Chong SUN, Jinping YUAN, Zhnxi ZHU

a School of Astronautics, Northwestern Polytechnical University, Xi’an 710100, China

b Research and Development Institute of Northwestern Polytechnical University in ShenZhen, Shenzhen, 518057, China

KEYWORDS Active disturbance rejection control;Adaptive controller;Ground experiment;Micro-gravity;Obstacle avoidance;Spacecraft proximity

Abstract In this study, a neural adaptive controller is developed for a ground experiment with a spacecraft proximity operation. As the water resistance in the experiment is highly nonlinear and can significantly affect the fidelity of the ground experiment,the water resistance must be estimated accurately and compensated using an active force online. For this problem, a novel control algorithm combined with Chebyshev Neural Networks (CNN) and an Active Disturbance Rejection Control (ADRC) is proposed. Specifically, the CNN algorithm is used to estimate the water resistance. The advantage of the CNN estimation is that the coefficients of the approximation can be adaptively changed to minimize the estimation error. Combined with the ADRC algorithm, the total disturbance is compensated in the experiment to improve the fidelity. The dynamic model of the spacecraft proximity maneuver in the experiment is established. The ground experiment of the proximity maneuver that considers an obstacle is provided to verify the efficiency of the proposed controller. The results demonstrate that the proposed method outperforms the pure ADRC method and can achieve close-to-real-time performance for the spacecraft proximity maneuver.

1. Introduction

The spacecraft’s autonomous proximity maneuver has received considerable attention due to its wide applications in space missions, including rendezvous and docking, debris removing,and spacecraft repairing or refueling. Ground experiments before the space operations are essential to space-enabling technologies before the actual space missions. For ground demonstrations of these technologies, a ground system must provide a long-duration, accurate, controllable, and almost real micro-gravity environment to simulate the space operation maneuvers. Several micro-gravity systems have been reported.1-4For spacecraft orbital servicing in space,two main approaches are used to simulate the micro-gravity effect. The first one is the environment simulation approach that can provide a comparable micro-gravity environment, such as zero-g airplanes and drop towers. Zero-g airplanes, such as IL-76MGK (Russia), KC-135 (USA), C-9 (USA), Caravell, A-300 (France), and MU-300 (Japan), can only provide a short-time micro-gravity environment (10-30 s). The Bremen drop tower can provide 10-6g for 4.74 s5; China’s national micro-gravity laboratory drop tower can achieve 10-6g for 3.5 s6;some drop towers in the USA and Japan have the height over 700 m and can achieve 10-6g for 10 s.7Overall,zero-g airplanes and drop towers have the following drawbacks:(A)the micro-gravity level can only be maintained for a short time,and (B) the volume of the test body is limited by the aircraft or the drop tower and is always very small. The second approach is to use an effect simulation method that maintains the apparent gravity of the test body at zero, including the sling suspension system, air-bearing suspension system, and neutral buoyancy system. Sling suspension systems can generate compensation force of the same size but in the opposite direction of gravity to simulate zero-g motion in the threedimensional space.8,9However, it can only provide the micro-gravity statically,and the suspension cables cannot provide extra tension to the test body in a non-vertical direction.Air-bearing suspension systems use a flat floor with air-bearing pads to realize near-frictionless relative motion,which can provide micro-gravity in the two-dimensional space. The first airbearing suspension system developed by NASA can provide yaw of ±120 deg and carry 400 kg.10,11However, the drawback of the air-bearing suspension system is that it cannot simulate micro-gravity in the vertical direction. Thus, complex operations in 3-D space cannot be simulated using this system.Neutral buoyancy systems use the buoyancy force to balance the gravity of the test body;the advantage is that it can afford long-term 6-Degree-of-Freedom (DOF) micro-gravity for a large-volume test body. MIT in the USA tested a teleported manipulator using NASA’s Neutral Buoyancy Lab.12The multi-arm free-flying robot experiment to test arm-based interaction was performed in the buoyancy system at Maryland University.13However, the simulation accuracy highly depends on the balancing results:if the gravity of the test body is well balanced, it can maintain good accuracy and high fidelity for the ground experiment.Yet,the balancing result is difficult to adjust online, especially for a new test body. For this problem, a novel magnetism-buoyancy hybrid Micro-G simulation system was proposed.14The system combines neutral buoyancy with electromagnetic force to balance the gravity of the test body and provide a micro-gravity environment.As the electromagnetic force on the test body can be adjusted in the electromagnetic system, it can simplify the balancing process using a micro-gravity testbed compared with the neutral buoyancy system.

For the spacecraft proximity maneuver mission, care is needed to efficiently handle the collision avoidance between the spacecraft and the target. In addition, the control method should also handle complex environmental perturbations in the space.15-17Thus,real-time control is a challenge,and multiple studies have addressed it.Considering obstacle avoidance constraints in the autonomous rendezvous and docking process, an artificial potential field-based terminal approaching guidance law was proposed,15in which the impulse control magnitude, direction, and switching time can be analytically expressed. However, the unknown parameters for the nonlinear dynamics of the proximity maneuver were ignored. Based on the model predictive control system, a nonlinear guidance law was developed for a chaser spacecraft rendezvous and docking with a passive target spacecraft in an elliptical or circular orbit.16However, sampling a time step using the MPC method is usually set small in a close proximity maneuver to avoid violation of path constraints between sampling time intervals. Therefore, a trajectory optimization problem with a large number of decision variables must be solved, which may make this approach too computationally intensive. In Ref.17, a Proportion-Integrative-Derivative (PID) control law was applied for spacecraft rendezvous and docking experiments.However,the contradiction of the fast tracking and the overshoot in the PID method results in low tracking performance. Besides, for a spacecraft proximity operation, the PID controller cannot handle uncertainty dynamics and nonlinear external disturbance.

In this study, a neural adaptive control approach is proposed for the ground experiment of a six-DOF spacecraft proximity operation in a micro-gravity testbed. Specifically,majority of the gravity of the test body is balanced by the buoyancy,and the small residual gravity is offset by an electromagnetic force. The main difference between the dynamics of the ground experiment and the dynamics in the space is that the spacecraft in the ground experiment suffers the water resistance.Thus,the water resistance has to be estimated and compensated online. In the proposed control algorithm, the CNN algorithm is applied to estimate the highly nonlinear disturbance in the ground experiment, and the Active Disturbance Rejection Control (ADRC) is used to compensate for the water resistance and improve the experiment fidelity.

The remainder of this paper is organized as follows. Section 2 introduces the experimental system. Section 3 presents the dynamics of the ground experiment for a spacecraft proximity mission. Section 4 proposes a neural adaptive controller for the ground experiment, combined with CNN and ADRC.Subsequently, a six-DOF spacecraft proximity maneuver in a several meters range of a target object is performed, and the experiment results demonstrate the efficiency of the proposed control method. Section 5 concludes the paper.

2. Facilities of the Micro-G test-bed experiment

As illustrated in Fig. 1,18,19the micro-gravity test body includes six parts: an electromagnetic system, Experimental Electromagnetic Balancing Platform (EEBP) test body,Liquid-Float System (LFS), control system, measurement system,and communication system.As depicted in Fig.2,the test body is a spacecraft prototype with a Vision Positioning System(VPS)and four different light-emitting signs.Six propeller thrusters U=[U1，U2，...，U6]Tare installed around the spacecraft: two of the（U1，U5）m are mounted along the Z axis;（U2，U4） are mounted along the Y axis; （U3，U6） are mounted along the X axis. The test body communicates and exchanges data with a master computer through the optical fiber.The liquid float system is a water tank; its size is approximately 2.5 m×2.5 m×1.5 m. The electromagnetic system produces a distributed electromagnetic force on EEBP to precisely balance the residual gravity of the test body.The electromagnetic force can be adjusted according to real-time measurements of the VPS. The electromagnetic force can always be applied to the test body in a downward vertical direction through the universal joint to balance the residual gravity. The measurement system consists of the VPS and an Inertial Measurement Unit(IMU). The precision of the speed measurement of the VPS is±3 mm/s.As the VPS is used to detect the target’s position,it cannot measure the speed directly. Fig. 3 depicts the software and hardware of the measurement system and the data flowchart. The control system includes a master computer and an industrial control computer. The master computer fuses output data of the VPS and the IMU, displays final information on navigation, handles human-machine interaction, and feeds control instructions into the industrial control computer through the optical fiber. The complete experimental system is depicted in Fig. 4.

Fig. 1 Structure of the hybrid micro-gravity simulation system.18,19

Fig. 2 Test body of EESP.18,19

In the ground experiment system,the force generated in the experimental environment includes the buoyancy, electromagnetic force, gravity, and perturbation force produced by the environment. Our method can balance gravity with buoyancy and electromagnetic force:the liquid buoyancy can offset most of the gravity,and the controlled electromagnetic force is used to accurately balance the small residual gravity. The similarities between the spatial experimental system and the ground experimental system have been described in Ref.14,19.

3. Dynamics of the spacecraft proximity

3.1. Dynamics equation of a proximity maneuver

In this study, a ground experiment of a spacecraft proximity maneuver is performed in a six-DOF micro-gravity simulation system.In the experiment,the docking port is fixed at the edge of the water tank, as depicted in Fig. 5. As the distance between the chaser spacecraft and the target spacecraft is extremely small in comparison with the orbital height in space,it is assumed that environment disturbances (such as the Earth’s oblateness or solar radiation pressure) can be ignored.20First,for the ground experiment with the spacecraft approaching maneuver, an approaching trajectory is planned considering obstacle-avoidance constraints. Next, the water disturbance is estimated and compensated online, and the chaser spacecraft is driven to the target spacecraft following the planned trajectory. The ground experiment is described with the following assumptions:

Fig. 3 Software and hardware of the measurement system.

Fig. 4 Components of the experimental system.

(1) The approaching velocity of the spacecraft is in the range of [-10 cm/s， 10 cm/s]; then, the waterresistance perturbation can be estimated and compensated by the active control.

(2) the motion parameters of the target spacecraft and the servicing spacecraft are given, and the communication delay is ignored.

(3) the Earth’s oblateness perturbations for the target spacecraft and the chaser satellite are ignored.

Three reference frames are considered to derive the spacecraft rendezvous dynamics. The first one is the Earthcentered reference frame ｛OEXEYEZE｝, the second one is the body-fixed reference frame of the chaser spacecraft｛OCXCYCZC｝ , and the third one is the body-fixed reference frame of the target spacecraft ｛OTXTYTZT｝. In the Earthcentered reference frame, the position and attitude of the target spacecraft are denoted as ηt=[xt，yt，zt，φt，θt，ψt], and the position and attitude of the chaser spacecraft are represented as ηc=[xc，yc，zc，φc，θc，ψc]. In the body-fixed reference frame of the chaser spacecraft, the linear and angular velocity of the chaser spacecraft is denoted by vector vc=[v1，v2]=[vx，vy，vz，p，q，r]T,vc∈R6.Thus, the relative position and relative attitude of the chaser spacecraft and the target is

Fig. 5 Illustration of the on-ground experiment.

3.2. Reference trajectory generation using the shaping-based method

As depicted in Fig. 6, for the spacecraft proximity maneuver,the chaser spacecraft should be driven from initial point A to target docking point D; meanwhile, the attitude of the chaser spacecraft is required to align with the docking line of the target spacecraft. Here, the shaping-based method is applied to generate a reference trajectory. The relative position reference trajectory is defined as

Fig. 6 Reference trajectory for the proximity maneuver.

By analogy,the relative attitude reference trajectory can be obtained using the three-order polynomial shaping method.Assuming the position vector of the obstacle spacecraft is xo（x，y，z）∈A,and the position vector of the chaser spacecraft is xc（x，y，z）∈B, then the boundary constraints of trajectory planning considered obstacle is A ∩B=. In this study, the aim is to test the proximity maneuver of the spacecraft in a range of approximately several meters;thus,the relative speed is very small.20

4. Controller design for the on-ground experiment

Given a reference trajectory xr, the control objective is to design adaptive output feedback control signals ui， i=1，2，...，6 to minimize tracking error e=x-xr. The dynamics equation of the spacecraft motion is shown in Eq.(7). To follow the given reference trajectory, the objective is to design a robust adaptive controller to handle uncertainty dynamics and guarantee that the linear position error and angle error converge to zero. In this experiment, the Neural-ADRC approach is applied to estimate the water resistance and compensate for it.

4.1. CNN approximation for the nonlinear function

In this study,the Chebyshev neural network is applied to estimate the nonlinear disturbance (F（t） as defined in Eq. (11)).CNN is a function-link neural network based on Chebyshev polynomials.21,22The basic functions of single-layer CNN are composed by a set of orthogonal polynomials, derived from the solutions of Chebyshev differential function as

where γ denotes the optimal weight matrix of the CNN, and ε is the bounded CNN approximation error. In the experiment,adaptive parameter ^γ will estimate γonline with an error defined as γ～=^γ-γ. By Eqs. (15)-(17), the CNN is employed to approximate the nonlinear water resistance in the experiment. The weight matrix of the approximation expression can be tuned to minimize the approximated error.

4.2. ESO designer for the velocity construction

Fig. 7 Scheme of the proposed controller.

Table 1 Experiment conditions.

In this study, ESO is designed in the delta domain to obtain a relative velocity signal of the chaser spacecraft in the system dynamics of Eq. (10). Because VPS only detects the position of the target, it cannot obtain speed information directly. In some cases,it cannot obtain the position signal because the line sight of the camera is covered by an obstacle.In the IMU system, the signal of the angular velocity is obtained through the differentiating signal of the angle, but it usually drifts as the time increases.Thus,the ESO is applied to obtain the velocity signal of the spacecraft.From Ref.23,the structure of the ESO is

Here ^x1and ^x2are the observations of x1and x2,and β1,β2,and β3are observer parameters.

4.3. Adaptive controller design

The scheme of the controller is depicted in Fig. 7.

Fig. 8 Layout of the ground experiment.

Fig. 9 X-axis position tracking error.

Fig. 10 Y-axis position tracking error.

Step 3. Controller design

Using the CNN technique,an unknown nonlinear function can be approximated as

Fig. 11 Z-axis position tracking error.

5. Experimental validation and analysis

Fig. 12 Attitude φ tracking error.

Fig. 13 Attitude θ tracking error.

In this section, an adaptive neural controller is applied to the proximity maneuver in the micro-gravity testbed. In the ground experiment, the neutral buoyancy and the electromagnetic force must balance the gravity of the test body, and the control system based on the Neural-ADRC method is applied to estimate the total external disturbance and compensate it online.The VPS system and the IMU system provide the position data, attitude data, position velocity, and attitude velocity. The experimental settings are provided in Table 1. The experiments are conducted at 500 mm under-Earth the water,where the buoyancy can generate approximately 98.5% of gravity, whereas the electromagnetic force provides the residual gravity. In the simulation, an industrial control computer works as the controller, and the proposed tracking control algorithm is implemented in C#. The parameters of the controller used in the experiment are listed in the appendix. Six nodes are selected for CNN in the experiments. The results demonstrate that an increased number of nodes cannot improve the tracking performance.

Fig. 14 Attitude Ψ tracking error.

The layout of the ground experiment is depicted in Fig. 8.The system includes the following parts: a docking port of the target spacecraft is located on one side of the water tank,and the chaser spacecraft is located nearby within the range of several meters, as depicted in Fig. 8(a). The docking port of the target spacecraft is illustrated in Figs. 8(b) and (c).The target spacecraft, obstacle, and chaser spacecraft are depicted in Fig. 8(d). The shaping-based method is applied to the plan proximity trajectory as follows.

where aXi,ε i[1,6],aYi,i ε[1,5],aZi,i ε[1,4],aφi,i ε[1,3],are the coefficients of the shaping linear trajectory in X axis, Y axis and attitude trajectory in Z axis.

Given the initial point and the terminal point, coefficients of the shaping function can be obtained as shown in the appendix. Suppose that yaw φ ensures that the X axis of the test body points to the target throughout an experimental simulation, whereas the pitch and the roll angle are zero. As the test body has an autonomous control ability, the Neural-ADRC algorithm and ADRC control algorithm in Ref.24,25are applied to this 6-DOF trajectory tracking problem. The simulation results are compared in terms of accuracy and tracking effects. The position tracking trajectory and the reference trajectory are depicted in Figs.9-11.The attitude tracking experimental results are depicted in Figs. 12-14. See Figs. 15 and 16(N1-N6 is the thrust of propellers assembled on test body).

Fig. 15 Control signal using Neural-ADRC.

Fig. 16 Control signal using ADRC.

Simulation results demonstrate that the position tracking errors obtained using the proposed algorithm are lower than those obtained using the ADRC algorithm. For the X axis,the tracking error is approximately 1 cm using the Neural-ADRC based controller, whereas it is approximately 3 cm using the controller based on the ADRC algorithm. For the Y axis and Z axis, the position tracking errors are approximately 1.5 cm and 0.8 cm using the proposed controller,whereas the result using ADRC is approximately 2 cm and 2.5 cm. For the attitude tracking problem, the Neural-ADRC controller also outperforms the ADRC algorithm:for-X axis,Y axis,and Z axis,tracking errors are less than 1°using the Neural-ADRC algorithm; the errors for the X axis and Y axis are less than 1.5 deg and the error of Z axis is approximately 2°. The main reason is that the water resistance and external disturbance are highly nonlinear. Although both the ESO and the CNN can estimate the nonlinear disturbance,the control precision and tracking effect are better with the Neural-ADRC approach than with the ADRC method.Besides, in the experiment, a longer time is required to tune ESO coefficients to estimate the highly nonlinear disturbance using pure ADRC. As the coefficients in the ESO algorithm are fixed, it is difficult to estimate the water resistance accurately. However, if the CNN algorithm is used to estimate the water resistance,the coefficients can be updated with more experimental data. Thus, the estimation error can be reduced as more data arrive(see Figs.15 and 16).However,the number of nodes in the CNN algorithm should be carefully selected using the proposed method, as an increased number of nodes can improve the accuracy of the disturbance estimation but increase the computational cost. The micro-gravity level is depicted in Fig. 17, and we observed that it can reach 6×10-4m/s2using the proposed ground experiment; hence,the proposed testbed can provide a long-time and high-level micro-gravity effect.

Fig. 17 Micro-gravity level.

6. Conclusions

In this study,a neural adaptive control method is proposed for the ground experiment with the spacecraft proximity maneuver in the micro-gravity testbed. In the proposed control method,the CNN algorithm is used to estimate the water resistance.As the coefficients of the approximation can be tuned adaptively with an increase of the experiment data, the estimation error can be reduced significantly compared with the pure ADRC method.A six-DOF spacecraft approaching maneuver considering the obstacle is applied in the ground experiment to demonstrate the efficiency of the proposed controller.The simulation results demonstrate that the micro-gravity level can reach approximately 6×10-4g, and the tracking error can reach 1 cm for the position tracking and 1 deg for the attitude tracking, which is better than in the pure ADRC approach.Hence, the proposed control method can improve the fidelity of the ground experiment and can be used for other ground experiments, such as on-orbit servicing and space debris capturing, in the future.

Acknowledgement

This research was supported by the National Natural Science Foundation of China (No. 11802238).

Appendix A

(1) Six-DOF dynamics equation is represented as follows

Table 2 Parameters of Neural-ADRC controller.

Table 3 Coefficients of shape-based function.

Table 4 The parameters in TD ESO CNN and control signal.

Table 5 The parameters that influence the control performance.

Remark 1.From Eqs. (A21) and (A24), it can find that, if design parameters ci，Γ，σ, and κm，riare properly selected, the Lyapunov is tracking error converge to a compact set around zero. Eq. (A24) guarantees transient and steady-state performance of whole system,the detail proof can be found in Ref.26.