Abstract
An adaptive saturated neural network (NN) controller is developed for 6 degree-of-freedom (6DOF) spacecraft tracking, and its hardware-in-the-loop experimental validation is tested on the ground-based test facility. To overcome the dynamics uncertainties and prevent the large control saturation caused by the large tracking error at the beginning operation, a saturated radial basis function neural network (RBFNN) is introduced in the controller design, where the approximate error is counteracted by an adaptive continuous robust term. In addition, an auxiliary dynamical system is employed to compensate for the control saturation. It is proved that the ultimate boundedness of the closed-loop system is achieved. Besides, the proposed controller is implemented into a testbed facility to show the final operational reliability via hardware-in-the-loop experiments, where the experimental scenario describes that the simulator is tracking a planar trajectory while synchronizing its attitude with the desired angle. Experimental results illustrate that the proposed controller ensures that the simulator can track a preassigned trajectory with robustness to unknown inertial parameters and disturbances.
1. Introduction
In recent few decades, spacecraft control has been attracting widespread interest because of its typical orbit applications, such as formation flying, construction of space station, and space surveillance and capturing, rendezvous, and docking. Since the highly nonlinear character arising from 6DOF dynamics in the presence of disturbance and parametric uncertainty would bring more difficulties, it is still challenging to design high-performance controllers for spacecraft. However, most previous work has only focused on the controller design without perhaps the most critical experimental validations [1].
In previous studies, different controllers have been found to be related to 6DOF spacecraft control problems, such as PD+ controller [2], sliding surface controller [3], and disturbance observer-based controller [4, 5]. However, controllers in [2–5] have only been carried out in the presence of exact inertial parameters or disturbances (or their bounds). A number of techniques have been developed to solve the control problems of 6DOF spacecraft with unavailable uncertainties [6–13]. For 6DOF spacecraft operations subject to unknown parameters and disturbances, adaptive controllers [6, 14, 15] are synthesized. In addition, adaptive saturated controllers [7, 8] are designed by using different saturation compensating methods. To achieve 6DOF spacecraft maneuvers in the presence of control saturations and dynamics uncertainties, disturbance observer-based saturated controllers are studied in [9, 10].
The neural network, which is an alternative solution with highly approximate capacity [11], also draws attention to 6DOF spacecraft controls. For formation flying control problems with parametric uncertainties and disturbances, a NN-based adaptive sliding mode controller is proposed in [12]. For cooperative rendezvous and docking maneuvers, a NN-based switching saturated control is investigated in [13]. Besides, adaptive NN controllers are also studied in helicopter [16] and marine surface vessel [17]. Nevertheless, the abovementioned studies have failed to demonstrate the designed controllers by hardware-in-the-loop experimental validations.
To realize the theoretical results in practical applications, the air-bearing ground test facilities have been developed [18, 19] and some controllers have been validated by hardware-in-the-loop experiments [20–26]. For spacecraft operations subject to parametric uncertainties, adaptive controllers are designed and the related experimental validations are conducted on the ground test facilities [20, 21]. For spacecraft operations, PID [22] and LQR [23] controllers are designed and experimentally tested, respectively. In addition, several controllers [24–26] are developed and validated on the testbed facility. Despite the fact that experimental results can be found in [20–26], the NN-based controllers are rarely validated by hardware-in-the-loop experiments in previous results.
This paper seeks to address the controller design for 6DOF spacecraft tracking operations subject to unknown inertial parameters and disturbances. A saturated NN is designed to approximate the unknown dynamics, and the approximate error is counteracted by an adaptive robust compensating term. An auxiliary dynamical system is introduced to ensure the designed controller satisfying the magnitude constraints. The most remarkable result in this work is that the proposed NN saturated controller is validated by hardware-in-the-loop experiments which are conducted on the ASTERIX facility. Compared with the aforementioned works, the main contributions of this paper are threefold. First, in contrast to the NN employed in [12, 13, 16, 17], a saturated NN is developed to approximate the dynamics uncertainties while avoiding the long time saturation arising from the large tracking errors at the beginning of the operation. Second, to satisfy the magnitude constraints of the actuators installed in spacecraft, an auxiliary variable generated by a dynamical system is introduced in the control design. Compared with the dynamical system designed in [17], the proposed one gives a simpler structure which makes it easier to be realized in practical engineering applications. Finally, different from the numerical simulation validation of NN controllers in [12, 13], the proposed controller is experimentally validated on the ASTERIX facility, where the practical impacts including parametric uncertainties, disturbances, and measurement errors are affecting. The experimental results demonstrate that the proposed controller works and that the results meet theoretical predictions, within a margin of error.
The remaining sections are arranged as follows. Mathematical preliminaries are formulated in Section 2. The control problem to be solved is stated in Section 3. The main results including the controller design and stability proof are given in Section 4. Basic hardware characteristics of the ASTERIX facility and experimental results are presented in Section 5. Finally, the conclusions are summarized in Section 6.
2. Preliminaries
2.1. Notations
In what follows, denotes the Euclidean norm of a vector , and denote the minimum and maximum eigenvalues of a square matrix , respectively. For , superscript represents the matrix form of the cross product satisfying . For and positive constants and , the saturation function is defined as follows:
If no confusion arises, always presents throughout this paper. For , define the saturation function vector .
2.2. RBFNN Approximation
Let be an unknown smooth function. According to [11], we can approximate on a compact set by employing the following RBFNN: where is the bounded approximation error, is the weight vector, and is the node number. is defined by where is the estimate of , is the RBF vector, and is the Gaussian function satisfying with its center and its spread .
3. Problem Formulation
3.1. Reference Coordinate Frames
Assume that the spacecraft and simulator are rigid bodies, respectively. To formulate their dynamics, the following coordinate frames are defined. The Earth inertial frame (): its origin is located at the Earth center, axes and point to the direction of the vernal equinox and toward the north pole, respectively, and three axes satisfy the Right-Hand-Rule (RHR) frame. The spacecraft (desired) body fixed frame (): its origin () coincides with its center of mass (c.m.), and three axes coincide with its three inertial principal axes, respectively. The desired orbit frame (local vertical local horizontal (LVLH) frame) : its origin is the center of the desired target, axis points from the earth center to , axis is perpendicular to the orbit plane, and axis is in the orbit plane complying with the RHR. The testbed centred frame : its origin is the geometric center of the testbed surface, two axes are along the edges of the bed forming an orthogonal plane coordinate frame.
3.2. 6DOF Dynamics Model
3.2.1. Attitude Error Dynamics
The attitude of spacecraft is represented by using Modified Rodrigues Parameters (MRPs) , where is the principal rotation axis and is the principal rotating angle. According to [27], by introducing a switching condition at the surface , the unique and nonsingular description can be guaranteed. This further ensures that . Let and be the MRPs of the spacecraft and the desired, respectively. The error MRPs are given by
According to [27], the attitude error kinematics satisfies where , is the error angular velocity expressed in frame , and , respectively, represent the angular velocities of the spacecraft and the desired, and is the rotation matrix defined as
In addition, according to [27] and , we have the following attitude error dynamics: where is the inertia matrix, and are the control torque and disturbance torque, respectively.
3.2.2. Orbit Error Dynamics
Let and be the position and velocity tracking errors, where is the position vector error between the spacecraft and desired target, is the velocity error, and is the desired trajectory referenced in frame . In terms of [28], the orbit error kinematics and dynamics satisfy where is the mass of the spacecraft,
is the rotation matrix from frame to frame , is the rotation matrix between frames and satisfying with the true anomaly , the radial distance between the desired target and the Earth , the right ascension of ascending node , the argument of latitude , the argument of perigee , and the orbit inclination , is the control force in frame , and is the disturbance force. and stand for cosine and sine, respectively.
3.2.3. Actuator Distribution
Consider that there are pairs of thrusters installed in the spacecraft controlling of attitude and orbit simultaneously. Each thruster in one pair is assembled symmetrically with respect to the rotation center for the dynamics. The control torque direction and the spin axis satisfy the right-hand principle. In particular, each pair of thrusters consists of thrusters and , where provides control force along one axis and control torque around another spin axis of frame simultaneously and provides the opposite control force and torque generated by . Let be the outputs of all the pairs of thrusters and be the distribution matrix. To ensure the control system non-under-actuated case, the distribution matrix should satisfy .
Due to the fact that the thrusters are generally constrained by power supply in practice, we further consider saturations in the command controls. Generally, the saturated control is expressed by using the saturation function . In the studied case, we have that with where is the command control, and , , are the magnitude constraints. Let and define . Then, we have where . In addition, to ensure that the command control of each thrust should be nonnegative, i.e., , the command control signals are arranged as follows:
3.2.4. 6DOF Error Dynamics
Let and . In view of equation (6) and equation (9), we have where . In terms of equation (8), equation (10), and equation (14), the integrated 6DOF error dynamics is derived as follows: where
3.3. Control Objective
Consider the 6DOF tracking error dynamics consisting of equation (16) and equation (17). Suppose that the full motion information of the desired and the spacecraft is available to the spacecraft. The control objective is to design controller for each thruster assembled in the spacecraft such that and converge to small sets around zero.
Assumption 1. The desired angular velocity and its derivative are bounded. The desired position and its derivatives and are bounded, respectively.
Assumption 2. and are unknown constant parameters with known nominal values and , respectively.
4. Main Results
In this section, the main results are presented. An adaptive controller is first proposed such that the 6DOF tracking objective is realized. Then, the closed-loop stability analysis is undertaken.
4.1. Controller Development
In the following, an adaptive controller is synthesized by introducing a saturated RBFNN and a feasible auxiliary dynamical system, which results in the high accuracy tracking.
Define a variable where with and . In terms of equation (16) and equation (17), the derivative of equation (20) satisfies where .
Since the exact knowledge of inertial parameters cannot be obtained easily, we use their nominal values and to calculate the value of . Define as the value of when and . Let . It follows that where . It can be easily known that consists of the variables , , , , , and . To approximate , we introduce the following RBFNN where is the constant weight matrix, is the number of nodes, is the RBF vector, is the input of the RBFNN, and is the approximation error. During the overshoot and transient at the beginning of the control process, large tracking errors may lead to large output of the RBFNN, which would increase the burden of the actuators. To overcome this disadvantage, we use a saturated RBFNN instead of the direct output from RBFNN. Let be the estimate of . Define and the estimate error , respectively. Then, we rewrite equation (23) as where .
Assumption 3. is bounded by an unknown constant , i.e., .
Let and be the estimates of and , respectively. Design the following controller and the adaptation laws where with and , , is a small positive constant, , is a positive constant, is the initial value of , and is an auxiliary signal generated from the following dynamical system where .
By substituting equation (20) into equation (16), and equation (24) and (25) into equation (22), a 6DOF closed-loop system is derived as follows
4.2. Stability Analysis
Since the controller for the 6DOF tracking error dynamics has been developed, we next focus on the closed-loop stability analysis. In particular, the following theorem summarizes the accomplishment of the 6DOF tracking objective.
Theorem 1. Consider the 6DOF closed-loop dynamics (28) and (29). Suppose that Assumptions 1–3 hold. If the control parameters are selected as where , is an arbitrary positive constant, and , the proposed controller (25) and adaptation laws (26) ensure that the 6DOF tracking objective is achieved, i.e., and converge to small sets around zero, resectively.
Proof. Define the estimate error . Choose the following Lyapunov function candidate Its time derivative along equation (27), (28), and (29) satisfies In view of and , we have where . Note that , , . It follows that where , , . Substituting adaptation laws equation (26) gives Note that and . Let . Then, we have where . In view of equation (31) and equation (36), then where , , and . It follows that So eventually converges to . It follows that are bounded. In particular, let and . With equation (20) and , we can obtain that and eventually converge to and , respectively. This completes the proof.
Remark 1. Different from the conventional NN used in [12, 13, 16, 17], a saturated NN output is designed to reduce the large values caused by the large tracking errors during the overshoot and transient at the beginning of the control operation. An adaptive robust term is employed in the controller to compensate for the nonlinear term arising from the NN approximation error. The saturated NN and robust compensating term work together not only to counteract the dynamics uncertainties efficiently but also to improve the control accuracy.
Remark 2. In this paper, we can easily determine the controller parameters according to equation (30). Careful analysis implies that increasing would result in a smaller set which illustrates that the ultimate convergent sets of and can be tuned by choosing appropriate control parameters. Moreover, larger control parameters could contribute to faster convergence rate but longer control saturation times which would bring the burden of thrusters. To this end, a reasonable compromise among practical control objectives is necessary.
5. Experimental Results
In this section, the hardware-in-the-loop experiments are tested to evaluate the proposed controller. The experiment scenario shows that a 3DOF simulator tracks a desired trajectory on the testbed surface while synchronizing its rotation angle with the desired attitude. First, basic hardware characteristics of the test facility are presented. Then, experimental results are illustrated by using the proposed controller.
5.1. ASTERIX Facility Description
The hardware experimental validation of the proposed control strategy is conducted in the ASTERIX facility [19]. The test facility has three main elements including an operational arena, two 5DOF spaceflight simulators, and tracking (measurement) systems. An overview of this test facility is presented in Figure 1.
Some basic and important characteristics of the facility are listed as follows. A operational arena consists of four aluminum plates organized side by side. Due to the limitation of the manufacturing process, the inclination is inevitable. From the measurement by an SPI TRONIC PRO 3600, the maximum local inclination turned out to be −0.07° and the maximum local displacement with respect to the lowest point of the arena is 0.863 mm. The 5DOF spaceflight simulator consists of a 2DOF translation stage and a 3DOF attitude stage. The mass of each simulator is 147.22 kg, and the moment of inertia of the attitude stage presented in frame is
In addition, the spaceflight simulator is equipped with 3 reaction wheels and 16 thrusters. Each reaction wheel is set to provide torque around each axis of frame . The 16 thruster assemblies are symmetrically arranged with respect to the CR. Each set of four thrusters is set to expel exhaust gas in a fixed direction of the body frame, and . The attitude of the simulator is obtained by an attitude and heading reference system (AHRS). The attitude measurement noise satisfies a random normal distribution where is a random scalar, , . The position of the simulator is measured by the tracking system. Similarly, the measurement error also satisfies the following random normal distribution where is a random scalar, , and .
5.2. Experimental Results
In this subsection, we show the experimental results.
The desired rotation angle is governed as where is the initial angle of the simulator and is the time of the experiment. The desired trajectory in each phase is given in Table 1.
In this experiment, we use 4 pairs of thrusters generating the control efforts. The distribution of thrusters are shown in Table 2 and Figure 2. The distribution matrix is given by , where is the arm of force of each pair of thrusters. The model parameters of the simulator are listed in Table 3, respectively.
The saturation constraints of the NN with nodes are chosen as . The parameter estimates are initialized as and . The parameters of the controller and adaptation laws are chosen as , , , , , and . The experiment time is . Four groups of experiments are performed by using different control parameters and nominal values of inertial parameters which are summarized in Table 4, where . Experimental results are presented in Figures 3–10.
Figure 3 collects the rotation angle error of each case. Figure 4 presents the trajectory of each case versus the desired. Figures 5 and 6 show the position tracking errors in -axis and -axis of each case, respectively. Table 5 summarizes all the standard derivations of tracking errors in different cases. From the experimental results, it can be seen that the simulator tracks the desired trajectory while synchronizing its rotation angle with a good performance in spite of measurement errors and a large range of parameter uncertainty. During the first phase of the experiment, the transient behavior arises because the initial position of the simulator does not coincide with the initial value of the desired trajectory. After that, the simulator tracks the desired trajectory fairly well. Apart from the slight discrepancy of the tracking errors, the high control accuracy is confirmed by the experimental results in spite of the parameter uncertainty from and .
We next show the detailed experimental result of Case 2 to evaluate the proposed controller. Figure 7 illustrates the top view of the experiment result, where the orange line is the initial position with an -axis in frame representing initial angle of the simulator and the blue line is the final position of the simulator. From this result, we can see that the simulator tracks the desired trajectory with a good performance. In particular, there is no obvious tracking error after the simulator enters the second phase. In addition, the command control of each thruster is collected in Figure 8. It can be seen that all the control signals are within their magnitude constraints, respectively. Besides, Figure 9 describes the output of the NN versus time, and Figure 10 presents the estimate and norm of the auxiliary signal versus time, respectively. At the beginning of the experiment, it can be seen that the output of the NN is much larger than the magnitude constraint of the thruster. Instead, the saturated value is applied in the controller which contributes to reduce the burden of the thrusters. It is also evidenced that the estimate and auxiliary signal remain bounded during the experiment.
6. Conclusions
We develop an NN-based saturated adaptive controller for 6DOF spacecraft tracking operations in this paper. To estimate the dynamics uncertainties with high accuracy while avoiding the long time control saturation during the initial operation, an adaptive saturated NN is designed with its approximate error being compensated by a continuous robust term. An auxiliary dynamical system is introduced to overcome the effects caused by control saturation. The ultimate boundedness of the closed-loop dynamics has been proved. Hardware-in-the-loop experimental validation is conducted on the test facility ASTERIX, where a 3DOF tracking scenario of the planar and rotational maneuver is performed in the presence of unknown inertial parameters, disturbances, and measurement errors. Experimental results demonstrate that the proposed controller ensures the simulator tracking a desired trajectory with a good performance and that the rotation angle and position tracking errors are within degree and millimeter during the final phase.
Data Availability
The simulation 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 KASI (Korea Astronomy and Space Science Institute) and Yonsei research collaboration program for the frontiers of astronomy and space science. This work was also supported by the Space Basic Technology Development Program through the National Research Foundation of Korea funded by the Ministry of Science and ICT of Republic of Korea (2018MIA3A3A02065610).