Deployment of Hub-Spoke Tethered Satellite Formation with Adaptive Sliding Mode Tension Control

This paper studies the deployment control of a spinning hub-spoke tethered satellite formation, which is a challenging issue due to the strong nonlinear coupling between the hub and sub-satellites, and the underactuated nature of the system if no thrust is used for control. The mathematical model of the formation system is established based on the assumption of rigid body of the hub, inextensible tether, and lumped masses of the sub-satellites. Two novel formation deployment controllers are proposed based on tension control and hybrid tension-thrust control strategies, where underactuated sliding mode control and nonsingular terminal sliding mode control method are used, respectively. The adaptive control theory is adopted to estimate the unknown upper bound of the gravitational perturbation caused by the rotation of the system around the hub. It can be proven by the Lyapunov theory that the close-loop systems have bounded and asymptotic stability under these two deployment controllers, respectively. Finally, numerical simulations are conducted to validate the effectiveness and robustness of the proposed controllers.


Introduction
Satellite formation flying is a promising technology that can increase the redundancy and robustness of space missions by using multiple small satellites, thereby replacing a large single satellite in the future [1]. However, in order to keep the formation, regular orbit corrections are needed for satellites in the constellation to eliminate orbital drift, which accelerates fuel consumption and reduces the on-orbit life of satellites.
As a new type of satellite formation flying, tethered satellite formation (TSF) has generated a lot of interest in recent years. The satellites in TSF are connected by tethers to keep a specific constellation in orbit [2], leading high accuracy of formation with much less fuel consumption.
Several TSF constellations are proposed in the literatures based on mission requirements, such as straight chain [3], hub-spoke [4], triangular ring [5], and double pyramid [6], just to name a few.
Among them, the hub-spoke tethered satellite formation (HS-TSF) has attracted extensive attention, especially in the solar wind sail [7] and multi-point measurement missions [8]. Zhao and Cai [9] studied the dynamics of hub-spoke constellation in halo orbits and showed that the HS-TSF performs better than free formation in terms of long-term stability of constellation. Avanzini and Fedi [10,11] discussed the spin stability of hub-spoke constellation in Earth orbits by considering the effects of orbital eccentricity and elasticity of tethers. They found that the constellation can keep rotational stability in the orbit plane with a larger rotation speed to reduce the effect of gravitational perturbations and external disturbances. However, the strong nonlinearity and state coupling of HS-TSF make tether deployment very challengeable, which is critical for any space tether missions. The merit of rotational stability of HS-TSF by the centrifugal effect makes rotation deployment widely adopted. One feasible method is the deployment strategy of Yo-Yo de-spin systems, where the tethers connecting two payloads are pre-wound around the central body [12]. However, it is difficult to conveniently deploy multiple payloads due to the complicated design of the Yo-Yo mechanism containing multiple tethers. Another popular deployment strategy is the radial deployment [13,14]. The deployer of each payload is independent and is placed in the central body together with tethers. This design can be easily extended to multiple payloads, which is more suitable for HS-TSF systems.
Significant research has been undertaken to study the motion of TSF systems located around the second Lagrangian point, because the gravitational and centripetal accelerations at this point are balanced, which is considered to be a fairly suitable condition for a spinning TSF. Wong et al. [15] proposed a linear length law and exponential length law to control the tether deployment of HS-TSF at the L2 point. Zhao et al. [16] employed the presented length laws in [15], and studied the stability of the formation during three-dimensional tether deployment/retrieval. It was found that under the constant tether law the libration motions of the studied linear three-body TSF system are stable during deployment, while under the exponential tether law the out-plane libration departs from the desired position, which means it is difficult for the system to maintain its initial configuration under the latter control law. The mentioned velocity control strategy for the deployment of tethers is characterized by simplicity in terms of the control form. However, it is not sufficient to ensure the stability of libration motions, more control effort is required to guarantee a successful deployment. One feasible approach is to employ thrusters in the deployment control strategy. In [17], apart from the constant tether length law, thrust forces, acting on the subsatellites of a TSF system at the L2 point, are applied to keep the flexible tethers taut and enhance the control of libration motions during tether deployment.
Compared with the TSF system at the Lagrangian points or Halo orbits, when the formation system spins on the Earth orbit, it will always be affected by the gravitational perturbations. The dynamics and control problems of such systems have also been studied for a wild range of cases.
Chong and Misra [18] found that the hub-spoke constellation can spin stably in both the orbital plane and the plane normal to the orbit. Zhai et al. [19] presented different deployment strategies for a HS-TSF system in the Earth orbit plane depending on if there is active thrust to compensate perturbations or not, such as deployment with constant spinning rate and libration angle. To get better deployment results, the optimal control method was widely used to control the tether deployment of HS-TSF with specific criteria. Without the consideration of gravity gradient and with the constant deployment velocity, the criterion of optimal tether deployment control, solved by the Gauss pseudospectral method, was to minimize the torque acting on the central satellite of the two-body HS-TSF [20]. With the consideration of the gravity perturbation in the Earth orbit plane, the optimal deployment control based on the Pontryagin's minimum principle was to control tether tension and thrusts to obtain the deployment profiles with minimized power consumption [21]. The deployment problem based on optimal control method treats the desired deployment state as one of the constraints, so the libration angles during deployment could strictly converge to zero. Although effective, these optimal controllers are computationally heavy and are usually done off-line. This makes the tether deployment of HS-TSF an open-loop control problem, which cannot response to environment disturbances effectively. Particularly, the situation is further complicated by the periodic gravitational perturbances in the tether deployment of HS-TSF on the Earth orbit.
To address the above challenges, close-loop deployment controllers should be developed to guarantee the global stability of the tethered formation in the presence of perturbances. Sliding mode control (SMC) is regarded as one of the widely used methods due to its strong robustness, and has been successfully applied to the deployment of two-body tethered system. An adaptive sliding mode controller was proposed in [22] to deploy an electro-dynamic tether with the consideration of partial unknown parameters. For a more extreme situation, when the nonlinear dynamic model has uncertainty and there exist external disturbances, the terminal sliding mode control (TSMC) approach combined with neural network was employed to deploy a dual-body tethered system [23]. In addition, in order to achieve better dynamic responses, fractional order SMC method, characterized by its favorable historical memory effect, was applied to control the process of tether deployment [24,25]. Although the SMC method-based deployment control has been extensively studied in dual-body tethered systems, few studies have been presented on its application in multi-body TSF systems. An adaptive hierarchical SMC was adopted to stabilize the attitude motions of a rotational multi-satellite inline system [26]. Hallaj et al. [27] proposed a robust SMC controller for the reconfiguration and keeping stage of an electromagnetic TSF system.
The deployment control problem of a spinning HS-TSF system within the orbital plane is studied in the current work. The system is assumed to consist of a central satellite modeled as a rigid body and sub-satellites modeled as point masses. The main novelty of this paper is to propose two robust deployment controllers to ensure that the system has stronger anti-interference ability.
First, a pure tether tension control strategy based on adaptive underactuated SMC is presented to make full use of the structural characteristics of the considered formation system, thereby leading to lower fuel consumption. Next, a hybrid tension-thrust control strategy based on adaptive TSMC is proposed to reduce the risk of tether entanglement during deployment. The TSMC method adopted in the hybrid controller design is expected to achieve rapid convergence of the control system. Besides, the adaptive laws for these two deployment controllers can estimate the upper bounds of periodic gravitational perturbations in system.

Mathematical Modeling
The multi-tether system under consideration, as shown in Fig.1, consists of a central satellite and n sub-satellites, individually connected to the central satellite by n tethers. It is assumed that all the tethers are rigid and of negligible mass. The central satellite is treated as a symmetrical rigid body, while the sub-satellites are simplified as a set of point masses. Moreover, the junction points of the tethers are evenly distributed around the central satellite.
Consider the system is rotating in a Keplerian circular orbit. O-XYZ denotes the Earth-centered inertial frame, in which the origin O is located at the center of the Earth, the OX axis points to the vernal equinox, the OZ axis aligns with the rotation axis of the Earth, and the OY axis completes the right-hand triad. The unit direction vectors of the OX, OY and OZ axis are defined by i, j and k, respectively. Thus, the orbital angular velocity of the system has the form: The motion of the multi-tether system is described with respect to two rotating coordinate systems: the orbital coordinate system -C xyz , and the body- where c i and c j are the unit vectors of the Cx and Cy axes, and r  i r because of the symmetry property of the central satellite.
Then, the velocity vector of the i-th external satellite with respect to the central satellite can be obtained by Thus, the total kinetic energy of the system can be written as [18]: where mc is the mass of the central satellite, mi is the mass of the i-th sub-satellite, J represents the moment of inertial of the central satellite, vc is the orbital velocity of CM of the central satellite The total potential energy of the system can be expressed as: where e  is the geocentric gravitational constant.
The equations that govern the rotation motion of the central satellite, and the deployment of tethers are derived by using the Lagrangian method: where j q are the generalized coordinates, and j q Q are the corresponding generalized forces.
For the i-th tether, the generalized coordinates are selected as the spinning angle of the central body i  , the length i l and the libration angle i  . The equations of motion can be expressed as follows: sin cos cos 3 cos 3 cos cos It has been common practice in dynamic analysis and control-system design to employ the simplified mathematical model for multi-tether system, in which the central body is assumed to be a mass point [16]. However, such a model is insufficient to demonstrate the coupling dynamics of the central body and the tethers, especially in the deployment phase of a spinning tethered constellation. This is one of the main concerns of the current work, and that's why the central satellite considered here is modeled as a rigid body. But in actual fact, overwhelming complexities exist when it comes to the dynamics behaviors of the system. On the other hand, the attitude motion of the central satellite is often expected to be stable in orbit, and it is promising to reach such a goal by employing proper active control [21]. That's where the approximation condition that the central body spins constantly to simplify the dynamic model and make sense for control design during tether deployment come in. In view of all the considerations above, we assume that And, the equations governing the motion of tethers during deployment can be simplified to some extent and studied separately.
As the non-dimensional equations are more convenient, define the non-dimensional time t   . In order to avoid ambiguity, it is important to note that in the rest of the paper all the derivatives in equations represents derivatives with respect to the non-dimensional time  . Thus, the equations of tether length and librations can be rewritten as follows:

Deployment strategies and controller design
The major goal of the current work is to demonstrate effective control for the deployment process of the HS-TSF using tether tension and thrust force. In this section, first of all, the deployment control problem is described, and then, two controllers are derived using sliding mode control technique.

Control Problem Description
The deployment problem for the considered multi-body tethered system is to deploy the tethers fully to form and maintain the desired configuration. After the tethers are deployed to full length at the final time eq t , one has       eq eq eq eq ,0 Furthermore, the equation governing the tether libration motion reduces to: .
As shown in Eq. (8) it can be found that phase trajectories match with the results of the analytical analysis above. That's because the magnitude of the periodic perturbation terms is small [21], so it's reasonable to make an approximation by simply ignoring the perturbations in preliminary analytical analysis or regard them as bounded disturbances in the following controller design. are much more crossover movements between the centers, which means the tether deployment will affect the libration motion and may even cause tether twining around the central satellite. Therefore, a proper deployment control law is required to ensure the stability of the process.

Controller Design
Notice that the dynamic equations of each sub-satellite have the same form, without loss of generality, the subscript i on any symbol is ignored in the following sections.
Define the system state vector as   T  T  1  2  3  4 , , , , , , . After full deployment of tethers, the formation system is expected to achieve the final state: T eq eq 0,0, ,0 l    x . In order to make it convenient for controller design, we introduce the transformation eq L l l  , so that all the system states will converge to 0 at the final time of the deployment. In this way, the new system state vector becomes   T  T  1 2  3  4 , , , , , , And the control task can be described as making the system move from its initial states Then, the dynamic model of the formation system can be expressed in a reduced-order form: where the functions f  and l f are: l eq and the control inputs are: To make sure that the tethers can be deployed successfully, the initial value for tether length needs to be greater than 0, that is 0 0 eq Ll  , which guarantees the perturbation terms are always (I) Tension Controller The advantage of employing tension control is that no fuel consumption is required to control the sub-satellites in the deployment process, except for the tether tension. The presented control problem can be solved using adaptive underactuated sliding mode control method, and the controller with anti-disturbance capacity is proposed to guarantee the stability of the deployment process.
The desired nonlinear sliding mode surface of the proposed control system can be designed as: is added to the previous perturbation term g  , and together form the new perturbation term g  . It is worth noting that the parameter  shouldn't be too small, otherwise the accuracy of tether length control cannot be ensured, but on the other, if  is too large, then the new perturbation term will be greatly enhanced, which may exceed the anti-interference ability of the tension controller.
Taking derivative of the sliding mode surface in Eq. (12) yields: is the sum of the terms related to the perturbations. It can be found from the underactuated formation deployment model, as shown in Eq. (11), that The underactuated tension controller is proposed as: where the parameter 0

Remark 2:
The pure tension control method provides the possibility to deploy the tethers without consuming fuel on the sub-satellites. The tension controller is designed based on the nonlinear underactuated SMC method [28], besides, the term 3 x  is introduced into the system during the controller design to guarantee the accuracy of the tether length control. Furthermore, considering the inherent periodic perturbations in the dynamic system, an adaptive law is combined with the controller to enhance the anti-disturbance capability of the system.

(II) Hybrid Tension-Thrust Controller
It is well-known that keeping the libration angle within an allowable range during the tether deployment is hardly guaranteed using pure tension control strategy. In order to reduce the risk of entanglement between tethers and the central satellite, a hybrid tension-thrust control strategy is presented in this section, and the controller is designed based on the terminal sliding mode control method.
Considering the system model shown in Eq. (10), define the deployment error signals as: (21) Then, the nonsingular terminal sliding surfaces are given as follows [29]: According to the value ranges of the parameters in Eqs. (22) and (23), it can be found that the The parameter j  could be selected to be a value greater than the estimation error of the upper boundary of the perturbation terms, that is, set ) are fulfilled in finite time [29]. For the region outside the vicinity, the sliding mode surface would converge to zero in finite time according to the conclusion drawn for the first case.
In summary, the sliding mode surface 0 j s  can be reached in finite time from anywhere of the phase plane under the proposed deployment controller with the adaptive law. After that, the finite-time convergence of the system states will be presented.
When the system reaches the sliding surface, one obtains: where   F  represents the Gauss's Hypergeometric function [31].
Hence, Theorem 2 is proven completely, which means that the proposed hybrid deployment controller shown in Eq. (23) and the adaptive law in Eq. (24) can ensure that the hub-spoke formation system can be deployed to the desired equilibrium state in finite time.

Remark 3:
Taking thrust as one of the control forces could improve control accuracy for the libration motion of tethers, thereby effectively reducing the risk of tether entanglement. The hybrid tension-thrust controller proposed based on the nonsingular TSMC method is able to deploy tethers fully in finite time without singularity of the control input. Besides, in this section an adaptive law is also developed to enhance the tolerance of the control system to the inherent perturbations.

Numerical Simulations and Discussions
In this section the effectiveness of the proposed two deployment controllers is validated by numerical simulations. All simulations were performed in MATLAB/SIMULINK environment.

Numerical Simulation Settings
The orbit altitude of the HS-TSF in simulation studies is set as h = 1000km. The central satellite is assumed as a uniform sphere with the radius r = 1m and mass mc = 300kg. Thus, the inertial moment of the central body is tether tension T and thrust force t F respectively in order to make the control inputs more intuitive. which is nearly 1.5 rad, and then swings back toward the equilibrium state. Although there wouldn't entanglement between tethers and the central body under the two controllers, the risk of tether entanglement is obviously larger without thrust. Besides, Fig. 5 and Fig. 7 show that the curves of tether length rate and libration angle rate fluctuate slightly around zero after the tethers are deployed completely, which is partly caused by the persistent periodic perturbations existed in the system. And it is worth noting that the fluctuation of the libration angle rate is smaller under the hybrid tension-thrust controller.    The control inputs are shown in Figs. 8 and 9. The tether tension in both controllers is always larger than zero and eventually stabilize at about 10.3N, which means that there is no slack in the tethers during deployment process. The thrust force increases rapidly to about 0.4N at the initial moment of deployment to stabilize the libration angle, and gradually converges to zero as the tethers are fully deployed.  Besides, since the magnitude of the relative potential energy is much smaller than that of the relative kinetic energy, the curves of T and r E are almost the same.   The above results demonstrate that both the proposed controllers are capable of deploying the formation system in different initial states. Next, take the first set of the initial states         Without loss of generality, the initial system state here is set as   1 x .

(I) Simulations with different initial system states
It can be seen from the dynamic model of the formation system, shown in Eq. (10), that a higher spin rate of the central body will result in a higher frequency of the inherent periodic perturbations. It is expected that the proposed controllers combined with adaptive laws could deal with the influence of the perturbations and thus improve the anti-disturbance ability of the system.
The simulations results, shown in Figs. 18 and 19, indicate that the formation system is successfully deployed under the presented controllers despite the spin rate varies. When using the tension controller, if the spin rate of the central satellite increases, the time required for the deployment process decreases. And the maximum libration angle also slightly decreases as tethers are deployed from the central body spinning at a higher speed. While the deployment process is almost unchanged with the different spin rate when the tension-thrust controller is employed in system. This indicates that the deployment under the hybrid controller is less sensitive to changes of the spin rate.  The control inputs of the proposed controllers are shown in Figs. 20 and 21. It is clear from the figures that a larger control tension is required to deploy the formation system into a configuration with a relatively high rotation rate. This may be explained from the aspect of the centrifugal forces generated by the rotation of the system. In addition, when it comes to the hybrid controller, if the central body spins at a higher rate, the maximum thrust during the deployment process will becomes larger, as shown in Fig. 21, for the reason that a greater thrust force is needed to provide rotational tangential acceleration.

Conclusions
Deployment control of a spinning HS-TSF system has been studied. The dynamic model, treating the central satellite as a rigid body and the sub-satellites as point masses, was used for controller design. First, a tension controller based on the nonlinear underactuated SMC method was derived, enabling to deploy the HS-TSF system to the desired configuration, which shows the possibility of controlling the deployment of a spinning HS-TSF without consuming fuel of the subsatellites. Then, a hybrid controller that takes tether tensions and thrusts as control inputs was obtained using the nonsingular TSMC method. In addition, adaptive laws were introduced in the controller design to estimate the unknown upper bounds of perturbations. Using the Lyapunov stability analysis, the closed-loop system of the HS-TSF is stable in a neighborhood of the equilibrium state under the tension controller, while the tension-thrust controller could guarantee the finite-time stability of the system. The simulation case studies show the validity of the proposed controllers in different initial system states and different spin rates of the central satellite. The results reveal that successful deployment of the HS-TSF can be obtained by either only tension or hybrid controller. In particular, the hybrid tension-thrust controller shows advantages when the demands for safety and rapidity are more concerned, because it demonstrates better performance in terms of deployment velocity and reducing the risk of tether entanglement.