# Adaptive Saturated Neural Network Tracking Control of Spacecraft: Theory and Experimentation

Authors: Kewei Xia; Taeyang Lee; Sang-Young Park
Journal: International Journal of Aerospace Engineering
 (2019)

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## 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.


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## Body

## 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 Section2. 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,x denotes the Euclidean norm of a vector x∈ℝn, λminX and λmaxX denote the minimum and maximum eigenvalues of a square matrix X∈ℝn×n, respectively. For x=x1,x2,x3T∈ℝ3, superscript × represents the matrix form of the cross product satisfying x×=0,−x3,x2;x3,0,−x1;−x2,x1,0. For x∈ℝ and positive constants a and b, the saturation function is defined as follows:
(1)sata,bx=a,ifx>a,x,if−b≤x≤a,−b,ifx<−b.If no confusion arises,satx always presents sata,bx throughout this paper. For x=x1,x2,⋯,xnT∈ℝn, define the saturation function vector satx=satx1,satx2,⋯,satxNT.


### 2.2. RBFNN Approximation

Letfx:ℝn→ℝ be an unknown smooth function. According to [11], we can approximate fx on a compact set Ω⊆ℝn by employing the following RBFNN:
(2)fx=wTΦx+ϵ,where ϵ is the bounded approximation error, w⊆ℝl is the weight vector, and l is the node number. w is defined by
(3)w=argminw^supx∈Ωfx−w^TΦx,where w^ is the estimate of w, Φx=ϕ1x,ϕ2x,⋯,ϕlxT:Ω→ℝl is the RBF vector, and ϕix is the Gaussian function satisfying
(4)ϕix=exp−x−μiηi2,i=1,2,⋯,l,with its center μi∈ℝn and its spread ηi>0.


## 2.1. Notations

In what follows,x denotes the Euclidean norm of a vector x∈ℝn, λminX and λmaxX denote the minimum and maximum eigenvalues of a square matrix X∈ℝn×n, respectively. For x=x1,x2,x3T∈ℝ3, superscript × represents the matrix form of the cross product satisfying x×=0,−x3,x2;x3,0,−x1;−x2,x1,0. For x∈ℝ and positive constants a and b, the saturation function is defined as follows:
(1)sata,bx=a,ifx>a,x,if−b≤x≤a,−b,ifx<−b.If no confusion arises,satx always presents sata,bx throughout this paper. For x=x1,x2,⋯,xnT∈ℝn, define the saturation function vector satx=satx1,satx2,⋯,satxNT.


## 2.2. RBFNN Approximation

Letfx:ℝn→ℝ be an unknown smooth function. According to [11], we can approximate fx on a compact set Ω⊆ℝn by employing the following RBFNN:
(2)fx=wTΦx+ϵ,where ϵ is the bounded approximation error, w⊆ℝl is the weight vector, and l is the node number. w is defined by
(3)w=argminw^supx∈Ωfx−w^TΦx,where w^ is the estimate of w, Φx=ϕ1x,ϕ2x,⋯,ϕlxT:Ω→ℝl is the RBF vector, and ϕix is the Gaussian function satisfying
(4)ϕix=exp−x−μiηi2,i=1,2,⋯,l,with its center μi∈ℝn and its spread ηi>0.


## 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 (FI=O,xi,yi,zi): its origin O is located at the Earth center, axes xi and zi 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 FB=B,xB,yB,zB (FB0=B0,xB0,yB0,zB0): its origin B (B0) 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) FL=L,xL,yL,zL: its origin L is the center of the desired target, xL axis points from the earth center to L, zL axis is perpendicular to the orbit plane, and yL axis is in the orbit plane complying with the RHR. The testbed centred frame FT=T,xT,yT: 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)σ∈σ∈ℝ3∣σ=n^tanη/4, where n^∈ℝ3 is the principal rotation axis and η∈−2π,2π is the principal rotating angle. According to [27], by introducing a switching condition at the surface σ=1, the unique and nonsingular description can be guaranteed. This further ensures that 0≤σTσ≤1. Let σ and σd be the MRPs of the spacecraft and the desired, respectively. The error MRPs are given by
(5)σ~=σdσTσ−1+σ1−σdTσd−2σd×σ1+σdTσdσTσ+2σdTσ.According to [27], the attitude error kinematics satisfies
(6)σ~̇=Gσ~ω~,where Gσ~=1/41−σ~Tσ~I3+2σ~×+2σ~σ~T, ω~=ω−R~ωd is the error angular velocity expressed in frame FB, ω and ωd, respectively, represent the angular velocities of the spacecraft and the desired, and R~ is the rotation matrix defined as
(7)R~≜Rσ~=I3−41−σ~Tσ~1+σ~Tσ~2σ~×+8σ~×21+σ~Tσ~2.In addition, according to [27] and R~̇=R~ω~×, we have the following attitude error dynamics:
(8)Jω~̇=−ω×Jω+τ+τd+Jω~×R~ωd−R~ω̇d,where J∈ℝ3×3 is the inertia matrix, τ∈ℝ3 and τd∈ℝ3 are the control torque and disturbance torque, respectively.


#### 3.2.2. Orbit Error Dynamics

Letr~=r−δ and v~=v−δ̇ be the position and velocity tracking errors, where r=rx,ry,rzT is the position vector error between the spacecraft and desired target, v is the velocity error, and δ is the desired trajectory referenced in frame FL. In terms of [28], the orbit error kinematics and dynamics satisfy
(9)r~̇=v~,(10)mv~̇=−mCtv−mnt+RLBf+fd−mδ,¨where m is the mass of the spacecraft,
(11)Ct=2ν̇0−10100000,nt=−ν¨ry−ν̇2rx+μr0+rxr0+rx2+ry2+rz23/2−μr02ν¨rx−ν̇2ry+μryr0+rx2+ry2+rz23/2μrzr0+rx2+ry2+rz23/2.RLB=RLRTσ is the rotation matrix from frame FB to frame FL, RL is the rotation matrix between frames FI and FL satisfying
(12)RL=cΩ0cθ0−sΩ0sθ0ci0sΩ0cθ0+cΩ0sθ0ci0sθ0si0−cΩ0sθ0−sΩ0cθ0ci0−sΩ0sθ0+cΩ0cθ0ci0cθ0si0sΩ0si0−cΩ0si0ci0,with the true anomaly ν, the radial distance between the desired target and the Earth r0, the right ascension of ascending node Ω0, the argument of latitude θ0=ωp+ν, the argument of perigee ωp, and the orbit inclination i0, f∈ℝ3 is the control force in frame FB, and fd∈ℝ3 is the disturbance force. c and s stand for cosine and sine, respectively.


#### 3.2.3. Actuator Distribution

Consider that there areN 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 Tj and Pj, where Tj provides control force along one axis and control torque around another spin axis of frame FBi simultaneously and Pj provides the opposite control force and torque generated by Tj. Let p∈ℝN be the outputs of all the pairs of thrusters and A∈ℝ6×N be the distribution matrix. To ensure the control system non-under-actuated case, the distribution matrix should satisfy rankA=6.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 functionsat·. In the studied case, we have that p=satξ=satξ1,satξ2,⋯,satξNT with
(13)satξj=ξ¯j+,ifξ>ξ¯j+,ξj,if−ξ¯j−≤ξj≤ξ¯j+,−ξ¯j−,ifξj<−ξ¯j−,where ξ∈ℝN is the command control, ξ¯j+>0 and ξ¯j−>0, j=1,2,⋯,N, are the magnitude constraints. Let u=τT,RLBfTT and define Δξ=satξ−ξ. Then, we have
(14)u=DAsatξ=DAξ+Δξ,where D=diagI3,RLB. In addition, to ensure that the command control of each thrust should be nonnegative, i.e., Tj,Pj≥0,j=1,2,⋯,N, the command control signals are arranged as follows:
(15)Tj=satξj,Pj=0,ifξj≥0,Pj=−satξj,Tj=0,ifξj<0.


#### 3.2.4. 6DOF Error Dynamics

Letx1=σ~T,r~TT and x2=ω~T,v~TT. In view of equation (6) and equation (9), we have
(16)ẋ1=Λx2,where Λ=diagGσ~,I. In terms of equation (8), equation (10), and equation (14), the integrated 6DOF error dynamics is derived as follows:
(17)Mẋ2=ς+Bξ+Δξ+d,where
(18)ς=−ω×Jω+Jω~×R~ωd−R~ω̇d−mCtv−mnt−mδ¨,M=diagJ,mI,B=DA,(19)d=τdT,fdTT.


### 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 ξ for each thruster assembled in the spacecraft such that x1 and x2 converge to small sets around zero.Assumption 1.
The desired angular velocityωd and its derivative ω̇d are bounded. The desired position δ and its derivatives δ̇ and δ¨ are bounded, respectively.Assumption 2.
J and m are unknown constant parameters with known nominal values J0 and m0, respectively.


## 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 (FI=O,xi,yi,zi): its origin O is located at the Earth center, axes xi and zi 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 FB=B,xB,yB,zB (FB0=B0,xB0,yB0,zB0): its origin B (B0) 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) FL=L,xL,yL,zL: its origin L is the center of the desired target, xL axis points from the earth center to L, zL axis is perpendicular to the orbit plane, and yL axis is in the orbit plane complying with the RHR. The testbed centred frame FT=T,xT,yT: 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)σ∈σ∈ℝ3∣σ=n^tanη/4, where n^∈ℝ3 is the principal rotation axis and η∈−2π,2π is the principal rotating angle. According to [27], by introducing a switching condition at the surface σ=1, the unique and nonsingular description can be guaranteed. This further ensures that 0≤σTσ≤1. Let σ and σd be the MRPs of the spacecraft and the desired, respectively. The error MRPs are given by
(5)σ~=σdσTσ−1+σ1−σdTσd−2σd×σ1+σdTσdσTσ+2σdTσ.According to [27], the attitude error kinematics satisfies
(6)σ~̇=Gσ~ω~,where Gσ~=1/41−σ~Tσ~I3+2σ~×+2σ~σ~T, ω~=ω−R~ωd is the error angular velocity expressed in frame FB, ω and ωd, respectively, represent the angular velocities of the spacecraft and the desired, and R~ is the rotation matrix defined as
(7)R~≜Rσ~=I3−41−σ~Tσ~1+σ~Tσ~2σ~×+8σ~×21+σ~Tσ~2.In addition, according to [27] and R~̇=R~ω~×, we have the following attitude error dynamics:
(8)Jω~̇=−ω×Jω+τ+τd+Jω~×R~ωd−R~ω̇d,where J∈ℝ3×3 is the inertia matrix, τ∈ℝ3 and τd∈ℝ3 are the control torque and disturbance torque, respectively.


### 3.2.2. Orbit Error Dynamics

Letr~=r−δ and v~=v−δ̇ be the position and velocity tracking errors, where r=rx,ry,rzT is the position vector error between the spacecraft and desired target, v is the velocity error, and δ is the desired trajectory referenced in frame FL. In terms of [28], the orbit error kinematics and dynamics satisfy
(9)r~̇=v~,(10)mv~̇=−mCtv−mnt+RLBf+fd−mδ,¨where m is the mass of the spacecraft,
(11)Ct=2ν̇0−10100000,nt=−ν¨ry−ν̇2rx+μr0+rxr0+rx2+ry2+rz23/2−μr02ν¨rx−ν̇2ry+μryr0+rx2+ry2+rz23/2μrzr0+rx2+ry2+rz23/2.RLB=RLRTσ is the rotation matrix from frame FB to frame FL, RL is the rotation matrix between frames FI and FL satisfying
(12)RL=cΩ0cθ0−sΩ0sθ0ci0sΩ0cθ0+cΩ0sθ0ci0sθ0si0−cΩ0sθ0−sΩ0cθ0ci0−sΩ0sθ0+cΩ0cθ0ci0cθ0si0sΩ0si0−cΩ0si0ci0,with the true anomaly ν, the radial distance between the desired target and the Earth r0, the right ascension of ascending node Ω0, the argument of latitude θ0=ωp+ν, the argument of perigee ωp, and the orbit inclination i0, f∈ℝ3 is the control force in frame FB, and fd∈ℝ3 is the disturbance force. c and s stand for cosine and sine, respectively.


### 3.2.3. Actuator Distribution

Consider that there areN 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 Tj and Pj, where Tj provides control force along one axis and control torque around another spin axis of frame FBi simultaneously and Pj provides the opposite control force and torque generated by Tj. Let p∈ℝN be the outputs of all the pairs of thrusters and A∈ℝ6×N be the distribution matrix. To ensure the control system non-under-actuated case, the distribution matrix should satisfy rankA=6.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 functionsat·. In the studied case, we have that p=satξ=satξ1,satξ2,⋯,satξNT with
(13)satξj=ξ¯j+,ifξ>ξ¯j+,ξj,if−ξ¯j−≤ξj≤ξ¯j+,−ξ¯j−,ifξj<−ξ¯j−,where ξ∈ℝN is the command control, ξ¯j+>0 and ξ¯j−>0, j=1,2,⋯,N, are the magnitude constraints. Let u=τT,RLBfTT and define Δξ=satξ−ξ. Then, we have
(14)u=DAsatξ=DAξ+Δξ,where D=diagI3,RLB. In addition, to ensure that the command control of each thrust should be nonnegative, i.e., Tj,Pj≥0,j=1,2,⋯,N, the command control signals are arranged as follows:
(15)Tj=satξj,Pj=0,ifξj≥0,Pj=−satξj,Tj=0,ifξj<0.


### 3.2.4. 6DOF Error Dynamics

Letx1=σ~T,r~TT and x2=ω~T,v~TT. In view of equation (6) and equation (9), we have
(16)ẋ1=Λx2,where Λ=diagGσ~,I. In terms of equation (8), equation (10), and equation (14), the integrated 6DOF error dynamics is derived as follows:
(17)Mẋ2=ς+Bξ+Δξ+d,where
(18)ς=−ω×Jω+Jω~×R~ωd−R~ω̇d−mCtv−mnt−mδ¨,M=diagJ,mI,B=DA,(19)d=τdT,fdTT.


## 3.2.1. Attitude Error Dynamics

The attitude of spacecraft is represented by using Modified Rodrigues Parameters (MRPs)σ∈σ∈ℝ3∣σ=n^tanη/4, where n^∈ℝ3 is the principal rotation axis and η∈−2π,2π is the principal rotating angle. According to [27], by introducing a switching condition at the surface σ=1, the unique and nonsingular description can be guaranteed. This further ensures that 0≤σTσ≤1. Let σ and σd be the MRPs of the spacecraft and the desired, respectively. The error MRPs are given by
(5)σ~=σdσTσ−1+σ1−σdTσd−2σd×σ1+σdTσdσTσ+2σdTσ.According to [27], the attitude error kinematics satisfies
(6)σ~̇=Gσ~ω~,where Gσ~=1/41−σ~Tσ~I3+2σ~×+2σ~σ~T, ω~=ω−R~ωd is the error angular velocity expressed in frame FB, ω and ωd, respectively, represent the angular velocities of the spacecraft and the desired, and R~ is the rotation matrix defined as
(7)R~≜Rσ~=I3−41−σ~Tσ~1+σ~Tσ~2σ~×+8σ~×21+σ~Tσ~2.In addition, according to [27] and R~̇=R~ω~×, we have the following attitude error dynamics:
(8)Jω~̇=−ω×Jω+τ+τd+Jω~×R~ωd−R~ω̇d,where J∈ℝ3×3 is the inertia matrix, τ∈ℝ3 and τd∈ℝ3 are the control torque and disturbance torque, respectively.


## 3.2.2. Orbit Error Dynamics

Letr~=r−δ and v~=v−δ̇ be the position and velocity tracking errors, where r=rx,ry,rzT is the position vector error between the spacecraft and desired target, v is the velocity error, and δ is the desired trajectory referenced in frame FL. In terms of [28], the orbit error kinematics and dynamics satisfy
(9)r~̇=v~,(10)mv~̇=−mCtv−mnt+RLBf+fd−mδ,¨where m is the mass of the spacecraft,
(11)Ct=2ν̇0−10100000,nt=−ν¨ry−ν̇2rx+μr0+rxr0+rx2+ry2+rz23/2−μr02ν¨rx−ν̇2ry+μryr0+rx2+ry2+rz23/2μrzr0+rx2+ry2+rz23/2.RLB=RLRTσ is the rotation matrix from frame FB to frame FL, RL is the rotation matrix between frames FI and FL satisfying
(12)RL=cΩ0cθ0−sΩ0sθ0ci0sΩ0cθ0+cΩ0sθ0ci0sθ0si0−cΩ0sθ0−sΩ0cθ0ci0−sΩ0sθ0+cΩ0cθ0ci0cθ0si0sΩ0si0−cΩ0si0ci0,with the true anomaly ν, the radial distance between the desired target and the Earth r0, the right ascension of ascending node Ω0, the argument of latitude θ0=ωp+ν, the argument of perigee ωp, and the orbit inclination i0, f∈ℝ3 is the control force in frame FB, and fd∈ℝ3 is the disturbance force. c and s stand for cosine and sine, respectively.


## 3.2.3. Actuator Distribution

Consider that there areN 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 Tj and Pj, where Tj provides control force along one axis and control torque around another spin axis of frame FBi simultaneously and Pj provides the opposite control force and torque generated by Tj. Let p∈ℝN be the outputs of all the pairs of thrusters and A∈ℝ6×N be the distribution matrix. To ensure the control system non-under-actuated case, the distribution matrix should satisfy rankA=6.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 functionsat·. In the studied case, we have that p=satξ=satξ1,satξ2,⋯,satξNT with
(13)satξj=ξ¯j+,ifξ>ξ¯j+,ξj,if−ξ¯j−≤ξj≤ξ¯j+,−ξ¯j−,ifξj<−ξ¯j−,where ξ∈ℝN is the command control, ξ¯j+>0 and ξ¯j−>0, j=1,2,⋯,N, are the magnitude constraints. Let u=τT,RLBfTT and define Δξ=satξ−ξ. Then, we have
(14)u=DAsatξ=DAξ+Δξ,where D=diagI3,RLB. In addition, to ensure that the command control of each thrust should be nonnegative, i.e., Tj,Pj≥0,j=1,2,⋯,N, the command control signals are arranged as follows:
(15)Tj=satξj,Pj=0,ifξj≥0,Pj=−satξj,Tj=0,ifξj<0.


## 3.2.4. 6DOF Error Dynamics

Letx1=σ~T,r~TT and x2=ω~T,v~TT. In view of equation (6) and equation (9), we have
(16)ẋ1=Λx2,where Λ=diagGσ~,I. In terms of equation (8), equation (10), and equation (14), the integrated 6DOF error dynamics is derived as follows:
(17)Mẋ2=ς+Bξ+Δξ+d,where
(18)ς=−ω×Jω+Jω~×R~ωd−R~ω̇d−mCtv−mnt−mδ¨,M=diagJ,mI,B=DA,(19)d=τdT,fdTT.


## 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 ξ for each thruster assembled in the spacecraft such that x1 and x2 converge to small sets around zero.Assumption 1.
The desired angular velocityωd and its derivative ω̇d are bounded. The desired position δ and its derivatives δ̇ and δ¨ are bounded, respectively.Assumption 2.
J and m are unknown constant parameters with known nominal values J0 and m0, 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(20)s=x2+K1x1,where K1=diagα1I3,α2I3 with α1>0 and α2>0. In terms of equation (16) and equation (17), the derivative of equation (20) satisfies
(21)Mṡ=ς+Bξ+Δξ+d+MK1Λx2=ρ+Bξ+BΔξ+d,where ρ=ς+MK1Λx2.Since the exact knowledge of inertial parameters cannot be obtained easily, we use their nominal valuesJ0 and m0 to calculate the value of ρ. Define ρ0 as the value of ρ when J=J0 and m=m0. Let Δρ=ρ0−ρ. It follows that
(22)Mṡ=ρ0+Bξ+BΔξ+d∗,where d∗=d−Δϱ. It can be easily known that d∗ consists of the variables x1, x2, ω̇d, δ¨, ν̇, and ν¨. To approximate d∗, we introduce the following RBFNN
(23)d∗=WTΦy+ϵ,where W∈ℝl×6 is the constant weight matrix, l is the number of nodes, Φy∈ℝl is the RBF vector, y=x1T,x2T,ω̇dT,δ¨T,ν̇,ν¨T is the input of the RBFNN, and ϵ∈ℝ6 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 W^ be the estimate of W. Define Δϵ=satW^TΦy−W^TΦy and the estimate error W~=W^−W, respectively. Then, we rewrite equation (23) as
(24)d∗=satW^TΦy−W~TΦy+ϵ∗,where ϵ∗=ϵ−Δϵ.Assumption 3.
ϵ∗ is bounded by an unknown constant ϵ¯, i.e., ∣ϵ∗∣≤ϵ¯.LetW^ and ϵ^ be the estimates of W and ϵ¯, respectively. Design the following controller
(25)ξ=BTBBT−1−ΛTx1−K2s−ρ0−satW^TΦy−ϵ^ss+ϖ+κ1Bχ,and the adaptation laws
(26)W^̇=ϑ1ΦysT−ϑ2W^,ϵ^̇=ϑ3s2s+ϖ−ϑ4ϵ^−ϵ0,where K2=diagα3I3,α4I3 with α3>0 and α4>0, κ1>0, ϖ is a small positive constant, ϑi,i=1,2,3,4, is a positive constant, ϵ0 is the initial value of ϵ^, and χ∈ℝN is an auxiliary signal generated from the following dynamical system
(27)χ̇=−κ2χ+Δξ,where κ2>0.By substituting equation (20) into equation (16), and equation (24) and (25) into equation (22), a 6DOF closed-loop system is derived as follows
(28)ẋ1=Λs−K1x1,(29)Mṡ=−ΛTx1−K2s+BΔξ+κ1Bχ−W~TΦy−ϵ^ss+ϖ+ϵ∗.


### 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
(30)minα3,α4≥κ14θ1+θ2,κ2≥β¯θ1κ1+θ3,where θi,i=1,2,3, is an arbitrary positive constant, and β¯=supB2, the proposed controller (25) and adaptation laws (26) ensure that the 6DOF tracking objective is achieved, i.e., x1 and x2 converge to small sets around zero, resectively.Proof.
Define the estimate errorϵ~=ϵ^−ϵ¯. Choose the following Lyapunov function candidate
(31)V=12x1Tx1+12sTMs+12χTχ+12ϑ1trW~TW~+12ϑ3ϵ~2.
Its time derivative along equation (27), (28), and (29) satisfies
(32)V̇=x1Tẋ1+sTMṡ+χTχ̇+1ϑ1trW~TW^̇+1ϑ3ϵ~ϵ^̇=−x1TΛK1x1−sTK2s−κ2χTχ+κ1sTBχ+sTBΔξ+χTΔξ−sTW~TΦy−ϵ^sTss+ϖ+sTϵ∗+1ϑ1trW~TW^̇+1ϑ3ϵ~ϵ.^̇
In view of0≤σ~Tσ~≤1 and σ~TGσ~=1+σ~Tσ~σ~T/4, we have
(33)x1TΛK1x1=x1T1+σ~Tσ~4α1I300α2I3x1≥c1x1Tx1,where c1=minα1/4,α2. Note that κ1sTBχ≤κ1/4θ1sTs+β¯θ1κ1χTχ, sTBΔξ≤θ2sTs+β¯/4θ2ΔξTΔξ, χTΔξ≤θ3χTχ+1/4θ3ΔξTΔξ. It follows that
(34)V̇≤−c1x1Tx1−λ2−κ14θ1−θ2sTs−κ2−β¯θ1κ1θ3χTχ+β¯4θ2+14θ3Δξ2−sTW~TΦy+1ϑ1trW~TW^̇+ϵ¯s−ϵ^sTss+ϖ+1ϑ3ϵ~ϵ^̇≤−c1x1Tx1−c2sTs−c3χTχ+β¯4θ2+14θ3Δξ2+trW~T1ϑ1W^̇−ΦysT+ϵ~1ϑ3ϵ^̇−s2s+ϖ,where λ2=minα3,α4, c2=λ2−κ1/4θ1−θ2, c3=κ2−β¯θ1κ1−θ3. Substituting adaptation laws equation (26) gives
(35)V̇≤−c1x1Tx1−c2sTs−c3χTχ+β¯4θ2+14θ3Δξ2−ϑ2trW~TW^−ϑ4ϵ~ϵ^−ϵ0.
Note that−ϑ2trW~TW^≤−ϑ2/2trW~TW~+ϑ2/2trWTW and −ϑ4ϵ~ϵ^−ϵ0≤−ϑ4/2ϵ~2+ϑ4/2ϵ¯−ϵ02. Let Δ¯=supΔξ2. Then, we have
(36)V̇≤−c1x1Tx1−c2sTs−c3χTχ−ϑ22trW~TW~−ϑ42ϵ~2+k∗,where k∗=β¯/4θ2+1/4θ3Δ¯+ϑ2/2trWTW+ϑ4/2ϵ¯−ϵ02>0. In view of equation (31) and equation (36), then
(37)V̇≤−c∗V+k∗,where c∗=η/φ>0, η=minc1,c2,c3,ϑ2/2,ϑ4/2, and φ=1/2max1,λmaxM,1/ϑ1,1/ϑ3. It follows that
(38)Vt≤V0−k∗c∗e−c∗t+k∗c∗.
SoV eventually converges to ΩV=V∣V≤k∗/c∗. It follows that x1,x2,χ,W~,ϵ~ are bounded. In particular, let λm=λminM and km=maxα1,α2. With equation (20) and ΩV, we can obtain that x1 and x2 eventually converge to Ω1=x1∣x1≤2k∗/c∗ and Ω2=x2∣x2≤1/λm+km2k∗/c∗, 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 αjj=1,2,3,4 would result in a smaller set Ω1 which illustrates that the ultimate convergent sets of σ~ and r~ 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.


## 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(20)s=x2+K1x1,where K1=diagα1I3,α2I3 with α1>0 and α2>0. In terms of equation (16) and equation (17), the derivative of equation (20) satisfies
(21)Mṡ=ς+Bξ+Δξ+d+MK1Λx2=ρ+Bξ+BΔξ+d,where ρ=ς+MK1Λx2.Since the exact knowledge of inertial parameters cannot be obtained easily, we use their nominal valuesJ0 and m0 to calculate the value of ρ. Define ρ0 as the value of ρ when J=J0 and m=m0. Let Δρ=ρ0−ρ. It follows that
(22)Mṡ=ρ0+Bξ+BΔξ+d∗,where d∗=d−Δϱ. It can be easily known that d∗ consists of the variables x1, x2, ω̇d, δ¨, ν̇, and ν¨. To approximate d∗, we introduce the following RBFNN
(23)d∗=WTΦy+ϵ,where W∈ℝl×6 is the constant weight matrix, l is the number of nodes, Φy∈ℝl is the RBF vector, y=x1T,x2T,ω̇dT,δ¨T,ν̇,ν¨T is the input of the RBFNN, and ϵ∈ℝ6 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 W^ be the estimate of W. Define Δϵ=satW^TΦy−W^TΦy and the estimate error W~=W^−W, respectively. Then, we rewrite equation (23) as
(24)d∗=satW^TΦy−W~TΦy+ϵ∗,where ϵ∗=ϵ−Δϵ.Assumption 3.
ϵ∗ is bounded by an unknown constant ϵ¯, i.e., ∣ϵ∗∣≤ϵ¯.LetW^ and ϵ^ be the estimates of W and ϵ¯, respectively. Design the following controller
(25)ξ=BTBBT−1−ΛTx1−K2s−ρ0−satW^TΦy−ϵ^ss+ϖ+κ1Bχ,and the adaptation laws
(26)W^̇=ϑ1ΦysT−ϑ2W^,ϵ^̇=ϑ3s2s+ϖ−ϑ4ϵ^−ϵ0,where K2=diagα3I3,α4I3 with α3>0 and α4>0, κ1>0, ϖ is a small positive constant, ϑi,i=1,2,3,4, is a positive constant, ϵ0 is the initial value of ϵ^, and χ∈ℝN is an auxiliary signal generated from the following dynamical system
(27)χ̇=−κ2χ+Δξ,where κ2>0.By substituting equation (20) into equation (16), and equation (24) and (25) into equation (22), a 6DOF closed-loop system is derived as follows
(28)ẋ1=Λs−K1x1,(29)Mṡ=−ΛTx1−K2s+BΔξ+κ1Bχ−W~TΦy−ϵ^ss+ϖ+ϵ∗.


## 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
(30)minα3,α4≥κ14θ1+θ2,κ2≥β¯θ1κ1+θ3,where θi,i=1,2,3, is an arbitrary positive constant, and β¯=supB2, the proposed controller (25) and adaptation laws (26) ensure that the 6DOF tracking objective is achieved, i.e., x1 and x2 converge to small sets around zero, resectively.Proof.
Define the estimate errorϵ~=ϵ^−ϵ¯. Choose the following Lyapunov function candidate
(31)V=12x1Tx1+12sTMs+12χTχ+12ϑ1trW~TW~+12ϑ3ϵ~2.
Its time derivative along equation (27), (28), and (29) satisfies
(32)V̇=x1Tẋ1+sTMṡ+χTχ̇+1ϑ1trW~TW^̇+1ϑ3ϵ~ϵ^̇=−x1TΛK1x1−sTK2s−κ2χTχ+κ1sTBχ+sTBΔξ+χTΔξ−sTW~TΦy−ϵ^sTss+ϖ+sTϵ∗+1ϑ1trW~TW^̇+1ϑ3ϵ~ϵ.^̇
In view of0≤σ~Tσ~≤1 and σ~TGσ~=1+σ~Tσ~σ~T/4, we have
(33)x1TΛK1x1=x1T1+σ~Tσ~4α1I300α2I3x1≥c1x1Tx1,where c1=minα1/4,α2. Note that κ1sTBχ≤κ1/4θ1sTs+β¯θ1κ1χTχ, sTBΔξ≤θ2sTs+β¯/4θ2ΔξTΔξ, χTΔξ≤θ3χTχ+1/4θ3ΔξTΔξ. It follows that
(34)V̇≤−c1x1Tx1−λ2−κ14θ1−θ2sTs−κ2−β¯θ1κ1θ3χTχ+β¯4θ2+14θ3Δξ2−sTW~TΦy+1ϑ1trW~TW^̇+ϵ¯s−ϵ^sTss+ϖ+1ϑ3ϵ~ϵ^̇≤−c1x1Tx1−c2sTs−c3χTχ+β¯4θ2+14θ3Δξ2+trW~T1ϑ1W^̇−ΦysT+ϵ~1ϑ3ϵ^̇−s2s+ϖ,where λ2=minα3,α4, c2=λ2−κ1/4θ1−θ2, c3=κ2−β¯θ1κ1−θ3. Substituting adaptation laws equation (26) gives
(35)V̇≤−c1x1Tx1−c2sTs−c3χTχ+β¯4θ2+14θ3Δξ2−ϑ2trW~TW^−ϑ4ϵ~ϵ^−ϵ0.
Note that−ϑ2trW~TW^≤−ϑ2/2trW~TW~+ϑ2/2trWTW and −ϑ4ϵ~ϵ^−ϵ0≤−ϑ4/2ϵ~2+ϑ4/2ϵ¯−ϵ02. Let Δ¯=supΔξ2. Then, we have
(36)V̇≤−c1x1Tx1−c2sTs−c3χTχ−ϑ22trW~TW~−ϑ42ϵ~2+k∗,where k∗=β¯/4θ2+1/4θ3Δ¯+ϑ2/2trWTW+ϑ4/2ϵ¯−ϵ02>0. In view of equation (31) and equation (36), then
(37)V̇≤−c∗V+k∗,where c∗=η/φ>0, η=minc1,c2,c3,ϑ2/2,ϑ4/2, and φ=1/2max1,λmaxM,1/ϑ1,1/ϑ3. It follows that
(38)Vt≤V0−k∗c∗e−c∗t+k∗c∗.
SoV eventually converges to ΩV=V∣V≤k∗/c∗. It follows that x1,x2,χ,W~,ϵ~ are bounded. In particular, let λm=λminM and km=maxα1,α2. With equation (20) and ΩV, we can obtain that x1 and x2 eventually converge to Ω1=x1∣x1≤2k∗/c∗ and Ω2=x2∣x2≤1/λm+km2k∗/c∗, 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 αjj=1,2,3,4 would result in a smaller set Ω1 which illustrates that the ultimate convergent sets of σ~ and r~ 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.Figure 1
The testbed facility: ASTERIX.Some basic and important characteristics of the facility are listed as follows. A6320mm×3060mm operational arena consists of four 3060mm×1580mm 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 FBi is
(39)J=7.2002−0.0882−0.1230−0.08827.01410.0493−0.12300.049311.8319kg·m2.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 frameFBi. 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, ±xBi and ±yBi. The attitude of the simulator is obtained by an attitude and heading reference system (AHRS). The attitude measurement noise satisfies a random normal distribution
(40)eaj=fυaj=12πσajexp−υaj−μaj22σaj2,j=x,y,z,where υaj is a random scalar, σa=σax,σay,σazT=0.0175,0.0170,0.0347Tdeg, μa=μax,μay,μazT=0.3,0.3,0.7Tdeg. The position of the simulator is measured by the tracking system. Similarly, the measurement error also satisfies the following random normal distribution
(41)ep=fυp=12πσpexp−vp−μp22σp2,where υp is a random scalar, σp=0.2950×10−3m, and μp=−19.7499×10−6m.


### 5.2. Experimental Results

In this subsection, we show the experimental results.The desired rotation angle is governed as(42)ψd=ψ0−ψ0Ttdeg,where ψ0 is the initial angle of the simulator and T is the time of the experiment. The desired trajectory in each phase is given in Table 1.Table 1
Desired trajectory.
Phase
Time (s)
x-axis rxd (m)
y-axis ryd (m)
1st
t∈0,T2
0.6cos4πTt
0.4sin4πTt
2nd
t∈T2,T
0.45cos4πTt+0.15
0.3sin4πTtIn this experiment, we use 4 pairs of thrusters generating the control efforts. The distribution of thrusters are shown in Table2 and Figure 2. The distribution matrix is given by A=l,l,−l,−l;1,0,0,1;0,1,1,0, where l is the arm of force of each pair of thrusters. The model parameters of the simulator are listed in Table 3, respectively.Table 2
Thruster distribution.
Control direction
Number of each pair
1
2
3
4
+xB
F1
F16
−xB
F4
F13
+yB
F5
F12
−yB
F8
F9Figure 2
Distribution of the thrusters.Table 3
Model parameters.
Parameter
Value
Unit
Massm
147.22
kg
Moment of inertiajz
11.8319
kg·m2
Gravitational accelerationg
9.8
m/s2
Arm of forcel
0.4875
M
Magnitude constraintξ¯i+i=1,5,12,16
1, 1, 1, 1
N
Magnitude constraintξ¯j−j=4,8,9,13
1, 1, 1, 1
NThe saturation constraints of the NN withl=80 nodes are chosen as a=b=0.2,0.8,0.8T. The parameter estimates are initialized as W^0=0 and ϵ^0=1. The parameters of the controller and adaptation laws are chosen as κ1=κ2=0.1, ϑ1=0.025, ϑ2=0.01, ϑ3=0.03, ϑ4=0.01, and ϖ=0.01. The experiment time is T=270s. Four groups of experiments are performed by using different control parameters and nominal values of inertial parameters which are summarized in Table 4, where M=diagjz,m,m. Experimental results are presented in Figures 3–10.Table 4
Control parameters and nominal values.
Case
Control parameter
Nominal value
1
α1=4, α2=11, α3=4, α4=6
0.65M
2
α1=4, α2=10, α3=4, α4=6
0.9M
3
α1=4, α2=2, α3=4, α4=4
1.3M
4
α1=4, α2=2, α3=4, α4=4
1.75MFigure 3
Experimental result: angle errors.Figure 4
Experimental result: trajectory of simulator.Figure 5
Experimental result: tracking error of simulator inx-axis.Figure 6
Experimental result: tracking error of simulator iny-axis.Figure 7
Experimental result: attitude and position of simulator in Case 2.Figure 8
Experimental result: command control force of each thruster in Case 2.Figure 9
Experimental result: output of NN in Case 2.Figure 10
Experimental result: estimate value and auxiliary signal in Case 2.Figure3 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 x-axis and y-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 −35% and +75%.Table 5
Standard derivations of tracking errors.
Case
1-2 phases
2nd phase
ψ (deg)
r~x (mm)
r~y (mm)
ψ (deg)
r~x (mm)
r~y (mm)
1
1.7275
16.8991
23.5340
0.4100
1.5233
1.2869
2
0.3983
5.7777
7.6600
0.2551
3.4071
1.7211
3
0.1810
7.1407
19.7165
0.1657
3.8055
2.7741
4
0.2636
9.3525
17.0123
0.2183
7.2070
3.7914We next show the detailed experimental result of Case 2 to evaluate the proposed controller. Figure7 illustrates the top view of the experiment result, where the orange line is the initial position with an x-axis in frame FB 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 ϵ^ and norm of the auxiliary signal χ 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 ϵ^ and auxiliary signal χ remain bounded during the experiment.


## 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.Figure 1
The testbed facility: ASTERIX.Some basic and important characteristics of the facility are listed as follows. A6320mm×3060mm operational arena consists of four 3060mm×1580mm 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 FBi is
(39)J=7.2002−0.0882−0.1230−0.08827.01410.0493−0.12300.049311.8319kg·m2.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 frameFBi. 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, ±xBi and ±yBi. The attitude of the simulator is obtained by an attitude and heading reference system (AHRS). The attitude measurement noise satisfies a random normal distribution
(40)eaj=fυaj=12πσajexp−υaj−μaj22σaj2,j=x,y,z,where υaj is a random scalar, σa=σax,σay,σazT=0.0175,0.0170,0.0347Tdeg, μa=μax,μay,μazT=0.3,0.3,0.7Tdeg. The position of the simulator is measured by the tracking system. Similarly, the measurement error also satisfies the following random normal distribution
(41)ep=fυp=12πσpexp−vp−μp22σp2,where υp is a random scalar, σp=0.2950×10−3m, and μp=−19.7499×10−6m.


## 5.2. Experimental Results

In this subsection, we show the experimental results.The desired rotation angle is governed as(42)ψd=ψ0−ψ0Ttdeg,where ψ0 is the initial angle of the simulator and T is the time of the experiment. The desired trajectory in each phase is given in Table 1.Table 1
Desired trajectory.
Phase
Time (s)
x-axis rxd (m)
y-axis ryd (m)
1st
t∈0,T2
0.6cos4πTt
0.4sin4πTt
2nd
t∈T2,T
0.45cos4πTt+0.15
0.3sin4πTtIn this experiment, we use 4 pairs of thrusters generating the control efforts. The distribution of thrusters are shown in Table2 and Figure 2. The distribution matrix is given by A=l,l,−l,−l;1,0,0,1;0,1,1,0, where l is the arm of force of each pair of thrusters. The model parameters of the simulator are listed in Table 3, respectively.Table 2
Thruster distribution.
Control direction
Number of each pair
1
2
3
4
+xB
F1
F16
−xB
F4
F13
+yB
F5
F12
−yB
F8
F9Figure 2
Distribution of the thrusters.Table 3
Model parameters.
Parameter
Value
Unit
Massm
147.22
kg
Moment of inertiajz
11.8319
kg·m2
Gravitational accelerationg
9.8
m/s2
Arm of forcel
0.4875
M
Magnitude constraintξ¯i+i=1,5,12,16
1, 1, 1, 1
N
Magnitude constraintξ¯j−j=4,8,9,13
1, 1, 1, 1
NThe saturation constraints of the NN withl=80 nodes are chosen as a=b=0.2,0.8,0.8T. The parameter estimates are initialized as W^0=0 and ϵ^0=1. The parameters of the controller and adaptation laws are chosen as κ1=κ2=0.1, ϑ1=0.025, ϑ2=0.01, ϑ3=0.03, ϑ4=0.01, and ϖ=0.01. The experiment time is T=270s. Four groups of experiments are performed by using different control parameters and nominal values of inertial parameters which are summarized in Table 4, where M=diagjz,m,m. Experimental results are presented in Figures 3–10.Table 4
Control parameters and nominal values.
Case
Control parameter
Nominal value
1
α1=4, α2=11, α3=4, α4=6
0.65M
2
α1=4, α2=10, α3=4, α4=6
0.9M
3
α1=4, α2=2, α3=4, α4=4
1.3M
4
α1=4, α2=2, α3=4, α4=4
1.75MFigure 3
Experimental result: angle errors.Figure 4
Experimental result: trajectory of simulator.Figure 5
Experimental result: tracking error of simulator inx-axis.Figure 6
Experimental result: tracking error of simulator iny-axis.Figure 7
Experimental result: attitude and position of simulator in Case 2.Figure 8
Experimental result: command control force of each thruster in Case 2.Figure 9
Experimental result: output of NN in Case 2.Figure 10
Experimental result: estimate value and auxiliary signal in Case 2.Figure3 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 x-axis and y-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 −35% and +75%.Table 5
Standard derivations of tracking errors.
Case
1-2 phases
2nd phase
ψ (deg)
r~x (mm)
r~y (mm)
ψ (deg)
r~x (mm)
r~y (mm)
1
1.7275
16.8991
23.5340
0.4100
1.5233
1.2869
2
0.3983
5.7777
7.6600
0.2551
3.4071
1.7211
3
0.1810
7.1407
19.7165
0.1657
3.8055
2.7741
4
0.2636
9.3525
17.0123
0.2183
7.2070
3.7914We next show the detailed experimental result of Case 2 to evaluate the proposed controller. Figure7 illustrates the top view of the experiment result, where the orange line is the initial position with an x-axis in frame FB 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 ϵ^ and norm of the auxiliary signal χ 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 ϵ^ and auxiliary signal χ 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 within0.5 degree and 8 millimeter during the final phase.



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*Source: 7687459-2019-11-20.xml*