Angular-Velocity Tracking with Unknown Dynamics ...

Angular-Velocity Tracking with Unknown Dynamics for Satellite Rendezvous and Docking Xiumin Diao, Jianxun Liang, and Ou Ma* Department of Mechanical and Aerospace Engineering New Mexico State University, P.O. Box 30001, MSC 3450, Las Cruces, NM 88003 ABSTRACT Autonomous satellite on-orbit servicing is a very challenging task when the satellite to be serviced is tumbling and has an unknown dynamics model. This paper addresses an adaptive control approach which can be used to assist the control of a servicing satellite to rendezvous and dock with a tumbling satellite whose dynamics model is unknown. A proximity-rendezvous and docking operation can be assumed to have three steps: 1) pre-dock alignment, 2) soft docking and latching/locking-up, and 3) post-docking stabilization. The paper deals with the first and third steps. Lyapunovbased tracking law and adaptation law are proposed to guarantee the success of the nonlinear control procedures with dynamics uncertainties. A dynamics simulation example is presented to illustrate the application of the proposed control approach. Simulation results demonstrated that the adaptive control method can successfully track any required angular velocity trajectory even when the dynamics model of the target satellite is unknown. Keywords: Rendezvous and docking, velocity tracking, satellite on-orbit servicing, adaptive control, parameter estimation, inertia estimation.

1. INTRODUCTION Satellite on-orbit servicing includes refueling, repairing and upgrading of a flying satellite on orbit. Such a mission is desirable for many important and expensive satellites, such as the International Space Station and Houble Telescope. Currently, satellite on-orbit servicing can only be carried out manually by astronauts. Such a manned space mission is usually very costly and bears tremendous safety concerns [1]. As an alternative approach, autonomous on-orbit servicing is gaining more and more interest in the space community recently due to its significant advantages in efficiency, cost, and safety (due to unmanned operation). Several major space agencies have launched technology demonstration missions for autonomous satellite on-orbit servicing. Japanese Aerospace Exploration Agency (JAXA) has completed a technology demonstration mission for Engineering Test Satellite – VII (ETS-7) [2-3]. Defense Advanced Research Projects Agency (DARPA) developed a technology demonstration mission for autonomous satellite rendezvous through the Orbital Express Program [4-5]. Germany Space Agency also sponsored the Dynamics of the Equatorial Ionosphere Over SHAR (DEOS) program for a similar purpose [6]. While the technologies demonstrated by these missions can be used for autonomous servicing certain satellites, their use is limited to the known and cooperative satellites. In reality, a satellite may be at faulty or out of control due to some malfunctions. The satellite may not behave as expected in space. Such a satellite is considered as a non-cooperative satellite. Rescuing a non-cooperative satellite may be of high interest to the satellite owners, satellite insurance companies, or other stakeholders for continuation of a critical mission, reduction of economical loss, or avoiding new space debris. However, such an effort is a tremendous technical challenge since the satellite to be rescued and/or serviced in space is not corporative. Researchers in spacerobotics community have initiated various research activities in this challenging subject. Michigan Aerospace Corporation developed an autonomous satellite docking system for an on-orbit demonstration of autonomous rendezvous and docking of two satellites to enable fluid/gas re-supply and payload exchange [7]. Qureshi and Terzopoulos presented a space robotic system capable of capturing a free-flying satellite for the purposes of on-orbit satellite servicing [8]. The system relies on vision and cognition to deal with an uncooperative satellite. A prototype system was also developed to control a robotic arm that autonomously captures a free-flying satellite in a realistic laboratory setting that faithfully *

Email: [email protected]; telephone: (575)646-6534; fax: (575)646-6111. Sensors and Systems for Space Applications III, edited by Joseph L. Cox, Pejmun Motaghedi Proc. of SPIE Vol. 7330, 733004 · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.818250

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mimics on-orbit conditions. Ma et al. proposed an optimal control strategy for a servicing spacecraft to rendezvous with a target satellite in a proximity range [9]. Based on Pontryagin’s maximum principle, the method can generate an optimal approaching trajectory to guide the servicing spacecraft to approach the target satellite. Sakawa also studied the optimal control problem of controlling a free-flying robot to fly from one position and orientation to another [10]. Using the Sakawa-Shindo algorithm, the optimal control was calculated and demonstrated for planar motion. Yoshida et al. investigated the dynamics and control during capturing a non-corporative satellite using a robot [11]. The contact motion and the dynamic conditions were formulated and the impedance control was introduced to realize a wide range of impedance characteristics. They also presented experimental results of using two robot manipulators as a motion simulator of the chaser and target satellite. Matsumoto et al. addressed how to plan a safe kinematic approach trajectory for robotic capture of an uncontrolled rotating satellite [12]. In order to conduct on-orbit service to a non-corporative satellite, the servicing spacecraft has to be able to track and rendezvous with the satellite first and then capture or dock to it. Most of the research work addressing docking or capture of a satellite use a robotic manipulator onboard the servicing satellite. Some of the on-orbit servicing tasks do not necessarily need a robotic arm to do, such as a refueling mission. Even with a very capable manipulator, the servicing satellite still has to rendezvous with the satellite to be serviced before a service task can be done. In other words, both the servicing satellite and the target satellite have to be in synchronized flight (no relative linear and angular velocities) before any subsequent service operation can meaningfully start. Therefore, rendezvous and docking with a noncorporative satellite are critical for the subsequent service operations. This is the motivation of the research described in this paper. The proposed method addresses two critical problems in a proximity rendezvous and docking operation: 1) a servicing satellite tracks the angular velocity of a tumbling target satellite and 2) the servicing satellite stabilizes the angular rotation of the two-satellite compound system after having docked to or captured the tumbling target satellite. Since the dynamics of the target satellite is assumed unknown, all the advanced control technologies that are based on known dynamics cannot be applied to the problem. Instead, adaptive control technology is employed. In the method, the unknown dynamics is estimated while the adaptive controller is tracking a required angular velocity profile. A Lyapunov-based tracking law and an adaptation law are designed in order to guarantee the convergence of the tracking and adaptation procedures. The method has been demonstrated using a three-dimensional dynamics simulation example. The remaining of the paper is organized as follows: the problem studied in this paper is defined in Section 2, which is followed by a description of the tracking law and adaptive law in Section 3. A dynamic simulation example of using the proposed control method is presented in Section 4. The paper is concluded in Section 5.

2.

PROBLEM DEFINITION

Although the major space agencies and the aerospace research community have realized the importance of on-orbit servicing to non-corporative satellites, much research work has to be done in order to have the enabling technologies become mature enough for real space missions. One of the most challenging open issues for autonomously servicing a non-corporative satellite which is tumbling in orbit is how to safely align with and capture the tumbling satellite. Since the angular velocity change of a non-corporative satellite could be drastic, the servicing satellite has to be able to follow such a velocity change before any service operations such as docking or capture can meaningfully start. The issue concerned in this paper is how to control a servicing satellite to rotate itself at the same angular velocity as the target satellite, so that it can eventually capture or dock to the tumbling target. As we all know, any subsequent physical service operation can be safely performed only after both flying vehicles are rigidly docked together (no relative linear and angular velocities between them) and the tumbling motion is stably eliminated. It is assumed in this paper that both the servicing and target satellites are rigid bodies flying and have been in a ready-to-dock distance in the same orbit, as shown in Fig. 1. In other words, the servicing satellite has completed its proximity rendezvous in translation with the target satellite but two satellites still have a significant difference in angular velocity. The target satellite is moving with spinning or tumbling motion in orbit. The motion state (position, velocity and acceleration) of both vehicles are completely known. The inertia properties (mass, mass center, inertia matrix) of the servicing satellite are known while those of the target satellite are not completely known. The servicing satellite is driven by several thrusters that are assumed to be continuously controllable. The locations and directions of the thrusters are completely known. Other than the motion state, all other dynamics and control properties of the target satellite may be unknown. Except the thrust forces (i.e., the control forces) of the servicing satellite, all other external forces are neglected. The task of the servicing satellite is to first track the angular velocity of the target satellite; then dock to it

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after they have the same angular velocity; and finally stabilize the angular rotation of the two-satellite compound system. The operation can be divided into the following three phases: Tracking phase: In this phase, the servicing satellite is controlled to rotate itself aimed at tracking the angular velocity of the target satellite (see Fig. 1). Actively controlled by thrust forces, the servicing satellite adjusts its own angular velocity to match that of the target satellite. At the end of the phase, there should be no relative rotation between the two satellites and the docking interfaces on both satellites have been aligned properly for docking. Docking phase: In this phase, the servicing satellite approaches to the target satellite to make physical contact (soft docking) with the target satellite until the two satellite are rigidly locked together (hard docking) and become a single rigid body (see Fig. 2). For safety consideration, this phase is usually done passively without an active vehicle control. In other words, the soft docking is accomplished passively by the relative momentum between the two satellites. Stabilization phase: In this phase the servicing satellite will use its attitude control system to slow down the tumbling motion of the newly connected two-satellite compound system and stabilize the attitude of the compound system. Since the dynamics model of the newly captured target satellite many not be known, the stabilization task is also very challenging because the control system must have a guaranteed stability even the dynamics model of the system is unknown. This paper deals with the control problems in the tracking phase and the post-docking stabilization phase.

Thrust force

ωs

Tumbling satellite

Servicing satellite

ωt

Docking interface

Fig. 1 A servicing satellite is tracking a tumbling target satellite

ω s = ωt

Thrust force

Tumbling satellite Servicing satellite

ωt

Fig. 2 The servicing satellite has docked to the target satellite

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In general, the attitude dynamics of a rigid satellite is governed by the following Euler equations: & + (ω × I )ω = τ Iω

(1)

& are the angular velocity and angular where I is the inertia tensor of the satellite about its center of mass; ω and ω acceleration of the satellite with respect to the inertia frame, respectively; and τ is the resultant of all the trust torques acting on the satellite. For the convenience of mathematics implementation, all the terms of equation (1) can be represented in the body fixed reference frame of the satellite. Using this dynamics equation, the proposed control method will be derived and discussed in the next section.

3.

CONTROL METHODOLOGY

As discussed in Section 2, the control strategy discussed in the paper covers two phases of a proximity rendezvous and docking mission: the angular–velocity tracking phase and the post-docking stabilization phase. Accordingly, two separate controllers are designed for the two phases. The detailed developments of the two controllers are presented in this section. Unless otherwise indicated, all the inertia, velocity, acceleration, and force terms appeared in the presented mathematical formulation are expressed in the body reference frame attached to the servicing satellite.

3.1 Tracking the Angular Velocity of the Target Satellite Let vectors ω s and ω t be the angular velocities of the servicing satellite and the target satellite, respectively. The relative angular velocity between the two satellites can be defined as ~ = ω −ω ω (2) s t A nonnegative Lyapunov function candidate can be chosen as follows V =

1 ~T ~ ω I sω 2

(3)

where I s is the inertia matrix of the servicing satellite. Differentiating the Lyapunov function with respect to time, one gets ~TI ω ~& = ω ~ T I (ω & t) V& = ω s s & s −ω

(4)

Since the inertia matrix I s is positive-definite, substituting (1) into (4), one has ~ T (τ − ω × I ω − I ω ~T ~ V& = ω s s s s & t ) = −ω ω ≤ 0

(5)

as long as ~ & t = −ω τ − ω s × I sω s − I sω

This condition can be satisfied if the following nonlinear tracking law ~ = ω ×I ω +I ω & t −ω τ = ω s × I sω s + I sω s s s s & t + ωt − ω s

(6)

(7)

is used as the reference of the thrust control. Based upon the Lyapunov theory, one can conclude that the control law given in (7) can guarantee the servicing satellite to track any angular velocity ω t (t ) of the target satellite. In other words, the servicing satellite is able to change its own angular velocity to match that of the target satellite. The only condition is that the angular velocity and angular acceleration of the target satellite must be known in real time because the nonlinear controller needs the information to determine the required thrust forces. Such a condition can be practically met with the currently available sensing and estimation technologies. It should be pointed out that the matching of the angular velocity of the two satellites is only a necessary condition for a subsequent docking operation. It is not a sufficient condition for docking. A sufficient condition would be that the servicing satellite not only matches the velocity of the target satellite but also aligns its docking interface with that of the target satellite. The alignment problem has been discussed in [9].

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3.2 Adaptation Law for Post-Docking Stabilization Right after the servicing satellite docked to the target satellite and physically locked up the two vehicles together, the two satellites become one compound satellite system tumbling. The system may still rotate or tumble with an unwanted angular velocity. Obviously, the next task for the servicing satellite is to slow down the rotating of the twosatellite combined system using its attitude control system. This task is also called post-docking stabilization. This is a challenging task because the inertia properties of the compound system are unknown due to the unknown inertia properties of the target satellite. In other words, the servicing satellite is manipulating an “object” with unknown dynamics. Therefore, the tracking law developed in Section 3.1 is no longer applicable because the inertia matrix of the compound system is unknown. Likely, any other advanced control technologies requiring a known dynamics model will not work either. To solve this problem, adaptive control technology is employed. In this case ω s becomes the angular velocity of the two-satellite compound system. Let ω r be a reference angular velocity which one wants the compound system to achieve eventually. Then the difference in angular velocity is ~ = ω −ω ω (8) s

r

Assume that a is a vector consisting of all the unknown dynamics parameters of the target satellite and aˆ is an estimated a vector. Then, the estimation error is defined as ~ a = aˆ − a (9)

A nonnegative Lyapunov function can be constructed as follows V =

1 ~ T ~ ~T ~ (ω Iω + a A a ) 2

(10)

Where I is the inertia matrix of the two-satellite compound system (i.e., a combination of I s and I t ). Apparently I is unknown because I t is unknown. Moreover, A is assumed to be a symmetric and positive-definite gain matrix. Differentiating (10) with respect to time yields ~ T Iω ~& + 1 ω ~ T I&ω ~ +~ V& = ω a T A~ a& 2 1 ~T & ~ T ( τ − ω × Iω − Iω ~ +~ & r)+ ω [I − 2ω s × I + 2ω s × I ]ω a T A~ a& =ω s s 2 ~ T ( τ − ω × Iω − Iω ~ T (ω × I )ω ~ +~ & r)+ω a T A~ a& =ω s s s ~ T [ τ − ω × Iω − Iω & + (ω × I )(ω − ω )] + ~ a T A~ a& =ω s

s

r

s

s

(11)

r

~ T ( τ − Iω & r − ω s × Iω r ) + ~ a T A~ a& =ω ~ T [I& − 2ω × I ]ω ~ = 0 . This is because [I& − 2ω × I ] is a skew In the above derivation, it has been applied that ω s s symmetric matrix for a multibody dynamical system [14].

Next, one can define the following tracking law ~ & r + ω s × Iˆω r − Kω τ = Iˆω

where matrix K is positive-definite and matrix Iˆ is an estimate of I . Then ~ ~ T (~I ω ~) + ~ & r + ω s × I ω r − Kω V& = ω a T A~ a&

(12)

(13)

where ~ ˆ I = I−I

Since the I matrix is linear in terms of the inertia parameters, one can write

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(14)

~ ~ & r + ω s × I ω r = Y~ Iω a

(15)

& t ) . Substituting (15) into (13) yields where Y = Y(ω s , ω t , ω ~ T Kω ~ +ω ~ T Y~ ~ T Kω ~ +~ ~) V& = −ω a+~ a T A~ a& = −ω a T ( A~ a& + Y T ω

(16)

which suggests us to choose the adaptation law such that ~ =0 A~ a& + Y T ω

(17)

~ aˆ& = − A −1 Y T ω

(18)

T ~ dt = − A −1 Y(ω , ω , ω aˆ = − A −1 Y T ω s t & t ) (ω s − ω t ) dt

(19)

which leads to

or the estimated dynamics parameters vector

∫

∫

a& because the unknown parameters are constants. The resulting expression of V& is Note that aˆ& = ~ ~ T Kω ~ ≤0 V& = −ω

(20)

which indicates that the tracking law given by (12) and the adaptation law by (19) yield a globally stable adaptive controller that can track any required angular velocity of the two-satellite compound system during a post-docking stabilization phase. The positive-definite gain matrix A has a large influence on the convergence speed and the weights on individual inertia parameters to be estimated. It should be pointed out that the estimate of the unknown dynamics parameters does not necessarily converge to their true values (i.e., aˆ or Iˆ does not necessarily converge to a or I ). This should not be a problem because the final goal of the mission is to stabilize the two-satellite compound system to a required angular velocity ω r as opposed to the precise identification of the unknown dynamics parameters. References [15] and [16] have good discussions about how to make more accurate estimation during an adaptive control of a general robotic system. It is done by planning a sufficient rich reference trajectory. In this case, the reference trajectory is ω r (t ) . The same ideas may be applied here but that will be a topic of future research.

4.

SIMULATION EXAMPLE

To demonstrate the proposed adaptive control strategy, an example using three-dimensional dynamics simulation is presented in this section. The simulation example was implemented and performed on Matlab/Simulink (the MathWorks Inc., Natick, MA). 4.1 Simulation of Tracking Angular Velocity using the Tracking Law The servicing satellite is assumed to be already in a proximity range with the target satellite but it still has a large relative angular velocity with respect to the target satellite, so that a docking operation cannot start before the relative angular velocity can be eliminated. The initial angular velocities of the two satellites are assumed to be ⎡5⎤ ω s (0) = ⎢ 8 ⎥ (deg/s), ⎢10⎥ ⎣ ⎦

⎡0⎤ ω t (0) = ⎢10⎥ (deg/s) . ⎢5⎥ ⎣ ⎦

(21)

The servicing satellite is supposed to have an inertia matrix of I s = diag (200 180 80) (Kg ⋅ m 2 ) in its body fixed frame. The satellite is controlled by twelve thrusters. The positions and directions of all the twelve thrusters with respect to the body-fixed reference frame are given in Table 1. The twelve thrusters are grouped into 6 pairs. Thrusters 1 and 7 are in a pair, 2 and 8 in another pair, and so on, as indicated in Table 1. Each pair of thrusters produces a couple moment in one direction about one of the three body axes.

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Therefore, the six pairs of thrusters can produce any required resultant couple moment in all three axes to fully control the attitude of the servicing satellite. Since the thrusters for attitude control have very limited thrust capability, each thruster is assumed to generate a thrust force ranging only from 0 to 2 N. Table 1. Positions and directions of thrusters in the body frame of the servicing satellite Thruster pair

Thruster number

1

1

Thruster position [-0.5 0.3 0.2] T

[1 0 0] T

T

2

2

[-0.5 0.3 -0.2]

3

3

[0.5 -0.3 0.2] T

4

[-0.5 -0.3 0.2]

5

5

[-0.5 0.3 -0.2] T

6

[-0.5 -0.3 -0.2]

7

Thruster position [0.5 0.3 -0.2] T

8

[0.5 0.3 0.2]

[0 1 0] T

9

[-0.5 0.3 0.2] T

T

T

10

[0.5 0.3 0.2]

[0 0 1] T

11

[-0.5 -0.3 0.2] T

[0 1 0]

T

Thruster number

T

[1 0 0]

T

4 6

Thruster direction

[0 0 1]

T

12

T

[-0.5 0.3 0.2]

T

Thruster direction [-1 0 0] T [-1 0 0]

T

[0 -1 0] T [0 -1 0]

T

[0 0 -1] T [0 0 -1]

T

Couple moment Positive y Negative y Positive z Negative z Positive x Negative x

Angular velocity (deg/s)

10

ωy

6

ωz

4 2 0

Angular acceleration (deg/s 2)

ωx

8

0

100

200

300

400

500 Time (s)

600

700

800

900

1000

0.02 0 -0.02

αx

-0.04

αy

-0.06 -0.08

αz 0

100

200

300

400

500 Time (s)

600

700

800

900

1000

Fig. 3 Time histories of the angular velocity and acceleration of the servicing satellite The tracking law defined in Section 3.1 is employed to control the servicing satellite in an attempt to reach the same angular velocity of the target satellite. The tracking results from the simulation are plotted in Figs. 3 and 4. Shown in Fig.3 are the time histories of the angular velocity and acceleration of the servicing satellite. As one can see from the plots, the angular velocity of the servicing satellite is eventually stabilized at [0 10 5]T (deg/s) which is exactly the

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angular velocity of the target satellite. In other words, both satellites have had the same angular velocity at the end. The thrust forces controlling the servicing satellite calculated by the tracking law are shown in Fig. 4. Since thrusters 7-12 have the same forces as their counterpart thrusters 1-6 except in opposite directions, their forces are not plotted. The tracking took a relatively long time to accomplish because each thrust force has been limited to no more than 2 N. 2.5 Thruster 1 Thruster 2 Thruster 3 Thruster 4 Thruster 5 Thruster 6

2

Thrust forces (N)

1.5

1

0.5

0

-0.5

0

100

200

300

400

500 Time (s)

600

700

800

900

1000

Fig. 4 Time histories of the thrust forces of the thrusters 1-6 of the servicing satellite 4.2 Simulation of a Post-Docking Stabilization using the Adaptive Control In this case the two-satellite compound system is assumed to have an inertia matrix of I = diag(250 230 150) (Kg ⋅ m 2 ) expressed in its body fixed reference frame. The initial angular velocity of the compound system prior to the post-docking stabilization is ⎡8⎤ ω s (0) = ⎢10⎥ (deg/s) . ⎢5⎥ ⎣ ⎦

(22)

The reference angular velocity for the satellite controller to track is ω r (t ) = e −α t ω s (0)

(23)

Apparently, the reference ω r is equal to the initial angular velocity of the compound system at the beginning and it becomes zero eventually. The exponent α can be selected based on how fast one wishes the stabilization will take place. To achieve a fast tracking, the 9 × 9 gain matrix A is set to A = diag (10 −6 ,10 −6 , L ,10 −6 ) in the example. The actual angular velocity ω s as a result of the adaptive control is plotted in Fig. 5. The expected angular velocity trajectory ω r (t ) ( α = 0.09 in this case) is also plotted in the same figure for reference. As one can see, the controller tracks the desired angular velocity trajectory very well. The required thrust forces for the tracking are plotted in Fig. 6. Note that each thrust force in this case has been allowed to vary from 0 to 4 N. The result of the three estimated principle

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moments of inertia of the compound system is plotted in Fig. 7. Apparently, the estimated moments of inertia have been adjusted by the adaptation law toward their true values but have not really reached to their true values. This is acceptable in this case because the goal of the online estimation is just to help the controller achieve the desired angular-velocity tracking or the angular-velocity stabilization as opposed to precisely identifying the true values of the estimated inertia parameters. If one really needs to accurately estimate the unknown inertia parameters, an adaptive control process may not be the right approach. Other techniques should be applied instead. For example, references [17-20] have good discussions regarding accurate identification of unknown satellite inertia parameters. Angular velocity (deg/s)

10 Ang.Velocity in X Ang.Velocity in Y Ang.Velocity in Z

8 6 4 2 0

Reference angular velocity (deg/s)

0

10

20

30

40

50 Time (s)

60

70

80

10

90

100

Ang.Velocity in X Ang.Velocity in Y Ang.Velocity in Z

8 6 4 2 0 0

10

20

30

40

50 Time (s)

60

70

80

90

100

Fig. 5 The actual and reference angular velocities of the compound satellite system 4 Thruster1 Thruster2 Thruster3 Thruster4 Thruster5 Thruster6

3.5 3

Thrust Forces (N)

2.5 2 1.5 1 0.5 0 -0.5 0

10

20

30

40

50 Time (s)

60

70

80

90

Fig. 6 Time histories of the thrust forces of the thrusters 1-6 of the servicing satellite

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100

260

Estimated principal moments of inertial (kg.m2)

240 220 200

Ixx Iyy Izz

180 160 140 120 100 80 0

10

20

30

40

50

60

70

80

90

100

Time (s)

Fig. 7 Time histories of the three estimated principal moments of inertia of the compound system

5.

CONCLUSION

This paper presented a control approach for tracking an arbitrary angular velocity of a tumbling satellite and for stabilizing the rotation of the two-satellite compound system right after docking. Using the proposed control method, the servicing satellite can rotate itself to reach the same angular velocity of the target satellite, so that the two flying vehicles will have no relative velocity, which makes the subsequent docking operation possible. This method, based on adaptive control technology, is also able to stabilize the angular velocity of the two-satellite compound system right after docking, assuming that the dynamics model of the target satellite is unknown. Lyapunov-based tracking law and adaptation law were designed and mathematically proven in order to guarantee the convergence of the tracking and adaptation procedures. A dynamic simulation example was provided and discussed to illustrate the application of the control approach. The simulation results demonstrated the effectiveness of the proposed control method. However, the method is presented and demonstrated only in theory. There are still difficult problems regarding the practical implementation of the method, which requires further research.

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