Fly-around Control of Space Tumbling Target under Multiple Constraints

Aiming at the practical problems of external unknown disturbance and internal modeling uncertainty, when space tumbling target flies around in close range, a sliding mode controller (SMC) based on active disturbance rejection control (ADRC) technology is proposed to realize real-time estimation and compensation of “total disturbance.” Firstly, according to the motion characteristics of space tumbling target, the relative motion equation in rotating line of sight (RLOS) coordinate system is established; Secondly, the compound controller is designed, and the convergence of the nonlinear state expansion observer and the stability of the closed-loop system are analyzed based on the root locus method and Lyapunov function method respectively; Finally, the simulation results show that the SMC based on ADRC technology can effectively suppress the disturbance and overcome the chattering problem of traditional sliding mode controller. It has a good control quality, and strong robustness is an easy method for engineering practice.


I. INTRODUCTION
In 1957, when the Soviet Union successfully launched the first man-made satellite, human beings began to explore the mysteries of space. However, the continuous frequency and deepening of space activities have also brought a large amount of space debris such as rocket ejectors, failed satellites, discards, and collision fragments to the space environment. As of November 4, 2021, according to the USA space surveillance network, there are 23522 space targets, including 7824 spacecraft and 15698 rocket bodies and debris [1]. Much space debris has become the main pollution source of the space environment and poses a great threat to the development and safety of the aerospace industry. Therefore, the development of on-orbit repair, fuel filling, capture and recovery, and other on-orbit services is of great significance for promoting the on-orbit service capability and maintaining the safety and stability of the space environment [2]- [9].
Space debris often presents rolling motion states, such as spin and nutation, without cooperative identification, and some uncertain factors, such as orbital maneuvers, exist. It is difficult to accurately model the relative motion of such space tumbling targets [10], [11]. The space environment is complex, and spacecraft will be disturbed by the outside world, such as gravity gradient and solar light pressure, especially in short-range relative motion, which puts forward higher requirements for the accuracy, robustness, and stability of the control technology.
When studying relative motion control with noncooperative targets, most of them follow the existing relative motion modeling methods of rendezvous, docking, and formation flights, which are mainly based on the localvertical local-horizontal (LVLH) coordinate system and the line-of-sight (LOS) coordinate system. In addition, for noncooperative targets, a few scholars have proposed a relative motion modeling method to improve the LOS coordinate system, such as the RLOS coordinate system and an integrated modeling method based on the dual quaternion [12]- [16].

A. RELATIVE MOTION CONTROL BASED ON THE LVLH COORDINATE SYSTEM
This method mainly includes the CW equation for a circular orbit proposed by Clohessy and Wiltshire [17] and the TH equation for an elliptical orbit further derived by Tschauner and Hempel [18]. The CW equation is a linearized equation to solve the relative motion of a spacecraft over a short distance; its precondition is that the target moves in a circular orbit, and the relative distance between two spacecraft is far less than the target's center distance. The TH equation is further derived to obtain analytical solutions related to the eccentricity and true near-point angle. The origin of the base coordinate system for the above two methods is established on the target spacecraft.
In [19], considering the motion characteristics of the space tumbling target, it is possible that the centroid and sensor do not coincide, the CW equation was modified, and a nonlinear suboptimal tracking controller was designed. In [20], an attitude orbit coupling equation was established based on the CW equation. The θ-D optimal control method was proposed to realize attitude orbit synchronization control of a largeangle maneuver. In [21], the TH equation was used to describe the autonomous rendezvous and approach mission of noncooperative targets in an elliptical orbit. The guidance error of tracking spacecraft caused by the dynamic characteristics of noncooperative targets was analyzed, and a guidance scheme based on a convex optimization algorithm was proposed. In [22], while studying the problem of short-range relative motion control, a robust adaptive controller was designed based on the CW equation. In [23], based on the CW equation, a terminal sliding mode adaptive controller considering disturbance was designed for the short-range rendezvous section of the space tumbling target. In [24] and [25], according to the analytical solution of the CW equation, a variety of fly around forms were proposed, and a sliding mode variable structure controller was designed to realize the six-degrees-of-freedom attitude orbit coupling control of spacecraft fly-around quickly. In [26], fast flyaround control was realized for noncooperative targets based on the CW equation. In [27], the fly-around and acquisition of a runaway rolling satellite were studied, with emphasis on the collision avoidance problem when the tracker flies over the target. In [28] and [29], the attitude orbit coupling model was established based on the TH equation and the error quaternion, and an adaptive method was designed to realize uncontrolled tumbling target tracking and approximation control to overcome external interference and system uncertainty.
In addition, many scholars have studied relative motion equations involving perturbations. In [30], a geometric method was proposed to study the relative motion using orbital element differences. This method can be easily used to study perturbation effects. In [31], the state transition matrix of relative motion was obtained using a geometric method that includes the influences caused by the eccentricity of the reference orbit, differential gravitational perturbations, and the equatorial bulge term J2. In [32] and [33], the CW equation was modified by Carter and Humi to include perturbations, and the dynamic equation of the relative motion in a central force field with linear drag was studied. In [34], a generalized analytical solution of relative motion dynamics with arbitrary perturbations was developed using orbital element difference.

B. RELATIVE MOTION CONTROL BASED ON THE LOS COORDINATE SYSTEM
In the relative motion control of noncooperative targets, a method based on the LOS coordinate system is widely used and has the most application prospects.
In [35], under the conditions of parameter uncertainty and external interference, a six-degrees-of-freedom relative motion equation was established based on the LOS coordinate system, and an adaptive finite-time tracking controller was designed to realize a noncooperative target fly-around control task. In [36], for a noncooperative target with maneuvering, a relative orbit dynamics equation based on the LOS coordinate system was established, and the θ-D method was designed for target tracking and approach control. In [37], aiming at the problem of forced fly-around noncooperative targets, a six-degrees-of-freedom attitude orbit coupling model considering control input and dynamic coupling factors was established based on the LOS coordinate system, and an adaptive control satisfying closedloop stability under multiple constraints was designed. In [38], in the case of external interference, unmodeled dynamics, and thrust saturation, an adaptive control law was proposed to realize relative position tracking control of noncooperative targets in the LOS coordinate system. In [39], the Lyapunov method was designed to realize noncooperative target-tracking control based on the LOS coordinate system. In [40], the surveillance fly-around of noncooperative targets was realized based on the LOS coordinate system. In [41], the problem of obstacle avoidance guidance for autonomous rendezvous and docking with noncooperative spacecraft was studied based on the LOS coordinate system.

C. RELATIVE MOTION CONTROL BASED ON THE RLOS COORDINATE SYSTEM
The above relative motion modeling methods have the problem of insufficient representation of the motion state for space tumbling targets. Differential geometry is an effective method for studying the motion law of space curves. The curvature and torsion were used to describe the rotation of the space curve. It was initially used to deduce the space pure proportional guidance law (PPN) in the guidance field. Based on the differential geometry theory, Chiou et al. [42] first constructed the relative motion equation of projectiles under ideal conditions in the arc-length domain. Meng et al. [43] further derived the relative motion equation when the approaching velocity changed against the background of the rendezvous task. However, these methods are still based on an inertial coordinate system. Li et al. [44], [45] further studied the time-domain method, proposing the concepts of the RLOS coordinate system and instantaneous rotation plane of LOS (IRPL), and reduced the relative motion of curves in three-dimensional space to that in two-dimensional space based on the Frenet-Serret active frame theory. On this basis, in [46]- [50], a new relative motion modeling method was proposed to solve the practical problems of the relative motion control of space-tumbling targets in the RLOS coordinate system. Augmented proportional navigation (APN), SMC, and so on were designed for this model.
At present, research on the control of the relative motion equation of space tumbling targets based on the RLOS coordinate system is still in its infancy, and there are few research results, especially on the control accuracy and robustness under multiple constraints, such as external interference, parameter uncertainty, unmodeled dynamic characteristics, and control input saturation.
In this study, the space tumbling target is considered as the research object, and the constraints of external disturbance and internal modeling uncertainty are considered to study the control problem of its close fly-around. In the closed fly-around stage, the tracker continuously measures the relative motion parameters with the space tumbling target by installing a microwave or optical sensor and guides the tracker to complete the fly-around mission. The thruster used in this study mainly adopts the PRISMA layout. There are six main thrusters, two in each group, which are placed in three directions, passing through the centroid of the tracker and perpendicular to each other. Among them, two groups are placed on the plane parallel to the solar panel to provide thrust in the Los direction, and the other group is placed in the direction perpendicular to the solar panel to provide thrust perpendicular to the LOS [48]. First, the relative motion equation in the RLOS coordinate system is derived and transformed into a form that is convenient for the controller design. Second, under the constraints of external disturbance and internal modeling uncertainty, the compound controller is designed based on two control methods: ADRC [51] - [56] and SMC [57] - [62]. The simulation results show that the controller designed in this study can effectively suppress the disturbance, overcome the chattering problem of the traditional SMC, and have good control quality and strong robustness.

II. RELATIVE MOTION MODEL
In this paper, the relative motion dynamic equation will be derived based on the RLOS coordinate system. First, the J2000 geocentric inertial coordinate system e e e e -O X Y Z is defined, as shown in Fig. 1. Geocentric e O is the coordinate origin, the epoch equatorial plane is taken as the datum plane, ee OZ is the normal vector direction of the datum plane, the direction of ee OX axis points from geocentric to J2000 spring equinox, and ee OY is determined by the right-hand rule. The second is the LOS coordinate system s s s s -O X Y Z , the coordinate origin s O is located at the centroid of the tracker, the ss OY axis is the LOS direction pointing to the target centroid, the ss OZ axis is perpendicular to the ss OY axis in the vertical plane, the ss OX axis forms a right-hand system with the ss OY axis and ss OZ axis. The third is the RLOS coordinate system sr θ ω O -e e e , the coordinate origin s O is located at the centroid of the tracking spacecraft, r e is the unit vector in the LOS direction,  e is the unit vector in the LOS angular velocity direction, and  e meets the righthand coordinate system. r is the relative position vector between the tracker and the target,  is the altitude angle,  is the azimuth, c r is the position vector of the tracker in J2000, and t r is the position vector of the target in J2000. The three-dimensional motion equation [44] established in the RLOS coordinate system is (1) where r e and  e constitute the IRPL;  e is the normal vector of IRPL; s   is the angular velocity of rotation of  e ( IRPL around r e ), s s r = The relative position vector between the tracker and the target is as follows： tc =− r r r (2) The LOS direction unit vector r e is as follows: where r  is the relative distance between the tracker and the target.
After deriving Eq. (2) and Eq. (3), the relative velocity vector between the tracker and the target is as follows: tc rr The relative acceleration between the tracker and the target is derived from Eq. (4), and is expressed as follows: 2 r r r r r r = = + + a v e e e (5) Substituting Eq. (1) into Eq. (5), the expression for relative acceleration is converted into Eq. (6).
Among them, t a and c a are the control acceleration of the target and the tracker respectively, and the subscripts "r,  , and  " are the components of the control acceleration in the three directions of the RLOS coordinate system r e ,  e , and  e respectively; The first two sub-equations determine the change law of r, r , and s ω in IRPL, and the third subequation determines the change law of s Ω , when r and s ω are constant. As shown in Fig. 2, in the IRPL, the rotation angle of the LOS is defined as the LOS angle q, and the rotation angle of the normal vector of the IRPL is defined as the IRPL angle η.
Two abstract variables in the equation of relative motion, the LOS rotation rate s ω and IRPL rotation angular velocity s Ω , are equivalently transformed and can be written as follows: ss ω = q Ω = η (8) Considering the noncooperative characteristics of the space tumbling target, problems such as whether there is a maneuver in the motion state are attributed to the uncertainty of internal modeling, which are recorded as r () Ft, () Ft  , and ω () Ft in the direction of each coordinate axis, and the gravity gradient and solar light pressure are attributed to external disturbances, which are recorded as r () wt, θ () wt, and () wt  in the direction of each coordinate axis. The uncertainty of the internal modeling and external disturbance is collectively referred to as the "total disturbance" of the system. Therefore, the relative motion equation can be rewritten as This study considers that in the actual fly-around process, the target orbit maneuvers ( s  is a constant value finally) or does not maneuver ( s 0  = ), and the flying around in the above two cases can be realized in the IRPL without applying the control of the  e direction [47][48][49]. To facilitate the design of the controller in the next step, the system object model should be transformed into a form of direct feedback, and new variables, are defined. According to the new variables, 1 x and 2 x , the relative motion equation can be rewritten as follows:

III. CONTROL PROBLEM DESCRIPTION
During the actual on-orbit operation and fly-around, there are the following practical problems: space debris often presents spin, nutation, and other motion states, without a cooperative logo, and some uncertain factors such as orbital maneuvers exist that make it difficult to accurately model the relative motion of such space tumbling targets, which are disturbed by gravity gradient, solar light pressure, and other external disturbances during on-orbit operation, which are also difficult to determine. To further design the controller and facilitate simulation verification, the following reasonable assumptions are made: Assumption 1: The tumbling target is a regular precession target whose precession angle (ψ) rate and spin angle (φ) rate are constant; the nutation angle (θ) rate is 0.
Assumption 2: External disturbances r () wt and θ () wt are unknown and bounded, and the boundary is an unknown constant.
Assumption 3: The internal unmodelled dynamics of r () Ft and () Ft  are unknown and bounded, and the boundary is an unknown constant.
Control purpose: Considering the constraints of unknown external disturbances and internal modeling uncertainty, a robust controller is designed to realize the fly-around control of the space tumbling target. The relative distance of the LOS direction flies around the radius, and the LOS rotation rate is consistent with the target precession rate.

A. TRACKING DIFFERENTIATOR (TD)
TD [63], [64] is used to arrange the transition process for the desired signal, effectively solve the contradiction between overshoot and rapidity while realizing the purpose of tracking the desired signal as soon as possible, and accurately extracting the differential signal of the desired signal. A TD has many forms, and its structure is usually divided into linear and nonlinear forms. For the relative motion equation in the RLOS coordinate system, the second-order TD is designed as follows: where   is the expected tracking signal of the system, d r is the expected flying radius, d q is the expected LOS angle, and 1 v is the tracking signal of the expected signal.
is the differential signal of the expected tracking signal of the system, d r is the expected relative speed, d q is the expected LOS rotation rate, and 2 v is the tracking signal of the expected differential signal.
where the function of ( ) fsg , xd [65] is an equivalent formula derived using the characteristics of symbolic function and parameter reduction when two conditional statements appear at the same time.

B. NONLINEAR EXTENDED STATE OBSERVER (NLESO)
ESO [66]- [68] is the key link in the entire system. In this study, the external disturbance and uncertainty inside the system are reduced to "total disturbance," and expanded into new state variables 3 x for real-time estimation and compensation. 3 x is expressed as follows: The third-order NLESO is designed as follows:  (14) where 0i   (i=1,2,3) are the parameters of the observer; i  e [69], [70] which can effectively avoid the occurrence of high-frequency flutter as follows:  x This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.

C. IMPROVED NONLINEAR STATE ERROR FEEDBACK LAW (NLSEF)
Based on the idea of "error elimination error," considering the form of the classical ADRC error feedback control law and the characteristics that the traditional SMC is prone to chattering and steady-state error, the integral sliding mode surface function is selected in NLSEF in Eq. (16), and its derivative is given by Eq. (17).  (17) can be rewritten as follows: The exponential approach law with finite-time convergence term is selected as follows: The NLSEF is as follows:  (21) The NLSEF after compensation is as follows: The structural diagram of the controller is shown in Fig. 3.

D. CONVERGENCE AND STABILITY ANALYSIS
First, the convergence of the ESO in the controller is analyzed.
The initial values of the error term and 12 ( , , ) ft xx assumed to be zero. The equivalent gain method is adopted The and in Eq. (14) can be expressed as follows：  [72]. Assuming: (24) Eq. (14) of NLESO can be rewritten as follows: The transfer function from input to output is as follows:  (28) Then, the system  0)) V x is the initial value of () V x .
(1 )  (22) will guarantee the the system is GPFB property of the closed-loop system if the design parameters are selected appropriately.
Proof. The positive-definite candidate of the Lyapunov function is selected as follows: (32) We take the derivative of Eq. (32), and substitute Eq. (17) and Eq. (22) In the case of bounded total interference, the observation error is bounded, and the value depends on the parameters of ESO [65]. It is assumed that 1 , the observation error is as follows: where 0 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.

V. SIMULATION VERIFICATION
The initial parameters of the tumbling target and tracking spacecraft are presented in Table 1.
The initial relative distance, initial relative speed, initial LOS angle, and initial LOS rotation rate are as follows: To verify that the controller designed in this study has strong robustness to external unknown disturbances and internal modeling uncertainty, a simulation comparison is carried out under the four conditions of "total disturbance:" step disturbance, sinusoidal disturbance, uniformly distributed random disturbance, and no disturbance. It should be noted that according to the relative motion equation and the characteristics of the ESO, the disturbance in the vertical LOS direction is set as the ratio of "total disturbance" to the relative distance in the LOS direction. Among them, step disturbance: the uniformly distributed random disturbance is a uniform random distribution function whose amplitude satisfies 0.05 m/s 2 and 0.01 m/s 2 respectively.

A. ESO CONVERGENCE AND OBSERVATION EFFECT SIMULATION VERIFICATION
Before studying the control problem with multiple constraints, the convergence of the ESO is verified by simulation when the "total disturbance" is sinusoidal. Fig. 4 shows the error root locus in the LOS direction and vertical LOS direction. It can be seen that when the error system changes with 1 e , the closed-loop pole is always located on the left half-plane of [S]; that is, any 1 e value system gradually tends to equilibrium, and the ESO has good convergence.

B. COMPARATIVE SIMULATION VERIFICATION
To further prove the superiority of the controller designed in this study, a compound controller is simulated and compared with traditional SMC. As shown in Fig. 6, within the acceptable overshoot range, the controller designed in this study can converge to the expected value faster by adjusting the TD parameters. As shown in Fig. 7, the compound controller designed in this study overcomes the chattering problem of the traditional SMC in both LOS and vertical LOS directions. For quantitative purposes, six popular performance specifications are employed to evaluate the control performance: the rise time, overshoot, settling time, mean absolute error (MAE) , mean absolute control input (MAI) and mean total variation (MTV) of the control [76]. The expressions for the last three performance specifications are as follows:  This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3186996

FIGURE 7. The acceleration comparison between the compound controller and the traditional SMC.
A quantitative comparison of the control performances is presented in Table 2. Note that the overshoot, settling time for the LOS rotation rate, and MAI for the LOS rotation rate (underlined in Table 2) are observed to be larger under the compound controller than under the traditional SMC. Even so, the compound controller is more reasonable when evaluating the performance indices from the overall point of view. System response is preferred in practice, considering the convergence response, disturbance rejection, energy consumption, and control curve smoothness.

C. SIMULATION VERIFICATION UNDER MULTIPLE CONSTRAINTS
Considering the existence of unknown external disturbances, internal unmodeled dynamics, and limited saturation of the control input, the variation curves of the spacecraft fly-around radius, relative speed in the LOS direction, LOS angle, and LOS rotation rate with time under the action of the compound controller designed in this study are shown in Fig. 8. Under the four disturbance conditions, the tracking spacecraft can track the desired radius within 25 s and remain stable; the relative velocity in the LOS direction first increases and then decreases, and converges to 0 when the relative distance in the LOS direction reaches the expected value; the LOS angle reaches the desired curve within 5 s and tracks stably; the LOS rotation rate first increases and then decreases, and converges to near 0.039 m/s after the LOS angle reaches the desired curve. From the locally enlarged view of the relative speed in the LOS direction and the LOS rotation rate, it can be observed that the control effect is slightly affected by the disturbance, and the control quality under different disturbances remains unchanged, which shows that the controller designed in this study has strong robustness.
Considering the existence of unknown external disturbances, internal unmodelled dynamics, and limited saturation of the control input, the change curves of the control acceleration in the LOS and vertical LOS directions are shown in Fig. 9. At the initial stage, they are limited by actuator saturation, but with a reduction in the control error, the saturation phenomenon gradually disappears, and finally achieves the purpose of control. It should be noted that when the "total disturbance" is a random disturbance, because of the phase lag in the estimation of the random disturbance, the random disturbance cannot be directly used to compensate for the control acceleration. In this study, the upper bound of the random disturbance estimation is used to compensate for the control acceleration, and the compensated control acceleration oscillates within an acceptably small range.

VI. CONCLUSION
Considering the on-orbit service task of a space tumbling target as the research background, this study investigates the short-range relative motion control of a space tumbling target under multiple constraints. The relative orbit dynamics model in the RLOS coordinate system is derived to solve the control problem of space tumbling target fly-around under the constraints of external unknown disturbances and internal modeling uncertainty. In this study, ADRC and SMC are combined to design a compound controller, which realizes the real-time estimation and compensation of "total disturbance," reduces the steady-state error of the system, overcomes the chattering problem of the traditional SMC, and effectively improves the robustness of the system. The simulation results show that, under the four conditions of no disturbance, step disturbance, sinusoidal disturbance, and random disturbance, the controller designed in this study can track and compensate for the disturbance in real time, ensure good control quality, and have strong robustness to external disturbances and internal modeling uncertainty. After reasonably arranging the transition process, the sensitivity of the control law to parameters decreases, and the adjustable range of parameters is large, which is a control method that is easy for engineering practice. This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.