motion in the 57-deg-inclined orbitduring the deorbit process.

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Fig. 15 Comparison of EDT deorbit efficiency in the 57-deg-inclinedorbit using different control law.

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Fig. 18 Time history of the pitch motion in the polar orbit during thedeorbit process.

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The deorbit efficiency of the proposed optimal control is comparedwith the simple current on–off regulation in Fig. 20. It shows theadvantage of the proposed optimal control in this case is smaller thanin the other two cases. However, the reduction of deorbit timeby roughly 100 days, or 28% improvement, is still encouraging.Moreover, the deorbit efficiency is still much higher than the non-EDT case, although in the polar orbit, which is considered as theworst case for EDT deorbit. The EDT without control will tumblewithin 30 days.

V. Conclusions

This paper investigated the optimal current switching control ofelectrodynamic tethers for fast space debris removal. An optimalcontrol strategy combining the current on–off switching optimizationand simple current on–off switch that is based on maximum librationamplitude monitoring is proposed to determine an optimal currenton–off switching profile that can achieve the fast deorbit and librationstability simultaneously. The numerical simulation shows the pro-posed optimal control is effective. The tether libration angles in bothpitch and roll motion are well bounded, whereas the deorbit rateincreases significantly compared with cases in which only the simplecurrent on–off switch is used without optimization. Besides, theproposed optimal current on–off switching control reduces com-putational effort significantly and has a high potential in onboardimplementation. Finally, it is worth noting that the current methodcan be applied in the orbit boost using EDT in principle if the cost

function is revised to include boosting requirements and cor-responding system constraints.

Acknowledgment

This work is supported by the Discovery Grant and DiscoveryAccelerator Supplements Grant of Natural Sciences and EngineeringResearch Council of Canada and the Fundamental Research Fundsfor the Central Universities (No. YWF-14-YHXY-021).

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0

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deorbit process.

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Orb

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non-EDT

Simple On-Off Control

EDTno-control

Fig. 20 Comparison of EDT deorbit efficiency in the polar orbit usingdifferent control law.

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