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Angular velocity tracking for satellite rendezvous and docking
Acta Astronautica 69 (2011) 1019–1028
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Angular velocity tracking for satellite rendezvous and docking Jianxun Liang, Ou Ma n Department of Mechanical and Aerospace Engineering, New Mexico State University, P.O. Box 30001, MSC 3450, Las Cruces, NM 88003, United States
a r t i c l e i n f o
abstract
Article history: Received 22 July 2010 Received in revised form 30 May 2011 Accepted 6 July 2011 Available online 9 August 2011
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) prior-dock alignment, (2) soft docking and latching/locking-up, and (3) post-docking stabilization. The paper deals with the first and third steps. Lyapunov-based tracking law and adaptation law are proposed to guarantee the success of the nonlinear control procedures with dynamics uncertainties. Dynamics simulation examples are 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. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Rendezvous and docking Angular velocity tracking Satellite on-orbit servicing Adaptive control Inertial parameter estimation
1. Introduction Satellite on-orbit servicing includes refueling, repairing, and upgrading a flying satellite on orbit. Such a mission is desirable for many important and expensive satellites, such as the International Space Station and Hubble 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 onorbit 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].
n
Corresponding author. Tel.: þ 1 575 646 6534; fax: þ1 575 646 6111. E-mail address: [email protected] (O. Ma).
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Defense Advanced Research Projects Agency (DARPA) developed a technology demonstration mission for satellite on-orbit refueling and repairing 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 autonomously servicing certain satellites, their applications are limited only to 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 noncooperative satellite. Rescuing a non-cooperative satellite may be of high interest to the satellite owner, satellite insurance company, 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 cooperative. Researchers in space-robotics community have initiated various research activities in this challenging subject.
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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 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 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-cooperative 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-cooperative 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 servicing 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 non-cooperative 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 communities have realized the importance of onorbit servicing to non-cooperative satellites, much research work has to be done in order to make the technologies mature enough for real space missions. One of the most challenging open issues for autonomously servicing a noncooperative 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-cooperative 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 and have been in a final approaching 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 the two satellites still have a significant difference in angular velocity. The target satellite is moving with spinning or tumbling motion in orbit. The motion states (position, velocity, and acceleration) of both vehicles are completely known. The inertia properties (mass, mass center, and 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
Fig. 1. A servicing satellite is tracking a target satellite.
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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 after the two satellites have the same angular velocity; and finally stabilize the angular rotation of the twosatellite compound system. The operation can be divided into the following three phases [13]: Tracking phase: In this phase, the satellite first aligns its docking interface with that of the target satellite. This is a very challenging task. Some research work has been done on this subject [9,10]. Even after the two docking interface are properly aligned, there still might be angular velocity difference between the two satellites. In this case, the servicing satellite has to adjust its own angular velocity to match that of the target satellite so that the impact during the docking process can be mitigated. At the end of this 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 usually 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 may 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 angular-velocity control problems in the tracking phase and the post-docking stabilization phase. In general, the attitude dynamics of a rigid satellite is governed by the following Euler equations: _ þ ðx IÞx ¼ s Ix
ð1Þ
where I is the centroid inertia tensor of the satellite with _ are the respect to its body-fixed coordinate frame; x and x angular velocity and angular acceleration of the satellite
Fig. 2. The servicing satellite has docked to the target satellite.
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with respect to the inertia frame, respectively; and s is the resultant of all the thrust torques acting on the satellite. For the convenience of mathematics implementation, all the terms of Eq. (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 operation: 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 xs and xt be the angular velocities of the servicing and target satellites, respectively. The relative angular velocity between the two satellites can be defined as
x~ ¼ xs xt
ð2Þ
A nonnegative Lyapunov function candidate can be chosen as follows: V¼
1 T x~ Is x~ 2
ð3Þ
where Is is the inertia matrix of the servicing satellite. Differentiating the Lyapunov function with respect to time, one gets _~ ¼ x ~ T Is x ~ T Is ðx _ s x _ tÞ V_ ¼ x
ð4Þ
Since the inertia matrix Is is positive-definite, substituting (1) into (4), one has _ t Þ ¼ x ~ r0 ~ T ðsxs Is xs Is x ~ Tx V_ ¼ x
ð5Þ
as long as
sxs Is xs Is x_ t ¼ x~
ð6Þ
This condition can be satisfied if the following nonlinear tracking law
s ¼ xs Is xs þ Is x_ t x~ ¼ xs Is xs þ Is x_ t þ xt xs
ð7Þ
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 xt ðtÞ of the target satellite if there is no constraint due to the limitation of thruster forces. In other words, the servicing satellite is able to change its own angular velocity to match that of the target satellite provided that the angular velocity and angular acceleration of the target satellite are known in real time. Such a condition can be practically met with the currently available sensing and estimation technologies.
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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 an end-to-end docking operation. A sufficient condition would be that the servicing satellite not only matches the angular 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].
In the above derivation, it has been applied that _ xs Ix _ xs I is a skew ~ ¼ 0. This is because ½I2 x~ T ½I2 symmetric matrix for a multibody dynamical system [14]. Next, one can define the following tracking law
s ¼ I^x_ r þ xs I^xr Kx~
ð11Þ
where matrix K is positive-definite and matrix I^ is an estimate of I. Then ~ T ðI~ x _ r þ xs I~ xr K x ~ Þ þ a~ T Aa_~ V_ ¼ x
ð12Þ
where
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. 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 two-satellite 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 xs becomes the angular velocity of the twosatellite compound system. Let xr be a reference angular velocity which one wants the compound system to achieve eventually. Then the difference in angular velocity is
^ I~ ¼ II
ð13Þ
Since the I matrix is linear in terms of the inertia parameters, one can write _ r þ xs I~ xr ¼ Ya~ I~ x
ð14Þ
_ r Þ. Substituting (14) into (12) yields where Y ¼ Yðxs , xr , x ~ T Kx ~Þ ~ þx ~ T Ya~ þ a~ T Aa_~ ¼ x ~ þ a~ T ðAa_~ þYT x ~ T Kx V_ ¼ x ð15Þ which suggests us to choose the adaptation law such that ~ ¼0 Aa_~ þ YT x
ð16Þ
which leads to ~ a^ _¼ A1 YT x
ð17Þ
or the estimated dynamics parameters vector Z Z ~ dt ¼ A1 Yðxs , xr , x _ r ÞT ðxs xr Þ dt a^ ¼ A1 YT x ð18Þ _ _ Note that a^ ¼ a~ because the unknown parameters are constants. The resulting expression of V_ is then
x~ ¼ xs xr
~ r0 ~ T Kx V_ ¼ x
Assume that a is a vector consisting of all the unknown dynamics parameters of the compound satellite system and a^ is an estimated a vector. Then, the estimation error is defined as
which indicates that the tracking law given by (11) and the adaptation law by (18) 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 estimated inertia parameters. It should be pointed out that the estimate of the unknown dynamics parameters does not necessarily converge to their true values (i.e., a^ or I^ does not necessarily converge to a or I). This should not be a problem because the final goal of the operation is to stabilize the twosatellite compound system to a required angular velocity xr as opposed to the precise identification of the unknown dynamics parameters. Refs. [15, 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 xr ðtÞ. The same ideas may be applied here but that will be a topic of future research.
^ a~ ¼ aa
ð8Þ
A nonnegative Lyapunov function can be constructed as follows V¼
1 T ~ ~ Ix ~ þ a~ T AaÞ ðx 2
ð9Þ
where I is the inertia matrix of the two-satellite compound system (i.e., a combination of Is and It ). Apparently I is unknown because It is unknown. Moreover, A is assumed to be a symmetric and positive-definite gain matrix. Differentiating (9) with respect to time yields
_~ þ 1 x ~ T Ix ~ T I_ x ~ þ a~ T Aa_~ V_ ¼ x 2 h i _~ þ 1 x _ xs Iþ 2xs I x ~ T Ix ~ T I2 ~ þ a~ T Aa_~ ¼x 2 _ rÞþx ~ T ðxs IÞx ~ þ a~ T Aa_~ ~ T ðsxs Ixs Ix ¼x ~ T ½sxs Ixs Ix _ r þ ðxs IÞðxs xr Þ þ a~ T Aa_~ ¼x T _ r xs Ixr Þ þ a~ T Aa_~ ~ ðsIx ¼x
ð19Þ
4. Simulation example
ð10Þ
To demonstrate the proposed adaptive control strategy, an example using three-dimensional dynamics simulation is presented in this section. The simulation example was
J. Liang, O. Ma / Acta Astronautica 69 (2011) 1019–1028
implemented and performed on Matlab (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
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large relative angular velocity with respect to the target satellite, so that a docking operation cannot start before the relative angular velocity is eliminated. The initial angular velocities of the two satellites are assumed to be 2 3 2 3 5 0 6 7 6 xs ð0Þ ¼ 4 8 5ðdeg=sÞ, xt ð0Þ ¼ 4 10 7 ð20Þ 5ðdeg=sÞ 10 5
Table 1 Positions and directions of the thrusters in the body frame of the servicing satellite. Thruster pair
Thruster number
Thruster position
Thruster direction
Thruster number
Thruster position
Thruster direction
Couple moment
1 2 3 4 5 6
1 2 3 4 5 6
[ 0.5 0.3 0.2]T [ 0.5 0.3 0.2]T [0.5 0.3 0.2]T [ 0.5, 0.3, 0.2]T [ 0.5 0.3 0.2]T [ 0.5 0.3 0.2]T
[1 0 0]T [1 0 0]T [0 1 0]T [0 1 0]T [0 0 1]T [0 0 1]T
7 8 9 10 11 12
[0.5 0.3 0.2]T [0.5 0.3 0.2]T [ 0.5 0.3 0.2]T [0.5 0.3 0.2]T [ 0.5 0.3 0.2]T [ 0.5 0.3 0.2]T
[ 1 0 0]T [ 1 0 0]T [0 1 0]T [0 1 0]T [0 0 1]T [0 0 1]T
Positive y Negative y Positive z Negative z Positive x Negative x
Fig. 3. Time histories of the angular velocities of the servicing satellite and target satellite.
Fig. 4. Time histories of the thrust forces of the thrusters 1–6 of the servicing satellite.
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The servicing satellite is supposed to have an inertia matrix of Is ¼ diagð 200 180 80 Þ ðKg m2 Þ in its body fixed frame, and the target satellite’s inertial matrix is Table 2 Positions and directions of the three thrusters in the body frame of the servicing satellite. Thruster number
Thruster position
Thruster direction
1 2 3
[0.5 0.3 0.2]T [ 0.5 0.3 0.2]T [ 0.5 0.3 0.2]T
[0 0 1]T [ 1 0 0]T [0 1 0]T
It ¼ diagð 300 250 150 Þ ðKg m2 Þ. The servicing 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. 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
Fig. 5. Time histories of the angular velocities of the target satellite and the servicing satellite with 3 thrusters.
Fig. 6. Time histories of the 3 thrusters of the servicing satellite.
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limited thrusting capability, each thruster is assumed to generate a thrusting force ranging from 0 to 5 N. 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 of the servicing satellite and the target satellite. As one can see from the plots, the angular velocity of the servicing satellite eventually converges to the angular velocity of the target satellite. The thrust forces controlling the servicing satellite calculated by the tracking law
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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 5 N. It should be pointed out that the number of the thrusters is not necessarily to be 12. Actually just 3 thrusters can do the job if they are allocated to proper positions and properly aligned. To demonstrate the versatility of the proposed tracking control law, another simulation example is presented with different parameters from the previous simulation example. The inertial matrix of the
Fig. 7. The actual and reference angular velocities of the compound satellite system.
Fig. 8. Time histories of the thrust forces of the thrusters 1–6 of the servicing satellite.
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servicing satellite is Is ¼ diag ð100 300 200Þ ðKgm2 Þ, and the target satellite’s inertial matrix is Is ¼ diagð300 25000 150Þ ðKgm2 Þ. The initial angular velocities of the two satellites are 2
5
3
2
10
3
7 6 7 xs ð0Þ ¼ 6 4 8 5ðdeg=sÞ, xt ð0Þ ¼ 4 15 5ðdeg=sÞ 10
ð21Þ
12
The allocations and orientations of the three thrusters are listed in Table 2. Figs. 5 and 6 show the time histories of both satellites’ angular velocities and the thruster forces when only 3 thrusters are used to control the servicing satellite’s attitude. Fig. 5 shows that the angular velocity of the servicing satellite eventually converges to the angular velocity of the target satellite with the help of just three thrusters. In
Fig.6, the magnitude of the thruster forces are much larger than the thruster forces shown in Fig.4, which is partly due to the different parameters setting in this simulation and partly due to the fact that there are fewer thrusters used to control the servicing satellite. It can also be noticed that the thruster forces take negative values over some time. This requires that the thruster can work on a reverse direction; otherwise at least three pairs of thrusters are required to fully control 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 ¼ diagð250 230 150Þ ðKg m2 Þ expressed in its body fixed reference frame. To ensure that the thruster forces do not exceed 5 N, 6
Fig. 9. Time histories of the three estimated principal moments of inertia of the compound system.
Fig. 10. Comparison of the angular velocities with different gain matrix A.
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Fig. 11. Comparison of thruster forces with different gain matrix A.
pairs of thrusters, as indicated in Table 1, are used to control the combined system. The initial angular velocity of the compound system prior to the post-docking stabilization is 2 3 8 6 xs ð0Þ ¼ 4 10 7 ð22Þ 5ðdeg=sÞ 5
matrix will not affect the thrusters’ force output. However, the choice of the gain matrix A affects the estimated inertial parameters. A comparison study is carried out with different choice of the gain matrix A. In one case the A is set to be, 10I9 where I9 is a 9 by 9 identity matrix; in the other case the matrix is 100I9. The angular velocity trajectories and thrusters forces with different gain matrix A are shown in Figs.10 and 11, respectively.
The reference angular velocity for the satellite controller to track is
5. Conclusion
xr ðtÞ ¼ ea t xs ð0Þ
ð23Þ
Apparently, the reference xr is equal to the initial angular velocity of the compound system at the beginning and it becomes zero eventually. The exponent a can be selected based on how fast one wishes the stabilization to take place. The actual angular velocity xs as a result of the adaptive control is plotted in Fig. 7. The expected angular velocity trajectory xr ðtÞ (a ¼ 0:09 in this case) is also plotted in the same figure for reference. The required thrust forces for the tracking are plotted in Fig. 8. The result of the three estimated principle moments of inertia of the compound system is plotted in Fig. 9. Note that the estimated inertial parameters do not converge to the 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 very good discussions regarding the accurate identification of unknown satellite inertia parameters. In the derivation of the adaptation law for post docking stabilization, a positive definite gain matrix A is introduced. The gain matrix A can be arbitrarily chosen as long as it is a positive definite matrix and the choice of this
This paper proposed a control approach for tracking an arbitrary angular velocity of a tumbling satellite right before docking and for stabilizing the rotation of the two-satellite compound system right after docking. Using this method, the servicing satellite can be controlled to 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 assuming the dynamics model of the target satellite is unknown. Lyapunov-based tracking law and adaptation law were designed and mathematically proven that they can 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 method. However, the method is presented and demonstrated only in theory. There are still difficult problems regarding the practical implementation of the method, which require further research in the future. References [1] N. Davinic, A. Arkus, S. Chappie, J. Greenberg, Cost-benefit analysis of on-orbit satellite servicing, Journal of Reducing Space Mission Cost 1 (1) (1988) 27–52.
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satellite, in: Proceedings of the 7th International Symposium on Artificial Intelligence, Robotics and Automation in Space, NARA, Japan, 2003. S. Matsumoto, S. Jacobsen, S. Dubowsky, Y. Ohkami, Approach planning and guidance for uncontrolled rotating satellite capture considering collision avoidance, in: Proceedings of the 7th International Symposium on Artificial Intelligence, Robotics and Automation in Space, NARA, Japan, 2003. L. Rekleitis, E. Martin, G. Rouleau, R. L’Archevˆeque, K. Parsa, E. Dupuis, Autonomous capture of a tumbling satellite, Journal of Field Robotics 24 (4) (2007) 275–296. M. Vidyasagar, Nonlinear Systems Analysis, SIAM, Philadephia, 2002. J.-J.E. Slotine, W. Li, On the adaptive control of robot manipulators, The International Journal of Robotics Research 6 (3) (1978) 49–59. J.-J.E. Slotine, M.D.D.I. Benedetto, Hamiltonian adaptive control of spacecraft, IEEE Transactions on Automatic Control 35 (7) (1990) 848–852. E.V. Bergmann, J. Dzielski, Spacecraft mass property identification with torque-generating control, Journal of Guidance, Control, and Dynamics 13 (1990) 99–103. S. Tanygin, T. Williams, Mass property estimation using coasting maneuvering, Journal of Guidance, Control, and Dynamics 20 (4) (1997) 625–632. E. Wilson, C. Lages, R. Mah, On-line, gyro-based, mass-property identification for thruster-controlled spacecraft using recursive least squares, 45th Midwest Symposium on Circuits and Systems, Institute of Electrical and Electronics Engineers, Tulsa, OK, August4–7, 2002, pp.334–337. O. Ma, H. Dang, K. Pham, Identification of spacecraft’s inertia property using robotics technology, AIAA Journal of Guidance, Control and Dynamics 31 (6) (2008) 1761–1771.
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Angular velocity tracking for satellite rendezvous and docking Jianxun Liang, Ou Ma n Department of Mechanical and Aerospace Engineering, New Mexico State University, P.O. Box 30001, MSC 3450, Las Cruces, NM 88003, United States
a r t i c l e i n f o
abstract
Article history: Received 22 July 2010 Received in revised form 30 May 2011 Accepted 6 July 2011 Available online 9 August 2011
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) prior-dock alignment, (2) soft docking and latching/locking-up, and (3) post-docking stabilization. The paper deals with the first and third steps. Lyapunov-based tracking law and adaptation law are proposed to guarantee the success of the nonlinear control procedures with dynamics uncertainties. Dynamics simulation examples are 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. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Rendezvous and docking Angular velocity tracking Satellite on-orbit servicing Adaptive control Inertial parameter estimation
1. Introduction Satellite on-orbit servicing includes refueling, repairing, and upgrading a flying satellite on orbit. Such a mission is desirable for many important and expensive satellites, such as the International Space Station and Hubble 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 onorbit 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].
n
Corresponding author. Tel.: þ 1 575 646 6534; fax: þ1 575 646 6111. E-mail address: [email protected] (O. Ma).
0094-5765/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2011.07.009
Defense Advanced Research Projects Agency (DARPA) developed a technology demonstration mission for satellite on-orbit refueling and repairing 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 autonomously servicing certain satellites, their applications are limited only to 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 noncooperative satellite. Rescuing a non-cooperative satellite may be of high interest to the satellite owner, satellite insurance company, 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 cooperative. Researchers in space-robotics community have initiated various research activities in this challenging subject.
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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 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 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-cooperative 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-cooperative 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 servicing 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 non-cooperative 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 communities have realized the importance of onorbit servicing to non-cooperative satellites, much research work has to be done in order to make the technologies mature enough for real space missions. One of the most challenging open issues for autonomously servicing a noncooperative 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-cooperative 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 and have been in a final approaching 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 the two satellites still have a significant difference in angular velocity. The target satellite is moving with spinning or tumbling motion in orbit. The motion states (position, velocity, and acceleration) of both vehicles are completely known. The inertia properties (mass, mass center, and 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
Fig. 1. A servicing satellite is tracking a target satellite.
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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 after the two satellites have the same angular velocity; and finally stabilize the angular rotation of the twosatellite compound system. The operation can be divided into the following three phases [13]: Tracking phase: In this phase, the satellite first aligns its docking interface with that of the target satellite. This is a very challenging task. Some research work has been done on this subject [9,10]. Even after the two docking interface are properly aligned, there still might be angular velocity difference between the two satellites. In this case, the servicing satellite has to adjust its own angular velocity to match that of the target satellite so that the impact during the docking process can be mitigated. At the end of this 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 usually 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 may 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 angular-velocity control problems in the tracking phase and the post-docking stabilization phase. In general, the attitude dynamics of a rigid satellite is governed by the following Euler equations: _ þ ðx IÞx ¼ s Ix
ð1Þ
where I is the centroid inertia tensor of the satellite with _ are the respect to its body-fixed coordinate frame; x and x angular velocity and angular acceleration of the satellite
Fig. 2. The servicing satellite has docked to the target satellite.
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with respect to the inertia frame, respectively; and s is the resultant of all the thrust torques acting on the satellite. For the convenience of mathematics implementation, all the terms of Eq. (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 operation: 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 xs and xt be the angular velocities of the servicing and target satellites, respectively. The relative angular velocity between the two satellites can be defined as
x~ ¼ xs xt
ð2Þ
A nonnegative Lyapunov function candidate can be chosen as follows: V¼
1 T x~ Is x~ 2
ð3Þ
where Is is the inertia matrix of the servicing satellite. Differentiating the Lyapunov function with respect to time, one gets _~ ¼ x ~ T Is x ~ T Is ðx _ s x _ tÞ V_ ¼ x
ð4Þ
Since the inertia matrix Is is positive-definite, substituting (1) into (4), one has _ t Þ ¼ x ~ r0 ~ T ðsxs Is xs Is x ~ Tx V_ ¼ x
ð5Þ
as long as
sxs Is xs Is x_ t ¼ x~
ð6Þ
This condition can be satisfied if the following nonlinear tracking law
s ¼ xs Is xs þ Is x_ t x~ ¼ xs Is xs þ Is x_ t þ xt xs
ð7Þ
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 xt ðtÞ of the target satellite if there is no constraint due to the limitation of thruster forces. In other words, the servicing satellite is able to change its own angular velocity to match that of the target satellite provided that the angular velocity and angular acceleration of the target satellite are known in real time. Such a condition can be practically met with the currently available sensing and estimation technologies.
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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 an end-to-end docking operation. A sufficient condition would be that the servicing satellite not only matches the angular 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].
In the above derivation, it has been applied that _ xs Ix _ xs I is a skew ~ ¼ 0. This is because ½I2 x~ T ½I2 symmetric matrix for a multibody dynamical system [14]. Next, one can define the following tracking law
s ¼ I^x_ r þ xs I^xr Kx~
ð11Þ
where matrix K is positive-definite and matrix I^ is an estimate of I. Then ~ T ðI~ x _ r þ xs I~ xr K x ~ Þ þ a~ T Aa_~ V_ ¼ x
ð12Þ
where
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. 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 two-satellite 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 xs becomes the angular velocity of the twosatellite compound system. Let xr be a reference angular velocity which one wants the compound system to achieve eventually. Then the difference in angular velocity is
^ I~ ¼ II
ð13Þ
Since the I matrix is linear in terms of the inertia parameters, one can write _ r þ xs I~ xr ¼ Ya~ I~ x
ð14Þ
_ r Þ. Substituting (14) into (12) yields where Y ¼ Yðxs , xr , x ~ T Kx ~Þ ~ þx ~ T Ya~ þ a~ T Aa_~ ¼ x ~ þ a~ T ðAa_~ þYT x ~ T Kx V_ ¼ x ð15Þ which suggests us to choose the adaptation law such that ~ ¼0 Aa_~ þ YT x
ð16Þ
which leads to ~ a^ _¼ A1 YT x
ð17Þ
or the estimated dynamics parameters vector Z Z ~ dt ¼ A1 Yðxs , xr , x _ r ÞT ðxs xr Þ dt a^ ¼ A1 YT x ð18Þ _ _ Note that a^ ¼ a~ because the unknown parameters are constants. The resulting expression of V_ is then
x~ ¼ xs xr
~ r0 ~ T Kx V_ ¼ x
Assume that a is a vector consisting of all the unknown dynamics parameters of the compound satellite system and a^ is an estimated a vector. Then, the estimation error is defined as
which indicates that the tracking law given by (11) and the adaptation law by (18) 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 estimated inertia parameters. It should be pointed out that the estimate of the unknown dynamics parameters does not necessarily converge to their true values (i.e., a^ or I^ does not necessarily converge to a or I). This should not be a problem because the final goal of the operation is to stabilize the twosatellite compound system to a required angular velocity xr as opposed to the precise identification of the unknown dynamics parameters. Refs. [15, 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 xr ðtÞ. The same ideas may be applied here but that will be a topic of future research.
^ a~ ¼ aa
ð8Þ
A nonnegative Lyapunov function can be constructed as follows V¼
1 T ~ ~ Ix ~ þ a~ T AaÞ ðx 2
ð9Þ
where I is the inertia matrix of the two-satellite compound system (i.e., a combination of Is and It ). Apparently I is unknown because It is unknown. Moreover, A is assumed to be a symmetric and positive-definite gain matrix. Differentiating (9) with respect to time yields
_~ þ 1 x ~ T Ix ~ T I_ x ~ þ a~ T Aa_~ V_ ¼ x 2 h i _~ þ 1 x _ xs Iþ 2xs I x ~ T Ix ~ T I2 ~ þ a~ T Aa_~ ¼x 2 _ rÞþx ~ T ðxs IÞx ~ þ a~ T Aa_~ ~ T ðsxs Ixs Ix ¼x ~ T ½sxs Ixs Ix _ r þ ðxs IÞðxs xr Þ þ a~ T Aa_~ ¼x T _ r xs Ixr Þ þ a~ T Aa_~ ~ ðsIx ¼x
ð19Þ
4. Simulation example
ð10Þ
To demonstrate the proposed adaptive control strategy, an example using three-dimensional dynamics simulation is presented in this section. The simulation example was
J. Liang, O. Ma / Acta Astronautica 69 (2011) 1019–1028
implemented and performed on Matlab (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
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large relative angular velocity with respect to the target satellite, so that a docking operation cannot start before the relative angular velocity is eliminated. The initial angular velocities of the two satellites are assumed to be 2 3 2 3 5 0 6 7 6 xs ð0Þ ¼ 4 8 5ðdeg=sÞ, xt ð0Þ ¼ 4 10 7 ð20Þ 5ðdeg=sÞ 10 5
Table 1 Positions and directions of the thrusters in the body frame of the servicing satellite. Thruster pair
Thruster number
Thruster position
Thruster direction
Thruster number
Thruster position
Thruster direction
Couple moment
1 2 3 4 5 6
1 2 3 4 5 6
[ 0.5 0.3 0.2]T [ 0.5 0.3 0.2]T [0.5 0.3 0.2]T [ 0.5, 0.3, 0.2]T [ 0.5 0.3 0.2]T [ 0.5 0.3 0.2]T
[1 0 0]T [1 0 0]T [0 1 0]T [0 1 0]T [0 0 1]T [0 0 1]T
7 8 9 10 11 12
[0.5 0.3 0.2]T [0.5 0.3 0.2]T [ 0.5 0.3 0.2]T [0.5 0.3 0.2]T [ 0.5 0.3 0.2]T [ 0.5 0.3 0.2]T
[ 1 0 0]T [ 1 0 0]T [0 1 0]T [0 1 0]T [0 0 1]T [0 0 1]T
Positive y Negative y Positive z Negative z Positive x Negative x
Fig. 3. Time histories of the angular velocities of the servicing satellite and target satellite.
Fig. 4. Time histories of the thrust forces of the thrusters 1–6 of the servicing satellite.
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The servicing satellite is supposed to have an inertia matrix of Is ¼ diagð 200 180 80 Þ ðKg m2 Þ in its body fixed frame, and the target satellite’s inertial matrix is Table 2 Positions and directions of the three thrusters in the body frame of the servicing satellite. Thruster number
Thruster position
Thruster direction
1 2 3
[0.5 0.3 0.2]T [ 0.5 0.3 0.2]T [ 0.5 0.3 0.2]T
[0 0 1]T [ 1 0 0]T [0 1 0]T
It ¼ diagð 300 250 150 Þ ðKg m2 Þ. The servicing 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. 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
Fig. 5. Time histories of the angular velocities of the target satellite and the servicing satellite with 3 thrusters.
Fig. 6. Time histories of the 3 thrusters of the servicing satellite.
J. Liang, O. Ma / Acta Astronautica 69 (2011) 1019–1028
limited thrusting capability, each thruster is assumed to generate a thrusting force ranging from 0 to 5 N. 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 of the servicing satellite and the target satellite. As one can see from the plots, the angular velocity of the servicing satellite eventually converges to the angular velocity of the target satellite. The thrust forces controlling the servicing satellite calculated by the tracking law
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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 5 N. It should be pointed out that the number of the thrusters is not necessarily to be 12. Actually just 3 thrusters can do the job if they are allocated to proper positions and properly aligned. To demonstrate the versatility of the proposed tracking control law, another simulation example is presented with different parameters from the previous simulation example. The inertial matrix of the
Fig. 7. The actual and reference angular velocities of the compound satellite system.
Fig. 8. Time histories of the thrust forces of the thrusters 1–6 of the servicing satellite.
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servicing satellite is Is ¼ diag ð100 300 200Þ ðKgm2 Þ, and the target satellite’s inertial matrix is Is ¼ diagð300 25000 150Þ ðKgm2 Þ. The initial angular velocities of the two satellites are 2
5
3
2
10
3
7 6 7 xs ð0Þ ¼ 6 4 8 5ðdeg=sÞ, xt ð0Þ ¼ 4 15 5ðdeg=sÞ 10
ð21Þ
12
The allocations and orientations of the three thrusters are listed in Table 2. Figs. 5 and 6 show the time histories of both satellites’ angular velocities and the thruster forces when only 3 thrusters are used to control the servicing satellite’s attitude. Fig. 5 shows that the angular velocity of the servicing satellite eventually converges to the angular velocity of the target satellite with the help of just three thrusters. In
Fig.6, the magnitude of the thruster forces are much larger than the thruster forces shown in Fig.4, which is partly due to the different parameters setting in this simulation and partly due to the fact that there are fewer thrusters used to control the servicing satellite. It can also be noticed that the thruster forces take negative values over some time. This requires that the thruster can work on a reverse direction; otherwise at least three pairs of thrusters are required to fully control 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 ¼ diagð250 230 150Þ ðKg m2 Þ expressed in its body fixed reference frame. To ensure that the thruster forces do not exceed 5 N, 6
Fig. 9. Time histories of the three estimated principal moments of inertia of the compound system.
Fig. 10. Comparison of the angular velocities with different gain matrix A.
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Fig. 11. Comparison of thruster forces with different gain matrix A.
pairs of thrusters, as indicated in Table 1, are used to control the combined system. The initial angular velocity of the compound system prior to the post-docking stabilization is 2 3 8 6 xs ð0Þ ¼ 4 10 7 ð22Þ 5ðdeg=sÞ 5
matrix will not affect the thrusters’ force output. However, the choice of the gain matrix A affects the estimated inertial parameters. A comparison study is carried out with different choice of the gain matrix A. In one case the A is set to be, 10I9 where I9 is a 9 by 9 identity matrix; in the other case the matrix is 100I9. The angular velocity trajectories and thrusters forces with different gain matrix A are shown in Figs.10 and 11, respectively.
The reference angular velocity for the satellite controller to track is
5. Conclusion
xr ðtÞ ¼ ea t xs ð0Þ
ð23Þ
Apparently, the reference xr is equal to the initial angular velocity of the compound system at the beginning and it becomes zero eventually. The exponent a can be selected based on how fast one wishes the stabilization to take place. The actual angular velocity xs as a result of the adaptive control is plotted in Fig. 7. The expected angular velocity trajectory xr ðtÞ (a ¼ 0:09 in this case) is also plotted in the same figure for reference. The required thrust forces for the tracking are plotted in Fig. 8. The result of the three estimated principle moments of inertia of the compound system is plotted in Fig. 9. Note that the estimated inertial parameters do not converge to the 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 very good discussions regarding the accurate identification of unknown satellite inertia parameters. In the derivation of the adaptation law for post docking stabilization, a positive definite gain matrix A is introduced. The gain matrix A can be arbitrarily chosen as long as it is a positive definite matrix and the choice of this
This paper proposed a control approach for tracking an arbitrary angular velocity of a tumbling satellite right before docking and for stabilizing the rotation of the two-satellite compound system right after docking. Using this method, the servicing satellite can be controlled to 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 assuming the dynamics model of the target satellite is unknown. Lyapunov-based tracking law and adaptation law were designed and mathematically proven that they can 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 method. However, the method is presented and demonstrated only in theory. There are still difficult problems regarding the practical implementation of the method, which require further research in the future. References [1] N. Davinic, A. Arkus, S. Chappie, J. Greenberg, Cost-benefit analysis of on-orbit satellite servicing, Journal of Reducing Space Mission Cost 1 (1) (1988) 27–52.
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