# Dynamic Modeling and Active Vibration Isolation of a Noncontact 6-DOF Lorentz Platform Based on the Exponential Convergence Disturbance Observer

Authors: Xubin Zhou; Weidong Chen; Fagang Zhao; Dapeng Sui; Qing Xiao; Xingtian Liu; Liping Zhou; Quan Zhang
Journal: Shock and Vibration
 (2021)

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## Abstract

In order to study the vibration isolation and positioning performance of the noncontact 6-DOF platform in the space microgravity environment, this paper presented a cosimulation model of a virtual prototype. Based on the model driven by biaxial noncontact Lorentz force actuators (NLFAs), an equivalent dynamic model has been established. In the meanwhile, the 6-DOF sliding mode robust controller with exponential convergence disturbance observer is developed. The mechanical system simulation model was designed using ADAMS, and the corresponding 6-DOF decoupling control system and disturbance observer programs were developed using MATLAB/Simulink. According to the mechatronics simulation results, the system can enable the floating platform to achieve micron-level posture positioning within 0.5 s. In vibration isolation simulation, the disturbance observer can predict the external disturbance input and compensate the control force more accurately so that the floating platform can effectively suppress low-frequency disturbance and step disturbance under the control of the sliding mode controller. And the displacement of the floating platform under the disturbance of 1–100 Hz frequency sweep is less than 1μm.


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## Body

## 1. Introduction

The existence of microvibration on spacecraft due to different sources has affected aerospace activities [1]. The requirements for suppressing and isolating the microvibration on high-precision spacecraft and sensitive loads are becoming more and more urgent. At present, the methods for isolating microvibration are mainly active vibration isolation and passive vibration isolation. Passive vibration isolation, although simple in structure and strong in reliability, has achieved certain vibration isolation effects limited in the relative high-frequency range, hence powerless for microvibration environments with low-frequency and complex disturbance sources [2–8]. The low-frequency microvibration in the space has a great influence on the accuracy of high-resolution pointing, the use of precision payload, and the accuracy of scientific experimental results of space stations [9]. For optical observation lenses, a relatively small vibration of 10 μm in space can achieve an observation error of 500 km on the ground [10]. Therefore, the research on spacecraft microvibration and its positioning technology has received great attention. Active vibration isolation was gaining more and more attention in the control applications for spatial microvibration disturbance due to its low-frequency control capability. The magnetic suspension vibration isolation platform was widely used in the fields of the ultraquiet platform, precision manufacturing, and precision positioning [11]. This type of platform is mainly used for space exploration, high-resolution Earth observation, remote laser communication, and other scientific and technological activities [12].At present, the research on positioning and vibration isolation platform is divided into two types, namely, micropositioning and vibration isolation platform and macro- and microcombined platform [13]. The most studied active vibration isolation platform is based on the Stewart configuration [14–16], followed by the noncontact vibration isolation platform based on Lorentz force actuators [12, 17–20]. In addition, there are microvibration active vibration isolation platforms designed according to other different working principles of actuators [21–25].The noncontact vibration isolation platform was more suitable for microvibration control in a space microgravity environment due to its nonmechanical contact characteristics, where Locke Martin proposed a magnetic-floating vibration isolation system, which realizes the noncontact connection of Stewart configuration through a 6-pole magnetic-floating mechanism, and improves the vibration isolation capability in the low-frequency band by 1∼2 orders of magnitude [26]. MVIS-II provided a high-stability platform for vibration-sensitive optical loads on the TacSat-2 satellite platform. The electromagnetic actuation technique separated the load from the platform and controls the load pointing through three electromagnetic actuators [27]. The ARIS electromagnetic actuation system was a full-cabinet active electromagnetic actuator developed under NASA’s ATD project. It was mainly composed of 8 single-degree-of-freedom actuators, 8 position sensors, 1 controller, and 3 sets of accelerometer components. ARIS was used as a rotating voice coil actuator [28]. The g-LIMIT was a vibration isolation device supplied to the Microgravity Scientific Glove Box MSG, based on the improved STABLE vibration isolation technology. Its biggest feature was the integration and modularization and divided the exciter and measurement system into three identical units. Each unit has a two-axis Lorentz force exciter and a two-axis accelerometer combined with a two-dimensional position. The measuring system greatly simplifies the machining process and assembly complexity [29]. The electromagnetically actuated 3-DOF precision planar motion system developed by Tsinghua University provided users with a high-precision motion positioning platform which can be applied in harsh environments [30]. Ren et al. used the same method to establish a 6-DOF motion differential equation in the stator coordinate system, but the further transformation was required when designing the control algorithm [31].For the active vibration isolation platform used in the on-orbit work, the ground tests were significant and necessary. The test methods of the prototype mainly included the drop tower method, parabolic flight method, water float method, suspension method, and air suspension method. However, the testing of related prototypes is very costly, and the prototype needs to be diagnosed and improved [32, 33].Aiming at the control problem of the 6-DOF noncontact platform, Li and others proposed PID and PD control methods and deduced the transfer function model [34]. Wu and others proposed a redundant degree of freedom platform, including 8 actuators, built a dynamic model of the maglev platform based on dynamic equations in MATLAB/Simulink and studied the control strategy using the sliding mode control of the saturation function [12, 35]. In recent years, Mobayen and others have also proposed many new control strategies in the field of sliding mode control [36–39].In order to solve the problem that the dynamic equation cannot fully reflect the characteristics of the prototype, this paper proposes a method of using ADAMS to build a dynamic model of the 6-DOF platform, shapes the microgravity environment, solves the difficult problem of the experimental environment, and makes the dynamic model closer to the actual prototype. At the same time, we use the S-function in MATLAB/Simulink to design the control program, thereby constructing a virtual prototype of the 6-DOF platform. This can make the simulation more suitable for the actual prototype. In terms of control methods, this paper uses a 6-DOF disturbance observer to minimize the chattering characteristics of sliding mode control and conducts joint simulations of multiple disturbance situations to provide theoretical support for the noncontact 6-DOF platform. The cosimulation model was used to simulate and analyze the positioning and vibration isolation performance of the platform, and the working capabilities of the platform were verified.The rest of the paper is organized as follows. The design of the noncontact 6-DOF vibration isolation platform is described in Section2. In Section 3, the dynamic equations of the noncontact 6-DOF vibration isolation platform are derived through the Kane method. Then, in Section 4, the general calculation method of the force on the floating platform is derived using the coordinate transformation matrix. The design of a 6-DOF sliding mode robust controller with disturbance observer based on exponential convergence is proposed in Section 5. The construction of the cosimulation model in ADAMS and MATLAB is introduced in Section 6. In Section 7, the results of the cosimulation are given and analyzed. And the conclusions are given in Section 8. Finally, the future research direction and outlook are proposed.


## 2. The Design of the Noncontact 6-DOF Vibration Isolation Platform

In Figure1(a), the noncontact 6-DOF platform was hoisted on the hoisting bracket through the latex tube. An enlarged view of the noncontact platform is shown in Figure 1(b). In order to study the vibration isolation and positioning performance of the noncontact 6-DOF platform in the space microgravity environment, this paper proposed a cosimulation model of a virtual prototype shown in Figure 2.Figure 1
Test device architecture. (a) Overall structure of test device. (b) Noncontact 6-DOF platform.
(a)(b)Figure 2
Simulation model of mechanical system in ADAMS.As shown in Figure2, the lower part is a fixed platform, which is generally connected to a large spacecraft, and the upper part is a floating platform. The floating platform and the fixed platform are connected by three biaxial NLFAs separated by 120°. Each biaxial NLFA can generate force along the tangent to the circular table and force along the z-axis. The disturbance force application point is set at a random position of the floating platform, and there are three intersecting disturbance forces. The six active control forces and three disturbance force action points are all fixedly connected to the floating platform, and they are all controlled by the external controller designed in MATLAB/Simulink. In addition, the position information and acceleration information of the floating platform measured in ADAMS are transmitted to MATLAB/Simulink as feedback information through output variables.


## 3. Dynamic Model of the Noncontact 6-DOF Vibration Isolation Platform

Due to the characteristics of the proposed noncontact 6-DOF vibration isolation platform used in a space environment with microgravity, Newton’s second law and the Kane method are adopted to develop the dynamic model.As shown in Figure3, N0 is the inertial coordinate system, S0 is the base coordinate system, and F0 represents the floating platform coordinate system. Fc∗ and Fc are, respectively, the mass of the floating platform in the initial stable state and the position of the mass center after the floating platform moves, while m is the mass of the floating platform. Taking the floating platform as the research object, the vector of six degrees of freedom is used to describe its posture. Define a state-space array X=rθT, where r=xyzT represents the position of the floating platform relative to the base table, and θ=θxθyθzT represents the rotation angle of the floating platform around x, y, and z axes, respectively. Then, the resultant forces and moments on the floating platform can be regarded as F=FxFyFzT and M=MxMyMzT, respectively.Figure 3
Coordinate system relationship.In addition, based on the Euler angles and coordinate transformation laws, the conversion matrix from the inertial platform coordinate system to the floating platform can be derived as(1)ΓS/F=cosycoszsinxsinycosz−cosxsinzcosxsinycosz+sinxsinzcosysinzsinxsinysinz+cosxcoszcosxsinysinz−sinxcosz−sinysinxcosycosxcosy.The Newtonian equations for the floating platform are formulated as(2)F=m⋅I3×3⋅X¨.Write the rotation equation of the floating platform as matrix form related to state variableX by using the Kane method:(3)M=03×3ΓS/F⋅JF3×6⋅X¨.In summary, the differential equation of motion of the floating platform can be reorganized as follows:(4)FM=m⋅I3×3003×3ΓS/F⋅JF⋅X¨.


## 4. Force Derivation of the Floating Platform

### 4.1. Calculation of the Driving Force

According to the Lorentz actuator used, the driving force of the actuator is as follows:(5)Fa=ILBsinI,B,where I is the magnitude of the current, L is the length of the winding coil, and B is the magnetic field strength. Considering that the driving displacement of the actuator is a minimum value, and when the positions of the coil and the magnetic field are determined, the output force and current remained relatively good linearity, and then the formula is simplified as follows:(6)Fa=KaIA.The six driving forces are generated by the three biaxial NLFAs, and the matrix form is shown as follows:(7)FA=F1zF1xF2zF2xF3zF3xT.The coil current intensity matrix corresponding to the six driving forces is(8)IA=I1zI1xI2zI2xI3zI3xT.In Figure4, the direction matrix of the six drivers in the floating platform coordinate system F0 is(9)PT=PF1zTPF1xTPF2zTPF2xTPF3zTPF3xT=001010001−32−12000132−120.Figure 4
Force analysis of the floating platform.Assuming that the vertical distance between the electromagnetic driving force acting point and the floating platform coordinate system isdv and the horizontal distance is rf, then the position matrix of the six driving force acting points in the floating platform coordinate system F0 is(10)RT=RF1zTRF1xTRF2zTRF2xTRF3zTRF3xT=rf0−dvrf0−dv−rf23rf2−dv−rf23rf2−dv−rf2−3rf2−dv−rf2−3rf2−dv.Then, the resultant force of the three two-axis actuators can be expressed in the inertial coordinate system as(11)FSA=∑a=16Fa=ΓS/F⋅P⋅FA.The total torque can be expressed as(12)MSA=∑a=16ΓS/F⋅RF×Fa.


### 4.2. Calculation of the Disturbance Force

Define the disturbance force acting directly on the floating platform asfd; then, the disturbance force in the inertial coordinate system can be expressed as(13)Fd=ΓS/F⋅fd,where fd=fdxfdyfdzT.The torque acting directly on the floating platform can be expressed as(14)Md=rd−rc×fd=ΓS/F⋅rfd×ΓS/F⋅fd,where rfd is the position vector from the center of mass of the floating platform to the point of disturbance force.


### 4.3. The Resultant Force and Moment on the Floating Platform

The resultant force acting directly on the floating platform can be expressed as(15)F=FSA+Fd.The total torque acting directly on the floating platform can be expressed as(16)M=MSA+Md.In summary, the dynamic equation of the system is(17)FSA+FdMSA+Md=mI3×3003×3ΓS/F⋅JFX¨.According to the 3D modeling data in the CREO software, we can know from Table1 the model’s main parameters.Table 1
Model’s main parameter table.
ParametersNumerical valueUnitrf0.19232mdv0.01574mJxx0.17391kg⋅m2Jyy0.17391kg⋅m2Jzz0.34681kg⋅m2rfd[0.1, 0, 0]—m11kg


## 4.1. Calculation of the Driving Force

According to the Lorentz actuator used, the driving force of the actuator is as follows:(5)Fa=ILBsinI,B,where I is the magnitude of the current, L is the length of the winding coil, and B is the magnetic field strength. Considering that the driving displacement of the actuator is a minimum value, and when the positions of the coil and the magnetic field are determined, the output force and current remained relatively good linearity, and then the formula is simplified as follows:(6)Fa=KaIA.The six driving forces are generated by the three biaxial NLFAs, and the matrix form is shown as follows:(7)FA=F1zF1xF2zF2xF3zF3xT.The coil current intensity matrix corresponding to the six driving forces is(8)IA=I1zI1xI2zI2xI3zI3xT.In Figure4, the direction matrix of the six drivers in the floating platform coordinate system F0 is(9)PT=PF1zTPF1xTPF2zTPF2xTPF3zTPF3xT=001010001−32−12000132−120.Figure 4
Force analysis of the floating platform.Assuming that the vertical distance between the electromagnetic driving force acting point and the floating platform coordinate system isdv and the horizontal distance is rf, then the position matrix of the six driving force acting points in the floating platform coordinate system F0 is(10)RT=RF1zTRF1xTRF2zTRF2xTRF3zTRF3xT=rf0−dvrf0−dv−rf23rf2−dv−rf23rf2−dv−rf2−3rf2−dv−rf2−3rf2−dv.Then, the resultant force of the three two-axis actuators can be expressed in the inertial coordinate system as(11)FSA=∑a=16Fa=ΓS/F⋅P⋅FA.The total torque can be expressed as(12)MSA=∑a=16ΓS/F⋅RF×Fa.


## 4.2. Calculation of the Disturbance Force

Define the disturbance force acting directly on the floating platform asfd; then, the disturbance force in the inertial coordinate system can be expressed as(13)Fd=ΓS/F⋅fd,where fd=fdxfdyfdzT.The torque acting directly on the floating platform can be expressed as(14)Md=rd−rc×fd=ΓS/F⋅rfd×ΓS/F⋅fd,where rfd is the position vector from the center of mass of the floating platform to the point of disturbance force.


## 4.3. The Resultant Force and Moment on the Floating Platform

The resultant force acting directly on the floating platform can be expressed as(15)F=FSA+Fd.The total torque acting directly on the floating platform can be expressed as(16)M=MSA+Md.In summary, the dynamic equation of the system is(17)FSA+FdMSA+Md=mI3×3003×3ΓS/F⋅JFX¨.According to the 3D modeling data in the CREO software, we can know from Table1 the model’s main parameters.Table 1
Model’s main parameter table.
ParametersNumerical valueUnitrf0.19232mdv0.01574mJxx0.17391kg⋅m2Jyy0.17391kg⋅m2Jzz0.34681kg⋅m2rfd[0.1, 0, 0]—m11kg


## 5. Design of 6-DOF Sliding Mode Robust Controller with Disturbance Observer Based on Exponential Convergence

In this section, the disturbance observer based on exponential convergence is used to observe the disturbance force of the floating platform in six degrees of freedom combined with the six degrees of freedom sliding mode robust controller to design the control system. In addition, we use PID control for comparison.


### 5.1. Design of Disturbance Observer Based on Exponential Convergence

According to the dynamic differential equation of the floating platform,(18)MxX¨=Fu+Fv.During the control process, if the size of the disturbanceFv can be effectively observed, the control effect will be greatly improved. The basic idea of designing an observer is to correct the estimated value by the difference between the estimated output and the actual output.TakeFv^⋅=KFv−Fv^, where K>0.The auxiliary parameter vector is defined as(19)z=Fv^−KMxX˙,and then,(20)z˙=F^⋅v−KMxX¨.Since(21)Fv^⋅=KFv−Fv^=−KFv^+KMxX¨−Fu,the following equation is obtained:(22)z˙=−KFv^+KMxX¨−Fu−KMxX¨=−KFu−KFv^.The disturbance observer is designed as(23)z˙=−KFu−KFv^,Fv^=z+KMxX˙.Then,(24)z˙=−KFu−Kz+KMxX˙=K−Fu−KMxX˙−Kz.It is generally believed that there is no prior knowledge of disturbanceFv. Relative to the dynamic characteristics of the observer, the change of disturbance Fv is slow; that is, F˙v=0. Then, the observation error is(25)Fv˜=Fv−Fv^.So,(26)Fv˜⋅=F˙v−Fv^⋅=−F^⋅=−z˙−KMxX¨=−KFu−Kz+KMxX˙=KFu+KMxX˙+Kz−KMxX¨=Kz+KMxX˙−KMxX¨−Fu=KFv^−KFv=−KFv^−Fv=−KFv˜.Thus, the equation for the observation error can be obtained:(27)Fv˜⋅+KFv˜=0.Solving the differential equation, we have(28)Fv˜t=Fv˜t0e−Kt.Since the value ofFv˜t0 is determined, it can be seen that the convergence accuracy of the 6-DOF observer depends on the parameter matrix K. By designing the parameter matrix K, the estimated value Fv^ approaches Fv exponentially.


### 5.2. Sliding Mode Control Based on Approach Rate

The main reasons for the wide application of sliding mode control are the good control performance for nonlinear systems, the applicability to MIMO systems, and the establishment of good design standards for discrete-time systems. The important advantage of sliding mode control is its robustness. When the system is in sliding mode, it has excellent insensitivity to the model error of the controlled object, changes in object parameters, and external disturbances.The errors and their derivatives areeD=X0D−XD and e˙D=X˙0D−X˙D, respectively, where X0D is the desired state of the floating platform after vibration isolation.The slip face is defined as(29)s=ceD+e˙D,where c>0 satisfies the Hurwitz condition.According to Lyapunov’s second criterion, letV=1/2s2≥0; then, V˙=ss˙≤0.Using the exponential approach law, one has(30)s˙=−εsgns−ks,where ε>0, k>0; construct V˙=ss˙=−ks2≤0; then,(31)s˙=ce˙D+e¨D=ce˙D+X¨0D−Fu+FvMx.Then, combining formulas (39) and (40), one can get(32)ce˙D+X¨0D−Fu+FvMx=−εsgns−ks.Then, the sliding mode control law can be defined as(33)Fu=MxceD+X0D+εsgns+ks−Fv.Obviously, because the disturbanceFv is unknown, the above control law cannot be realized. In order to solve this problem, the control law can be designed using the boundary of disturbance.The design sliding mode control law is(34)Fu=MxceD+X0D+εsgns+ks−Fv˜,where Fv˜ is a positive real number to be designed related to the boundary of disturbance.Then,(35)s˙=−εsgns−ks+Fv˜−Fv.We select the appropriateFv˜ to ensure the stability of the control system, that is, to meet the sliding mode arrival conditions.Suppose thatFvlower≤Fv≤Fvupper, where Fvlower and Fvupper are the underworld and upper bounds of disturbance, respectively.The selection principles ofF˜ are as follows:(1)
Whens>0, s˙=−ε−ks+F˜v−Fv; in order to ensure s˙<0, take Fv˜=Fvlower.(2)
Whens<0, s˙=ε−ks+Fv˜−Fv; in order to ensure s˙<0, take Fv˜=Fvupper.TakingFv1=Fvupper−Fvlower/2 and Fv2=Fvupper+Fvlower/2, you can design Fv˜ that meets the above two conditions:(36)Fv˜=Fv2−Fv1sgns.The coefficient matrix in the simulation is set as follows:(37)c=150000001500000015000000150000001500000015,k=100000001000000010000000100000001000000010,ε=10−500000010−500000010−500000010−500000010−500000010−5.


### 5.3. PID Controller Design

LeteD=X0D−XD (the superscript D is the absolute movement), where X0D is the absolute displacement after vibration isolation. The differential equation of the system can be written as(38)MxX¨=Fu+Fv.In order to facilitate the analysis, let the displacement disturbanceRd0=R003×1T, according to the relationship between the absolute quantity and the relative quantity, we can get(39)X=XD−Rd0=X0D−eD−Rd0.Bringing the erroreD into the kinetic equation will give(40)Mx−e¨D+R¨d0=Fu+Fv.According to the PID control principle, set the control law as(41)Fu=Kae¨D+Kde˙D+KpeD+Ki∫eDdt−MxR¨0−Fv.At this time, since the movement displacement of the fixed platform relative to the inertial coordinate system is a minimum value, in order to verify the control performance, it can be assumed that the fixed platform coordinate system and the inertial platform coordinate system are fixed. Then.R0 is a zero matrix.Then, the dynamic equation with PID control then becomes(42)Mx+Kae¨D+Kde˙D+KpeD+Ki∫eDdt+Fv=0.Since the system is a microvibration isolation platform, its disturbance inputFv is bounded. After adding the control force, the value of Kp is equivalent to increasing the elastic stiffness of the system, which can effectively suppress the low-frequency vibration of the system; Kd is equivalent to increasing the damping of the system, so that the vibration of the system gradually decreases; Ki is the displacement. The coefficient of deviation integral is helpful to reduce the positional static error of the system; Ka is equivalent to increasing the mass and rotational inertia of the floating platform, making the system insensitive to external disturbance. Then, the whole system belongs to a system where the external input is bounded and the energy continues to decay. According to the definition of Lyapunov’s second law, it can be determined that the PID control algorithm can make the system effectively converge.The coefficient matrix of 6-DOF PID control selected in this paper is as follows:(43)Kp=100000000100000000100000000100000000100000000100,Kd=200000002000000020000000200000002000000020,Ki=150000001500000015000000150000001500000015,Ka=1000000008000000080000000100000000100000000100.


## 5.1. Design of Disturbance Observer Based on Exponential Convergence

According to the dynamic differential equation of the floating platform,(18)MxX¨=Fu+Fv.During the control process, if the size of the disturbanceFv can be effectively observed, the control effect will be greatly improved. The basic idea of designing an observer is to correct the estimated value by the difference between the estimated output and the actual output.TakeFv^⋅=KFv−Fv^, where K>0.The auxiliary parameter vector is defined as(19)z=Fv^−KMxX˙,and then,(20)z˙=F^⋅v−KMxX¨.Since(21)Fv^⋅=KFv−Fv^=−KFv^+KMxX¨−Fu,the following equation is obtained:(22)z˙=−KFv^+KMxX¨−Fu−KMxX¨=−KFu−KFv^.The disturbance observer is designed as(23)z˙=−KFu−KFv^,Fv^=z+KMxX˙.Then,(24)z˙=−KFu−Kz+KMxX˙=K−Fu−KMxX˙−Kz.It is generally believed that there is no prior knowledge of disturbanceFv. Relative to the dynamic characteristics of the observer, the change of disturbance Fv is slow; that is, F˙v=0. Then, the observation error is(25)Fv˜=Fv−Fv^.So,(26)Fv˜⋅=F˙v−Fv^⋅=−F^⋅=−z˙−KMxX¨=−KFu−Kz+KMxX˙=KFu+KMxX˙+Kz−KMxX¨=Kz+KMxX˙−KMxX¨−Fu=KFv^−KFv=−KFv^−Fv=−KFv˜.Thus, the equation for the observation error can be obtained:(27)Fv˜⋅+KFv˜=0.Solving the differential equation, we have(28)Fv˜t=Fv˜t0e−Kt.Since the value ofFv˜t0 is determined, it can be seen that the convergence accuracy of the 6-DOF observer depends on the parameter matrix K. By designing the parameter matrix K, the estimated value Fv^ approaches Fv exponentially.


## 5.2. Sliding Mode Control Based on Approach Rate

The main reasons for the wide application of sliding mode control are the good control performance for nonlinear systems, the applicability to MIMO systems, and the establishment of good design standards for discrete-time systems. The important advantage of sliding mode control is its robustness. When the system is in sliding mode, it has excellent insensitivity to the model error of the controlled object, changes in object parameters, and external disturbances.The errors and their derivatives areeD=X0D−XD and e˙D=X˙0D−X˙D, respectively, where X0D is the desired state of the floating platform after vibration isolation.The slip face is defined as(29)s=ceD+e˙D,where c>0 satisfies the Hurwitz condition.According to Lyapunov’s second criterion, letV=1/2s2≥0; then, V˙=ss˙≤0.Using the exponential approach law, one has(30)s˙=−εsgns−ks,where ε>0, k>0; construct V˙=ss˙=−ks2≤0; then,(31)s˙=ce˙D+e¨D=ce˙D+X¨0D−Fu+FvMx.Then, combining formulas (39) and (40), one can get(32)ce˙D+X¨0D−Fu+FvMx=−εsgns−ks.Then, the sliding mode control law can be defined as(33)Fu=MxceD+X0D+εsgns+ks−Fv.Obviously, because the disturbanceFv is unknown, the above control law cannot be realized. In order to solve this problem, the control law can be designed using the boundary of disturbance.The design sliding mode control law is(34)Fu=MxceD+X0D+εsgns+ks−Fv˜,where Fv˜ is a positive real number to be designed related to the boundary of disturbance.Then,(35)s˙=−εsgns−ks+Fv˜−Fv.We select the appropriateFv˜ to ensure the stability of the control system, that is, to meet the sliding mode arrival conditions.Suppose thatFvlower≤Fv≤Fvupper, where Fvlower and Fvupper are the underworld and upper bounds of disturbance, respectively.The selection principles ofF˜ are as follows:(1)
Whens>0, s˙=−ε−ks+F˜v−Fv; in order to ensure s˙<0, take Fv˜=Fvlower.(2)
Whens<0, s˙=ε−ks+Fv˜−Fv; in order to ensure s˙<0, take Fv˜=Fvupper.TakingFv1=Fvupper−Fvlower/2 and Fv2=Fvupper+Fvlower/2, you can design Fv˜ that meets the above two conditions:(36)Fv˜=Fv2−Fv1sgns.The coefficient matrix in the simulation is set as follows:(37)c=150000001500000015000000150000001500000015,k=100000001000000010000000100000001000000010,ε=10−500000010−500000010−500000010−500000010−500000010−5.


## 5.3. PID Controller Design

LeteD=X0D−XD (the superscript D is the absolute movement), where X0D is the absolute displacement after vibration isolation. The differential equation of the system can be written as(38)MxX¨=Fu+Fv.In order to facilitate the analysis, let the displacement disturbanceRd0=R003×1T, according to the relationship between the absolute quantity and the relative quantity, we can get(39)X=XD−Rd0=X0D−eD−Rd0.Bringing the erroreD into the kinetic equation will give(40)Mx−e¨D+R¨d0=Fu+Fv.According to the PID control principle, set the control law as(41)Fu=Kae¨D+Kde˙D+KpeD+Ki∫eDdt−MxR¨0−Fv.At this time, since the movement displacement of the fixed platform relative to the inertial coordinate system is a minimum value, in order to verify the control performance, it can be assumed that the fixed platform coordinate system and the inertial platform coordinate system are fixed. Then.R0 is a zero matrix.Then, the dynamic equation with PID control then becomes(42)Mx+Kae¨D+Kde˙D+KpeD+Ki∫eDdt+Fv=0.Since the system is a microvibration isolation platform, its disturbance inputFv is bounded. After adding the control force, the value of Kp is equivalent to increasing the elastic stiffness of the system, which can effectively suppress the low-frequency vibration of the system; Kd is equivalent to increasing the damping of the system, so that the vibration of the system gradually decreases; Ki is the displacement. The coefficient of deviation integral is helpful to reduce the positional static error of the system; Ka is equivalent to increasing the mass and rotational inertia of the floating platform, making the system insensitive to external disturbance. Then, the whole system belongs to a system where the external input is bounded and the energy continues to decay. According to the definition of Lyapunov’s second law, it can be determined that the PID control algorithm can make the system effectively converge.The coefficient matrix of 6-DOF PID control selected in this paper is as follows:(43)Kp=100000000100000000100000000100000000100000000100,Kd=200000002000000020000000200000002000000020,Ki=150000001500000015000000150000001500000015,Ka=1000000008000000080000000100000000100000000100.


## 6. Construction of the Cosimulation Model in ADAMS and MATLAB

As shown in Figure5, the dynamic equations module is a 6-DOF vibration isolation platform model established based on dynamic equations, and the ADAMS_submodule is a 6-DOF system model obtained by modeling by the ADAMS software. The disturbance observer module is a packaged disturbance observer based on exponential convergence. The controller module contains two methods of PID and sliding mode variable structure control.Figure 5
The cosimulation model of ADAMS and MATLAB.


## 7. Cosimulation Results and Analysis of ADAMS and MATLAB

### 7.1. Research on Platform Position and Posture Control

The position and posture positioning results of millimeter level and micrometer level are, respectively, shown in Figures6 and 7. Sliding mode control takes much less time to reach the desired position than PID control. Among them, the adjustment time of PID control is about 5 s, while the adjustment time of sliding mode control is only less than 10% of PID control, and the position adjustment of the floating platform can be completed within 0.5 s. Moreover, the adjustment amplitude of sliding mode control is very small, only less than 5% of PID control on average. Although the sliding mode control will have a slight jitter after reaching the desired position, the magnitude is small and will not affect the control effect. The simulation results show that sliding mode control can adjust the position and attitude of the floating platform greatly.Figure 6
Large displacement (millimeter level) positioning test.
(a)(b)(c)(d)(e)(f)Figure 7
Tiny displacement (micron level) positioning test: (a)θx, (b) θy, (c) θz, (d) x, (e) y, and (f) z.
(a)(b)(c)(d)(e)(f)


### 7.2. Research on Platform Vibration Control

#### 7.2.1. Vibration Isolation Scenario 1

The vibration source is set at the position of the floating platform coordinate system (0, 0.1, 0), and thex, y, and z axes of the vibration source are, respectively, set to have an amplitude of 3mN and a frequency of 0.1 Hz, 1 Hz, and 10 Hz. Sinusoidal excitation disturbance signal is used to analyze the movement state of the six degrees of freedom before and after the control of the platform.Disturbance observers are added to both PID control and sliding mode control. The disturbance observer designed in this paper can estimate the six degrees of freedom disturbance based on the motion information on the floating platform and the input of the control force. In order to facilitate the analysis of the relationship between the observation results of the disturbance observer and the actual input disturbance, the 6-DOF disturbance observation results can be used in combination with the location information of the disturbance source to solve thex, y, and z axes’ observation disturbance forces at the input of the disturbance source. Then, we compare the observed disturbance force with the actual input disturbance signal.It can be seen in Figure8 that the observer has a strong ability to observe low and medium frequency disturbance, and the phase difference between the observed signal and the actual signal is about 0.2s, which can fit the actual disturbance signal well.Figure 8
Test results of disturbance observer: (a)fdx, (b) fdy, and (c) fdz.
(a)(b)(c)Figure9 shows the displacement and attitude control of the floating platform. The results show that both control methods can control the displacement and rotation angle around 0.Figure 9
Floating platform position and attitude control: (a)θx, (b) θy, (c) θz, (d) x, (e) y, and (f) z.
(a)(b)(c)(d)(e)(f)Figure10 shows the control of floating platform acceleration and angular acceleration. The results show that the two control methods can reduce the acceleration of each axis and the angular acceleration by more than 90%. The enlarged diagram shows that, after the control, the static characteristics of PID will be better than the sliding mode control.Figure 10
Floating platform acceleration and angular acceleration control.
(a)(b)(c)(d)(e)(f)According to Figure11, from a macroperspective, the control force of the actuators of the two control methods is basically the same, but the partially enlarged view shows that the control force of the sliding mode control will have a certain degree of chatter.Figure 11
Comparison of PID control and slip mode control force input: (a)F1x, (b) F1z, (c) F2x, (d) F2z, (e) F3x, and (f) F3z.
(a)(b)(c)(d)(e)(f)


#### 7.2.2. Vibration Isolation Scenario 2

A 3mNstep disturbance signal is applied to the eccentric vibration source in the x, y, and z directions.In Figure12, it can be seen that the disturbance observer can better fit the actual disturbance signal in the x, y, and z directions of the disturbance source. At 0 and 1 s, when the disturbance signal suddenly changes, the disturbance observer can be adjusted within 0.2 s, so that the observed signal better fits the actual disturbance signal.Figure 12
Test results of disturbance observer: (a)fdx, (b) fdy, and (c) fdz.
(a)(b)(c)In Figure13, PID control and sliding mode control can well control the position and attitude of the 6-DOF platform. However, it can be seen from Figure 14 that, in the face of sudden disturbance, the sliding mode control is robust and can better control the acceleration of the platform, making the acceleration floating range only less than 20% of the PID control. In the meantime, Figure 15 shows the control force input required for PID control and sliding mode control. Therefore, sliding mode control is more suitable for irregular and tiny vibrations in space.Figure 13
Floating platform displacement and posture control: (a)θx, (b) θy, (c) θz, (d) x, (e) y, and (f) z.
(a)(b)(c)(d)(e)(f)Figure 14
Floating platform acceleration and angular acceleration control: (a)θ¨x, (b) θ¨y, (c) θ¨z, (d) x¨, (e) y¨, and (f) z¨.
(a)(b)(c)(d)(e)(f)Figure 15
Comparison of PID control with slip mode control force input.
(a)(b)(c)(d)(e)(f)


#### 7.2.3. Vibration Isolation Scenario 3

There is a 0.1–100 Hz frequency sweep disturbance system response. Without control, the platform movement scale is relatively large, so here only PID control and sliding mode control are compared.In the frequency sweep disturbance simulation, it can be seen from Figure16 that the two control methods can effectively control the pose of the floating platform to keep it relatively static with the base station, and the accuracy can reach the micron level.Figure 16
Changes in floating platform position and posture under sweep disturbance: (a)θx, (b) θy, (c) θz, (d) x, (e) y, and (f) z.
(a)(b)(c)(d)(e)(f)


#### 7.2.4. Vibration Isolation Scenario 4

Thex, y, and z axes of the disturbance force application point all have a disturbance force of 1 N and a frequency of 20 Hz. This part mainly explores the influence of the different coefficients of the disturbance observer on the z-axis displacement of the floating platform.The coefficient matrixK of the disturbance observer with six degrees of freedom can be expressed as(44)K=k1000000k2000000k3000000k4000000k5000000k6.In order to facilitate analysis, the values ofk1∼k6 in the coefficient matrix are all equal and set to k. Figure 17 shows the displacement change diagram of the z-axis in the main vibration direction of the floating platform. It can be observed from the figure that the larger the k value of the disturbance observer based on exponential convergence, the smaller the convergence of the z-axis displacement of the floating platform. However, the simulation shows that too large a value of k leads to a divergent trend of z-axis displacement and control failure. In summary, the coefficient k used in this article is generally set to 60.Figure 17
z-axis displacement of the floating platform.


## 7.1. Research on Platform Position and Posture Control

The position and posture positioning results of millimeter level and micrometer level are, respectively, shown in Figures6 and 7. Sliding mode control takes much less time to reach the desired position than PID control. Among them, the adjustment time of PID control is about 5 s, while the adjustment time of sliding mode control is only less than 10% of PID control, and the position adjustment of the floating platform can be completed within 0.5 s. Moreover, the adjustment amplitude of sliding mode control is very small, only less than 5% of PID control on average. Although the sliding mode control will have a slight jitter after reaching the desired position, the magnitude is small and will not affect the control effect. The simulation results show that sliding mode control can adjust the position and attitude of the floating platform greatly.Figure 6
Large displacement (millimeter level) positioning test.
(a)(b)(c)(d)(e)(f)Figure 7
Tiny displacement (micron level) positioning test: (a)θx, (b) θy, (c) θz, (d) x, (e) y, and (f) z.
(a)(b)(c)(d)(e)(f)


## 7.2. Research on Platform Vibration Control

### 7.2.1. Vibration Isolation Scenario 1

The vibration source is set at the position of the floating platform coordinate system (0, 0.1, 0), and thex, y, and z axes of the vibration source are, respectively, set to have an amplitude of 3mN and a frequency of 0.1 Hz, 1 Hz, and 10 Hz. Sinusoidal excitation disturbance signal is used to analyze the movement state of the six degrees of freedom before and after the control of the platform.Disturbance observers are added to both PID control and sliding mode control. The disturbance observer designed in this paper can estimate the six degrees of freedom disturbance based on the motion information on the floating platform and the input of the control force. In order to facilitate the analysis of the relationship between the observation results of the disturbance observer and the actual input disturbance, the 6-DOF disturbance observation results can be used in combination with the location information of the disturbance source to solve thex, y, and z axes’ observation disturbance forces at the input of the disturbance source. Then, we compare the observed disturbance force with the actual input disturbance signal.It can be seen in Figure8 that the observer has a strong ability to observe low and medium frequency disturbance, and the phase difference between the observed signal and the actual signal is about 0.2s, which can fit the actual disturbance signal well.Figure 8
Test results of disturbance observer: (a)fdx, (b) fdy, and (c) fdz.
(a)(b)(c)Figure9 shows the displacement and attitude control of the floating platform. The results show that both control methods can control the displacement and rotation angle around 0.Figure 9
Floating platform position and attitude control: (a)θx, (b) θy, (c) θz, (d) x, (e) y, and (f) z.
(a)(b)(c)(d)(e)(f)Figure10 shows the control of floating platform acceleration and angular acceleration. The results show that the two control methods can reduce the acceleration of each axis and the angular acceleration by more than 90%. The enlarged diagram shows that, after the control, the static characteristics of PID will be better than the sliding mode control.Figure 10
Floating platform acceleration and angular acceleration control.
(a)(b)(c)(d)(e)(f)According to Figure11, from a macroperspective, the control force of the actuators of the two control methods is basically the same, but the partially enlarged view shows that the control force of the sliding mode control will have a certain degree of chatter.Figure 11
Comparison of PID control and slip mode control force input: (a)F1x, (b) F1z, (c) F2x, (d) F2z, (e) F3x, and (f) F3z.
(a)(b)(c)(d)(e)(f)


### 7.2.2. Vibration Isolation Scenario 2

A 3mNstep disturbance signal is applied to the eccentric vibration source in the x, y, and z directions.In Figure12, it can be seen that the disturbance observer can better fit the actual disturbance signal in the x, y, and z directions of the disturbance source. At 0 and 1 s, when the disturbance signal suddenly changes, the disturbance observer can be adjusted within 0.2 s, so that the observed signal better fits the actual disturbance signal.Figure 12
Test results of disturbance observer: (a)fdx, (b) fdy, and (c) fdz.
(a)(b)(c)In Figure13, PID control and sliding mode control can well control the position and attitude of the 6-DOF platform. However, it can be seen from Figure 14 that, in the face of sudden disturbance, the sliding mode control is robust and can better control the acceleration of the platform, making the acceleration floating range only less than 20% of the PID control. In the meantime, Figure 15 shows the control force input required for PID control and sliding mode control. Therefore, sliding mode control is more suitable for irregular and tiny vibrations in space.Figure 13
Floating platform displacement and posture control: (a)θx, (b) θy, (c) θz, (d) x, (e) y, and (f) z.
(a)(b)(c)(d)(e)(f)Figure 14
Floating platform acceleration and angular acceleration control: (a)θ¨x, (b) θ¨y, (c) θ¨z, (d) x¨, (e) y¨, and (f) z¨.
(a)(b)(c)(d)(e)(f)Figure 15
Comparison of PID control with slip mode control force input.
(a)(b)(c)(d)(e)(f)


### 7.2.3. Vibration Isolation Scenario 3

There is a 0.1–100 Hz frequency sweep disturbance system response. Without control, the platform movement scale is relatively large, so here only PID control and sliding mode control are compared.In the frequency sweep disturbance simulation, it can be seen from Figure16 that the two control methods can effectively control the pose of the floating platform to keep it relatively static with the base station, and the accuracy can reach the micron level.Figure 16
Changes in floating platform position and posture under sweep disturbance: (a)θx, (b) θy, (c) θz, (d) x, (e) y, and (f) z.
(a)(b)(c)(d)(e)(f)


### 7.2.4. Vibration Isolation Scenario 4

Thex, y, and z axes of the disturbance force application point all have a disturbance force of 1 N and a frequency of 20 Hz. This part mainly explores the influence of the different coefficients of the disturbance observer on the z-axis displacement of the floating platform.The coefficient matrixK of the disturbance observer with six degrees of freedom can be expressed as(44)K=k1000000k2000000k3000000k4000000k5000000k6.In order to facilitate analysis, the values ofk1∼k6 in the coefficient matrix are all equal and set to k. Figure 17 shows the displacement change diagram of the z-axis in the main vibration direction of the floating platform. It can be observed from the figure that the larger the k value of the disturbance observer based on exponential convergence, the smaller the convergence of the z-axis displacement of the floating platform. However, the simulation shows that too large a value of k leads to a divergent trend of z-axis displacement and control failure. In summary, the coefficient k used in this article is generally set to 60.Figure 17
z-axis displacement of the floating platform.


## 7.2.1. Vibration Isolation Scenario 1

The vibration source is set at the position of the floating platform coordinate system (0, 0.1, 0), and thex, y, and z axes of the vibration source are, respectively, set to have an amplitude of 3mN and a frequency of 0.1 Hz, 1 Hz, and 10 Hz. Sinusoidal excitation disturbance signal is used to analyze the movement state of the six degrees of freedom before and after the control of the platform.Disturbance observers are added to both PID control and sliding mode control. The disturbance observer designed in this paper can estimate the six degrees of freedom disturbance based on the motion information on the floating platform and the input of the control force. In order to facilitate the analysis of the relationship between the observation results of the disturbance observer and the actual input disturbance, the 6-DOF disturbance observation results can be used in combination with the location information of the disturbance source to solve thex, y, and z axes’ observation disturbance forces at the input of the disturbance source. Then, we compare the observed disturbance force with the actual input disturbance signal.It can be seen in Figure8 that the observer has a strong ability to observe low and medium frequency disturbance, and the phase difference between the observed signal and the actual signal is about 0.2s, which can fit the actual disturbance signal well.Figure 8
Test results of disturbance observer: (a)fdx, (b) fdy, and (c) fdz.
(a)(b)(c)Figure9 shows the displacement and attitude control of the floating platform. The results show that both control methods can control the displacement and rotation angle around 0.Figure 9
Floating platform position and attitude control: (a)θx, (b) θy, (c) θz, (d) x, (e) y, and (f) z.
(a)(b)(c)(d)(e)(f)Figure10 shows the control of floating platform acceleration and angular acceleration. The results show that the two control methods can reduce the acceleration of each axis and the angular acceleration by more than 90%. The enlarged diagram shows that, after the control, the static characteristics of PID will be better than the sliding mode control.Figure 10
Floating platform acceleration and angular acceleration control.
(a)(b)(c)(d)(e)(f)According to Figure11, from a macroperspective, the control force of the actuators of the two control methods is basically the same, but the partially enlarged view shows that the control force of the sliding mode control will have a certain degree of chatter.Figure 11
Comparison of PID control and slip mode control force input: (a)F1x, (b) F1z, (c) F2x, (d) F2z, (e) F3x, and (f) F3z.
(a)(b)(c)(d)(e)(f)


## 7.2.2. Vibration Isolation Scenario 2

A 3mNstep disturbance signal is applied to the eccentric vibration source in the x, y, and z directions.In Figure12, it can be seen that the disturbance observer can better fit the actual disturbance signal in the x, y, and z directions of the disturbance source. At 0 and 1 s, when the disturbance signal suddenly changes, the disturbance observer can be adjusted within 0.2 s, so that the observed signal better fits the actual disturbance signal.Figure 12
Test results of disturbance observer: (a)fdx, (b) fdy, and (c) fdz.
(a)(b)(c)In Figure13, PID control and sliding mode control can well control the position and attitude of the 6-DOF platform. However, it can be seen from Figure 14 that, in the face of sudden disturbance, the sliding mode control is robust and can better control the acceleration of the platform, making the acceleration floating range only less than 20% of the PID control. In the meantime, Figure 15 shows the control force input required for PID control and sliding mode control. Therefore, sliding mode control is more suitable for irregular and tiny vibrations in space.Figure 13
Floating platform displacement and posture control: (a)θx, (b) θy, (c) θz, (d) x, (e) y, and (f) z.
(a)(b)(c)(d)(e)(f)Figure 14
Floating platform acceleration and angular acceleration control: (a)θ¨x, (b) θ¨y, (c) θ¨z, (d) x¨, (e) y¨, and (f) z¨.
(a)(b)(c)(d)(e)(f)Figure 15
Comparison of PID control with slip mode control force input.
(a)(b)(c)(d)(e)(f)


## 7.2.3. Vibration Isolation Scenario 3

There is a 0.1–100 Hz frequency sweep disturbance system response. Without control, the platform movement scale is relatively large, so here only PID control and sliding mode control are compared.In the frequency sweep disturbance simulation, it can be seen from Figure16 that the two control methods can effectively control the pose of the floating platform to keep it relatively static with the base station, and the accuracy can reach the micron level.Figure 16
Changes in floating platform position and posture under sweep disturbance: (a)θx, (b) θy, (c) θz, (d) x, (e) y, and (f) z.
(a)(b)(c)(d)(e)(f)


## 7.2.4. Vibration Isolation Scenario 4

Thex, y, and z axes of the disturbance force application point all have a disturbance force of 1 N and a frequency of 20 Hz. This part mainly explores the influence of the different coefficients of the disturbance observer on the z-axis displacement of the floating platform.The coefficient matrixK of the disturbance observer with six degrees of freedom can be expressed as(44)K=k1000000k2000000k3000000k4000000k5000000k6.In order to facilitate analysis, the values ofk1∼k6 in the coefficient matrix are all equal and set to k. Figure 17 shows the displacement change diagram of the z-axis in the main vibration direction of the floating platform. It can be observed from the figure that the larger the k value of the disturbance observer based on exponential convergence, the smaller the convergence of the z-axis displacement of the floating platform. However, the simulation shows that too large a value of k leads to a divergent trend of z-axis displacement and control failure. In summary, the coefficient k used in this article is generally set to 60.Figure 17
z-axis displacement of the floating platform.


## 8. Conclusion

In this paper, a cosimulation virtual prototype model was designed to study the vibration isolation and positioning performance of the noncontact 6-DOF platform in the space microgravity environment. Based on the Kane dynamics method combined with the coordinate conversion method to carry on the dynamic modeling of the platform, a 6-DOF sliding mode robust controller based on the approach rate was designed, which has the advantages of strong antidisturbance and fast response speed. A 6-DOF disturbance observer based on exponential convergence was designed to predict the disturbance force and solve the chattering problem of sliding mode control to the greatest extent. In order to further verify the proposed model, a cosimulation model was established using ADAMS combined with MATLAB/Simulink, and 6-DOF PID control, sliding mode control, and disturbance observer programs were developed throughS-function. The simulation results show that the control system can be well decoupled to achieve 6-DOF independent control, and the floating platform can be accurately positioned at the millimeter and micrometer levels. Among them, the adjustment time of sliding mode control is shorter than that of PID control by more than 90%. In the simulation of low-frequency vibration isolation, it is found that the 6-DOF disturbance observer can well fit the actual disturbance effect, indicating that the disturbance observer can well compensate the disturbance force in the low-frequency range (0.01–10 Hz) and thus effectively reduce the chatter of sliding mode control. In the step disturbance simulation, it can be seen that, in the face of sudden disturbance, the sliding mode control is robust and can better control the acceleration of the platform, making the acceleration floating range only less than 20% of the PID control. In the frequency sweep disturbance simulation with a magnitude of 1 mN, the sliding mode control keeps the floating platform absolutely still, and the maximum displacement produced is less than 1 μm, to achieve vibration isolation control. In summary, the 6-DOF sliding mode robust controller based on approach rate combined with the 6-DOF disturbance observer based on exponential convergence can complete the pointing and vibration isolation functions of the floating platform greatly.


## 9. Future Recommendation

Future research work will focus on the following two aspects:(1)
Improve the 6-DOF control algorithm in this paper, use the new sliding mode control algorithm in recent years (such as global SMC or SMC based on a genetic algorithm) to extend to six degrees of freedom, and use it on the virtual prototype of the noncontact 6-DOF platform(2)
If the conditions are ripe, a prototype will be built to conduct field experiments to verify the virtual prototype model and theoretical algorithms and continue to improve



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*Source: 6641863-2021-03-25.xml*