Abstract
This paper presents a robust controller with an extended state observer to solve the Synchronous Fly-Around problem of a chaser spacecraft approaching a tumbling target in the presence of unknown uncertainty and bounded external disturbance. The rotational motion and time-varying docking trajectory of tumbling target are given in advance and referred as the desired tracking objective. Based on dual quaternion framework, a six-degree-of-freedom coupled relative motion between two spacecrafts is modeled, in which the coupling effect, model uncertainties, and external disturbances are considered. More specially, a novel nonsingular terminal sliding mode is designed to ensure the convergence to the desired trajectory in finite time. Based on the second-order sliding mode, an extended state observer is employed to the controller to compensate the closed-loop system. By theoretical analysis, it is proved that the modified extended-state-observer-based controller guarantees the finite-time stabilization. Numerical simulations are taken to show the effectiveness and superiority of the proposed control scheme. Finally, Synchronous Fly-Around maneuvers can be accomplished with fast response and high accuracy.
1. Introduction
Synchronous Fly-Around (SFA) technique represents the concept of proximity operation for driving a chaser spacecraft to fly around a space tumbling target with attitude synchronized. With the number of failed spacecraft and space debris increasing, on-orbit service operations are urgently required to extend life of the failed spacecrafts, such as assembly, repairing, module replacement, detumbling, refueling, and orbital debris removal [1, 2]. Under the influence of terrestrial gravitational perturbation, most of the failed spacecrafts have evolved into space tumbling targets, developing the noncooperative feature [3]. The SFA approach (see Figure 1) can be identified as a significant technology for on-orbit service missions since it performs high accuracy in close-range proximity to noncooperative tumbling target. Some studies have been performed with the intention to solve the SFA problem [4–7].
As pointed out in [8], it also introduces a series of problems and challenges in SFA process. Different from traditional cooperative space missions, rendezvous and docking (RVD), and spacecraft formation flying (SFF), the noncooperative characteristics of SFA will result in deficiency of shape structure and quality information, which causes low accuracy of navigation in close proximity. Another problem is complicated modeling. It is obvious that SFA is a six-degree-of-freedom (6-DOF) relative motion, in which both the translational motion and rotational one are included. Mutual coupling between the two motions will also lead to complexity in modeling. Therefore, high precise modeling is an essential step to overcome the highly coupled drawback during SFA mission. Ma et al. [9] used C-W dynamics equation to establish the relative translational motion and adopted modified Rodrigues parameters to describe the relative rotational dynamics. Similar modeling method has also been performed in [10]. More recently, Xu et al. [11] considered the dynamical coupling inside the 6-DOF model and presented a nonlinear suboptimal control law for SFA problem. However, all the abovementioned studies gave the two-part relative motion and ignored the nonlinear coupling effect [12], such that it would cause the problem that rotational and translational control will not actuate simultaneously. Therefore, it is significant to establish integrated 6-DOF relative motion considering coupled term. Dual quaternion [13–15] gives us inspiration to solve SFA problem since it has been an effective tool for motion description in physical problems and mathematical calculation. It combines traditional Euler quaternions with dual numbers and inherits elegant properties of both. Based on dual quaternion, the coupled 6-DOF dynamics motion between two spacecrafts can be derived successfully. Wang et al. [12] made attempt to use dual quaternion to describe the coupled dynamics of rigid spacecraft and investigated the coordinated control problem for SFF problem. Wu et al. [16, 17] proposed a nonlinear suboptimal control based on dual quaternion for synchronized attitude-position control problem. Under dual-quaternion framework, Filipe and Tsiotras [18] proposed an adaptive tracking controller for spacecraft formation problem and ensured global asymptotical stability in the presence of unknown disturbance. For the safety problem of RVD, Dong et al. [19] proposed a dual-quaternion-based artificial potential function (APF) control to guarantee the arrival at the docking port of the target with desired attitude. The above research results illustrate the effectiveness of 6-DOF modeling using dual-quaternion framework and also provide the motivation of this paper.
The guidance, navigation, and control (GNC) systems for space proximity operations should be taken into account during on-orbit SFA missions [20]. Due to the nonlinear and highly coupled dynamics, designing a controller for the accurate SFA operation is still an open problem. In addition to the preceding interests in 6-DOF modeling, uncertainties and disturbances are another key issue that should be addressed in the control system, which will severely reduce the performance of the controller or lead to instability of the closed-loop system. Therefore, it is urgent to realize accurate and fast SFA tracking control, under strong coupling effects and multisource interferences in practical process. In view of this problem, various nonlinear control methods have been carried out to estimate the disturbance and compensate the system, such as state-dependent Riccati equation (SDRE) control [21], adaptive control [22] and suboptimal control [11]. Among these methods, terminal sliding mode (TSM) control is an effective technique due to its robustness to system uncertainties and can provide finite-time convergence. Wang and Sun [23] carried out an adaptive TSM control method for spacecraft formation flying problem within dual-quaternion framework. However, the major disadvantage of initial TSM is the singularity problem. In order to eliminate this problem, Zou et al. [24] proposed a nonsingular TSM control (NTSMC) to realize the finite-time stability of satellite attitude control. Extended-state-observer (ESO) methods have recently been introduced to compensate for unmodeled dynamics and system with external disturbance and uncertainty. Conventional ESO has shown fine performance in disturbance rejection. Based on a new technique known as the second-order sliding mode (SOSM) [25], a novel ESO can guarantee high accuracy and robustness of the estimation with finite-time convergence. Some studies have recently combined conventional TSM and ESO for spacecraft control problem [26, 27]. Moreover, the combination of NTSMC and ESO is also widely used, especially in aerospace engineering, such as attitude tracking control [28, 29], underactuated spacecraft hovering [30], and reusable launch vehicle [31]. In Reference [28], an ESO-based third-order NTSMC was proposed for the attitude tracking control problem. Ran et al. [32] proposed an adaptive SOSM-based ESO with fault-tolerant NTSMC for the model uncertainty, external disturbance, and limited actuator control during spacecraft attitude control. Zhang et al. [33] presented an adaptive fast NTSMC using estimated information by SOSM-based ESO to solve the problem of spacecraft 6-DOF coupled tracking maneuver in the presence of model uncertainties and actuator misalignment. Moreno and Osori [25] gave a Lyapunov finite-time stability proof for a closed-loop system combining SOSM controller and ESO. To the best of the authors’ knowledge, ESO-based NTSMC is still an open problem for SFA mission within dual-quaternion framework.
This paper aims to investigate the Synchronous Fly-Around maneuver problem in the presence of model uncertainties and external disturbances. Compared to existing results, the major contributions and differences of this study are summarized as follows:(1)To date, it is the first time to use dual quaternion to describe 6-DOF relative dynamics for tumbling target-capturing problem, in which the nonlinear coupled effect, model uncertainties, and external disturbances are taken into account. Moreover, the error kinematics and dynamics are given to improve modeling accuracy significantly.(2)Motivated by the concept of SOSM, a novel ESO is proposed to reconstruct the model uncertainties and external disturbances with finite-time convergence. Based on the estimated information of ESO, a fast NTSMC is carried out to eliminate the total disturbances and ensure the finite-time stable of the closed-loop system. The proposed method can gain faster response and higher accuracy than existing methods.
This paper is organized as follows. Some necessary mathematical preliminaries of dual quaternion and useful lemmas are given and coupled kinematics and dynamics are derived for the relative motion problem in Section 2. In Section 3, NTSMC and ESO strategies are elaborated, and the finite-time stability analysis of the system is performed in Section 4. Following that, numerical simulations are illustrated and discussed in Section 5 to show the effectiveness of the proposed method. Finally, some appropriate conclusions are presented in Section 6.
2. Mathematical Preliminaries
This study considers the leader-follower spacecraft formation, including a chaser and a tumbling target (Figure 2). In order to facilitate the description of the model, the following coordinate system is used: is the Earth-centered inertial coordinate system, with the Cartesian right-hand reference frame; is LVLH (local-vertical, local-horizontal) reference frame with the origin at the center of mass of target, is pointing to the spacecraft radially outward; is the axis normal to the target orbital plane, and is the axis established by right-hand rule; (respectively, the tumbling target, ) is the body-fixed coordinate system of the chaser, with its origin in the center of mass of the chaser and axes pointing to the axis of inertial; is the desired tracking reference frame, with the capturing point of the tumbling target. The control objective is to control the chaser () synchronize with desired state () so that the on-orbit service can be guaranteed in the next step.
2.1. Dual Quaternion
Dual quaternion can be regarded as a quaternion whose element is dual numbers and can be defined as [14, 15, 23].where is the dual scalar part; is the dual vector part; and are the traditional Euler quaternions, representing the real part and dual part of dual quaternion, respectively; and is the called dual unit with and . In this paper, is employed to denote the set of dual quaternions and denotes the set of dual numbers. Therefore, it is supposed that , , and .
Similar to the traditional quaternion and dual vector (see Appendix), the basic operations for dual quaternions are given as follows:where and , is a scalar, and represents the conjugate of .
Using dual quaternion, the frame with respect to frame can be defined aswhere and represent the dual quaternion of frame of the target and the chaser, respectively; is the relative quaternion; and are the relative position vectors of the chaser relative to the target, expressed in frames and , respectively.
2.2. Relative Kinetics and Coupled Dynamics
Referring to [12, 18, 23], the kinematics and coupled dynamics equation of with respect to using dual quaternion representation can be described as follows:where is the relative velocity-expressed frame ; is the relative angular velocity; and is the relative linear velocity between two frames. As can be seen from the definition, can also be written as . Besides, the force is the total dual force; is the dual gravity; is the dual control force; is the dual disturbance; and is the dual gravity force and is the dual gravity torque referred to the chaser ( and , respectively). is the dual inertia matrix of chaser, in which and are the mass and moment of inertia of chaser.
2.3. Lemmas
The lemmas useful for this study are introduced as follows. Firstly, the following system is considered:where is the system status and is the control input.
Lemma 1. (see [34]). Suppose is a positive definite function and there exists scalar , , and satisfiesthen there is an field so that the can reach equilibrium point in finite time. And the convergence time is satisfied as
Lemma 2. (see [35]). Suppose is a positive definite function and there are scalars and , , and satisfiesthen there is a field so that the can reach equilibrium point in finite time. And the convergence time is satisfied as
2.4. Problem Formulation
The control objective of this paper is to design a controller so that the relative motion state of the chaser converges to the desired state in finite time. In order to describe the relative motion between the chaser and the capturing point of tumbling target, the relative motion between the reference frame and frame is established. Firstly, the description of target’s attitude is given which is considered as the general Euler attitude motion. The kinematics and dynamic are shown as follows, respectively:where is the quaternion of the tumbling target; is the angular velocity expressed in ; is the moment of inertia of tumbling target; and is the total disturbance torque acting on target. The operation represents the multiplication of traditional quaternion.
The position vector of reference point with respect to the centroid on the tumbling target in the frame is represented as follows:where is the attitude quaternion of the chaser relative to the target and is the position of the noncentroid point with respect to the centroid at the initial moment.
Remark 1. Referring to [36], some information of relative motion can be measured. The model of 3D reconstruction which is obtained by the mathematical alignment of the image plane and binocular stereo matching is established to determine the relative position and pose of the target’s capturing point. In such a way, relative rotational and translational motion between two spacecrafts can be obtained.
Considering the external disturbances and model uncertainty of the spacecraft, the error kinematics and dynamics equations based on and can be obtained aswhere and are the dual tracking error; and are the rotation error and translation error, respectively; and are the angular velocity error and linear velocity error, respectively; and are the nominal and uncertain parts of the dual matrix inertia matrix; is the bounded disturbance; and are the dual gravity associated with and ; and is the uncertainty of the dual inertia matrix and the value of which is
3. Controller Design
This section mainly studies the control problem, and modified controller of ESO-NTSMC is given. By defining the dual quaternion tracking error as with , the controller is designed to make the tracking error converge to in finite time.
3.1. Nonsingular Terminal Sliding Mode Controller Design
In order to achieve the control objective, a fast terminal sliding surface is defined aswhere and is a constant which satisfies and and are both positive normal numbers. The time derivative of the sliding surface is
As can be seen, when , , and , the singularity may occur in equation (15). Motivated by [24, 37], an alternative form of sliding surface is proposed as follows to avoid singularity:where is
In the above formula, and are the parameters; is the nominal sliding surface; and is a positive small constant. Therefore, the time derivative of sliding surface (16) is
Remark 2. The conventional nonsingular sliding mode (NTSM) in Reference [38]was proposed aswhere ; ; and and are the positive odd integers satisfying . In Reference [39], it has been proved that the NTSM (19) is still singular when a dual sliding mode surface control method is designed. Furthermore, as explained in Reference [24], the NTSM (19) has slower convergence rate than the linear sliding mode when the sliding mode is reached. Therefore, in order to overcome this disadvantage, the fast NTSM is proposed in this paper. The singularity of TSM is avoided by switching from terminal to general sliding manifold, which is function in equation (17).
Equation (18) is continuous, and singularity is avoided. According to above derivation, the paper propose a fast NTSM surface aswhere is the positive dual constant, with and , so does ; the operation of is defined in Appendix; is designed as follows:where ; ; is a positive value; and ; and .
Taking derivation of the sliding surface with respect to time yieldswhere isThe NTSMC method iswhere is the switching term, with , satisfying ; ; and is the equivalent term and the value of which is
3.2. Extended-State-Observer Design
The ESO method shows very fine performance in disturbance rejection and compensation of uncertainties that the accuracy of the controller can be improved. Based on the SOSM method mentioned in References [25, 32], and by combining the linear and nonlinear correction terms, a novel extended state observer is proposed as follows:where and are the outputs of the observer; is the total disturbance and also the extended state variable; and represent the observer measurement; ; are the observer designed parameters, respectively; and is
The proposed ESO (26) can not only obtain the feature of SOSM, such as finite-time convergence but also weaken the chattering of conventional first-order sliding mode also. In addition, it inherits the best properties of linear and nonlinear correction terms. Then, the error dynamics can be established from equation (26) in the scalar form:where is the derivative of and the amplitude of is assumed to be bounded by a positive value , that is .
Based on the NTSMC and ESO designed above, the integral controller is designed aswhere , , and are given by
Under the proposed controller in equations (29) and (30), the control objective can be achieved for tracking tumbling target. The closed-loop system for coupled rotational and translational control is shown in Figure 3.
4. Stability Analysis
In this section, the stability analysis is divided into two parts. In the first part (Theorem 1), the finite-time convergence of ESO is proved, which implies that the proposed ESO can estimate the exact state and total disturbances within a fixed time. In the second part (Theorem 2), the finite-time stability of the closed-loop system is proved, which indicates that the finite-time convergence of the tracking error can be obtained. The proofs of finite-time convergence of ESO and total closed-loop system will be given using Lyapunov stability theory.
Theorem 1. Suppose the parameters of ESO in equation (26) satisfy the following:
Then, the observer errors will converge to the following region in finite-time:where , , and the matrix satisfies the following expression:
The convergence time is upperbounded by , where is the initial Lyapunov state and and are the constants depending on the gains.
Proof. To verify the finite-convergence performance, in accordance with the idea of Moreno and Osorio [25], a Lyapunov candidate is proposed aswhere and the matrix is defined asIt is obvious that is positive definite and radially unbounded if and are positive. The range of is obtained as follows:where is the Euclidean norm of and and represent the minimum and maximum eigenvalues of the matrix, respectively.
The time derivative of can be obtained asSubstituting equation (28) into equation (37) yieldswhere the matrices , , and are given byIt is obvious that is a symmetric matrix, and if and are positive, it is positive definite. In addition, calculate the determinant of asSubstituting equation (31) into equation (40), it yields . Therefore, is the positive definite matrix. According to the definition of , it can be obtained thatAccording to equations (34)–(41), one obtainswhere , , and . Based on equation (36), we assume that , which implies that . And it is obvious that . By substituting and into equation (42), one obtainsTherefore, according to Lemma 2, it is obvious that is definitely negative and is positive. Therefore, observer error will converge to the small set in finite time and the convergence time satisfies . The proof of Theorem 1 is completed.
Remark 3. As seen in Theorem 1 and its proof, the reconstruction error of the model uncertainties and external disturbances can be driven to the neighborhoods of the origin in finite time. The parameter gets larger, and the region will be smaller by choosing adequate ESO parameters, , , , , and , which implies that the proposed ESO has high accuracy. In this way, the output of observer will not influence the controller. Therefore, the finite-time convergence of the closed-loop system, mentioned in Theorem 2, is proved on the basis of Theorem 1. Some studies containing ESO-NTSMC analyzed the total finite-time convergence of the closed-loop system in the same way [32, 33].
Theorem 2. Based on the estimated state in observer (26), and with the application of the controller in equations (29) and (30), the close-loop system states will be driven to a neighborhood of the sliding surface in finite time. Moreover, the tracking errors and will converge to a small region in finite time.
Proof. The proof includes two consecutive steps:
Firstly, it is to be proved that the sliding surface will converge to in finite time. Define the form of the Lyapunov function as follows:Taking the time derivative of in equation (20), it yieldsWith the application of equations (29) and (30), can be obtained asTake the time derivative of yieldswhere is the total external disturbance and model uncertainty and is the error state of ESO. Recalling from the stability in Theorem 1, the ESO errors can enter the small set in fixed time . After finite-time , the ESO errors satisfy . In consequence, it can be obtained aswhere , is the positive scalar. For equation (48), if , that is, or , then one can obtainwhere is a positive number. Therefore, it can be proved by Lemma 1 that the sliding surface will converge to the set in finite time.
Secondly, when the close-loop system state reaches the neighborhood of , it should be proved that the tracking error and will converge into small regions containing the origin in finite time. Due to different conditions of and in (23), the following cases should be taken into consideration.
Case 1. If , equation (20) becomesConsider the following Lyapunov functionTaking the time derivative of yieldswhere and are the parameters.
Therefore, according to Lemma 2, the error states and will converge to in finite time.
Case 2. If , which indicates that converged into the set . One can obtainTherefore, will eventually converge to the region as follows:
Case 3. If , it implies thatOnce and , then is still kept in the form of terminal sliding mode, which implies that will be bounded as .
In conclusion, and will be bounded as and in finite time, where and . Based on the analysis above, it can be concluded that the control objective can be accomplished in finite time. The proof of Theorem 2 is completed.
Remark 4. The total finite-time stability of closed-loop is achieved on the basis of the above proof. With the application of ESO and its estimation of total disturbances, the controller is reconstructed by adding the disturbance rejection term in equation (30). Therefore, the robust controller can guarantee the finite-time stability of the overall closed-loop system with high accuracy. By choosing suitable controller parameters and such that is smaller enough, the convergence set can be a sufficiently small set. Therefore, it can be seen that, under the proposed controller method, the closed-loop system can achieve finite-time convergence with satisfactory performance.
Remark 5. Compared with the existing NTSMC methods, the structure of proposed controller scheme in this research combines ESO and NTSMC techniques, which can obtain higher tracking accuracy and achieve finite-time convergence. The overall controller is developed based on the estimated information of external disturbances and model uncertainties by ESO. Therefore, the controller does not need the priori information of perturbations. Moreover, based on the proof in equation (48), the ESO-based controller can achieve smaller steady-state error when the estimation of ESO is applied to develop the robust controller, which indicates that the system has better performance and higher accuracy.
5. Simulation Results
In this section, numerical simulations are given to illustrate the effectiveness of the proposed ESO-NTSMC for the chaser tracking the tumbling target. It is assumed that the orbital elements of tumbling target are presented in Table 1. The target body-fixed frame completely coincides with the orbital coordinate system. The control objective is to keep attitude of chaser consistent with the target, and the relative position of two can be kept at a certain distance.
The rotational angular velocity of the tumbling target is considered to be slow speed as , and the capturing point in is set as . The inertia matrix is supposed to be measured as . The initial conditions of the relation motion between two spacecraft are given as ; ; and .
In addition, proposed NTSMC with linear-ESO, LESO-NTSMC, designed in Reference [25], and proposed ESO with conventional NTSMC, ESO-TSM, designed in Reference [38] are also implemented for purpose of comparison. The gains of proposed and above controllers are given in Table 2.
The preset uncertainties and disturbances are set in accordance with Reference [12]. The real mass and inertia matrix of the chaser are assumed to be and . And the nominal mass and inertia matrix of the chaser is assumed to be and , with the model uncertainties bounded by and , respectively. Moreover, it is assumed that the disturbance forces and torques are set aswhere is a target orbit period constant with . The control force is limited to the range of , and the control torque is limited to the range of .
Figures 4–13 show the time responses of the simulation results using the proposed ESO-NTSMC and compared control schemes, LESO-NTSMC and ESO-TSM, respectively. The results shown in Figure 4 illustrate the effectiveness of the ESO where the observer reconstruction errors of translational term and rotational term converge to a small range in finite time. In such a situation, the accuracy of the system can be enhanced when the model uncertainties and bounded external disturbances are estimated by ESO and the system can be compensated. Figures 5(a) and 6(a) shows the time responses of the tracking error of relative translation and rotation, respectively. It can be seen that the rotation error falls to tolerance within 10 s and converges to the range of while the translation tracking error converges to the range of 0.02 m in around 20 s, which illustrates the performance of accuracy and rapidity of ESO-NTSMC. The time histories of relative velocity and angular velocity tracking errors are given in Figures 7(a) and 8(a), respectively. The steady-state behavior shows that the angular velocity error converges to in 10 s, and the velocity error converges to at 20 s, which confirms the control objective of tracking the velocity of target accurately. It can be seen that, under the bounded sliding mode control force and torque shown in Figures 9(a) and 10(a), the sliding surfaces converge to the range of and in finite time successfully, which are shown in Figures 11(a) and 12(a).
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The simulations of LESO-NTSMC are also carried out, which are shown in Figures 5(b)–12(b). From these figures, it can be seen that LESO-NTSMC is also tolerant to model uncertainties and external disturbances when the system states can be driven to a neighborhood of origin. As shown in Figures 11(b) and 12(b), the sliding surface accuracy of LESO-NTSMC is much lower than of ESO-NTSMC, which can illustrate that the proposed method obtains better control ability and high control accuracy in comparison with LESO-NTSMC scheme. Figures 5(c)–12(c) show the simulation results using the ESO-based conventional NTSMC method. Severe chattering and large overshoot exist in ESO-TSM during the control process, and the convergence time of ESO-TSM is larger than the proposed one. As seen, the proposed ESO-NTSMC has faster convergence rate than ESO-TSM and can also avoid singularity. From the tracking trajectory shown in Figure 13, the chaser can successfully track the desired trajectory of docking port of tumbling target. In conclusion, it is obvious that the control object is achieved under the control strategy of proposed ESO-NTSMC while the performance and robustness can be guaranteed.
6. Conclusion
In this study, a finite-time stabilization problem of a chaser spacecraft approaching a tumbling target in presence of unknown uncertainty and bounded external disturbance has been handled. Based on the 6-DOF coupled relative motion described by dual quaternion, a combination of NTSMC and ESO is proposed to drive the translational and rotational motion to the desired state. Meanwhile, the finite-time convergence and stability property of the proposed method is rigorously proven. It was shown that the tracking error can be driven to the equilibrium point with fast response and high accuracy. Numerical simulation results show the effectiveness and performance of the proposed controller and its robustness to the model uncertainty and external disturbance.
Appendix
For a given quaternion , and are the scalar part and the vector part, respectively. The conjugate of is defined as and its norm is calculated by . It is called unit quaternion if . Its logarithm is defined as
The equivalent of the unit quaternion referred to Euler angle and the unit vector is defined as
As seen, when the angle , there is , and when , correspondingly . Because of , it is obvious that a unit quaternion can be transformed to a three-dimensional vector by the logarithm operation.
The definition of dual numbers iswhere and are, respectively, called real part and dual part. is called the dual unit, satisfying and . And the operation of the dual numbers is defined aswhere is a scalar.
The dual vector is that both the dual part and real part are vectors. For two dual vectors and , their cross multiplication is
In addition to this, dual number and dual vector have the following operation:
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Major Program of National Natural Science Foundation of China under Grant nos. 61690210 and 61690213.