Control Strategies for Stable Orbits Around Phobos

Stephen Takacs and Christopher J. Damaren

Abstract— This study compares three linear optimal con-trollers for stationkeeping with respect to reference orbitsaround Phobos. The dynamics of the Mars-Phobos system areconstrained to the synodical plane of the circular restrictedthree-body problem (CRTBP) with Phobos modelled as anellipsoid. The controllers rely on periodic orbits which permitthe dynamics to be expressed as a linear system with periodiccoefficients. A novel method of determining the necessaryconditions for periodic orbits is formulated through nonlinearoptimization techniques, where the function to be minimizedis the vector norm of the difference between the initial andfinal conditions of the orbit. The optimization algorithm is aNelder-Mead simplex and is shown to outperform any gradient-based methods as well as other techniques for determining suchorbits. Two controllers, constant feedback and scheduled, aredeveloped from the algebraic Riccati equation (ARE), whichis solved at specific points on the reference orbits. Thesecontrollers are then compared to the optimal solution whichuses the time-varying Riccati equation. At high orbits, theperiodicity of the linearized system is very small and thecontrollers are nearly identical in performance. Closer orbitsreveal increases in the periodicity of the dynamics, leading to anincrease in performance of the time-varying Riccati equation-based controller over the scheduled and constant feedback gaincases.

I. INTRODUCTION

The incentive to study the Mars-Phobos system can bederived from two important practical outcomes of a missionto Phobos. The first being a lunar soil sample, and the secondto provide useful resources to missions to Mars. It is widelyconsidered that the moon is either a captured asteroid or Marsejecta [1]. In either case, the composition of the moon is ofgreat interest and the results of a soil analysis could providesome insight into the history of Phobos and its primary. Theother practical outcome involves the supply of resources toMars missions. Since the orbit of Phobos is very low and itorbits Mars just over 3 times per Sol, a lunar outpost on thenear side of the moon could provide valuable remote sensinginformation and communication relay from the surface ofMars with a longer operational life expectancy and at a lowerrisk and cost than an orbital device.

Initial investigations into Phobos involved determiningaccurate shape models of the moon. Thomas [2] and Duxbury[3] have produced both numerical and analytical models,both of which are in agreement considering measurementerrors. The dynamics of this system were first approachedby Dobrovolskis and Burns [4] where ejecta trajectories

S. Takacs and C. J. Damaren are with the University of Toronto Institutefor Aerospace Studies, 4925 Dufferin Street, Toronto, ON, M3H 5T6,Canada.takacs@utias.utoronto.cadamaren@utias.utoronto.ca

and Hill’s curves are developed with Phobos modelled as atriaxial ellipsoid. This was followed by work by Wiesel [5]to determine periodic orbits in both the circular and eccentriccase of the motion of Phobos. Stable retrograde orbits existwith periods approaching that of Phobos as the distance fromthe moon increases. Position and velocity sensitivity studieswere performed and concluded that while injection positionsshow little change over long-term projections, injection ve-locities must be tightly controlled. Prieto-Llanos and Gomez-Tierno [6] have simulated stationkeeping scenarios about theL1 libration point of Phobos using a discrete-time modalcontroller. Although the controller is robust in the presenceof model errors in mass, the L1 point of Phobos is naturallyunstable and allows limited observation of the moon.

This paper studies the control of a satellite about stableperiodic orbits in the Mars-Phobos system. A model isdeveloped following the above mentioned works and periodicorbits are determined through previous methods as well asa novel method involving optimization techniques. Severalcontrol strategies are compared and conclusions are drawnfor a Phobos orbiter.

II. EQUATIONS OF MOTION

Phobos’ low orbital inclination and eccentricity allow areasonable approximation of the equations of motion bythe circular restricted three-body problem (CRTBP). Further-more, the developments of this study are constrained to thetwo-dimensional in-plane motion of a body in this system.The equations of motion are formulated with respect to arotating (synodical) frame with angular velocity ω. Sincethe mass ratio for this system is very low, the origin of therotating frame is taken at the center of Mars (Fig. 1). Theframe’s x-axis faces away from Mars, its z-axis points outof the synodical plane, and its y-axis completes the system.Although extensive studies on Phobos’ irregular shape haveproduced some accurate forms of its gravity potential, thesehave mostly concluded that it can be reasonably approx-imated by that of a triaxial ellipsoid of uniform density[2], [7]. Here, MacCullagh’s approximation for the gravitypotential of an arbitrary body with moments of inertia A, B,and C is used. The reader is referred to Battin’s work for athorough derivation of this formula [8]. Table I shows valuesfor the Mars-Phobos system taken in this study. The constantdistance rphob between Phobos and Mars is the distance atperimartem. The equations of motion can be formulated as[9]:

x − 2ωy =∂U

∂x(1)

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

MC6.1

0-7803-9354-6/05/$20.00 ©2005 IEEE 553

r1 r

2

rphobMars

Phobos

SpacecraftDirection of

motion

y

x

Fig. 1. Coordinate System

TABLE I

MARS-PHOBOS SYSTEM PARAMETERS [5]

Gmmars, km3/s2 42,828.44Gmphob, km3/s2 7, 206 × 10−4

a, km 9,378.53856e 0.015

rphob, km 9,237.86048X axis, km 13.4Y axis, km 11.2Z axis, km 9.2

GA, km5/s2 3.027 × 10−2

GB, km5/s2 3.807 × 10−2

GC, km5/s2 4.395 × 10−2

y + 2ωx =∂U

∂y(2)

The pseudopotential U is given by:

U(x, y) =ω2

2(x2 + y2) +

Gmmars

r1+

Gmphob

r2

+Gmphob

2r32

(A + B + C)

− 3Gmphob

2r52

(Ax2 + By2), (3)

where

r1 =√

x2 + y2 (4a)

r2 =√

(x − rphob)2 + y2 (4b)

The terms r1 and r2 refer to the relative distance of thesatellite to Mars and Phobos, respectively. It should be notedthat terms involving motion in the z-direction have beenremoved from (3) and (4). The equations of motion canbe solved using any differential equation numerical solver.MATLABTM ’s ODE45 is an explicit 4th-5th order Runge-Kutta solver and is used in this analysis.

III. PERIODIC ORBITS

Periodic solutions to the CRTBP are of great concern tothe control of a satellite. They not only represent poten-tially stable solutions to the problem, but also provide ameans to apply various linear control methods through thelinearization of a reference trajectory. Periodic orbits in theCRTBP have been extensively studied and work done bySzebehely [10] and Poincare [11] should be referenced forfurther information. A function is T -periodic if [12]:

f(t + kT ) = f(t), ∀ t > 0, k ∈ Z (5)

Since no analytical solution to the equations of motion of theCRTBP exist, periodic orbits must be found using numericalanalysis. Several methods are available and are described in[5], [10] and [13]. However, only two are described here:the equations of variation method and a novel method ofoptimization

A. Equations of Variation Method

Families of stable periodic orbits for the Mars-Phobossystem are reported by Wiesel [5]. His approach involvesassembling the linearized equations of motion into statevector form x = F(x, t) and determining the state transitionmatrix (also called the matrizant [14]) over a specified periodT . The linearized equations take the form:

δx(t) = A(t)δx(t), (6)

where the state vector is

δx(t) =

⎡⎢⎢⎣

δx(t)δy(t)δx(t)δy(t)

⎤⎥⎥⎦ =

⎡⎢⎢⎣

x(t) − xref (t)y(t) − yref (t)x(t) − xref (t)y(t) − yref (t)

⎤⎥⎥⎦ , (7)

and represents deviations from a reference trajectory. Thestate transition matrix is described by [5]:

Φ(t, to) = A(t)Φ(t, to), Φ(to, to) = 1, (8)

where A(t) = ∇xF. Φ(T, to) is called the monodromy matrix[15] and is not only used in the methodology here, but alsoin stability analysis. Following the developments of [5], thespecified orbit is restricted to being symmetric about the x-axis in the rotating frame (i.e. y(0) = x(0) = 0, x(0) �=0, y(0) �= 0). The symmetry of the orbit allows for thecalculation of the variation in the states δx and δy betweent = 0 and t = T (using the monodromy matrix) neededto force δy(T ) and δx(T ) to zero, followed by iteratingthe updated initial conditions (i.e. x(0)i+1 = x(0)i − δxi,y(0)i+1 = y(0)i−δyi). The state transition matrix is numer-ically computed in conjunction with the equations of motionand the algorithm will usually converge to a periodic orbitwhen initial conditions are within 5% of a T -periodic orbit.This method is not only restricted in robustness with respectto initial conditions, but is also numerically ill-conditionedwhen the residual error becomes very small, leading to anunstable solution [16]. The principle design parameter (i.e.the parameter which governs the solution) is the period of theorbit, which is inconvenient for determining the ∆v requiredto stabilize the orbit of a satellite at a specific location.The algorithm is amenable to changes in its principle designparameter such as using the x-axis position instead of theperiod [10]. However, this modification does not eliminatethe issues of robustness and numerical instability. In lightof these drawbacks, a novel method of optimization todetermine periodic orbits is developed in the next section.

B. Novel Method of Optimization

In this section, the use of optimization methods is con-sidered to determine the conditions necessary for periodic

554

orbits. For reasons of comparison, the problem is stated asin section III-A with the initial conditions lying on the x-axis.The objective function is:

J (x(0)) = ∆xT ∆x, (9)

∆x = x(0) − x(τ), (10)

where x(0) is the state vector at to = 0 and x(τ) is thestate vector at some later time τ . It should be noted thatproper scaling of the terms in (9) is required for relativelyequal sensitivity. Minimizing (9) determines the conditionswhereby the initial and final states of the solution are equal,satisfying the conditions in (5) for a periodic orbit. Thefunction J is nonlinear since the relationships between theinitial and final states are the equations of motion which mustbe solved numerically. This necessitates the use of nonlinearoptimization techniques.

The method mentioned in section III-A requires infor-mation regarding the period a priori as well as two initialconditions, x(0) and y(0). To avoid inputting information onthe period here, an event function is introduced in the nu-merical integration of the states. An event is described as thecurrent position crossing the positive x-axis in the negativedirection during integration. The optimization algorithm canbe performed as follows:

1) Input of the desired position on the x-axis for a periodicorbit with initial guess for y(0). x(0) can be fixed at0 for a simpler single-variable optimizer.

2) Integration of the equations of motion until someterminal time or the occurrence of an event. The periodof the synodical frame is suggested as an approximatevalue for the terminal time.

3) Evaluation of the objective function in (9).4) Solution of the optimization subproblem to determine

the search direction.5) Update of the initial conditions.6) Repetition of steps 1 to 5 until a desired tolerance for

J is reached.

Since the previous method involves determining the zeroof a linearized system (i.e. finding the residual to forceδy(T ) and δx(T ) to zero), it is analogous to Newton’smethod of root finding. From an optimization point of view,Newton’s method is perhaps one of the least effective waysof determining the minimum of a nonlinear function withvarying curvature.

Various optimization algorithms have been tested: theBroyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newtonmethod [17], the Davidon-Fletcher-Powell (DFP) method[18], a trust region method using preconditioned conjugategradients (PCG) [19] and a Nelder-Mead Simplex (NMS)method [20]. The latter is a gradient-free optimization al-gorithm, while the rest are all gradient-based. The gradientof the objective function cannot be supplied to the algo-rithm due to its nonlinearity and must be approximated.This greatly inhibits the performance of the gradient-basedalgorithms. Furthermore, the event function introduces dis-continuities in the design space, which also discourages the

use of gradient-based methods. It is important to note thateach of these methods converge to a local minimum whichmay or may not be a periodic orbit. Careful review of theresulting orbits is therefore necessary. The above formulationdescribes a single-variable optimization method. However,the method can easily be expanded to a multivariable case.This implies a fixed position on the x-axis and the iterationof both x and y to determine new local minima representingother families of periodic orbits.

C. Stability

The stability of periodic orbits can be determined throughthe Floquet representation theorem [21]. The state transitionmatrix of (6) can be factored into:

Φ(t, to) = F(t)etM, (11)

where F(t) is a T -periodic matrix and M is constant.The eigenvalues of M are called the Floquet exponents orPoincare exponents and are analogous to the characteristicroots of dynamic systems. If the Floquet exponents lie inthe open left-half plane, then the linearized orbit is asymp-totically stable, which guarantees asymptotic stability in thenonlinear case. Any eigenvalues in the open right-half planeresult in an unstable orbit. Any on the imaginary axis resultin marginal stability for the linear case and nonlinear stabilityis inconclusive.

IV. CONTROL FORMULATION

The stability of periodic orbits described in section III areat best marginally stable in the linear sense. Long term de-viations from reference trajectories and velocity sensitivitiesare reported in [5]. This, coupled with model errors, requirethe use of control in orbital motion. While the nonlinearsystem can prove to be a difficult control problem, thelinearized system in (6) does lend itself to linear periodiccontrol applications through the periodicity of A(t). Byadding control forces, (6) becomes:

δx(t) = A(t)δx(t) + Bu(t),

B =[01

], u(t) =

[u1

u2

], (12)

where the control vector will be determined as

u(t) = K(t)δx(t) (13)

The matrix K is the gain matrix and need not be time-varying. Three control laws will be used: optimal constantgain control, optimal scheduled control and optimal periodiccontrol.

A. Optimal Constant Gain

Although the system in (6) is time-varying and periodicwith period T , a constant feedback control law can still beapplied to acquire asymptotic stability. An optimal controlproblem is solved and the constant gain matrix is then foundby:

K = −R−1BT P, (14)

555

where P is the solution of the algebraic Riccati equation att = 0:

AT (t)P + PA(t) − PBR−1BT P + CT C = 0 (15)

R is the control effort penalty matrix and C is from theoutput equation:

y(t) = Cx(t), C =[1 0

], (16)

where

y(t) =[

y1(t)y2(t)

]=

[x(t) − xref (t)y(t) − yref (t)

](17)

The closed loop system becomes:

x(t) = (A(t) + BK)x(t), (18)

and is also periodic.

B. Optimal Scheduled Control

Although the control law in the previous section is ableto produce asymptotic stability, it does not recognize theperiodicity of the system. Intuitively, a better approach wouldbe to apply a time-varying feedback that closer representsthe periodic nature of the system. A scheduled controllerevaluates the optimal control solution at specific points inthe reference orbit and linearly interpolates the feedbackgain between those solutions. This provides a time-varyingfeedback with period T . By producing a predetermined arrayof scheduled points in the reference trajectory, the ARE in(15) is solved at each point and the interpolated feedbackgain at any time t is:

K(t) =(

t − titi+1 − ti

)K(ti+1) +

(ti+1 − t

ti+1 − ti

)K(ti),

ti ≤ t < ti+1, 0 ≤ t < T, (19)

where ti is the time at the i-th scheduled point. The designis flexible in regard to the number of scheduled points to useand where their respective feedback gains are solve in thereference trajectory.

C. Optimal Periodic Control

The solution to the Periodic Ricatti Equation (PRE) yieldsthe optimal time-varying feedback for a linear periodicsystem. It is solved by reverse-time integration over manyperiods of the orbit (i.e. t : kT → 0, k ∈ Z) until a steady-state solution is obtained. The continuous-time optimal con-trol problem yields the time-varying control vector in (13)which minimizes the function:

H =12

∫ ∞

0

(δxT CT Cδx + uT Ru

)dt (20)

The equations of motion for the reference trajectory areintegrated in conjunction with the PRE [22]:

P(t) = − AT (t)P(t) − P(t)A(t)

+ P(t)BR−1BT P(t) − CT C, (21)

which is integrated backwards from t = Tf with P(Tf ) =0. We are interested in the periodic solution obtained as

0 1 2 3 4 5 6 7 8

x 104

0

1000

2000

3000

4000

5000

6000

T 2T 3T 4T

Time (s)

P11

(τ)

Tf

Fig. 2. Reverse-time periodic Riccati solution

Tf → ∞. Fig. 2 displays a solution of the reverse-timeintegration of an element of the P(t) matrix and sectionsin time denoting the period T of the orbit. The steady-statesolution of P(t) can be represented by a Fourier series overone period of the motion. The periodic gain matrix K(t) isthen found from:

K(t) = −R−1BT P(t) (22)

and describes the feedback over one period.

V. NUMERICAL RESULTS

A. Periodic Orbits

A comparison between the various methods for determin-ing periodic orbits is made by determining the differencebetween their final converged values and the true solution fora periodic orbit. By varying the initial guess in each method,the final convergence of y(0) is compared to the true solutionof orbit A at 20km on the x-axis, shown in Fig. 3. The resultsin Fig. 4 clearly show the NMS algorithm outperforming theother gradient-based methods. It converges to the desired

solution at nearly every test point (near zero values in

A) B)

C)

Fig. 3. Periodic orbits for 20km design space

556

−40 −30 −20 −10 0 10 20 30 4010

−10

10−5

100

105

1010

1015

1020

% Deviation from Initial Conditions for a Periodic Orbit

|| F

inal

Con

verg

ence

− T

rue

Sol

utio

n ||

BFGSDFPPCGNMSEq. of Variation

Fig. 4. Method convergence values

−15 −10 −510

−25

10−20

10−15

10−10

10−5

100

105

1010

1015

Velocity [m/s]

J(x(

0))

AB

C

Fig. 5. 20km orbit design space

Fig. 4). The PCG method is second in performance to theNMS method, followed by the BFGS and DFP methods.The equations of variation method is also displayed andconverges only when initial guesses are within 5% of aperiodic orbit. It should be noted that many values near100 represent solutions to other periodic orbits. Methodsthat converge to these values have not necessarily failedin finding a periodic orbit but simply did not converge tothe one being orbit A. Fig. 5 displays the single-variabledesign space for an orbit at 20 km from the center ofPhobos and the resulting orbits for specific minima. Manydiscontinuities are present and can mostly be attributed tosudden terminations in integration from the event function.The design space continues to increase beyond the range ofvelocities displayed and no other minima are present. Thisis because the satellite begins to acquire too much energy tosuccessfully orbit Phobos and it’s low gravitational influence.Orbit A is the only permissible motion for a satellite aroundPhobos given the restrictions on initial conditions mentionedin section III-A. While orbits B and C are periodic, they

include motion inside the moon and are of no practical use.Other permissible orbits are possible as the search algorithmis extended to the multivariable case to include x and out-of-plane motion.

B. Control

The three controllers mentioned in section IV are simu-lated at various reference orbit altitudes. Insertion deviationsare kept at 5% of the maximum reference trajectory values.The initial conditions of each simulation describe the satelliteon an approach vector to the reference trajectory from ahigher retrograde orbit. Controller performance is based onthe total cost H and the RMS thrust required to maintainorbit over 5 orbital periods. The value of the control penaltymatrix R is chosen so that the maximum thrust during theentire maneuver never exceeds 20 mN. Hall plasma thrusters,for example, could accommodate this design specification fora 100 kg spacecraft.

Fig. 6 shows the transient norm of the control effortat an orbital distance of 20 km for one period of themotion. The control effort is smooth and the time requiredto reach a steady state error of less than 2% of δx(0) isapproximately 2 hours. The area under each function in Fig.6 is the total thrust required for the maneuver. The optimalperiodic controller has the best performance followed by thescheduled controller and finally the constant gain controller.The total cost H is defined in (20) with the limits t = 0 tot = 5T . It is a function of both control effort and statedeviation from a reference trajectory. From Fig. 7, it isapparent that the variation in performance for all controllersis indistinguishable at higher orbits. This result displays animportant trait of the intrinsic periodicity of the system.Closer orbits exhibit more periodicity in their dynamicsdue to an increase in the higher order terms defining thegravity potential of the ellipsoidal nature of Phobos. Asorbits increase in altitude, higher order terms die out andthe system approaches a point mass model. The behaviour

0 5000 10000 150000

0.5

1

1.5

2

2.5

3x 10

−4

Time [s]

||u(t

)||

Constant GainScheduled GainOptimal Gain

Fig. 6. Control effort for 20km orbit

557

15 20 25 30 35 401

2

3

4

5

6

7

8

9

10x 10

9

H

Reference Orbit Altitude from Phobos [km]

Constant GainScheduled GainOptimal Gain

Fig. 7. Cost vs. Orbital Distance

15 20 25 30 35 404

4.5

5

5.5

6

6.5

7

7.5x 10

−5

RM

S T

hrus

t per

uni

t mas

s [N

/kg]

Reference Orbit Altitude from Phobos [km]

Constant GainScheduled GainOptimal Gain

Fig. 8. RMS Thrust vs. Orbital Distance

of the controllers directly reflects this, and Fig. 8 confirms itas well. The RMS thrusts of each controller approach eachother as orbital distance increases. RMS thrust is calculatedby:

RMS Thrust =

√1

5T

∫ 5T

0

uT u dt (23)

Another observation is an optimal distance for control costat approximately 25 km. This minimum is explained by thenatural characteristics of the dynamics in this model. Themaximum change in x exhibits the same behaviour.

VI. CONCLUSIONS AND FUTURE WORK

A. Conclusions

A dynamic model was developed for the Mars-Phobossystem and periodic quasi-stable satellite motions were de-termined using a novel method of optimization. This methoduses a Nelder-Mead simplex and is shown to outperform anygradient-based methods as well as a previous method fordetermining periodic orbits. Three controllers were simulated

for the tracking of a light Phobos orbiter with injection errorsaround a predetermined reference orbit. At larger orbits, theperiodicity of the system is small and asymptotic stabilityis achievable through a simple constant gain controller withminimal loss in performance. Closer orbits require a time-varying periodic feedback due to larger fluctuations in thegravity potential of Phobos. However, at these distances theellipsoid approximation to Phobos is no longer valid andmore accurate forms of the potential are required.

B. Future Work

Future work involves an increase in model accuracy byincluding the inclination and eccentricity of Phobos, theoblateness of Mars and any other common spacecraft dis-turbances. Also, further studies into optimization techniquesfor determining stable orbits is necessary. In particular,developing algorithms that include the three dimensionalmotion of a spacecraft would be quite useful.

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