Sloshing-Aware Attitude Control of Impulsively Actuated ...

Sloshing-aware attitude control of impulsively actuated spacecraft

Pantelis Sopasakisa, Daniele Bernardinia,c, Hans Strauchb, Samir Bennanid and Alberto Bemporada,c

Abstract— In this tutorial paper we present a novel modelingmethodology to derive a nonlinear dynamical model whichadequately describes the effect of fuel sloshing on the attitudedynamics of a spacecraft. We model the impulsive thrustersusing mixed logic and dynamics leading to a hybrid formulation.We design a hybrid model predictive control scheme for theattitude control of a launcher during its long coasting period,aiming at minimising the actuation count of the thrusters.

Index Terms— Attitude Control, Sloshing, Hybrid ModelPredictive Control, Aerospace.

I. INTRODUCTION

Upper stages of launchers sometimes have a control modeknown as “long coasting phase” which can last up to fivehours. The spacecraft, already in orbit with the main engineswitched off, drifts with its payloads toward the point onthe orbit where the separation shall take place. The stageslowly rotates around its roll axis in order to avoid heatingup (barbecue mode). The control torques are generated bythrusters. For some types of thrusters the accepted totalnumber of actuation is limited and the long duration of thecoasting period makes this problem rather challenging.

Although the spin rate is low (1 to 5deg/sec), the gyro-scopic coupling cannot be neglected and the plant dynamicsmust be treated in a multiple-input/multiple-output (MIMO)fashion [1]. Sometimes the main engine will be re-ignitedjust prior to the payload release in order to change the orbitparameters. Therefore, considerable amount of propellant isleft in the tank (up to 1 to 2 tons). Compared to the dry-mass (in the order of 4 to 6 tons) the torques generatedby the propellant motion cannot be neglected (sloshingphenomenon).

The classical way to model the fluid motion are pendu-lum or spring/damper models (see e.g., [2] and referencestherein). Such models are fairly representative if a sufficientacceleration (either from the main engine or in a gravitationalfield) is present. In a near zero-g environment no constantacceleration is present, which could generate the restoringforce, responsible for the oscillating behavior of the fluidand the motion of the fluid is only dominated by the surfacetension. Unlike these two cases the barbecue mode during

a IMT Institute for Advanced Studies Lucca, Piazza SanPonziano 6, 55100 Lucca, Italy. Emails: {pantelis.sopasakis,daniele.bernardini,alberto.bemporad}@imtlucca.it.

b Airbus DS, Airbus-Allee 1, 28199 Bremen, Germany. Email:[email protected].

c ODYS Srl, via della Chiesa XXXII trav. I n. 231, 55100 Lucca, Italy.d European Space Agency (ESA), Keplerlaan 1, Noordwijk, The Nether-

lands, Email: [email protected] part, this work has been carried out in connection with ESA’s Future

Launcher Preparatory Program (FLPP) study “Upper Stage Attitude ControlDesign Framework” under the lead of Adriana Sirbi.

the long coasting flight generates a special accelerationenvironment.

In the case of a cylindrical, centrally placed tank thespin rate generates, via the centrifugal force, a rotationalsymmetrical acceleration field. In principle, an oscillatingbehaviour could be expected again, however, a spinning bodyin free-fall condition will exhibit a motion combining thespinning around the body axis plus a slower rotation of theaxis itself (nutation and precession, see [3] and Figure 1).The fluid collects as a bulge and, following the rotatingacceleration vector, slowly rotates along the tank wall as it isalso reported by Veldman and Vogels [4]. This motion createslarge, time-varying off-diagonal elements in the inertia ten-sor. Computational fluid dynamics (CFD) analyses have beenproposed and are best suited to model the sloshing effect,but the difficulty to perform such simulations in real timerenders them unsuitable for [5]. In this paper we describea control-oriented model in analytic form whose parametersare determined offline based on CFD computations.

A noteworthy burgeoning interest in applications of modelpredictive control (MPC) in aerospace and, in particular,in attitude control can be observed. The use of MPC forattitude control has been proposed by Manikonda et al. [6],Vieira et al. [7] and other authors. Hegrenæs et al. proposean explicit MPC control scheme for attitude control [8],[9]. Other attitude control approaches have been proposedin the literature. Simpler control solutions such as PD andLQR have also been proposed without, however, beingable to take consistently into account the constraints thatapply on the system [10], [11]. Nonlinear model predictivecontrol approaches for constrained attitude control have beenproposed by Kalabic et al. [12].

The thrusters of the spacecraft that are used to control itsattitude are subject to a minimum impulse bit, that is, onceactivated they will apply a minimum torque to the spacecraft.This effect leads to a hybrid description of the dynamics ofthe spacecraft and, eventually, to a hybrid MPC problemwhich is formulated as a mixed-integer quadratic problem.Mixed integer programming has been used by Richardset al. [13] and Mellinger et al. [14] for offline trajectoryplanning. The proposed approach takes trajectory planningonline and applies the control actions in a receding horizonmanner.

Recent developments in optimization theory enable the de-sign of fast embedded MPC controllers with guaranteed con-vergence in fixed-point arithmetic [15]. These results havemade their appearance in the field of attitude control [16].Frick et al. [17] proposed certain heuristics to considerablyspeed-up the solution of hybrid MPC optimization problemsand yield near-optimal solutions. Evidently, optimization and

2015 European Control Conference (ECC)July 15-17, 2015. Linz, Austria

978-3-9524269-3-7 ©2015 EUCA 1376

Fig. 1: CFD simulation of upper stage with two cylindrical tanks.Two types of propellant in red and blue. From left to right: (i) endof spin-up phase, started from initially flat propellant distribution,(ii) fluid is collected as a bulge, (iii) the bulge has slowly rotatedwith the tank, (iv) illustration of spinning and nutation; precessionnot shown.

control theory offer the tools for the use of of MPC in realaerospace applications.

In this paper we derive a state-space continuous-timemodel of the spacecraft attitude employing a simplifiedsloshing model (see Section II). The impulsive nature of thethrusters is modeled using binary variables to yield a hydriddynamical system. In Section III, we introduce an extendedKalman filter to reconstruct the unmeasured sloshing-relatedstates of the systems which we combine with a hybrid modelpredictive controller which makes use of a linearized versionof the system which is updated online using estimated stateinformation (Section IV). Finally, we provide simulationresults to demonstrate the performance of the closed-loopsystem.

Notation. Let N, R, Rn, Rm⇥n be the sets of natural m-by-n matrices. Let P be a logical proposition. We denoteby [P ] its truth value, i.e., [P ] = 1 if P is true and[P ] = 0 otherwise. We use the notation x(t) with t � 0

for continuous-time signals and xk

with k 2 N for discrete-time ones. For any nonnegative integers k

1

, k2

with k1

k2

,the finite set {k

1

, . . . , k2

} is denoted by N[k1,k2]

.

II. ATTITUDE MODEL WITH ROTATING MASS

In this paper we study the attitude dynamics using abody-fixed (BF) frame which is a right-handed, orthonormalreference frame fixed to the spacecraft so that the x-axisis aligned to its principal axis and the rotation about it isdenoted by � and is called the roll angle [10, Sec. 1.1.2].The rotational displacement about the y-axis defines the pitchangle ⇥ and the rotation about the z-axis is the yaw angle .

In this section we provide a detailed discussion on thederivation of a dynamical model that captures the upper stageattitude dynamics in light of the additional torques causedby the sloshing of fuel and the impulsive thrusters which areused to control the attitude of the spacecraft.

In the following the dynamics of motion is derived via theLagrange formalism (see [18]). The system is modeled as arigid body (stage plus payloads) and a ring within which apoint mass m

f

can rotate (see Figure 2). The parameters p,r and ↵ are determined via CFD simulations and represent

Fig. 2: (Up) Multibody model for the bulge phenomenon. Thevalues of p and r define the circular rotation which the bulge canmove (see Fig. 1) and ↵ describe the current fluid position. (Down)Body-fixed body frame with the x-axis, defined by e

bfx

, aligned withthe principal axis of the spacecraft.

the position of the fluid within the tank for a specific spinrate.

The generalized coordinates q of the system are theangular rates ! = (!

x

, !y

, !z

) and the propellant position↵. The energy T of the system is:

T =

1

2

3X

i,j=1

Jij

!i

!j

+

1

2

mf

r2↵2. (1)

The solution of the Lagrangian equation of the system, thatis d

dt

(

@T

@qi)� @T

@qi= 0, provides the equation of motion, thus,

the motion of the mass mf

is given by

↵ = �↵ +

✓(↵, !)

mf

r2, (2)

where > 0 is a constant representing the wall friction and✓ is given as

✓(↵, !) = r2(!2

y

� !2

z

) sin ↵ cos ↵ + pr!x

!y

sin ↵

� pr!x

!z

cos ↵ � r2!y

!z

cos(2↵). (3)

Equation (3) is the coupling of the main body motion ontothe moving mass m

f

, i.e., the excitation caused by thecombined motion of precession and nutation. Equation (2)is the sum of the accelerations acting on m

f

and must beadapted such that relative motion of m

f

resembles the bulgemotion computed via CFD. Equation (2) can be written as

↵ = �, (4)

˙� = �� +

✓(↵, !)

mf

r2. (5)

The inertia tensor J of the upper stage is a function ofthe sloshing state ↵ as J(↵) = J

0

+ Jmf (↵), where J

0

is the inertia tensor of the spacecraft without the effect ofsloshing (which is a diagonal matrix) and J

mf

(↵) is the

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contribution of the moving mass to the overall inertia givenby the symmetric matrix

Jmf (↵)=m

f

2

4r2 �pr cos ↵ �pr sin ↵⇤ p2 + r2 sin

2 ↵ �r2 sin ↵ cos ↵⇤ ⇤ p2 + r2 cos

2 ↵

3

5 . (6)

For convenience let us define

⌦(!) =

2

40 !

z

�!y

�!z

0 !x

!y

�!x

0

3

5 . (7)

The torque ⌧ given by

⌧ = ⌧ext

+ ⌦(!) · l, (8)

where l = J! is the angular momentum and ⌧ext

is thetorque applied by the thrusters. Differentiating l and by virtueof (8) and using the notation J 0

(↵) = dJ/d↵ we have

˙l =

˙J! + J ! = J 0(↵)↵dt + J !, (9)

and given that ⌧ = dl/dt we have that

⌧ = J 0(↵)↵! + J !. (10)

The attitude dynamics, by virtue of (8) and (10), is nowdescribed by

! = J�1

⌦J! + J�1⌧ext

� J�1J 0(↵)�!. (11)

The right-hand side of (11) is a complex function of ! mainlybecause of the involvement of the inverse J�1; its derivationin explicit form was carried out using the Symbolic Toolboxof MATLAB. The pitch and yaw errors, denoted by ✏

y

(t)and ✏

z

(t) respectively, follow the dynamics:

✏y

(t) = !y

(t) + ✏z

(t)!x

(t), (12a)✏z

(t) = !z

(t) � ✏y

(t)!x

(t). (12b)

The attitude dynamics is described by the state vec-tor z(t) = (✏

y

, ✏z

, !x

, !y

, !z

, ↵, �)

0 with input u(t) =

(⌧ext,x

, ⌧ext,y

, ⌧ext,z

)

0. The overall dynamics given in equa-tions (4), (11) and (12) can then be written concisely in theform

z(t) = F (z(t), u(t)), (13a)y(t) = Cz(t), (13b)

where in particular F : R7 ⇥ R3 ! R7 has the input-affineform F (z, u) = f(z) + g(z)u and C 2 R5⇥7 is the matrixC = [

I5 0

], i.e., the sloshing states ↵ and � cannot bemeasured directly in real time.

The above system is discretized with sampling periodh > 0 to give:

zk+1

= zk

+ hF (zk

, uk

), (14a)yk

= Czk

, (14b)

which will be used as the nominal plant model in theformulation of the state estimation and model predictivecontrol problems in what follows. For convenience we definefh

(z, u) = z + hF (z, u).

III. STATE ESTIMATION & ONLINE LINEARIZATION

We employ an extended Kalman filter (EKF) to estimatethe state of the discrete-time system (14). The extendedKalman filter is the nonlinear version of the Kalman filterwhich makes use of the nominal nonlinear system dynamicsto predict the evolution of the state while it uses updated locallinearizations of the nonlinear system at the current estimatedstate to estimate the covariance of the state vector [19]. Here,the state estimates z

k

are updated according to the nonlinearequation

zk+1

= (I � Kk

Cfh

(zk

, uk

)) + Kk

yk

, (15)

where Kk

2 R7⇥7 is determined by Kk

=Gk

C 0(CG

k

C 0+

R)

�1 with Pk+1

=(I � Kk

C)Pk

, and

Fk

=

@fh

@z

����(zk,uk)

and Gk

= Fk

Pk

F 0k

+ Q. (16)

The matrix Q in the above equations is the covariance matrixof a term w

k

acting as a zero-mean additive noise on thesystem dynamics, that is, z

k+1

= zk

+hF (zk

, uk

)+wk

, andPk

is a covariance estimate for the current state estimatezk

. The matrix R is the covariance matrix of a zero-meanadditive measurement noise n

k

, that is, yk

= Czk

+nk

. Theestimates of the EKF as in (15) are expected to converge tothe extent the initial estimate z

0

is sufficiently close to theactual initial state.

At time kj

we linearize the discrete time model (14)around the current estimated state z

kj and input ukj to arrive

at the following affine dynamical model

zk+1

= Akjzk + B

kjuk

+ fkj , (17)

for k>kj

, where Akj , B

kj and fkj are functions of z

kj andukj as A

kj = I +h @F

@z

��(zkj

,ukj)

, Bkj = h @F

@u

��(zkj

,ukj)

, andfkj = z

kj+F (zkj , ukj ). The linearization is updated every

Nl

sampling periods, that is kj

= jNl

and the resulting,time-varying, estimated system matrices are given to theMPC controller.

IV. HYBRID MODELING & PREDICTIVE CONTROL

A. Hybrid Modeling

In this section we model the hybrid behaviour of the ac-tuators using the mixed logic and dynamics framework [20].The torques exerted by the thrusters are subject to a minimumimpulse bit (MIB), meaning that, once the thrusters areswitched on they cannot be turned off immediately andthere is a fixed minimum period of time for which theyremain open. This entails a minimum exerted torque on thecorresponding axis which can be modeled by constraints ofthe form u

k

2 U , with

U = [�umax

,�umin

] [ {0} [ [umin

, umax

], (18)

where umin

2 R3 denotes the minimum impulse bit andumax

2 R3 denotes the maximum torque that can beprovided in a sampling interval on each axis. In order to

1378

translate this constraint to a computationally tractable form,we consider the convex constraints

�umax

uk

umax

, (19)

introduce the binary vectors �k

2 {0, 1}3 and ✓k

2 {0, 1}3which are written as �

k

= [�1k

, �2k

, �3k

]

0 and ✓k

= [✓1k

, ✓2k

, ✓3k

]

0,and we establish the correspondence

�ik

= [ui

(k) �umin,i

] and ✓ik

= [ui

(k) � umin,i

], (20)

for i = 1, 2, 3. Notice that whenever �i = 1 or ✓i = 1,the control action u is outside the bounds defined by {u |�u

min

u umin

}, so in light of (19) it can be appliedto the system. We define the following propositional logicconstraints on the auxiliary continuous variables ⌘

k

2 R3:

⌘i,k

= ui,k

· [�ik

_ ✓ik

], i 2 {1, 2, 3} (21)

where _ denotes the logical disjunction (OR) operator;clearly ⌘

k

2 U . We introduce the auxiliary variables vk

2 R3

to trace whether a thruster activation takes place at time k

vi,k

= [�ik

_ ✓ik

], i 2 {1, 2, 3} (22)

The system dynamics subject to the thrusters constraints canbe described by the linear discrete-time model

zk+1

= Axk

+ B⌘k

+ f (23a)�k+1

= �k

+ [

1 1 1

] vk

, (23b)

where the additional state variable �k

2 R, namely theactivation count, if necessary, can be bounded by the numberof maximum activations allowed �

max

according to:

�k

�max

. (24)

This constraint is likely to become active only if the pre-diction horizon is long enough to foresee the exhaustion ofavailable actuations or �

k

is close to �max

. In (23a), A, Band f are provided by the linearization step explained inSubsection III.

B. Model Predictive Control

MPC is an optimization-based control methodology whereat each time instant a performance index is optimized usinga discrete-time model of the controlled process taking intoaccount the constraints on the state and input variables ofthe system. This optimization yields a sequence of controlactions whose first element is applied to the system as inputwhile other elements are discarded [21]. As already men-tioned, the proposed control scheme aims at (i) steering thepitch and yaw errors and the spin rate to desired set-pointswhile (ii) accounting for the aforementioned constraints and(iii) exhibiting a sparse actuation profile.

The model used by MPC here is an affine model of the sys-tem obtained by linearization and the proposed hybrid MPC

HMPC%

EKF%

Online%Linearisa2on%%

(A, B, f)

z

u y

z

Fig. 3: The proposed control scheme with the hybrid MPC con-troller, the EKF and the online linearization.

scheme, illustrated in Figure 3, is formulated as follows:

P(x0

, �0

, A, B, f) : min

⇡N

VN

(⇡N

, �0

) (25a)

s.t. x(0) = x0

, �(0) = �0

, (25b)Constraints (19) – (24), for k 2 N

[0,Nu], (25c)

zk+1

= Azk

+B⌘k

+f, for k 2 N[Nu,N�1]

, (25d)� u

max

⌘k

umax

, for k 2 N[Nu,N�1]

(25e)

where Nu

N defines the hybrid control horizon, i.e., thenumber of time steps for which the minimum impulse bitis taken into account as in (25c). Let us denote the setof optimization variables of the MPC problem by ⇡

N

=

{{uk

, vk

, �k

}k2N[0,Nu]

, {⌘k

}k2N[0,N�1]

}, where N � 1 is theprediction horizon and let x

0

be the current state, �0

be thetotal number of activations up to the current time instant kand let A, B and f be estimated linearization matrices of thesystem derived as in Section III. The MPC control action iscomputed in a receding horizon fashion: At every samplingtime instant, optimization problem (25) is solved to yield theoptimal solution ⇡?

N

and the first control action ⌘?

0

is appliedto the system. The MPC controller commands admissibletorques to the thrusters which will be activated for a certaintime between t

min

and Ts

, where tmin

is the minimum timefor which the thrusters can remain open (and corresponds toa u

min

torque) and Ts

is the sampling time. The proposedmethodology accounts for the hybrid nature of the thrustersand offers a clear advantage over other approaches to attitudecontrol that issue merely on/off commands [22]. In fact, withthe proposed approach the sampling time T

s

can be muchlarger than the minimum impulse time t

min

, as a result, theMPC controller can have a greater foresight of the systemevolution at a much lower computational cost.

A short hybrid control horizon compared to the predictionhorizon is typically employed to reduce the complexity ofthe resulting optimization problem. In the proposed formula-tion (25), the input is assumed to satisfy all constraints givenby equations (19)–(24) for all k 2 N

[0,Nu]. For time instants

after Nu

, we relax the hybrid constraints and we assume thatthe input can take any value subject to the constraints (25e)and the system dynamics (25d).

1379

0 20 40 60 80 100

1

2

3

4

5

6

7

8

9

10x 10

−4

Time [s]

Sta

te e

stim

ate

ion e

rror

(norm

)

Fig. 4: Norm of the observer error for various initial state estimatesz0.

For the chosen sampling time of 0.5s, a large predictionhorizon is required for the MPC controller to have enoughforesight given that the rotational dynamics of the upperstage is relatively slow. The computational complexity canbe mitigated by choosing a short hybrid control horizon.

The cost function VN

in (25) is defined as

VN

(⇡N

, �0

)=Vf

(zN

, �N

)+

N�1X

k=0

`(zk

, ⌘k

) (26)

where ` is the stage cost `(z, u) = kQzkp

+ kRukp

, and Vf

is the terminal cost defined as Vf

= kQN

zN

kp

+⇢(�N

��0

).Matrices Q, Q

N

, R and ⇢ are used to strike a desiredtrade-off between pointing accuracy and usage of thrusters.Any p-norm (1 p 1) can be used for the stage andterminal cost functions. Here we used p = 1 so that theresulting problem is a mixed-integer linear problem (MILP).

V. SIMULATION RESULTS

Firstly we assess the performance of the EKF observer.We define the observer error as e

k

= zk

� zk

. The normof e

k

for different initial state estimates is presented in Fig-ure 4. The observer, tested in closed-loop with the proposedhybrid MPC controller, was found to converge in a smallneighbourhood about the actual state of the system.

In order to assess the performance of the closed-loop sys-tem using different tuning parameters we introduce certainperformance indicators. First, the pointing-accuracy indica-tor

JK

pa

=

TsimX

k=Tsim�K

✏2y,k

+ ✏2z,k

, (27)

where Tsim

is the simulation time and K is the number oftime instants before the end of the simulation to be consid-ered. We also introduce the total squared error indicator

Jtse

=

TsimX

k=0

✏2y,k

+ ✏2z,k

, (28)

the number of thruster actuations along the y and z axes,denoted by Jy

act

and Jz

act

and the total actuation count onthe y and z axes, J

act

. In all cases !x

converges fast to

TABLE I: Evaluation of the closed-loop performance by simulationsover a period Tsim = 300s.

⇢ Jy

act Jy

act Jact J40pa Jtse

0.005 37 131 168 1.14 · 10�4 0.0510

0.05 28 56 84 0.0018 0.0834

0.1 32 48 80 0.0349 0.2988

0.2 34 45 79 0.1279 1.6862

its set-point with few actuations along the x axis (see forexample Figures 6 and 7).

In Table I we summarise the evaluation results of theclosed-loop system for different values of ⇢ having fixedQ = diag(7, 7, 4, 1, 1, 1, 1), Q

N

= 2Q, R = 0 and p = 1.The linearization is updated with N

l

= 10, i.e., every 5s.The prediction horizon is fixed to N = 20 and N

u

= 8; thisparticular choice of the prediction and hybrid control horizonwas found to offer a good trade-off between optimality andcomputational complexity. It is interesting to see that if wedecrease the hybrid control horizon to N

u

= 2, the pointingaccuracy worsens significantly leading to J40

pa

= 1.1669 andJtse

= 52.98 with Jact

= 144. The respective simulationsare presented in Figure 8. This important observation justifiesthe use of hybrid MPC for the control of spacecraft withimpulsive thrusters. Moreover, hybrid control horizon valueslarger than 8 were not found to improve the closed-loopperformance, thus, were avoided for the sake of retainingthe complexity as low as possible. The average computationtime for the derivation of the control action was 0.12s ona 2.2GHz Intel Core i7 machine. Simulations were carriedout in MATLAB 2013a, using YALMIP [23] and the MILPsolver of Gurobi.

We can notice that higher values of ⇢ lead to a sparseractuation profile reducing the number of y and z actuations atthe cost of a lower pointing accuracy and overall performance(in terms of J

pa

and Jtse

). The state trajectory for ⇢ = 0.05

is illustrated on the ✏y

-✏z

-plane in Figure 5 where we observethat the state moves into a small neighbourhood of the set-point. The HMPC commands are shown in Figure 6 and thespin rate is presented in Figure 7.

REFERENCES

[1] J. Wertz, ed., Spacecraft attitude determination and control. Dor-drecht: Kluwer Academic Publishers, 1978.

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[3] P. Hughes, Spacecraft attitude dynamics. Dover Publications Inc.,1986.

[4] A. Veldman and M. Vogels, “Axisymmetric liquid sloshing under low-gravity conditions,” Acta Astronautica, vol. 11, no. 10–11, pp. 641 –649, 1984.

[5] J. Hervas and M. Reyhanoglu, “Thrust-vector control of a three-axis stabilized upper-stage rocket with fuel slosh dynamics,” ActaAstronautica, vol. 98, no. 0, pp. 120 – 127, 2014.

[6] V. Manikonda, P. Arambel, M. Gopinathan, R. Mehra, and F. Hadaegh,“A model predictive control-based approach for spacecraft formationkeeping and attitude control,” in American Control Conference, vol. 6,pp. 4258–4262, 1999.

[7] M. Vieira, R. Galvao, and K. Heinz Kienitz, “Attitude stabilizationwith actuators subject to switching-time constraints using explicitMPC,” in IEEE Aerospace Conference, pp. 1–8, 2011.

1380

50 100 150 200 250 300

0

0.2

0.4

0.6

0.8

Pitc

h e

rror

[deg]

50 100 150 200 250 300

0

0.5

1

Yaw

err

or

[deg]

Time [s]

Fig. 5: Controlled trajectory in the time domain with EKF andHMPC in the loop for 300s with ⇢ = 0.05.

0 50 100 150 200 250 3000

1000

2000

Tx

0 50 100 150 200 250 300−2000

0

2000

4000

Ty

0 50 100 150 200 250 300−2000

0

2000

4000

Tz

Time [s]

Fig. 6: Torques (in Nm) applied by the impulsive thrusters (⇢ =0.05).

0 50 100 150 200 250 300−6

−4

−2

0

2

4

6

Time [s]

Spin

rate

[deg/s

]

Fig. 7: Spacecraft in barbecue mode: the spin rate !

x

tracks thedesired set-point.

50 100 150 200 250 300

−1

0

1

Pitc

h e

rror

[deg]

50 100 150 200 250 300

−1

0

1

Yaw

err

or

[deg]

Time [s]

Fig. 8: Closed-loop trajectory with N = 20 and N

u

= 2.

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