Robust Adaptive Relative Position and Attitude Control for Noncooperative Spacecraft Hovering under Coupled Uncertain Dynamics

Liu, Jianghui; Li, Haiyang; Zhang, YaKun; Zhou, Jianyong; Lu, Lin; Li, Fuqi

doi:https://doi.org/10.1155/2019/8678473

Mathematical Problems in Engineering

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 8678473 | https://doi.org/10.1155/2019/8678473

Robust Adaptive Relative Position and Attitude Control for Noncooperative Spacecraft Hovering under Coupled Uncertain Dynamics

Jianghui Liu,¹Haiyang Li,¹YaKun Zhang,²Jianyong Zhou,¹Lin Lu,¹and Fuqi Li³

Academic Editor: Xuping Zhang

Received22 Apr 2019

Revised22 Sept 2019

Accepted15 Nov 2019

Published28 Dec 2019

Abstract

The control of body-fixed hovering over noncooperative target, as one of the key problems of relative motion control between spacecrafts, is studied in the paper. The position of the chaser in the noncooperative target’s body coordinate system is required to remain unchanged, and the attitude of the chaser and the target must be synchronized at the same time. Initially, a six-degrees-of-freedom-coupled dynamic model of a chaser relative to a target is established, and relative attitude dynamics is described through using modified Rodrigues parameters (MRP). Considering the model uncertainty and external disturbances of the noncooperative target system, an adaptive nonsingular terminal sliding mode (NTSM) controller is designed. Adaptive tuning method is used to overcome the effects of the model uncertainty and external disturbances. The upper bounds of the model uncertainty and external disturbances are not required to be known in advance. The actual control law is continuous and chatter-free, which is obtained by integrating the discontinuous derivative control signal. Finally, these theoretical results are verified by numerical simulation.

1. Introduction

On-orbit spacecraft is of great value. When breaking down in space, on-orbit repair, component replacement, and refueling can significantly prevent further cost of replacing a new one [1–6]. Therefore, it has drawn much attention from researchers. Relative hovering usually indicates that a spacecraft stays fixed in its position and attitude in the body coordinate system of another spacecraft. The state of relative hovering greatly facilitates space surveillance and inspections.

Historically, related research studies have mainly focused on the hovering over asteroids. Scheeres first proposed the concept of hovering orbit of the spacecraft relative to asteroids in 1999 [7]. In general, hovering over an asteroid mainly includes inertial hovering and body-fixed hovering [8, 9]. Broschart simulated the hovering control of an asteroid under slight gravity and determined the stable region of inertial hovering [8]. In the control of hovering over asteroids, the spacecraft needs to continuously apply the control thrust to counteract the gravity and rotational acceleration to maintain the desired position [9, 10]. This method is feasible on asteroids due to low nominal acceleration of the spacecraft [7, 8, 11]. Zeng proposed the solar sail spacecraft’s hover over an asteroid, which greatly extended the hover time and hover range without fuel consumption [12]. It should be pointed out that the research on spacecraft hovering over an asteroid in the early stage mainly concentrated on the relative position control of the spacecraft and asteroid irrespective of the relative attitude control [7–9, 11, 13–15]. Lee et al. put forward passive tracking control of the relative position and relative attitude of hovering over an asteroid in the framework of geometric mechanics [16]. Aiming at the orbit and attitude control of hovering over a rotating asteroid at a low speed, Lee et al. designed a continuous finite-time convergence control scheme [17, 18].

Strictly speaking, the situation of a spacecraft hovering over an asteroid is different from that of hovering over the target spacecraft. This is because the former is influenced by the gravity from both the asteroid and the sun, while the latter is only influenced by Earth’s gravity. Therefore, the dynamic equations in the two cases are also different.

In this paper, we mainly study the hovering control of one spacecraft relative to another spacecraft. Recently, some researchers have also published many findings concerning hovering control between two spacecrafts. Tan analyzed and solved the relative motions of the two spacecrafts in the near circle and elliptical orbits, respectively. The controller was designed using generalized inverse matrix transformation, and the guidance deviation was obtained by model prediction [19, 20]. Xue has established a hybrid system model for relative hovering between the two spacecrafts [21]. Based on the description of the state transfer matrix between the two spacecrafts, Cheng used a multipulse control method to study the relative hovering motion [22]. To solve the problem of hovering between the two spacecrafts at superclose distance, Xu proposed the control method of line of sight pointing tracking on the basis of relative orbit control and realized the joint control of the relative orbit and attitude [23]. Song studied the hovering closed-loop control method based on Hill equation [24]. Dang established a precise analytical solution for hovering between the two spacecrafts and deduced the minimum force and minimum fuel positions during the orbital period [25]. Huang studied the problem of finite-time hovering control in the absence of the radial or in-track thrust [26].

However, previous research studies on hovering control mainly focused on the control of the relative position between the two spacecrafts. There are few research studies on relative attitude control, and hovering control is mostly in the form of open-loop control. Even if the relative attitude control problem is considered, it is restricted to the condition of the stable attitude or slow attitude change of the target, and the relative position and relative attitude are also controlled separately, ignoring their coupling. Further complicating things, the model parameters of the spacecraft are not likely to be precisely known during hovering operations, and the spacecraft is always subject to external environment disturbances [27–29]. Sun investigates relative position and attitude control for spacecraft rendezvous and proximity operations subject to input saturation, kinematic couplings, parametric uncertainties, and unknown external disturbances. State feedback control method [30], disturbance-observer-based robust nonlinear control scheme [31], a six-degrees-of-freedom integrated adaptive fuzzy nonlinear control method [32], and robust adaptive control approach [33] are proposed. The terminal sliding mode control (TSMC) method based on the conventional sliding mode control (SMC) method has the advantage of finite-time convergence [34]. Its sliding mode is independent of the model parameters and external disturbances, and thus has been widely used in nonlinear control [35, 36]. However, the disadvantage is that the nonlinear term used in the TSMC may cause a singularity problem leading to a control magnitude to become unbounded. In order to solve this problem, Feng proposed the nonsingular terminal sliding mode control (NTSMC) method [37, 38]. Nonsingular terminal sliding mode controllers are now widely used in a variety of application areas like robotics [39–41], aerospace, and process control [42, 43].

Considering the limitations of previous studies, it is necessary to further study hovering control between the two spacecrafts. In this paper, control problem of the chaser hovering over the target with relatively rapid attitude change in space is studied. A nonlinear six-degrees-of-freedom-coupled dynamic model is established in the chaser’s body coordinate system. A chattering-free adaptive nonsingular terminal sliding mode (NTSM) controller is designed. Firstly, the conventional sliding surface is established, and then the nonsingular terminal sliding surface is constructed on this basis. The adaptive tuning method is used to deal with the model uncertainty and external disturbances. The upper bounds of the model uncertainty and external disturbances are not required to be known in advance [44–46]. The actual control law is continuous and chatter-free, which is obtained by integrating the discontinuous derivative control signal.

The rest of the sections are as follows. Section 2 introduces dynamics for the chaser and the target. Section 3 presents the designs of the adaptive nonsingular terminal sliding mode (NTSM) controller and demonstrates the stability of the closed-loop system. Simulation studies performed on the relative position and attitude control of the hovering example are presented in Section 4. Section 5 draws conclusions.

2. Dynamics for the Chaser and the Target

In this paper, the skew-symmetric matrix derived from a vector is defined asand it satisfies , , for any .

for any , where

2.1. Dynamics for Relative Orbit

The relative motion scenario between the chaser and the target is shown in Figure 1. is the Earth-centered inertial frame, with its coordinate origin at the Earth center and nonrotating with respect to the stars. The axis points in the direction of the Earth's vernal equinox. The axis points to the direction of North Pole. The direction of axis is determined by the right-hand rule. and are the body frames of the target and the chaser, respectively. The body frame is fixed onto the spacecraft body and rotates with it. is the position vectors in the frame , and is the position vectors in the frame .

The position and attitude dynamic model of the chaser relative to the frame is [47]where and are the chaser’s velocity and angular velocity, respectively. is the chaser’s position, and is the modified Rodrigues parameter (MRP) vector to describe the attitude of frame with respect to frame . and are the chaser’s inertial matrix and mass, respectively. and are the chaser’s control force and control torque, respectively. and are unknown bounded disturbance force and unknown bounded disturbance torque of the chaser, respectively. They are all expressed in the frame .

Ignoring the external forces and torques, the position and attitude dynamic model of the noncooperative target relative to the frame is [31]where and are the target’s velocity and angular velocity, respectively. is the target’s position, and is the attitude of frame with respect to frame . and are the target’s inertial matrix and mass, respectively. They are all expressed in the frame .

2.2. Relative Dynamic Model

The relative attitude of frame with respect to frame described by MRP is [48]

The corresponding attitude transfer matrix iswhere is a unit matrix. It can be seen from Figure 1 that the position and velocity of the hovering point described in the frame are

The relative angular velocity, relative position, and relative velocity of the two spacecrafts described in the frame are

Substituting equations (9)∼(11) into equation (3) and using the equations , and , the relative motion equations described in the frame can be derived:where is a nonsingular matrix. From equations (4) and (8)–(11) and , can be derived as

Also, in equation (13) can be calculated by equations (4) and (9):

Therefore, equation (12) can be rewritten aswhere

2.3. Integrated Dynamic Model

Defining state variables and from equation (15) yields

Remark 1. in model (17) and in model (18) reflect that the relative position motion between the two spacecrafts is affected by the relative attitude motion, which indicates that there is a strong coupling effect between relative attitude and relative position.
This paper aims to design a controller so that the chaser reaches a predetermined hovering point , and the relative attitude between the two spacecrafts is , is a constant. In order to facilitate the analysis, the relative attitude is , that is, the attitude between the two spacecrafts is synchronized. From equations (5), (9)–(11), (17), and (18), we can see that the control goal is equivalent to designing the control input to satisfy and , where and are any small positive numbers.

3. Controller Design and Stability Analysis

Considering the model uncertainty of the system, from equation (18) yieldswhere , , and are the nominal model of the system. , , and are model uncertainty terms of the system. Set as system compound disturbances including model uncertainty and external disturbances, and its expression is

Assumption 1. The chaser can obtain its own motion information and relative motion information through its own measurement device. The measured relative motion and relative attitude information are smooth and bounded.

Assumption 2. System compound disturbances and its time derivative are unknown but bounded.
The system compound disturbances satisfies the following equation [33]:where , , and are constants, and represents the 2 norm of the vector.
Substituting equation (21) into (20) yieldsFrom equation (23) yieldswhere is the generalized inverse matrix of . The expression of isThe time derivative of equation (17) givesThe time derivative of equation (24) giveswhere .
The sliding mode function is designed aswhere is a sixth-order positive-definite diagonal matrix,, . From equation (28), it can be obtained that when , then and as well.
The first derivative of the sliding mode function isSubstituting equations (17) and (24) into equation (29) yieldsIn this paper, the second-order nonsingular terminal sliding mode (NTSM) control theory is used to design the control law. In order to design a nonsingular terminal sliding mode (NTSM) controller, the nonsingular terminal sliding mode (NTSM) surface is defined aswhere , . and are odd integers and satisfy the condition . In order to facilitate subsequent derivation, the following definition is made.

Definition 1. for any two vectors and .
The time derivative of equation (31) givesThe time derivative of equation (29) giveswhere . According to equation (22) and Assumption 2 yieldwhere , , and are constants.

Proof. Lyapunov function is expressed aswhere , , and are constant parameters, , , and are adaptation errors, and , , and are the estimations of , , and , respectively.
The time derivative of equation (35) givesThe control law is designed aswhere , .
The adaptation law , , and is designed aswhere , , and are the tuning parameters.
Substituting equations (38)∼(41) into equation (36) yieldsthenwhereSo,where . According to equation (34), it can be known that . In order to ensure , , and , the condition , , and must be satisfied when designing parameters.
for any and only when . According to the accessibility condition of the sliding mode, the system can reach from any initial state within a finite time . Then, it will be proven that the sliding mode function will also reach the plane in the finite time on the nonsingular terminal sliding mode surface . First, the lemma for the stability of a nonlinear system is given [49].

Lemma 1. If there is a continuous function , the following conditions are met:(1) is positive(2)There is a real number and an open neighborhood so that

Then, the system is stable in finite time, and the convergence time satisfies

Lyapunov function is defined as

On the nonsingular terminal sliding surface , the following equation can be obtained:i.e.,

The time derivative of equation (48) gives

Substituting equation (50) in (51) yieldsi.e.,where is the smallest absolute component of , , .

Let the converge to within finite time . It can be deduced from Lemma 1 that when , and satisfies

Since the sliding mode function converges to within finite time , according to equation (49), will also converge to within finite time . It can be seen from equation (28) that if , then and . This case completes the proof.

The control system diagram with the adaptive nonsingular terminal sliding mode controller is shown in Figure 2.

From equation (38) yields

Remark 2. Equation (55) is essentially a low-pass filter with the right side of the equation as the input and as the output. Therefore, the adaptive nonsingular terminal sliding mode (ANTSM) controller designed in this paper can eliminate the chattering phenomenon of the conventional sliding mode.

Remark 3. The parameters , , and determine the rate at which the estimated values , , and converge to their respective boundaries. Larger , , and ensure that the estimated values , , and quickly converge to the actual boundary , , and . Considering , , and , the actual value is to weigh the convergence rate and constraints.

Remark 4. The parameter determines the convergence rate of the sliding surface. Larger means faster convergence rate, which requires larger input. In reality, the thrust of the spacecraft is limited. Therefore, the value of needs to be weighed between thrust condition and convergence rate.

Remark 5. The proposed methodology is applicable when is reachable. However, cannot become exactly 0 in a finite time due to nonlinear characteristics of the system and the switching delays. Therefore, the adaptive parameters , , and may become boundless. A simple method is to use the dead zone to overcome the above difficulties, and the adaptive law , , and of equations (39)∼(41) are modified as

4. Simulation Example

This section verifies the effectiveness of the designed controller through simulation of a chaser’s superclose distance hovering over a noncooperative target. The mass of the chaser is . The moments of inertia of the target and chaser, and , are [30]

The model uncertainties are and .

The hovering position in the frame is . The initial position and velocity are and , respectively. The initial attitude and angular velocity are and , respectively. The initial relative position and relative velocity are and , respectively. The initial relative attitude and relative angular velocity are and , respectively. The values of the above parameters are listed in Table 1.

Disturbance torque and disturbance force arewhere is the chaser’s average orbital angular velocity, and is the gravitational constant of the Earth.

The controller parameters are selected as , , , , , , , , , , , and . The simulation results are presented in Figures 3 and 4.

Figure 3 shows relative motion, control torque, and control force. The relative attitude drops from to after 58 s; the relative angular velocity increased rapidly from and then decreased to after reaching the peak value, which indicates that the chaser completes attitude capture for the noncooperative target and then maintains the attitude synchronization with the target. After 71 s, the relative position drops from to and the relative velocity drops from to . It is indicated that the chaser’s center of mass has reached the desired hovering position in the frame and has remained unchanged thereafter. The control torque required by the chaser is minimal after 73 s, and the maximum control torque in each direction does not exceed . The control force required by the chaser is minimal after 48 s, and the maximum control force in each direction does not exceed . The chaser's control torque and control force curve are smooth throughout the entire process, indicating no chattering occurs. Figure 4 shows estimated parameters , , and using the adaptive tuning method. , , and finally converge to 0.0085, 0.3925, and 0.0419.

In order to verify the effectiveness of the proposed controller, it is necessary to compare it with the conventional sliding mode controller. The conventional sliding mode controller is chosen as [50]

The parameter of the conventional sliding mode controller is , and all other simulation parameters are the same as those of the proposed sliding mode controller. The simulation results are presented in Figure 5. By further comparing Figure 5 with Figure 3, it can be concluded that under the effect of the conventional sliding mode controller, the chaser finishes the attitude synchronization with the target and reaches the hovering point need 59 s and 74 s , respectively, which is 1 s and 3 s longer than the proposed sliding mode controller. After reaching the hovering position and completing the attitude synchronization, the maximum control torque and maximum control force required by the chaser in each direction are still as high as and 0.8 N, respectively. Chattering phenomenon occurs in both control torque and control force of the conventional sliding mode controller.

In addition, for the mean performance error index , the tracking error is and the simulation time is . It can be calculated that the proposed controller is , and the conventional sliding mode controller is . Obviously, the proposed controller effectively improves the closed-loop system dynamic response performance.

In order to investigate the robustness of the proposed controller, the external bounded disturbance force and torque are increased to and , respectively. Other simulation parameters are the same. The simulation results are presented in Figures 6 and 7. By comparing Figures 6 and 7 with Figures 3 and 4, it can be found that the chaser finishes the position synchronization at the hovering point after 71 s and finishes the attitude synchronization with the target after 58 s. The estimated parameters , , and finally converge to 0.0085, 0.3926, and 0.0419. The control torque and control force curve are smooth without chattering, and the mean performance error index is . Compared with the small disturbance, the difference in control performance is very small, thus proving the strong robustness of the proposed sliding mode controller.

5. Conclusions

In this paper, an adaptive nonsingular terminal sliding mode controller is designed based on the measurable information when the chaser hovers over a noncooperative target, and the six-degrees-of-freedom-coupled relative position and relative attitude control are realized. Instead of the conventional control input, its time derivative is used in the controller. The derivative control contains the discontinuous sign function, and the actual control input is obtained by integration, so it is continuous and chatter-free. The adaptive tuning method is used to deal with the model uncertainty and external disturbances. Simulation results show that the controller can effectively overcome the model uncertainty and the influence of external disturbance factors. The chaser quickly reaches the hovering position of the noncooperative target, and both the hovering position error and the hovering velocity error converge to a smaller range. The required control torque and control force are minimally, continuously, and smoothly produced with no chattering occurring.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are grateful to Prof. Li and Scholar Hao for discussions. The paper was supported by the National Natural Science Foundation of China (no. 11472301) and the Foundation of National University of Defense Technology (no. ZK18-03-07).

References

D. Wang, P. Huang, and Z. Meng, “Coordinated stabilization of tumbling targets using tethered space manipulators,” IEEE Transactions on Aerospace and Electronic Systems, vol. 51, no. 3, pp. 2420–2432, 2015.
View at: Publisher Site | Google Scholar
S. Bandyopadhyay, S. J. Chung, and F. Y. Hadaegh, “Nonlinear attitude control of spacecraft with a large captured object,” Journal of Guidance, Control, and Dynamics, vol. 39, no. 4, pp. 754–769, 2016.
View at: Publisher Site | Google Scholar
L. Zhang, S. Qian, S. Zhang, and H. Cai, “Research on angles-only/SINS/CNS relative position and attitude determination algorithm for a tumbling spacecraft,” Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, vol. 231, no. 2, pp. 218–228, 2017.
View at: Publisher Site | Google Scholar
M. Wang, J. Luo, J. Yuan, and U. Walter, “Detumbling strategy and coordination control of kinematically redundant space robot after capturing a tumbling target,” Nonlinear Dynamics, vol. 92, no. 3, pp. 1023–1043, 2018.
View at: Publisher Site | Google Scholar
Y. She, J. Sun, S. Li, W. Li, and T. Song, “Quasi-model free control for the post-capture operation of a non-cooperative target,” Acta Astronautica, vol. 147, pp. 59–70, 2018.
View at: Publisher Site | Google Scholar
K. B. Li, W. S. Su, and L. Chen, “Performance analysis of differential geometric guidance law against high-speed target with arbitrarily maneuvering acceleration,” Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, vol. 223, no. 10, pp. 3547–3563, 2019.
View at: Publisher Site | Google Scholar
D. J. Scheeres, “Stability of hovering orbits around small bodies,” Advances in the Astronautical Sciences, vol. 102, pp. 855–875, 1999.
View at: Google Scholar
S. B. Broschart and D. J. Scheeres, “Control of hovering spacecraft near small bodies: application to asteroid 25143 Itokawa,” Journal of Guidance, Control, and Dynamics, vol. 28, no. 2, pp. 343–354, 2005.
View at: Publisher Site | Google Scholar
D. J. Scheeres, Orbital Motion in Strongly Perturbed Environments, Springer, Berlin, Heidelberg, Germany, 2012.
S. Sawai, D. J. Scheeres, and S. B. Broschart, “Control of hovering spacecraft using altimetry,” Journal of Guidance Control & Dynamics, vol. 25, no. 4, pp. 786–795, 2002.
View at: Publisher Site | Google Scholar
S. B. Broschart and D. J. Scheeres, “Boundedness of spacecraft hovering under dead-band control in time-Invariant systems,” Journal of Guidance, Control, and Dynamics, vol. 30, no. 2, pp. 601–610, 2007.
View at: Publisher Site | Google Scholar
X. Zeng, S. Gong, J. Li, and K. T. Alfriend, “Solar sail body-fixed hovering over elongated asteroids,” Journal of Guidance Control & Dynamics, vol. 39, no. 6, 2016.
View at: Publisher Site | Google Scholar
T. Kubota, T. Hashimoto, M. Uo, M. Maruya, and K. Baba, “Maneuver strategy for sta-tion keeping and global mapping around an asteroid AIAA/AAS spaceflight mechanics,” Advances in the Astronautical Sciences, vol. 108, pp. 769–780, 2001.
View at: Google Scholar
B. Gaudet and R. Furfaro, “Robust spacecraft hovering near small bodies in envi-ronments with unknown dynamics using reinforcement learning,” in Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, AIAA 2012-5072, Minneapolis, MN, USA, August 2012.
View at: Google Scholar
K. Berry, B. Suttery, A. Mayz et al., “OSIRIS-REx Touch-And-Go (TAG) mission design and analysis,” in Proceedings of the 36th Annual AAS Guidance and Control Conference, AAS 13-095, Breckenridge, CO, USA, February 2013.
View at: Google Scholar
D. Lee, A. K. Sanyal, E. A. Butcher, and D. J. Scheeres, “Almost global asymptotic tracking control for spacecraft body-fixed hovering over an asteroid,” Aerospace Science and Technology, vol. 38, no. 1, pp. 105–115, 2014.
View at: Publisher Site | Google Scholar
D. Lee, A. K. Sanyal, E. A. Butcher et al., “Spacecraft hovering control for body-fixed hovering over a uniformly rotating asteroid using geometric mechanics,” Advances in the Astronautical Sciences, vol. 150, pp. 1757–1773, 2014.
View at: Google Scholar
D. Lee, A. Sanyal, E. Butcher, and D. Scheeres, “Finite-time control for spacecraft body-fixed hovering over an asteroid,” IEEE Transactions on Aerospace and Electronic Systems, vol. 51, no. 1, pp. 506–520, 2015.
View at: Publisher Site | Google Scholar
T. Tan and H. Wu, “Analytical solution method for orbit rendezvous, hovering and fly-around control,” Journal of Astronautic, vol. 37, no. 11, pp. 1333–1341, 2016.
View at: Google Scholar
T. Tan, “Full state feedback control of rendezvous, hovering and fly-around in elliptical orbit,” Journal of Astronautic, vol. 37, no. 7, pp. 811–818, 2016.
View at: Google Scholar
B. Xue, Z. She, J. Yu et al., “Control on the spacecraft hovering based on hybrid systems,” Chinese Space Science & Technology, vol. 30, no. 2, pp. 61–67, 2010.
View at: Google Scholar
B. Cheng, J. Yuan, and W. Ma, “Spacecraft multiple pulse hovering method based on state transition matrix,” Chinese Space Science & Technology, vol. 36, no. 5, pp. 81–87, 2016.
View at: Google Scholar
W. Xu, H. Wu, S. Lu et al., “Spacecraft attitude-orbit combined control analysis for flying-around and hovering task in super-close relative distance,” in Proceedings of the 32nd Chinese Control Conference, Xi’an, China, July 2013.
View at: Google Scholar
X. Song, L. Fan, Y. Chen et al., “Study of close-loop hovering method at any selected position to space target,” Chinese Space Science & Technology, vol. 30, no. 1, pp. 41–45, 2010.
View at: Google Scholar
Z. Dang, Z. Wang, and Y. Zhang, “Modeling and analysis of relative hovering control for spacecraft,” Journal of Guidance, Control, and Dynamics, vol. 37, no. 4, pp. 1091–1102, 2014.
View at: Publisher Site | Google Scholar
X. Huang, Y. Yan, and Z. Huang, “Finite-time control of underactuated spacecraft hovering,” Control Engineering Practice, vol. 68, pp. 46–62, 2017.
View at: Publisher Site | Google Scholar
D. Lee and G. Vukovich, “Robust adaptive terminal sliding mode control on SE(3) for autonomous spacecraft rendezvous and docking,” Nonlinear Dynamics, vol. 83, no. 4, pp. 2263–2279, 2016.
View at: Publisher Site | Google Scholar
H. Yan, S. Tan, and Y. Xie, “Integrated translational and rotational control for the final approach phase of rendezvous and docking,” Asian Journal of Control, vol. 20, no. 5, pp. 1967–1978, 2018.
View at: Publisher Site | Google Scholar
S. Zhu, R. Sun, J. Wang, J. Wang, and X. Shao, “Robust model predictive control for multi-step short range spacecraft rendezvous,” Advances in Space Research, vol. 62, no. 1, 2018.
View at: Publisher Site | Google Scholar
L. Sun and Z. Zheng, “Adaptive relative pose control for autonomous spacecraft rendezvous and proximity operations with thrust misalignment and model uncertainties,” Advances in Space Research, vol. 59, no. 7, pp. 1861–1871, 2017.
View at: Publisher Site | Google Scholar
L. Sun, W. Huo, and Z. Jiao, “Disturbance-observer-based robust relative pose control for spacecraft rendezvous and proximity operations under input saturation,” IEEE Transactions on Aerospace and Electronic Systems, vol. 54, no. 4, pp. 1605–1617, 2018.
View at: Publisher Site | Google Scholar
L. Sun, W. He, and C. Sun, “Adaptive fuzzy relative pose control of spacecraft during rendezvous and proximity maneuvers,” IEEE Transactions on Fuzzy Systems, vol. 26, no. 6, pp. 3440–3451, 2018.
View at: Publisher Site | Google Scholar
L. Sun and Z. Zheng, “Adaptive relative pose control of spacecraft with model couplings and uncertainties,” Acta Astronautica, vol. 143, pp. 29–36, 2018.
View at: Publisher Site | Google Scholar
M. Zak, “Terminal attractors for addressable memory in neural networks,” Physics Letters A, vol. 133, no. 1-2, pp. 18–22, 1988.
View at: Publisher Site | Google Scholar
M. Zhihong, A. P. Paplinski, and H. R. Wu, “A robust MIMO terminal sliding mode control scheme for rigid robotic manipulators,” IEEE Transactions on Automatic Control, vol. 39, no. 12, pp. 2464–2469, 1994.
View at: Publisher Site | Google Scholar
K. B. Park and T. Tsuji, “Terminal sliding mode control of second-order nonlinear uncertain systems,” International Journal of Robust & Nonlinear Control, vol. 9, no. 11, pp. 769–780, 1999.
View at: Publisher Site | Google Scholar
Y. Feng, X. Yu, and Z. Man, “Non-singular terminal sliding mode control of rigid manipulators,” Automatica, vol. 38, no. 12, pp. 2159–2167, 2002.
View at: Publisher Site | Google Scholar
Y. Feng, X. Yu, and F. Han, “On nonsingular terminal sliding-mode control of nonlinear systems,” Automatica, vol. 49, no. 6, pp. 1715–1722, 2013.
View at: Publisher Site | Google Scholar
S. Mondal and C. Mahanta, “Adaptive second order terminal sliding mode controller for robotic manipulators,” Journal of the Franklin Institute, vol. 351, no. 4, pp. 2356–2377, 2014.
View at: Publisher Site | Google Scholar
R. Mohammadi Asl, Y. Shabbouei Hagh, and R. Palm, “Robust control by adaptive non-singular terminal sliding mode,” Engineering Applications of Artificial Intelligence, vol. 59, pp. 205–217, 2017.
View at: Publisher Site | Google Scholar
A. Safa, R. Y. Abdolmalaki, S. Shafiee, and B. Sadeghi, “Adaptive nonsingular terminal sliding mode controller for micro/nanopositioning systems driven by linear piezoelectric ceramic motors,” ISA Transactions, vol. 77, pp. 122–132, 2018.
View at: Publisher Site | Google Scholar
L. C. Maria and C. Andrea, “Nonsingular terminal sliding-mode control of nonlinear planar systems with global fixed-time stability guarantees,” Automatica, vol. 95, pp. 561–565, 2018.
View at: Publisher Site | Google Scholar
L. Haibo, W. Heping, and S. Junlei, “Attitude control for QTR using exponential nonsingular terminal sliding mode control,” Journal of Systems Engineering and Electronics, vol. 30, no. 1, pp. 195–204, 2019.
View at: Google Scholar
Y. Wang, X. Zhang, X. Yuan, and G. Liu, “Position-sensorless hybrid sliding-mode control of electric vehicles with brushless DC motor,” IEEE Transactions on Vehicular Technology, vol. 60, no. 2, pp. 421–432, 2011.
View at: Publisher Site | Google Scholar
Z. Xiaolei, Z. Yan, G. Kai, G. Li, and N. Deng, “An adaptive B-spline neural network and its application in terminal sliding mode control for a mobile satcom antenna inertially stabilized platform,” Sensors, vol. 17, no. 5, pp. 978–998, 2017.
View at: Publisher Site | Google Scholar
J. Liu, H. Li, Y. Luo, and J. Zhang, “Robust adaptive relative position and attitude integrated control for approaching uncontrolled tumbling spacecraft,” Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, pp. 1–14, 2019.
View at: Publisher Site | Google Scholar
M. Xin and H. Pan, “Nonlinear optimal control of spacecraft approaching a tumbling target,” Aerospace Science and Technology, vol. 15, no. 2, pp. 79–89, 2011.
View at: Publisher Site | Google Scholar
M. D. Shuster, “A survey of attitude representation,” The Journal of the Astronautical Sciences, vol. 41, no. 4, pp. 439–517, 1993.
View at: Google Scholar
E. Moulay and W. Perruquetti, “Finite time stability conditions for non-autonomous continuous systems,” International Journal of Control, vol. 81, no. 5, pp. 797–803, 2008.
View at: Publisher Site | Google Scholar
D. Feng, H. Jian, Y. Wang, J. Zhang, and S. He, “Sliding mode control with an extended disturbance observer for a class of underactuated system in cascaded form,” Nonlinear Dynamics, vol. 90, no. 4, pp. 2571–2582, 2017.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2019 Jianghui Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

683

Downloads

971

Citations