# 3-DOF Position and Orientation Control of an Air Flotation Platform for Spacecraft Ground Microgravity Simulation by Using Double Closed-Loop Cascade PIDnn

Authors: Siyue Wang; Qing Gao; Jinguo Liu; Jingkai Feng; Jingshu Liufu
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

Space assistant robots can help astronauts or their assistants perform certain tasks. A ground microgravity simulation environment is built for the space assistant robot AAR-2. The hardware requirements of the ground simulation by the 3-DOF microgravity air flotation platform. An algorithm is designed for this simulation system. By using momentum and RMSprop methods to improve the PID neural network, the challenging problem of strong coupling between system nonlinearity and variables is solved. Firstly, the paper introduces the hardware system and deduces the dynamic model of the system. Then, the algorithm is calculated and simulated. Through simulation, the effectiveness and feasibility of the algorithm are compared and proved. Finally, the control system is simulated by MATLAB/Simulink and compared with other advanced algorithms. The simulation results show that the designed neural network controller can quickly and accurately control the 3-DOF of freedom motion of AAR-2.


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## Body

## 1. Introduction

With the continuous development of aerospace, many countries have carried out the research and development of space robots to replace astronauts to perform some specific tasks, for example, space rendezvous and docking, space debris avoidance [1], fuel supply, on orbit maintenance [2], on orbit component reconstruction, etc. In the past, most of these tasks were completed by astronauts, mainly due to the complexity of the space environment and the arduousness of the tasks. The characteristics of space greatly limit the astronauts’ activities inside and outside the capsule, and the complex and cumbersome operations greatly increase the astronauts’ work and psychological pressure. Therefore, in recent years, more and more space operation tasks gradually turn to space robots.For the design, manufacture, and operation of space robots, simulation experiments are required in the ground microgravity environment for overall performance evaluation, component testing, key parameter determination, and various system verifications [3]. Therefore, it directly promotes the research on building a space microgravity environment. If you want to conduct a space robot simulation experiment, you need to build a microgravity experiment environment on the ground [4].The methods of constructing space microgravity environment mainly include tower dropping method [5, 6], water floating method, and air floating method [7, 8]. The falling tower method is a method to generate microgravity experimental environment by performing free falling motion in microgravity tower (well). It is usually carried out by building high towers or digging deep wells on the ground. Because of the special experimental environment and the limited weightlessness time, this method has great limitations, so it is not commonly used as a method to build a microgravity environment in the laboratory. The water floating method refers to a state in which when the density of the object is the same as that of the liquid. The object can be suspended at any point in the liquid. However, this method does not have real-time operability in the experimental process and needs to constantly adjust the counterweight to ensure balance and stability. On the other hand, it also requires that the experimental equipment can work normally under the liquid without affecting the accuracy. Therefore, this method will bring higher cost and experimental operation requirements to the test. However, the air flotation platform has the advantages of low cost, low operation difficulty, and sustainable real-time operation, so most experiments use the air flotation platform to simulate the microgravity environment [9, 10]. The core component of air flotation platform is gas bearing, which can be divided into plane gas bearing and spherical gas bearing. The compressed air flows out through the small hole or slit of the gas bearing and forms an air film between the workbench and the gas bearing, so as to achieve the purpose of frictionless movement. The air flotation platform has no contact with the worktable and produces a suspended effect, so it overcomes the influence of gravity to a great extent. Therefore, this paper selects the air flotation platform as the ground microgravity simulation platform of the space assistant robot and simulates the motion of the space assistant robot through the 3-DOF of air flotation platform.Many scholars have different views and studies on the position and attitude control of spacecraft in microgravity environment. Terui et al. proposed a sliding mode control and state-dependent Riccati equation method to achieve 6-DOF position and attitude maneuverability without uncertainty and disturbance [11, 12]. In the follow-up, high-pass filter and output feedback control law are used to solve the problem of translational and attitude coupling, but the stability and convergence time are unconstrained [13]. In order to solve the problem of model uncertainty and environmental disturbance, Wu et al. proposed a finite time controller based on nonsingular terminal sliding mode control technology, which can control the translation and attitude change of spacecraft [14]. Liu et al. realized the 3-DOF free attitude control of AAR-1. In order to meet the characteristics of multivariable, strong coupling, and nonlinear systems, the traditional PIDnn controller is used to quickly achieve stability and control effect [15]. Gao et al. upgraded the functions of AAR-1, designed AAR-2, and realized high-precision control of AAR-2 by combining synovial control and fuzzy control [16]. Malladi et al. adopted the nonlinear model predictive control (NMPC) method to solve the highly coupled and nonlinear problem of position and attitude [17]. Aiming at the fast attitude dynamics and slow position dynamics caused by time scale separation, a nonlinear hierarchical control law was proposed [18].The attitude and translational motion of spacecraft are coupled and highly nonlinear, so the model parameters of spacecraft cannot be accurately obtained, and the spacecraft is always affected by environmental disturbances. All these problems make it difficult for the spacecraft control system to achieve the desired control performance. Due to the high coupling and nonlinearity of the dynamic model, this paper uses the double closed-loop structure to solve the influence of strong coupling on the system and uses the improved PID neural network to solve the problem of nonlinear system. Because the uncertain environmental disturbance always exists in the spacecraft movement, the neural network in machine learning can better cater to the uncertainty to a certain extent and match the changing environment through continuous learning.The main innovations and contributions of this paper are as follows: (a) we use a double closed-loop system to solve the problem of the strong coupling and environmental disturbance in the model. This control structure can better overcome the impact of uncertainty on the system by matching with the neural network. (b) We optimized the PID neural network algorithm to solve the variable coupling problem and nonlinear problem when the spacecraft position and attitude change, so that the microgravity air flotation system can converge faster and reach the equilibrium position, better simulate the spacecraft motion and meet the experimental needs. (c) A complete control system and algorithm are built on the microgravity air flotation platform, which makes the flight simulation experiment of ground simulation spacecraft more efficient and convenient, and promote the development of ground microgravity environment simulation equipment.The rest of this paper is organized as follows: in Section2, the control system of the actual microgravity air flotation platform is designed and the motion principle is analyzed. In Section 3, the dynamic model of the microgravity air flotation platform is built and its motion characteristics are analyzed. In addition, the model is optimized to a certain extent according to the ideal environment of the actual space station. In Section 4, the control algorithm of the microgravity air flotation platform is improved on the basis of PIDnn. Meanwhile, the stability of the system is proved and the feasibility of the algorithm is simulated. In Section 5, the whole improved microgravity air flotation system is simulated. The feasibility of the scheme is verified by comparing other advanced algorithms and combining the theoretical simulation with the actual platform.


## 2. Design of the Control System Based on Air Flotation Platform

### 2.1. Control System Hardware Design

The microgravity simulation air flotation platform involved in this paper is shown in Figure1(a). It can realize two-dimensional microgravity simulation motion on the marble plane through three air flotation pads, gas cylinders, and air pressure transmission system. Four ducted fans are mounted on it to control the movement of the air flotation platform. The installation position is shown in Figure 1(b). Four ducted fans at 45° angles to the side of the air flotation platform are installed at the four corners of the platform. Microspacecraft, such as astronaut assistance robots AAR-1, AAR-2 [19, 20], etc., can be installed on the air flotation platform, as shown in Figure 2(a). By controlling the changes of the thrust of the four ducted fans, the two-dimensional plane motion of the spacecraft can be simulated in a microgravity environment.Figure 1
Structural model of air flotation platform.
(a)
Air flotation platform model(b)
Top view of modelFigure 2
(a) Schematic diagram of space station and ground simulation platform. The AAR-2 is a free flying robot used in the space station cabin to assist astronauts in some space missions. This robot can use the microgravity simulation air flotation platform to simulate the 3-DOF plane motion on the marble table. (b) Actual experimental platform.
(a)(b)In this paper, the actual platform is shown in Figure2(b). The STM series chip STM32f1 is selected as the main chip, equipped with two ultrasonic ranging modules HC-SR04, a six-axis gyroscope sensor MPU6050 (built-in DMP attitude calculation), and four ducted fans (brush less motor HL3508).Figure3 shows the top view of the air flotation platform, O−XnYnZn is the inertial coordinate system, O−XbYbZb is the geometric coordinate system, the arrow direction is the wind direction output by the culvert fan, and the four culvert fans are combined in pairs to control the forward and rotational degrees of freedom of the air flotation platform. The system forms a closed-loop structure through sensors, so as to achieve more efficient control effect.Figure 3
Top view of air flotation platform.


### 2.2. Principles of Movement

To correctly describe the motion of the microgravity simulated air flotation platform, the two coordinate systems involved in the system are analyzed and processed. The motion pose of the air flotation platform is determined by the angle between the base pointo and the axes Xb and Xn (the initial state is that the x-axis and the y-axis are parallel to each other), the coordinate of the base point is x0,y0, the rigid body gestures are determined by x0, y0, and θ. The basic motion unit is composed of translation and rotation in the directions of Xn and Yn (3-DOF), and the rotation is the rotation around the center of mass. F1, F2, F3, and F4 are the thrust (wind force) of the four ducted fans.
(1)FXb=mXb¨=22F1+F4−F2−F3,(2)FYb=mYb¨=22F1+F2−F3−F4,(3)mzb=Jcθ¨=22F1+F4−F2−F3r+rs.It is known that the aerial view of the actual air flotation platform is a square,r is the side length, and rs is the radius of the ducted fan. Let clockwise to be the positive direction, and mzb is the torque rotating around Zb with the center of mass of the air flotation platform.


## 2.1. Control System Hardware Design

The microgravity simulation air flotation platform involved in this paper is shown in Figure1(a). It can realize two-dimensional microgravity simulation motion on the marble plane through three air flotation pads, gas cylinders, and air pressure transmission system. Four ducted fans are mounted on it to control the movement of the air flotation platform. The installation position is shown in Figure 1(b). Four ducted fans at 45° angles to the side of the air flotation platform are installed at the four corners of the platform. Microspacecraft, such as astronaut assistance robots AAR-1, AAR-2 [19, 20], etc., can be installed on the air flotation platform, as shown in Figure 2(a). By controlling the changes of the thrust of the four ducted fans, the two-dimensional plane motion of the spacecraft can be simulated in a microgravity environment.Figure 1
Structural model of air flotation platform.
(a)
Air flotation platform model(b)
Top view of modelFigure 2
(a) Schematic diagram of space station and ground simulation platform. The AAR-2 is a free flying robot used in the space station cabin to assist astronauts in some space missions. This robot can use the microgravity simulation air flotation platform to simulate the 3-DOF plane motion on the marble table. (b) Actual experimental platform.
(a)(b)In this paper, the actual platform is shown in Figure2(b). The STM series chip STM32f1 is selected as the main chip, equipped with two ultrasonic ranging modules HC-SR04, a six-axis gyroscope sensor MPU6050 (built-in DMP attitude calculation), and four ducted fans (brush less motor HL3508).Figure3 shows the top view of the air flotation platform, O−XnYnZn is the inertial coordinate system, O−XbYbZb is the geometric coordinate system, the arrow direction is the wind direction output by the culvert fan, and the four culvert fans are combined in pairs to control the forward and rotational degrees of freedom of the air flotation platform. The system forms a closed-loop structure through sensors, so as to achieve more efficient control effect.Figure 3
Top view of air flotation platform.


## 2.2. Principles of Movement

To correctly describe the motion of the microgravity simulated air flotation platform, the two coordinate systems involved in the system are analyzed and processed. The motion pose of the air flotation platform is determined by the angle between the base pointo and the axes Xb and Xn (the initial state is that the x-axis and the y-axis are parallel to each other), the coordinate of the base point is x0,y0, the rigid body gestures are determined by x0, y0, and θ. The basic motion unit is composed of translation and rotation in the directions of Xn and Yn (3-DOF), and the rotation is the rotation around the center of mass. F1, F2, F3, and F4 are the thrust (wind force) of the four ducted fans.
(1)FXb=mXb¨=22F1+F4−F2−F3,(2)FYb=mYb¨=22F1+F2−F3−F4,(3)mzb=Jcθ¨=22F1+F4−F2−F3r+rs.It is known that the aerial view of the actual air flotation platform is a square,r is the side length, and rs is the radius of the ducted fan. Let clockwise to be the positive direction, and mzb is the torque rotating around Zb with the center of mass of the air flotation platform.


## 3. System Dynamics Model

### 3.1. Coordinate and Attitude Transformation

The pose of the air flotation platform is expressed as(4)ηb,nn=pb,nnθnb.The instantaneous speed is expressed as(5)Vb,nn=vb,nnwb,nn.Then, we can get(6)η´b,nn=p´b,nnθ´nbn=Rbnθnb03∗303∗3Tθθnbvb,nnwb,nn,whereηb,nn represents the attitude of b relative to the inertial system n under the inertial system n, Rbnθnb represents the rotation matrix, and θnb=ϕθψT is the geometric position b Euler angles to inertial position n.Rbnθnb and Tθθnb are the results of the transformation from the inertial coordinate system to the geometric coordinate system, and there are many ways to obtain them, which are explained in the order of Z-Y-X [21].
(7)Rbnθnb=RϕRθRϕ=1000cϕsϕ0−sϕcϕcθsθ0−sθcθ0001cψ0−sψ010sψ0cψ=cθcψsθ−cθsψsϕsψ−cϕsθcψcϕcθsϕcψ−cϕsθsψcϕsψ+cϕsθsψ−sϕcθcϕcψ−sϕsθsψ,(8)Tθθnb=10−sθ0cϕsϕcθ0−sϕcϕcθ−1=1sϕtθcϕtθ0cϕ−sϕ0sϕcθcϕcθ.The above sine, cosine, and tangent functions are replaced bys, c, and t. When the motion mode of the air flotation platform is fixed as translation and rotation, then
(9)vb,nn=Xḃ,Yḃ,0T,(10)θnb=00ψT,(11)Rbnθnb=cψ0−sψ010sψ0cψ,(12)Tθθnb=100010001.


### 3.2. Dynamics Model

To further accurately analyze the motion state of the microgravity simulated air flotation platform, the dynamic model of the air flotation platform is deduced and simulated in this paper. The dynamic model of the air flotation platform can be deduced as follows [22–24]:
(13)mv̇g,nb+wg,nb×vg,nb=Fg+fg,(14)Iẇg,nb+wg,nb×Ivg,nb=Mg+mg.Among them,g represents the center of gravity and Fg=Fxg,Fyg,FzgT is used to represent gravity, buoyancy, and other external forces acting on an air flotation platform. fg=fxg,fyg,fzgT is used to represent the thrust generated by the rotation of the ducted fan. Mg=Mxg,Myg,MzgT is used to represent all external moments acting on the air flotation platform except the moment generated by the ducted fan. mg=mxg,myg,mzgT indicates the moment acting on the air flotation platform generated by the rotation of the ducted fan.Since the air flotation platform is a rigid body and the additional mass is the same in all directions, the total inertia matrix about the center of gravity is(15)I=Ix000Iy000Iz.Since the air flotation platform needs to move freely on the marble tabletop, its motion mode is 3-DOF motion, and the yaw motion can be freely performed while translating on the two-dimensional marble tabletop:(16)vg,nb=vxg,nb,vyg,nb,0T,(17)wg,nb=0,0,wzg,nbT.When the air flotation platform rotates and flies forward at the same time, a lateral force is generated, which is represented bywg,nb×vg,nb. This term is offset by the air resistance created when the microgravity simulated air flotation platform moves. This force can also be ignored under the assumption that the spin speed is almost zero in forward flight and almost zero in spinning motion.In the inertia matrix, we can getIx=Iy, if the roll and pitch velocities are zero, no rotational moment will be generated. When the rolling and pitching moments are not zero, wg,nb×Ivg,nb has little effect on the roll and pitch moments, so they are easily offset by the resistance of gravity, so they are also ignored.The model can thus be simplified to(18)mv̇xg,nb=fxg+Fxg=fxg−kxx,̇(19)mv̇yg,nb=fyg+Fyg=fyg−kyẏ,(20)Izẇzg,nb=mzg+Mzg=mzg+Mzcorg+Mzdg−kzwzr+rs.Among them,Mzcorg refers to the Coriolis torque in the direction of rotation around the z-axis, Mzdg refers to the average disturbance torque on the z-axis, and kz represents the average torque coefficient in the air resistance on the z-axis.It is easy to design the controller with negligible Coriolis torque, disturbance torque, and air resistance. In the motion state, we are only interested in the dynamic equations of Euler angles in the geometric coordinate system in motion. Therefore, the dynamic model of the air flotation platform is further simplified as follows:(21)mXb¨=cψfxb,(22)mYb¨=fyb,(23)Izψ¨=mzb.


## 3.1. Coordinate and Attitude Transformation

The pose of the air flotation platform is expressed as(4)ηb,nn=pb,nnθnb.The instantaneous speed is expressed as(5)Vb,nn=vb,nnwb,nn.Then, we can get(6)η´b,nn=p´b,nnθ´nbn=Rbnθnb03∗303∗3Tθθnbvb,nnwb,nn,whereηb,nn represents the attitude of b relative to the inertial system n under the inertial system n, Rbnθnb represents the rotation matrix, and θnb=ϕθψT is the geometric position b Euler angles to inertial position n.Rbnθnb and Tθθnb are the results of the transformation from the inertial coordinate system to the geometric coordinate system, and there are many ways to obtain them, which are explained in the order of Z-Y-X [21].
(7)Rbnθnb=RϕRθRϕ=1000cϕsϕ0−sϕcϕcθsθ0−sθcθ0001cψ0−sψ010sψ0cψ=cθcψsθ−cθsψsϕsψ−cϕsθcψcϕcθsϕcψ−cϕsθsψcϕsψ+cϕsθsψ−sϕcθcϕcψ−sϕsθsψ,(8)Tθθnb=10−sθ0cϕsϕcθ0−sϕcϕcθ−1=1sϕtθcϕtθ0cϕ−sϕ0sϕcθcϕcθ.The above sine, cosine, and tangent functions are replaced bys, c, and t. When the motion mode of the air flotation platform is fixed as translation and rotation, then
(9)vb,nn=Xḃ,Yḃ,0T,(10)θnb=00ψT,(11)Rbnθnb=cψ0−sψ010sψ0cψ,(12)Tθθnb=100010001.


## 3.2. Dynamics Model

To further accurately analyze the motion state of the microgravity simulated air flotation platform, the dynamic model of the air flotation platform is deduced and simulated in this paper. The dynamic model of the air flotation platform can be deduced as follows [22–24]:
(13)mv̇g,nb+wg,nb×vg,nb=Fg+fg,(14)Iẇg,nb+wg,nb×Ivg,nb=Mg+mg.Among them,g represents the center of gravity and Fg=Fxg,Fyg,FzgT is used to represent gravity, buoyancy, and other external forces acting on an air flotation platform. fg=fxg,fyg,fzgT is used to represent the thrust generated by the rotation of the ducted fan. Mg=Mxg,Myg,MzgT is used to represent all external moments acting on the air flotation platform except the moment generated by the ducted fan. mg=mxg,myg,mzgT indicates the moment acting on the air flotation platform generated by the rotation of the ducted fan.Since the air flotation platform is a rigid body and the additional mass is the same in all directions, the total inertia matrix about the center of gravity is(15)I=Ix000Iy000Iz.Since the air flotation platform needs to move freely on the marble tabletop, its motion mode is 3-DOF motion, and the yaw motion can be freely performed while translating on the two-dimensional marble tabletop:(16)vg,nb=vxg,nb,vyg,nb,0T,(17)wg,nb=0,0,wzg,nbT.When the air flotation platform rotates and flies forward at the same time, a lateral force is generated, which is represented bywg,nb×vg,nb. This term is offset by the air resistance created when the microgravity simulated air flotation platform moves. This force can also be ignored under the assumption that the spin speed is almost zero in forward flight and almost zero in spinning motion.In the inertia matrix, we can getIx=Iy, if the roll and pitch velocities are zero, no rotational moment will be generated. When the rolling and pitching moments are not zero, wg,nb×Ivg,nb has little effect on the roll and pitch moments, so they are easily offset by the resistance of gravity, so they are also ignored.The model can thus be simplified to(18)mv̇xg,nb=fxg+Fxg=fxg−kxx,̇(19)mv̇yg,nb=fyg+Fyg=fyg−kyẏ,(20)Izẇzg,nb=mzg+Mzg=mzg+Mzcorg+Mzdg−kzwzr+rs.Among them,Mzcorg refers to the Coriolis torque in the direction of rotation around the z-axis, Mzdg refers to the average disturbance torque on the z-axis, and kz represents the average torque coefficient in the air resistance on the z-axis.It is easy to design the controller with negligible Coriolis torque, disturbance torque, and air resistance. In the motion state, we are only interested in the dynamic equations of Euler angles in the geometric coordinate system in motion. Therefore, the dynamic model of the air flotation platform is further simplified as follows:(21)mXb¨=cψfxb,(22)mYb¨=fyb,(23)Izψ¨=mzb.


## 4. Algorithmic Controller Design

### 4.1. PIDnn Controller Design

PIDnn is a kind of PID-type controller that relies on the self-adaptation and learning ability of the neural network algorithm [25]. There are various neural network structures that can be designed. This paper adopts a single-layer forward propagation network, in which the input layer, hidden layer, and output layer contain two, three, and one neurons (perceptrons), respectively [26].The input of the PID neural network system has two parts, which are the expected input and the real-time output of the system. After each iteration, the neural network outputs new values ofP, I, and D as three parameters of the PID neural network controller [27, 28]. After processing by the dynamic model, as the new input of the system continues to iterate, the system tends to stabilize after the number of sampling N approaches a larger value [29]. The structure of the PIDnn control algorithm is shown in Figure 4.Figure 4
PIDnn control algorithm structure.


#### 4.1.1. Forward Algorithm

When iterating to thenth time, the input and output of neurons in each layer are expressed as follows:Input layer:(24)v1n=zn,(25)v2n=yn,(26)xin=NETvin,i=1,2,whereNETn is an activation function. To reduce the actual hardware system code calculation amount, it can be set as an identity activation function here, namely,
(27)NETvin=vin.Hidden layer:(28)vh′n=∑i=12wihxin,h=1,2,3.The activation function of the hidden layer is special. Considering that the neural network serves the PID system, the three neurons here perform their proportional, integral, and differential functions, respectively. After discretizing the continuous system, we get the following:Proportion:(29)x1′n=v1′n.Integral:(30)x2′n=x2′n−1+v2′n.Derivative:(31)x3′n=v3′n−v3′n−1.Output layer:(32)v1′′n=∑h=13Whoxh′n,o=1,(33)xo′′n=vo′′n.


#### 4.1.2. Back-Propagation Algorithm

The back-propagation algorithm can achieve the effect of learning and memory by modifying the weight valueωihandWho of the neural network. First, calculate the systematic error (the ideal value minus the actual value) and then update the weight of each layer of the neural network according to the conventional gradient descent method. When taking the n+1th sampling, the error E=zn−yn. Let learning step size be η, after the nth learning, the weight value between the hidden layer and the output layer is changed to
(34)Whon+1=Whon−ηdEdWho,(35)dEdWho=dEdyodyodx1′′dx1′′dv1′′dv1′′dWho.The weight value between the input layer and the hidden layer is changed to(36)ωihn+1=Wihn−ηdEdωih,(37)dEdωih=dEdyodyodx1′′dx1′′dv1′′dv1′′dxh′dxh′dvh′dvh′dωih.The weights between hidden layer neurons and output layer neurons are real-timeP, I, and D coefficients.


### 4.2. Momentum and RMSprop Jointly Improve the Algorithm

The iterative principle of the standard gradient descent method is(38)Wn=Wn−1−η∆Wn,(39)∆Wn=∂EWn−1∂W,n=1,2,3⋯.The momentum method changes the weight update law in the standard gradient descent method and introduces an intermediate quantity:(40)Vn=βVn−1+1−β∆Wn.And replacing the originalΔWn with Vn can get
(41)Wn=Wn−1−ηVn=Wn−1−ηβVn−1+1−β∆Wn,(42)Wn=Wn−1−η1−βΔWn+β∆Wn−1+β2∆Wn−2+⋯+βn−1∆W1.The value ofβ around 0.9 can make the algorithm achieve better results. Doing so is equivalent to a weighted sum of historically processed data. From the nth iteration, the farther historical data has a relatively small impact on the weight update, but the weight updated in the nearby iterations has a greater impact.The method of adaptive adjustment of Adagrad learning rate is to convertη (learning rate) in Equation (43) into η/Sn+ε.
(43)Wn=Wn−1−η∆Wn.To prevent the denominator from being zero, setε to a particularly small number.
(44)Sn=Sn−1+ΔWn·ΔWn.Then, we can get(45)Wn=Wn−1−η∆W12+⋯+∆Wn2+εΔWn.If the weights are modified too much, the learning rate will decrease more. Then, in the process of gradient descent training, after a rapid change period, and then a platform area, the change of the learning rate in the platform area will be very slow. When it enters the fast period again after the plateau period, the decline is still very slow and cannot be modified, and all the historical data of the weights will be taken into account, and then, there is a problem.So, the RMSprop method introduces Momentum on the learning rate, so that the formula becomes(46)Sn=αSn−1+1−αΔWn·ΔWn.It is derived from Equation (46) that the gradient descent method in the PID neural network is finally optimized as
(47)Wn=Wn−1−η1−ββn−1∆W1+⋯+β0ΔWn1+ααn−1∆W12+⋯+α0∆Wn2+ε,whereα and β have the same effect, so we let α=β=0.9.


### 4.3. Proof of System Stability

The system control input of the PID neural network controller is(48)un=un−1+ηn∑h=13xh′nWhon,(49)En=12rn−Yn2,(50)dEndxh′n=dEndendendYndYndzndzndxh′n=−endyndznWhonηn.To analyze the stability of the system, a Lyapurov functionVn=1/2en2 is constructed, where en=zn−yn.
(51)∆Vn=Vn+1−Vn=12en+12−en2.According to the differential theorem,(52)∆en=∑j=13dendxh′n∆xh′n.According to Lyapurov function,(53)dendxh′n=dendYndYndundundxh′n=−sgndyndunηnWhon,(54)∆en=−∑j=13sgndYndunηnWhon×ηkeksgndyndunWhon.Because ofsgndyn/dun2=1,
(55)∆en=−∑j=13ηn2Whon2en,(56)en+1=en+∆en,(57)∆Vn=12en+12−en2=12∆en∆en+2en=en∆en+12∆en2.So we can come to a conclusion. The closed loop system is stable at∆Vn<0.
(58)en∆en<−12∆en2,(59)−en∑h=13ηn2Whon2en<−12−∑h=13ηn2Whon2en2,(60)2∑h=13Whon2−1>ηn2.Because the Lyapunov function has a lower bound and∆Vn<0, the function monotonically decreases, then when n⟶∞, Vn+1=Vn=0,∆Vn=0. Therefore, the neural network learning rate is set to a small value to make the system stable. Since the learning rate will affect the optimization speed of the gradient descent method, we have improved the neural network by momentum and RMSprop. By constantly changing the learning rate, the system can be stable and quickly reduced to the best.


## 4.1. PIDnn Controller Design

PIDnn is a kind of PID-type controller that relies on the self-adaptation and learning ability of the neural network algorithm [25]. There are various neural network structures that can be designed. This paper adopts a single-layer forward propagation network, in which the input layer, hidden layer, and output layer contain two, three, and one neurons (perceptrons), respectively [26].The input of the PID neural network system has two parts, which are the expected input and the real-time output of the system. After each iteration, the neural network outputs new values ofP, I, and D as three parameters of the PID neural network controller [27, 28]. After processing by the dynamic model, as the new input of the system continues to iterate, the system tends to stabilize after the number of sampling N approaches a larger value [29]. The structure of the PIDnn control algorithm is shown in Figure 4.Figure 4
PIDnn control algorithm structure.


### 4.1.1. Forward Algorithm

When iterating to thenth time, the input and output of neurons in each layer are expressed as follows:Input layer:(24)v1n=zn,(25)v2n=yn,(26)xin=NETvin,i=1,2,whereNETn is an activation function. To reduce the actual hardware system code calculation amount, it can be set as an identity activation function here, namely,
(27)NETvin=vin.Hidden layer:(28)vh′n=∑i=12wihxin,h=1,2,3.The activation function of the hidden layer is special. Considering that the neural network serves the PID system, the three neurons here perform their proportional, integral, and differential functions, respectively. After discretizing the continuous system, we get the following:Proportion:(29)x1′n=v1′n.Integral:(30)x2′n=x2′n−1+v2′n.Derivative:(31)x3′n=v3′n−v3′n−1.Output layer:(32)v1′′n=∑h=13Whoxh′n,o=1,(33)xo′′n=vo′′n.


### 4.1.2. Back-Propagation Algorithm

The back-propagation algorithm can achieve the effect of learning and memory by modifying the weight valueωihandWho of the neural network. First, calculate the systematic error (the ideal value minus the actual value) and then update the weight of each layer of the neural network according to the conventional gradient descent method. When taking the n+1th sampling, the error E=zn−yn. Let learning step size be η, after the nth learning, the weight value between the hidden layer and the output layer is changed to
(34)Whon+1=Whon−ηdEdWho,(35)dEdWho=dEdyodyodx1′′dx1′′dv1′′dv1′′dWho.The weight value between the input layer and the hidden layer is changed to(36)ωihn+1=Wihn−ηdEdωih,(37)dEdωih=dEdyodyodx1′′dx1′′dv1′′dv1′′dxh′dxh′dvh′dvh′dωih.The weights between hidden layer neurons and output layer neurons are real-timeP, I, and D coefficients.


## 4.1.1. Forward Algorithm

When iterating to thenth time, the input and output of neurons in each layer are expressed as follows:Input layer:(24)v1n=zn,(25)v2n=yn,(26)xin=NETvin,i=1,2,whereNETn is an activation function. To reduce the actual hardware system code calculation amount, it can be set as an identity activation function here, namely,
(27)NETvin=vin.Hidden layer:(28)vh′n=∑i=12wihxin,h=1,2,3.The activation function of the hidden layer is special. Considering that the neural network serves the PID system, the three neurons here perform their proportional, integral, and differential functions, respectively. After discretizing the continuous system, we get the following:Proportion:(29)x1′n=v1′n.Integral:(30)x2′n=x2′n−1+v2′n.Derivative:(31)x3′n=v3′n−v3′n−1.Output layer:(32)v1′′n=∑h=13Whoxh′n,o=1,(33)xo′′n=vo′′n.


## 4.1.2. Back-Propagation Algorithm

The back-propagation algorithm can achieve the effect of learning and memory by modifying the weight valueωihandWho of the neural network. First, calculate the systematic error (the ideal value minus the actual value) and then update the weight of each layer of the neural network according to the conventional gradient descent method. When taking the n+1th sampling, the error E=zn−yn. Let learning step size be η, after the nth learning, the weight value between the hidden layer and the output layer is changed to
(34)Whon+1=Whon−ηdEdWho,(35)dEdWho=dEdyodyodx1′′dx1′′dv1′′dv1′′dWho.The weight value between the input layer and the hidden layer is changed to(36)ωihn+1=Wihn−ηdEdωih,(37)dEdωih=dEdyodyodx1′′dx1′′dv1′′dv1′′dxh′dxh′dvh′dvh′dωih.The weights between hidden layer neurons and output layer neurons are real-timeP, I, and D coefficients.


## 4.2. Momentum and RMSprop Jointly Improve the Algorithm

The iterative principle of the standard gradient descent method is(38)Wn=Wn−1−η∆Wn,(39)∆Wn=∂EWn−1∂W,n=1,2,3⋯.The momentum method changes the weight update law in the standard gradient descent method and introduces an intermediate quantity:(40)Vn=βVn−1+1−β∆Wn.And replacing the originalΔWn with Vn can get
(41)Wn=Wn−1−ηVn=Wn−1−ηβVn−1+1−β∆Wn,(42)Wn=Wn−1−η1−βΔWn+β∆Wn−1+β2∆Wn−2+⋯+βn−1∆W1.The value ofβ around 0.9 can make the algorithm achieve better results. Doing so is equivalent to a weighted sum of historically processed data. From the nth iteration, the farther historical data has a relatively small impact on the weight update, but the weight updated in the nearby iterations has a greater impact.The method of adaptive adjustment of Adagrad learning rate is to convertη (learning rate) in Equation (43) into η/Sn+ε.
(43)Wn=Wn−1−η∆Wn.To prevent the denominator from being zero, setε to a particularly small number.
(44)Sn=Sn−1+ΔWn·ΔWn.Then, we can get(45)Wn=Wn−1−η∆W12+⋯+∆Wn2+εΔWn.If the weights are modified too much, the learning rate will decrease more. Then, in the process of gradient descent training, after a rapid change period, and then a platform area, the change of the learning rate in the platform area will be very slow. When it enters the fast period again after the plateau period, the decline is still very slow and cannot be modified, and all the historical data of the weights will be taken into account, and then, there is a problem.So, the RMSprop method introduces Momentum on the learning rate, so that the formula becomes(46)Sn=αSn−1+1−αΔWn·ΔWn.It is derived from Equation (46) that the gradient descent method in the PID neural network is finally optimized as
(47)Wn=Wn−1−η1−ββn−1∆W1+⋯+β0ΔWn1+ααn−1∆W12+⋯+α0∆Wn2+ε,whereα and β have the same effect, so we let α=β=0.9.


## 4.3. Proof of System Stability

The system control input of the PID neural network controller is(48)un=un−1+ηn∑h=13xh′nWhon,(49)En=12rn−Yn2,(50)dEndxh′n=dEndendendYndYndzndzndxh′n=−endyndznWhonηn.To analyze the stability of the system, a Lyapurov functionVn=1/2en2 is constructed, where en=zn−yn.
(51)∆Vn=Vn+1−Vn=12en+12−en2.According to the differential theorem,(52)∆en=∑j=13dendxh′n∆xh′n.According to Lyapurov function,(53)dendxh′n=dendYndYndundundxh′n=−sgndyndunηnWhon,(54)∆en=−∑j=13sgndYndunηnWhon×ηkeksgndyndunWhon.Because ofsgndyn/dun2=1,
(55)∆en=−∑j=13ηn2Whon2en,(56)en+1=en+∆en,(57)∆Vn=12en+12−en2=12∆en∆en+2en=en∆en+12∆en2.So we can come to a conclusion. The closed loop system is stable at∆Vn<0.
(58)en∆en<−12∆en2,(59)−en∑h=13ηn2Whon2en<−12−∑h=13ηn2Whon2en2,(60)2∑h=13Whon2−1>ηn2.Because the Lyapunov function has a lower bound and∆Vn<0, the function monotonically decreases, then when n⟶∞, Vn+1=Vn=0,∆Vn=0. Therefore, the neural network learning rate is set to a small value to make the system stable. Since the learning rate will affect the optimization speed of the gradient descent method, we have improved the neural network by momentum and RMSprop. By constantly changing the learning rate, the system can be stable and quickly reduced to the best.


## 5. Control System and Simulation Experiment

### 5.1. Control System

According to the actual experimental requirements of fixed-point and fixed-speed, the desired air flotation platform positionx,y, angle ψ, and speed-error curve VEx,VEy,WEψ are designed. For three control variables xn,yn,ψn are designed three cascade double-loop control models. The curve is sent to the algorithm controller for processing and then sent to the algorithm controller through the dynamic model for repeated iterations. The experiment combined the sensor to form a closed loop and combined with the cascade double-loop control to realize the stable control of the air flotation platform to move freely. The air flotation platform control system and cascade control structure are illustrated in Figures 5 and 6.Figure 5
Air flotation platform control system.Figure 6
Cascade control structure.Considering that the actual trajectory may have sharp turns or circular motion, the following two motion models are designed. The first is a right-angled square and the second is a circular motion. In the first one, the motion curve of the air flotation platform starts from point0,5 and returns to the origin 0,5 after moving counterclockwise for a week, and the overall consumption time is 10 seconds. And the initial direction of the air flotation platform is -90°, and the angle calculation is based on the angle between the direction of the air flotation platform and the x-axis. In the second one, the starting point, ending point, and elapsed time conditions of the movement curve of the air flotation platform remains unchanged. The motion curve presents a circular structure with a center of 5,5 and a radius of 5.


### 5.2. Simulation

To verify the effectiveness of the control algorithm designed for the 3-DOF air flotation platform, MATLAB/Simulink is used for simulation in this paper, and the overall effect is shown in Figure7. The predicted position curve of the trajectory was input into the system, and then, the algorithm was controlled by the cascade double closed-loop PIDnn system. The obtained control results are introduced into the dynamics model and then returned to the controller for further calculation.Figure 7
Simulink simulation structural diagram.Considering the various movement and work of space robot, there will be a variety of movement curves such as straight line, curve, or sharp angle. So, we designed two kinds of curves, one is the motion curve of the square, and the other is the motion curve of the circle.As shown in Figures8 and 9, for each curve, we give the system’s 3-DOF input.Figure 8
Square motion curve design.
(a)
x-axis input curve(b)
y-axis input curve(c)
Yaw-axis input curveFigure 9
Circular curve design.
(a)
x-axis input curve(b)
y-axis input curve(c)
Yaw-axis input curveAfter effective control by the controller, the system simulation curve is obtained as shown in Figure10.Figure 10
Comparison between control effect and expected effect.In the simulation experiment, as shown in Figures11 and 12, we get the error tracking of the three degrees of freedom curve.Figure 11
Error tracking of the first motion curve.
(a)
Error curve ofx-axis(b)
Error curve ofy-axis(c)
Error curve in yaw-axisFigure 12
Error tracking of the second motion curve.
(a)
Error curve ofx-axis(b)
Error curve ofy-axis(c)
Error curve in yaw-axisWe can see from the error curve that the error effect of moving distance can reach the level of10−3 on average in the simulation experiment. In yaw, the maximum error in motion is only 0.17°. It can be observed that using the improved PIDnn based on momentum and RMSprop, the position and attitude control of the air flotation platform has a good control effect. It can quickly and accurately track the input signal.


### 5.3. Comparison of Advanced Algorithms

In order to analyze the optimality of space robot control system, we compared other advanced control algorithms according to the above two trajectory models. We can see the error comparison curve in Figures13 and 14. We compare it with three other advanced control algorithms in recent years. The blue line represents the fuzzy slider mode control algorithm [16], the red line represents the nonlinear model predictive control algorithm (NMPC) [17], and the green line represents the robot integral terminal sliding mode control law (RITSMC) [30].Figure 13
Motion trajectory comparison of advanced algorithms (square).
(a)
Error curve ofx-axis(b)
Error curve ofy-axis(c)
Error curve in yaw-axisFigure 14
Motion trajectory comparison of advanced algorithms (circle).
(a)
Error curve ofx-axis(b)
Error curve ofy-axis(c)
Error curve in yaw-axisThe improvement and variation of the control effect can be obtained from the error curve. The average control error of position control can reach7.59×10−3 unit distance, and the error control accuracy of yaw can reach 8.36×10−2°. The improved PID neural network using momentum and RMSprop can be proved by simulation results. It can effectively control the high-precision movement of the space robot, so that the space robot can have smaller error with the preset curve and achieve accurate control.


## 5.1. Control System

According to the actual experimental requirements of fixed-point and fixed-speed, the desired air flotation platform positionx,y, angle ψ, and speed-error curve VEx,VEy,WEψ are designed. For three control variables xn,yn,ψn are designed three cascade double-loop control models. The curve is sent to the algorithm controller for processing and then sent to the algorithm controller through the dynamic model for repeated iterations. The experiment combined the sensor to form a closed loop and combined with the cascade double-loop control to realize the stable control of the air flotation platform to move freely. The air flotation platform control system and cascade control structure are illustrated in Figures 5 and 6.Figure 5
Air flotation platform control system.Figure 6
Cascade control structure.Considering that the actual trajectory may have sharp turns or circular motion, the following two motion models are designed. The first is a right-angled square and the second is a circular motion. In the first one, the motion curve of the air flotation platform starts from point0,5 and returns to the origin 0,5 after moving counterclockwise for a week, and the overall consumption time is 10 seconds. And the initial direction of the air flotation platform is -90°, and the angle calculation is based on the angle between the direction of the air flotation platform and the x-axis. In the second one, the starting point, ending point, and elapsed time conditions of the movement curve of the air flotation platform remains unchanged. The motion curve presents a circular structure with a center of 5,5 and a radius of 5.


## 5.2. Simulation

To verify the effectiveness of the control algorithm designed for the 3-DOF air flotation platform, MATLAB/Simulink is used for simulation in this paper, and the overall effect is shown in Figure7. The predicted position curve of the trajectory was input into the system, and then, the algorithm was controlled by the cascade double closed-loop PIDnn system. The obtained control results are introduced into the dynamics model and then returned to the controller for further calculation.Figure 7
Simulink simulation structural diagram.Considering the various movement and work of space robot, there will be a variety of movement curves such as straight line, curve, or sharp angle. So, we designed two kinds of curves, one is the motion curve of the square, and the other is the motion curve of the circle.As shown in Figures8 and 9, for each curve, we give the system’s 3-DOF input.Figure 8
Square motion curve design.
(a)
x-axis input curve(b)
y-axis input curve(c)
Yaw-axis input curveFigure 9
Circular curve design.
(a)
x-axis input curve(b)
y-axis input curve(c)
Yaw-axis input curveAfter effective control by the controller, the system simulation curve is obtained as shown in Figure10.Figure 10
Comparison between control effect and expected effect.In the simulation experiment, as shown in Figures11 and 12, we get the error tracking of the three degrees of freedom curve.Figure 11
Error tracking of the first motion curve.
(a)
Error curve ofx-axis(b)
Error curve ofy-axis(c)
Error curve in yaw-axisFigure 12
Error tracking of the second motion curve.
(a)
Error curve ofx-axis(b)
Error curve ofy-axis(c)
Error curve in yaw-axisWe can see from the error curve that the error effect of moving distance can reach the level of10−3 on average in the simulation experiment. In yaw, the maximum error in motion is only 0.17°. It can be observed that using the improved PIDnn based on momentum and RMSprop, the position and attitude control of the air flotation platform has a good control effect. It can quickly and accurately track the input signal.


## 5.3. Comparison of Advanced Algorithms

In order to analyze the optimality of space robot control system, we compared other advanced control algorithms according to the above two trajectory models. We can see the error comparison curve in Figures13 and 14. We compare it with three other advanced control algorithms in recent years. The blue line represents the fuzzy slider mode control algorithm [16], the red line represents the nonlinear model predictive control algorithm (NMPC) [17], and the green line represents the robot integral terminal sliding mode control law (RITSMC) [30].Figure 13
Motion trajectory comparison of advanced algorithms (square).
(a)
Error curve ofx-axis(b)
Error curve ofy-axis(c)
Error curve in yaw-axisFigure 14
Motion trajectory comparison of advanced algorithms (circle).
(a)
Error curve ofx-axis(b)
Error curve ofy-axis(c)
Error curve in yaw-axisThe improvement and variation of the control effect can be obtained from the error curve. The average control error of position control can reach7.59×10−3 unit distance, and the error control accuracy of yaw can reach 8.36×10−2°. The improved PID neural network using momentum and RMSprop can be proved by simulation results. It can effectively control the high-precision movement of the space robot, so that the space robot can have smaller error with the preset curve and achieve accurate control.


## 6. Conclusion and Future Work

In this paper, according to the needs of the ground preexperiment, the control system is designed and constructed in combination with the 3-DOF air flotation platform. The design allows it to move freely on the marble table. The design of the overall dynamic model and control algorithm is introduced, and the model is simplified and deduced according to the actual experimental environment. The control algorithm adopts the PID neural network to realize the 3-DOF motion of the air flotation platform. And the double-loop cascade control method is adopted to achieve better control effect. Finally, the designed control system is simulated by MATLAB/Simulink. The overall control effect is simulated, and it is known from the simulation effect that the system can control the air flotation platform quickly and effectively. In the era of rapid development of science and technology, the development of space operation technology and robot-assisted technology is gradually improving the hard power of human space exploration. As a ground-assisted experimental configuration, the air flotation platform plays an important role, and the improvement and research of experimental equipment will be further improved and further advance the development and progress of automation.In the future work, we mainly improve the following contents: (a) optimize the overall control structure and specific hardware equipment of the microgravity air flotation platform and (b) optimize the computing power and sensor accuracy of the single-chip microcomputer, so that it can perform more complex control algorithms and obtain data with higher precision.



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*Source: 9322534-2022-08-22.xml*