Research Article  Open Access
Jianghui Liu, Haiyang Li, YaKun Zhang, Jianyong Zhou, Lin Lu, Fuqi Li, "Robust Adaptive Relative Position and Attitude Control for Noncooperative Spacecraft Hovering under Coupled Uncertain Dynamics", Mathematical Problems in Engineering, vol. 2019, Article ID 8678473, 14 pages, 2019. https://doi.org/10.1155/2019/8678473
Robust Adaptive Relative Position and Attitude Control for Noncooperative Spacecraft Hovering under Coupled Uncertain Dynamics
Abstract
The control of bodyfixed 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 sixdegreesoffreedomcoupled 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 chatterfree, which is obtained by integrating the discontinuous derivative control signal. Finally, these theoretical results are verified by numerical simulation.
1. Introduction
Onorbit spacecraft is of great value. When breaking down in space, onorbit 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 bodyfixed 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 finitetime 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 closedloop 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 finitetime hovering control in the absence of the radial or intrack 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 openloop 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], disturbanceobserverbased robust nonlinear control scheme [31], a sixdegreesoffreedom 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 finitetime 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 sixdegreesoffreedomcoupled dynamic model is established in the chaser’s body coordinate system. A chatteringfree 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 chatterfree, 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 closedloop 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 skewsymmetric 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 Earthcentered 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 righthand 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 sixthorder positivedefinite 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 secondorder 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 lowpass 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