Passivity Based Controller-Observer Schemes for Relative Translation ...

Passivity based controller-observer schemes for relative translation of a formation of spacecraft Esten Ingar Grøtli and Jan Tommy Gravdahl

Abstract—In this paper we study the control of a formation of spacecraft in a leader-follower architecture. Two controllerobserver schemes are presented, that guarantee uniform global exponential stability of the closed loop system. Because of this strong stability property, and the fact that the controller-observer design is based on passivity properties, the overall system is robust to modeling errors and orbital perturbations.

determination systems and the control systems. At the same time, velocity measuring equipment is expensive and computationally demanding, and the communication channels between satellites are a scarce resource, hence the control law should show good performance with little information about other spacecraft in the formation.

I. INTRODUCTION

B. Previous work This section summarizes the previous work done on formations of spacecraft, using a relative position model similar to the one in Section III. Possibly the first solution to the control problem of this model was presented in [1]. Here, a nonlinear output feedback control law was developed guaranteeing global uniform ultimate boundedness (GUUB) of the position and velocity tracking errors in the presence of unknown spacecraft masses and disturbance force parameters. A filtering scheme was provided, to allow for the use of relative velocity in the controller. A similar result was given in [2]. In [3] the nonlinear tracking control problem for both translation and rotation was presented. The adaptive control law derived, ensure global asymptotic convergence in the presence of unknown mass and inertia parameters of the leader and follower spacecraft. In [4] a full state adaptive learning control algorithm was developed to give global asymptotic convergence of position and velocity tracking errors, in the presence of periodic disturbances and unknown spacecraft masses. Tracing the steps of [3], semiglobal asymptotic convergence of the spacecraft translational position and velocity tracking errors were ensured in [5], using an output feedback control law. Relative angular and translational velocities were provided to the controller, using a high-pass filter similar to the one in [1]. Assuming boundedness of orbital perturbations and the leader control force only, an adaptive controller was designed in [6] to prove the closed-loop system to be uniformly semiglobally practically asymptotically stable (USPAS). A velocity filter was used to provide sufficient knowledge about the relative velocity to solve the control problem. It should also be mentioned that the results in this paper highly builds on results achieved for the control of robot manipulators, e.g. [7].

A. Background The major reason for using formations of spacecraft, is the desire to place measuring equipment further apart than what is possible on a single spacecraft. Often the resolution of the observation from such measurements is inversely proportional to the baseline lengths, and hence the ability of small spacecraft to fly in precise formation make a wide array of new applications possible. These applications are mainly within surveillance and sensing, and include deep-space imaging and exploration, geodesy and environmental monitoring of the Earth and its surrounding atmosphere. Another advantage is the increased flexibility of spacecraft formations, compared to single spacecraft missions. The relative position and attitude of the formation can easily be changed throughout the lifetime of the system, so as to achieve the optimal configuration for different missions. It also brings redundancy into the systems, since it is sometimes a possibility that the functionality of a disabled spacecraft can be fulfilled by some other spacecraft in the formation. In worst case an obsolete or dead spacecraft can be replaced at a much lower cost than replacing one big spacecraft for a single spacecraft mission. There is also another economic aspect to this; often it is more expensive to place one large spacecraft with all the functions built-in, into orbit than several smaller ones of the same collective weight. Therefore, as the number of missions involving formation flying spacecraft, proposed or under development, still increases, one can imagine assembly lines of standardized spacecraft, thus lowering the production cost drastically. These standardized spacecraft will of course be fully equipped with the proper instruments for their mission. Though there are several advantages using spacecraft formations, these come at the cost of increased complexity and many technological challenges. Precise formation control in the presence of disturbances due to gravitational variations, nonspherical shapes of planets, atmospheric drag or solar radiation sets high requirements to both the relative position and velocity E. I. Grøtli and J. T. Gravdahl are with Department of Engineering Cybernetics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway [email protected],

[email protected]

C. Contribution This paper addresses the tracking control problem of a formation of spacecraft using a model for their relative position. Two different controller-observer schemes are stated, which in closed loop with the relative position model make the origin of the system uniformly globally exponentially stable (UGES) in the case of known model parameters and disturbances. Furthermore, for bounded disturbances the system is proven to be uniformly globally practically asymptotically stable (UGPAS).

II. N OTATION We use the notation x˙ for the time derivative of a vector x, i.e. x˙ = dx/dt. Moreover x ¨ = d2 x/dt2 . The identity matrix n×n in R is written In×n . We use | · | for the Euclidean norm of vectors. The minimum and maximum eigenvalue of a matrix A are denoted by λm (A) and λM (A), respectively. III. M ODEL The model used in this paper is found in [9], and can be traced back to [1]. Starting point for this model is the fundamental differential equation for the two-body problem µ (1) r¨ + 3 r = 0 |r| 3

where r ∈ R is the relative position of two point masses m1 and m2 and µ = G(m1 + m2 ) with G being the universal constant of gravity. This equation is generalized to include disturbance forces fl , ff ∈ R3 due to aerodynamic drag, third gravitating bodies, solar radiation, magnetic fields, etc., and actuator forces ul , uf ∈ R3 for the leader and follower spacecraft, respectively, such that ul fl µ + (2) r¨l = − 3 rl + rl ml ml ff uf µ + (3) r¨f = − 3 rf + rf mf mf By defining p ∈ R3 as the relative position, i.e. p = rf − rl , the relative position dynamics can be written in the following form (cf. [9])

M p¨ + C(ν) ˙ p˙ + D(¨ ν , ν, ˙ rl )p + n(rl, rf ) = U + F

(4)

where ν is the true anomaly of the leader spacecraft,

M = mf I3×3 ∈ R3×3

(5)

is a matrix with the mass of the follower satellite, mf , as elements along the diagonal,   0 −1 0 C(ν) ˙ = 2mf ν˙ 1 0 0 ∈ R3×3 (6) 0 0 0 is a skew-symmetric matrix, µ − ν˙ 2 rf3  ν¨ D(¨ ν , ν, ˙ rf ) = mf  0

µ rf3

−¨ ν

0

0

µ rf3

− ν˙ 2

and

n(rl, rf ) = mf µ

h

rl rf3

−

1 rl2

0 0



3×3 0 ∈R

i⊤

3

∈R

(7)

al (1 − e2l ) 1 + el cos ν

ν(t) ˙ = and

(8)

(10)

nl (1 + el cos ν(t))2 3

(1 − e2l ) 2

(11)

−2n2l el (1 + el cos ν(t))3 sin ν(t) (12) (1 − e2l )3 p respectively, with nl = µ/a3l as the mean motion of the leader. For a leader spacecraft in an elliptical orbit around the earth, the true anomaly rate ν(t) ˙ is bounded by a constant (cf. [6]) ν¨(t) =

αν˙ ≤ ν(t) ˙ ≤ βν˙

(13)

for all t, where αν˙ and βν˙ are positive constants.

IV. C ONTROLLER -O BSERVER D ESIGN A. Slotine and Li scheme In [7] the controller scheme of [10] is redefined for output feedback. A similar controller-observer scheme will be used in this section to prove uniform global exponential stability of the origin of the closed-loop system. We will first assume that the leader spacecraft is perfectly controlled to account for any perturbations, i.e. that ul − fl = 0, and that the follower spacecraft is not influenced by disturbances, so that ff = 0. These are severe assumptions, but as it will be shown in section IV-C, similar results can be obtained also when these assumptions are not satisfied. Proposition 1: Assume that the desired relative position pd (t) ∈ R3 , desired relative velocity p˙d (t) ∈ R3 and desired relative acceleration p¨d (t) ∈ R3 are all bounded functions. Assume that ul + fl = 0, and that ff = 0. Let the controller be given as uf

=

p˙r p˙0

= =

M p¨r + C(ν) ˙ p˙r + D(¨ ν , ν, ˙ rl )p + n(rl , rf ) −Kd (p˙0 − p˙r ) − Kp e (14) p˙d − Λ(ˆ p − pd ) (15) ˙pˆ − Λ˜ p (16)

where Kp , Kd , Λ ∈ R3×3 satisfies Kp = Kp⊤ > 0, Kd = kd I3×3 with kd > 0 scalar and Λ = Λ⊤ > 0. Furthermore, e = p − pd is the position error, p˜ = p − pˆ is the observer estimation error and pr , p0 ∈ R3 are auxiliary signals. Let the observer be given as pˆ˙ = z + Ld p˜ (17)

z˙

The composite disturbance force F ∈ R3 and the relative control force U ∈ R3 , is given by mf mf fl and U = uf − ul (9) F = ff − ml ml respectively. The radius of the elliptical orbit of the leader spacecraft with eccentricity el and semimajor axis al , is given by

rl =

The rate and rate of change of the true anomaly is

= p¨r + Lp2 p˜ − M −1 (Kp (ˆ p − pd ) +C(ν)( ˙ p˙0 − p˙r ))

(18)

3×3

where Ld , Lp2 ∈ R satisfies Ld = ld I3×3 + Λ and Lp2 = ld Λ, with ld > kd /mf scalar. Then, the origin of (4), in closed loop with controller (14)-(16) and observer (17)-(18) is UGES. Proof: The proof follows analogous to the one in [7], but the following should be noted: In robotics, the Coriolis matrix, usually denoted C , can be represented in terms of the Christoffel symbol. This is not the case here, and the stability analysis will be different. First define the sliding variables

s1 s2

= p˙ − p˙r = e˙ + Λ(ˆ p − pd ) = p˙ − p˙0 = p˜˙ + Λ˜ p

(19) (20)

From (14), (15), (16) and (4) we see that the tracking error dynamics are

M (¨ e + Λ(pˆ˙ − p˙d )) =

−C(ν)( ˙ e˙ + Λ(ˆ p − pd )) −Kd (p˙0 − p˙r ) − Kp e (21)

The time derivative of the Lyapunov function candidate can be upper bounded:

V˙ 1

which, by using (19), and that

p˙0 − p˙r = p˙ − s2 − (p˙ − s1 ) = s1 − s2

(22)

M s˙ 1 + C(ν)s ˙ 1 = −Kd (s1 − s2 ) − Kp e

(23)

becomes

Furthermore, from (17) and (18) the observer error dynamics are found to be

p−pd )+C(ν)(s ˙ 1 −s2 )) (24) p¨ˆ−Ld p˜˙ = p¨r +Lp2 p˜−M −1 (Kp (ˆ Premultiplication by M and inserting Ld = ld I3×3 + Λ and Lp2 = ld Λ gives

M (p¨ ˆ − Λp˜˙ − p¨r ) = M ld (p˜˙ + Λ˜ p) − Kp (e − p˜) −C(ν)(s ˙ 1 − s2 ) (25) where it has also been used that pˆ − pd = e − p˜. By using (16) and (22) we get that

M s˙ 2 +C(ν)s ˙ 2 +M ld s2 +Kp p˜ = M s˙ 1 +C(ν)s ˙ 1 +Kp e (26) Inserting the tracking error dynamics, (23), gives

M s˙ 2 + C(ν)s ˙ 2 = −M ld s2 − Kd (s1 − s2 ) − Kp p˜

(27)

Now, let the Lyapunov function candidate be given by

1 ⊤ x R1 x1 (28) 2 1 £ ¤⊤ e⊤ s⊤ p˜⊤ ∈ R12 and R1 = where x1 = s⊤ 1 2 12×12 diag(M, Kp , M, Kp ) ∈ R . The time derivative of the the Lyapunov function candidate along the error dynamics (23) and (27) is V1 (x1 , t) =

V˙ 1

=

s⊤ ˙ 1 − Kd (s1 − s2 ) − Kp e) 1 (−C(ν)s +e⊤ Kp e˙ +s⊤ ˙ 2 − M ld s2 − Kd (s1 − s2 ) − Kp p˜) 2 (−C(ν)s ⊤ ˙ +˜ p Kp p˜ (29)

Canceling terms, and using the fact that C(ν) ˙ is skew-symmetric gives

V˙ 1

=

⊤ ⊤ −s⊤ 1 Kd s1 + s2 Kd s2 − s2 ld M s2 +e⊤ Kp (e˙ − s1 ) + p˜⊤ Kp (p˜˙ − s2 )

(30)

Note that e˙ − s1 = −Λ(e − p˜) and p˜˙ − s2 = −Λ˜ p, hence

V˙ 1

=

⊤ ⊤ −s⊤ 1 Kd s1 + s2 Kd s2 − s2 ld M s2 −e⊤ Kp Λe + e⊤ Kp Λ˜ p − p˜⊤ Kp Λ˜ p

2

(32)

2

Since k1 |x1 | ≤ V1 ≤ k2 |x1 | with k1 = λm (R1 ) = min {mf , λm (Kp )} and k2 = λM (R1 ) = max{mf , λM (Kp )}, 2 and V˙ 1 ≤ −k3 |x1 | for ld > kd /mf where k1 , k2 and k3 are positive constants, the conditions of [11, Theorem 4.10] are satisfied, and the origin of the system is UGES. Note that it is easily verified that the closed-loop system is passive. Take for simplicity Kp = 0, and rewrite the errordynamics (23) and (27) as

¯ s˙ + C( ¯ ν)s M ˙ = v (34) v = −T¯s (35) £ ¤⊤ ¯ = diag(M, M ), C( ¯ ν) s⊤ where s⊤ = s⊤ , M ˙ = 1 2 diag(C(ν), ˙ C(ν)) ˙ and · ¸ Kd −Kd T¯ = (36) Kd M ld − Kd ¯ s as the storage function, we see that along Taking V = 12 s⊤ M ¯ + s⊤ v = s⊤ v , and according to [11, (34), V˙ = −s⊤ Cs Definition 6.3], the system is passive. For any vector y ∈ R6 we can write y ⊤ T¯y = 21 y ⊤ (T¯1 + T¯2 )y , where T¯1 = T¯ + T¯⊤ and T¯2 = T¯ − T¯⊤ . Since T¯2 is skew-symmetric, y ⊤ T¯2 y = 0, and T¯ is positive definite if T¯1 is positive definite. This is true for ld > kd /mf , which we have already assumed. Therefore, the block (35) is strictly passive from s to −v . One of the advantages of passivity-based control is its robustness. Assume that the model possesses the same passivity properties, regardless of the numerical values of the physical parameters. If the controller is designed in such a way that stability relies on passivity properties only, then the closed-loop system will be stable whatever the values of the physical parameters. B. Paden and Panja scheme In this section the controller scheme of [12] as redefined for output feedback in [7] will be used. The same assumptions as for the Slotine and Li scheme apply. Proposition 2: Assume that the desired relative position, velocity and acceleration, denoted pd (t), p˙d (t), p¨d (t) ∈ R3 are all bounded functions. Assume that ul + fl = 0, and that ff = 0. Let the controller be given as

uf

=

p˙r p˙0

= =

(31)

where the last three terms are bounded:

−e⊤ Kp Λe + e⊤ Kp Λ˜ p − p˜⊤ Kp Λ˜ p 1 ⊤ 1 ⊤ p ≤ − e Kp Λe − p˜ Kp Λ˜ 2 2

1 2 2 p| − (ld mf − kd ) |s2 | ≤ − λm (Kp ) λm (Λ) |˜ 2 1 2 2 −kd |s1 | − λm (Kp ) λm (Λ) |e| (33) 2

M p¨d + C(ν) ˙ p˙d + D(¨ ν , ν, ˙ rl )p + n(rl , rf ) −Kd (p˙0 − p˙r ) (37) p˙d − Λe (38) ˙pˆ − Λ˜ p (39)

where Λ = Λ√⊤ > 0, Kd = kd I3×3 ∈ R3×3 with kd > mf λM (Λ) + 12mf βν˙ , e = p − pd is the position error and

p˜ = p − pˆ is the observer estimation error. Let the observer be given as pˆ˙ = z + Ld p˜ z˙ = p¨d + Lp2 p˜

(40)

V˙ 2

(41)

where Ld , Lp2 ∈ R3×3 satisfies Ld = ld I3×3 + Λ and Lp2 = ld Λ, with ld > 2kd /mf scalar. Then the origin of (4) in closed loop with the controller (37)-(39) and observer (40)-(41) is UGES. Proof: By combining the dynamic equations of the formation with the equations for the proposed controller, the closedloop tracking error dynamics are found to be

M (¨ p − p¨d ) + C(ν)( ˙ p˙ − p˙d ) + Kd (p˙0 − p˙r ) = 0

2kd λm (Λ−1 ). The time derivative of the Lyapunov function candidate along the error dynamics (46) and (51) is

(42)

⊤ ⊤ ⊤ s⊤ 1 Kd s1 = e˙ Kd e˙ + 2e ΛKd e˙ + e ΛKd Λe

= p˙ − p˙r = e˙ + Λe = p˙ − p˙0 = p˜˙ + Λ˜ p

(43)

(44) (45)

one gets the tracking error dynamics

M s˙ 1 = M Λe˙ − C(ν) ˙ e˙ − Kd (s1 − s2 )

(46)

since p˙0 − p˙r = s1 − s2 . By differentiating (40) of the observer dynamics with respect to time, and combining with (41), the following expression is found

z˙ = p¨d + Lp2 p˜ = p¨ˆ − Ld p˜˙

(47)

Notice that from (38) and (39) it follows that

p¨d − p¨ˆ = p¨r + Λe˙ − (¨ p0 + Λp˜˙)

(48)

and

(49)

where it also has been used that s2 = p˜˙ + Λ˜ p , Ld = ld I3×3 + Λ and Lp2 = ld Λ. By premultiplication by M and using that p˙0 − p˙r = s1 − s2 , one gets that

M s˙ 2 = M s˙ 1 − M Λe˙ − M ld s2

(50)

Then the time derivative of the Lyapunov function candidate becomes

V˙ 2

=

e˙ ⊤ M Λe˙ + e⊤ ΛM Λe˙ − s⊤ ˙ e˙ − e˙ ⊤ Kd e˙ 1 C(ν) −2e⊤ ΛKd e˙ − e⊤ ΛKd Λe + s⊤ 1 Kd s2 +2e⊤ ΛKd e˙ − e⊤ ΛM Λe˙ − s⊤ ˙ e˙ − s⊤ 2 C(ν) 2 Kd s1 ⊤ ⊤ ⊤ ⊤ +s2 Kd s2 − s2 M ld s2 + s2 Kd s2 − p˜˙ Kd p˜˙ −˜ p⊤ ΛKd Λ˜ p (56)

£ ¤⊤ ∈ R12 as our state Define x3 = e˙ ⊤ (Λe)⊤ p˜˙⊤ (Λ˜ p vector, and note that by definition that x3 = T x2 , where  I 0 T = 0 0

−I I 0 0

0 0 I 0

 0 0  ∈ R12×12 −I  I

(57)

1 Here, I = I3×3 and 0¯ = 03×3 ¯ . Since ⊤3 |x2 | 2 ≤ 2 2 ⊤ ⊤ ⊤ λm (T T ) |x2 | ≤ |x3 | = ¯x2 T T x2 ¯ ≤ λM (T T ) |x2 | ≤ 2 3 |x2 | it is easily verified that 2

k1 |x3 | ≤ V2 ≤ k2 |x3 |

2

(58)

with k1 = 61 λm (R2 ) and k2 = 23 λM (R2 ). Furthermore, the time derivative of the energy function can be written ⊤ ⊤ ⊤ V˙ 2 = −x⊤ ˙ e˙ (59) 3 Qx3 − s2 (ld M − 2Kd )s2 − (s1 + s2 )C(ν)

Inserting the tracking error dynamics (46) gives:

M s˙ 2 = −C(ν) ˙ e˙ − Kd (s1 − s2 ) − M ld s2

(55)

2

Combining (47) and (48)

p¨r + Λe˙ − p¨0 = −ld Λ˜ p − ld p˜˙ = −ld s2

(54)

˜˙⊤ Kd p˜˙ + 2˜ s⊤ p⊤ ΛKd p˜˙ + p˜⊤ ΛKd Λ˜ p 2 Kd s2 = p

since p − pd = e. Now, defining

s1 s2

˙ e˙ − Kd (s1 − s2 )) s⊤ 1 (M Λe˙ − C(ν) ⊤ −1 +e (Λ(2Kd Λ − M )Λ)e˙ +s⊤ ˙ e˙ − Kd (s1 − s2 ) − M ld s2 ) 2 (−C(ν) ⊤ +2˜ p ΛKd p˜˙ (53)

Remember that s1 = e˙ + Λe and s2 = p˜˙ + Λ˜ p, hence

which can be written

M e¨ + C(ν) ˙ e˙ + Kd (p˙0 − p˙r ) = 0

=

(51)

Let the Lyapunov function candidate be given by (cf. [13], [14] and [7]) 1 R2 x2 (52) V2 (x2 , t) = x⊤ 2 2 £ ¤⊤ (Λe)⊤ s⊤ (Λ˜ p ∈ R12 and R2 = where x2 = s⊤ 1 2 −1 −1 diag(M, 2Kd Λ − M, M, 2Kd Λ ) ∈ R12×12 Note that 2 for Kd − M Λ > 0, we have that 21 λm (R2 ) |x2 | ≤ V2 ≤ 2 1 2 λM (R2 ) |x2 | , where λm (R2 ) = mf and λM (R2 ) =

where Q = diag(Kd − M Λ, Kd , Kd , Kd ) ∈ R12×12 . By using that ld ≥ 2kd /mf and (13) we get that

V˙ 2 ≤ −(kd − mf λM (Λ) −

√

2

2

12mf βν˙ ) |x3 | ≤ −k3 |x3 | (60) where k3 is a positive constant. It has also √been used that ¯ ¯ 2 ¯(s1 + s2 )⊤ C(ν) ˙ e˙ ¯ ≤ |(s1 + s2 )| |C(ν) ˙ e| ˙ ≤ 12mf βν˙ |x3 | ⊤ ⊤ where √ we wrote (s1 +s2 ) as y x3 and used that y x3 ≤ |y| |x3 | = 12 |x3 |. Hence, according to [11, Theorem 4.10], the origin of the system is UGES.

1 2 V˙ 2 ≤ − kd∗ |x3 | 2

(64)

lim kd∗ (δ)δ 2 = 0 and lim kd∗ (δ) 6= 0

(65)

δ→0

Position [m]

ex ey ez

20 0 −20 −40 −60 0

50

100

150

200

250

Time [s] Velocity [m/s]

2

e˙ x e˙ y e˙ z

1

0

−1

−2 0

50

100

150

200

250

Time [s] Fig. 1. scheme

Position and velocity tracking errors using the Paden and Panja

Position [m]

40

p˜x p˜y p˜z

20 0 −20 −40 −60 0

5

10

15

20

25

30

35

40

Time [s] 30

p˜˙ x p˜˙ y p˜˙ z

20 10 0 −10 −20 −30 0

and that δ→0

40

Velocity [m/s]

C. Unknown disturbances It is a well-known fact that exponential stability is robust to vanishing perturbations, see for instance [11, Lemma 9.1]. Let us now consider the case when there are non-vanishing orbital perturbations working on the formation. This problem was rigorously addressed in [6]. A simpler case is studied here to prove the efficiency of the controller-observer scheme proposed in the previous section. We assume that the disturbances of the follower spacecraft are bounded, i.e. |ff | ≤ βff , and also that the sum of thrust and external disturbances working on the leader spacecraft is bounded, such that |ul + fl | ≤ β(ul +fl ) . Proposition 3: The controller-observer scheme of Proposition 2 in closed-loop with (4) makes the resulting system UGPAS. Proof: The tracking error dynamics is now mf (ul +fl ) (61) M s˙ 1 = M Λe−C( ˙ ν) ˙ e−K ˙ d (s1 −s2 )+ff − ml and the observer error dynamics mf M s˙ 2 = −C(ν) ˙ e˙ − Kd (s1 − s2 ) − M ld s2 + fl − (ul + fl ) ml (62) Using (52) as the Lyapunov function candidate, we get that √ 2 V˙ 2 ≤ −(kd − mf λM (Λ) − 12mf βν˙ ) |x3 | √ mf β(ul +fl ) ) (63) + 12 |x3 | (βff + ml √ where it has been used that |s1 + s2 | ≤ 12 |x3 |. √ Let δ be any ∗ 12mf βν˙ + positive constant. Pick k ≥ k := 2m λ (Λ)+2 d f M d √ m 12(βff + mfl β(ul +fl ) )/δ . Then, for any |x3 | ≥ δ we have that

and hence all conditions of [6, Corollary 1] are satisfied with ∆ = ∞, and the model (4), in closed loop with the controller (37)-(39) and observer (40)-(41) is UGPAS.

V. S IMULATIONS In this section the performance of the Paden and Panja controller-observer scheme will be shown by simulations. Both the unperturbed case from section IV-B and the perturbed case from section IV-C will be considered. The leader spacecraft is assumed to be in an orbit with eccentricity el = 0.5, and semimajor axis al = 20000 km. Both spacecraft are of mass ml = mf = 100 kg. Furthermore, the thrust is assumed to be continuous and available in all directions of the leader spacecraft frame, but limited to max uf = 5 N. The desired trajectory of the follower spacecraft is given by £ ¤⊤ pd (t) = −10 cos ν 20 sin ν 0 , which means that the follower spacecraft evolves around the leader spacecraft in an ellipse during their orbit around the Earth. The initial position and velocity of the follower spacecraft is chosen as p(0) = £ ¤⊤ £ ¤⊤ −40 20 40 and p(0) ˙ = 1 0 −1 , where as the £ ¤⊤ initial states of the observer are pˆ(0) = 4 −4 1 and £ ¤⊤ z(0) = −1 4 2 . The controller and observer gains are

5

10

15

20

25

30

35

40

Time [s] Fig. 2. Position and velocity estimation errors using the Paden and Panja scheme

as follows: ld = 0.5, Kd = 20I3×3 , Λ = 0.06I3×3 . Figure 1 shows the position and velocity tracking errors using the Paden and Panja controller-observer scheme, where as Figure 2 shows the estimation errors. The control history are shown in Figure 3. The Slotine and Li scheme of section IV-A would show similar performance. We now consider the case were there are perturbations working on the formation. We use that ¤⊤ £ 1 1 1 t 2 sin 100 t 2 sin 1000 t and that 2 sin 10 ul + fl = ¤ £ ⊤ 1 1 t sin 10 t sin t . The controller- and observer ff = sin 100 gains are as above, but we use that mf = 95 kg in the controller, i.e. it doesn’t match the actual mass of mf = 100 kg. Figure 4 and 5 show the tracking and estimation errors, respectively. The control history is shown in Figure 6.

VI. C ONCLUSION This paper addresses the problem of relative position control for spacecraft in a leader-follower formation. Two controllerobserver schemes are proposed, that guarantee the origin of the closed-loop system to be globally uniformly exponentially

6

Fx Fy Fz

6

Fx Fy Fz

4

4

Force [N ]

2

Force [N ]

2

0

0

−2

−2 −4

−4 −6 0

100

200

300

400

500

600

700

800

900

1000

Time [s]

−6 0

50

100

150

200

250

Time [s] Fig. 6. Fig. 3.

Control history of the Paden and Panja scheme - with disturbances

Control history of the Paden and Panja scheme

stable. Furthermore, global uniform practical asymptotic stability is ensured for bounded non-vanishing disturbances. Position [m]

40

ex ey ez

20

R EFERENCES

0 −20 −40 −60 0

50

100

150

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Time [s] Velocity [m/s]

2

e˙ x e˙ y e˙ z

1

0

−1

−2 0

50

100

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Time [s] Fig. 4. Position and velocity tracking errors using the Paden and Panja scheme - with disturbances

Position [m]

40

p˜x p˜y p˜z

20 0 −20 −40 −60 0

5

10

15

20

25

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Time [s] Velocity [m/s]

30

p˜˙ x p˜˙ y p˜˙ z

20 10 0 −10 −20 −30 0

5

10

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20

25

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40

Time [s] Fig. 5. Position and velocity estimation errors using the Paden and Panja scheme - with disturbances

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