# Desired Compensation Adaptive Robust Control of an Active Vibration Isolation System

Authors: Bo Zhao; Weijia Shi; Ming Zhang; Jiaxin Li; Feng Li
Journal: Complexity
 (2018)

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## Abstract

Active vibration isolation system (AVIS) has attracted increasing attention of researchers in precision engineering. In this paper, a desired compensation adaptive robust controller (DCARC) is proposed for an AVIS developed in our laboratory. The AVIS composed of one platform and three active isolators is required to achieve high-performance vibration isolation as well as low trajectory tracking error for positioning. The vertical three degrees of freedom (DOFs) and horizontal three DOFs are decoupled by the joint bearing of the isolators. The dynamic model of the system is built and is simplified to three single-input-single-output (SISO) systems. The DCARC control scheme is then proposed, which contains a deterministic robust control (DRC) term and an adaptive control (AC) term. The high performance in vibration isolation and positioning can be subsequently achieved, even the actual load and system stiffness are unknown and there exists direct bounded disturbance on the platform. The AC term is designed to estimate the unknown parameters of the system. The DRC term can improve the robustness of the system, which is used to reject the direct disturbance and the parameter estimation error. Furthermore, the computing time and the influence of the measurement noise can be reduced effectively by reason of desired compensation. The numerical simulation and comparative experiments are carried out under the conditions of using DCARC, DRC, and AC controllers. The experimental results validate that the proposed DCARC control strategy outperforms other control method and possesses both high-performance vibration isolation and low tracking error.


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## Body

## 1. Introduction

Vibration isolation system is widely used in industrial equipment [1, 2], building structure [3, 4], and commercial vehicles [5–7], to protect the isolating object, such as human and structure, from being damaged by vibration with certain frequency regions. Especially in recent years, with the rapid development of nanoprecision technology, ultraprecision positioning devices, nanolevel measuring instruments, which are extensively used in medical research, biotechnology, semiconductor manufacturing, and space exploring, are demanding an increasingly need for high-performance vibration isolation system [8–14]. According to the source, vibration can be divided into two categories, the ground vibration which is caused by geodetic pulsation, wind blowing, automobile moving, people walking, and so forth, transferred via the suspension of the system, and the load vibration which is caused by inertial force from a moving load or other disturbance directly acted on the isolation system [15]. To reduce the ground vibration, an isolation system with low suspension stiffness is more susceptible, the zero stiffness is ideal, and the performance of transmissibility is the priority of the isolation system. However, the load capacity of the system will decrease with the reducing stiffness [16]. To reduce the load vibration, an isolation system with high suspension stiffness is more susceptible, the infinite stiffness is ideal, and the performance of compliance is the priority of the isolation system. However, the isolation system will be more sensitive to the ground vibration with the increasing stiffness. So in the conventional passive vibration isolation system, a trade-off between load capacity and isolation performance is inevitable [17].Active vibration isolation system (AVIS) has attracted significant research attention in recent years, which can give consideration to both load capacity and isolation performance [18]. The focus of AVIS research lies on three things, mechanical design without cross-coupling in each actuator, low-frequency vibration sensing, and control of the isolation system [19]. With the increasing demands to AVIS performance of broad frequency, high positioning accuracy, low transmissibility, and high compliance, the research interest is brought much to the control method [20]. Holterman and de Vries proposed an active damping method based on decoupled collocated control in combination with a passive multiple-input-multiple-output (MIMO) control law to avoid the modelling uncertainty [21]. The modal damping can be realized under the mild condition by using piezoelectric Smart Discs made by piezoelectric material [22], and the robust stability can be retained based on a simple kinematic model rather than a complex dynamic model with specific parameters of the system. Heertjes et al. proposed a switching control method with two regimes to enhance both the steady-state noise properties of the passive isolation system and disturbance rejection properties of transients [23]. The transient oscillations can be removed from the velocity response, and the steady-state noise can be avoided at small velocity by sky-hook damping. To resolve the subcritical vibration isolation problem, which is the natural frequency of the uncoupled source body larger than that of the uncoupled receiving body, Alujevic et al. used an inerter element to enhance the stability margins of the feedback disturbance rejection scheme of two degrees of freedom (DOFs) system [24]. The specific kinetic energy index monotonically decreased with the increase in the feedback gain, and the higher the gain, the lower the velocity response. Beijen et al. proposed a self-turning feedforward control method for an air mount systems to compensate the static damping and stiffness [25]. A higher-order weak integrator can limit the gain of the controller to prevent drift and actuator saturation, and an infinite impulse response (IIR) filter with fixed poles and self-turning zeros was implemented to online adjust the parameters by using filtered least mean squares (FLMS) optimization and residual noise shaping.In this paper, a desired compensation adaptive robust control (DCARC) strategy for an active vibration isolation system in the vertical three degrees of freedom is proposed. The focused indexes of the AVIS are transmissibility, ability to disturbance rejection, and positioning error. The DCARC control scheme is composed of a deterministic robust control (DRC) term and an adaptive control (AC) term. The AC term is designed to estimate the unknown parameters of the system. The DRC term can improve the robustness of the system, which is used to reject the direct disturbance and the parameter estimation error. The performance of DCARC is validated by both numerical simulation and experiment. The DRC and AC controllers are also tested as a comparison. The results show that the proposed DCARC control strategy outperforms other control method and possesses both high-performance vibration isolation and low tracking error.


## 2. System Description and Dynamic Modeling

### 2.1. System Description

The schematic illustration of the isolator is shown in Figure1. Each isolator contains two voice coil motor (VCM), which can provide a vertical force Fz and a horizontal force Fy. The inclination of the platform can be decoupled into vertical movement and horizontal movement by the joint bearing. So, the vertical 3 DOFs, which are Z,θx,andθy, and the horizontal 3 DOFs, which are X,Y,andθz, can be considered as two independents because the tilt angle of the platform is extreme small. The compressed air in the main air chamber from external source is used to support the platform and the load. The air spray out from the orifice can form a thin film which is the air bearing in this paper. There is no interference between the horizontal air bearing and the vertical air bearing, so the vertical movement and the horizontal movement of the AVIS are decoupled by the mechanical structure. The horizontal stiffness kh of the isolator is close to zero without the active control of the horizontal actuator, and the vertical stiffness kf is related to the load. The horizontal damping coefficient ch and the vertical damping coefficient cf are close to zero. The travel range along z-axis is 1 mm and that along x-axis and y-axis is 2 mm.Figure 1
Conceptual design of the isolator.The AVIS in this paper is composed of one platform and three isolators, as shown in Figure2. The centers of the three isolators are at the vertexes of an isosceles triangle, O represents the center of the platform, and O1, O2, and O3 are the supporting points of the isolators to the platform. The platform is controlled by three horizontal forces Fh1,Fh2,andFh3 and three vertical forces Fv1,Fv2,andFv3. As shown in Figure 2, the parameters LhandLv represent the arm of force to point O, for example, Lh2zandLv1x, respectively, represent the arm of horizontal force Fh2 to z-axis and the vertical forces Fv1 to x-axis, and each force has two arms. The parameters L1x and L1y represent the distances from O1 to XOZ plane and YOZ plane. The air pressure in the main chamber of AVI-2 is larger than the other two because of the configuration of the isolators. The platform will move suddenly at the moment when the high-pressure air source is open, because once the air bearings work, the friction coefficient changes from a large value to nearly zero. So, the mechanical limits are mounted around the horizontal air bearing to restrict the horizontal travel range. The linear travel ranges of the AVIS are 2 mm along the x-axis and y-axis and 0.8 mm along the z-axis. The rotational travel ranges are 179 arc second around x-axis, 541 arc second around y-axis, and 448 arc second around z-axis.Figure 2
Scheme of active vibration isolation system.


### 2.2. Dynamic Modeling

Due tokh, ch, and cf are close to zero, the dynamic equations of the six degrees of the AVIS are given by
(1)mx¨=−Fh2,my¨=Fh1+Fh3,Izθ¨z=Fh1Lh1z+Fh2Lh2z−Fh3Lh3z,mz¨+3kfz−z0=Fv1+Fv2+Fv3,Ixθ¨x+kfL1x2+L2x2+L3x2θx−θx0=Fh1Lh1x+Fh3Lh3x+Fv2Lv2x−Fv1Lv1x−Fv3Lv3x,Iyθ¨y+kfL1y2+L3y2θy−θy0=Fh2Lh2y+Fv2Lv2y+Fv3Lv3y−Fv1Lv1y,where (1) can be represented in matrix form as
(2)Mq¨+Kq−Kq0=Φ⋅f,where
(3)q=x,y,θz,z,θx,θyT,q0=x0,y0,θz0,z0,θx0,θy0T,f=Fh1,Fh2,Fh3,Fv1,Fv2,Fv3T,M=diagm,m,Iz,m,Ix,Iy,K=diag0,0,0,3kf,kfL1x2+L2x2+L3x2,kfL1y2+L3y2,Φ=0−10000101000Lh1zLh2z−Lh3z000000111Lh1x0Lh3x−Lv1xLv2x−Lv3x0Lh2y0−Lv1yLv2yLv3y.The driving forces of VCMs are given by(4)f=KFII.where KFI is the thrust coefficient of the VCM; I=I1,I2,I3,I4,I5,I6T is the control current vector.Single-input-single-output (SISO) controllers can be designed to generate control currents. Define the imaginary SISO controllers output asuc=u1,u2,u3,u4,u5,u6T. The relationship between uc and control currents can be derived as
(5)uc=Φ⋅f=KFIΦ⋅I.Substituting (5) into (2), the dynamic equation can be written as
(6)Mq¨+Kq−Kq0=uc.Equation (6) shows that the model of AVIS has been decoupled. As we concern the vertical three DOFs, the main object of this paper is to design control input uv=u4,u5,u6T to achieve active control of the vertical three DOFs of the system.


## 2.1. System Description

The schematic illustration of the isolator is shown in Figure1. Each isolator contains two voice coil motor (VCM), which can provide a vertical force Fz and a horizontal force Fy. The inclination of the platform can be decoupled into vertical movement and horizontal movement by the joint bearing. So, the vertical 3 DOFs, which are Z,θx,andθy, and the horizontal 3 DOFs, which are X,Y,andθz, can be considered as two independents because the tilt angle of the platform is extreme small. The compressed air in the main air chamber from external source is used to support the platform and the load. The air spray out from the orifice can form a thin film which is the air bearing in this paper. There is no interference between the horizontal air bearing and the vertical air bearing, so the vertical movement and the horizontal movement of the AVIS are decoupled by the mechanical structure. The horizontal stiffness kh of the isolator is close to zero without the active control of the horizontal actuator, and the vertical stiffness kf is related to the load. The horizontal damping coefficient ch and the vertical damping coefficient cf are close to zero. The travel range along z-axis is 1 mm and that along x-axis and y-axis is 2 mm.Figure 1
Conceptual design of the isolator.The AVIS in this paper is composed of one platform and three isolators, as shown in Figure2. The centers of the three isolators are at the vertexes of an isosceles triangle, O represents the center of the platform, and O1, O2, and O3 are the supporting points of the isolators to the platform. The platform is controlled by three horizontal forces Fh1,Fh2,andFh3 and three vertical forces Fv1,Fv2,andFv3. As shown in Figure 2, the parameters LhandLv represent the arm of force to point O, for example, Lh2zandLv1x, respectively, represent the arm of horizontal force Fh2 to z-axis and the vertical forces Fv1 to x-axis, and each force has two arms. The parameters L1x and L1y represent the distances from O1 to XOZ plane and YOZ plane. The air pressure in the main chamber of AVI-2 is larger than the other two because of the configuration of the isolators. The platform will move suddenly at the moment when the high-pressure air source is open, because once the air bearings work, the friction coefficient changes from a large value to nearly zero. So, the mechanical limits are mounted around the horizontal air bearing to restrict the horizontal travel range. The linear travel ranges of the AVIS are 2 mm along the x-axis and y-axis and 0.8 mm along the z-axis. The rotational travel ranges are 179 arc second around x-axis, 541 arc second around y-axis, and 448 arc second around z-axis.Figure 2
Scheme of active vibration isolation system.


## 2.2. Dynamic Modeling

Due tokh, ch, and cf are close to zero, the dynamic equations of the six degrees of the AVIS are given by
(1)mx¨=−Fh2,my¨=Fh1+Fh3,Izθ¨z=Fh1Lh1z+Fh2Lh2z−Fh3Lh3z,mz¨+3kfz−z0=Fv1+Fv2+Fv3,Ixθ¨x+kfL1x2+L2x2+L3x2θx−θx0=Fh1Lh1x+Fh3Lh3x+Fv2Lv2x−Fv1Lv1x−Fv3Lv3x,Iyθ¨y+kfL1y2+L3y2θy−θy0=Fh2Lh2y+Fv2Lv2y+Fv3Lv3y−Fv1Lv1y,where (1) can be represented in matrix form as
(2)Mq¨+Kq−Kq0=Φ⋅f,where
(3)q=x,y,θz,z,θx,θyT,q0=x0,y0,θz0,z0,θx0,θy0T,f=Fh1,Fh2,Fh3,Fv1,Fv2,Fv3T,M=diagm,m,Iz,m,Ix,Iy,K=diag0,0,0,3kf,kfL1x2+L2x2+L3x2,kfL1y2+L3y2,Φ=0−10000101000Lh1zLh2z−Lh3z000000111Lh1x0Lh3x−Lv1xLv2x−Lv3x0Lh2y0−Lv1yLv2yLv3y.The driving forces of VCMs are given by(4)f=KFII.where KFI is the thrust coefficient of the VCM; I=I1,I2,I3,I4,I5,I6T is the control current vector.Single-input-single-output (SISO) controllers can be designed to generate control currents. Define the imaginary SISO controllers output asuc=u1,u2,u3,u4,u5,u6T. The relationship between uc and control currents can be derived as
(5)uc=Φ⋅f=KFIΦ⋅I.Substituting (5) into (2), the dynamic equation can be written as
(6)Mq¨+Kq−Kq0=uc.Equation (6) shows that the model of AVIS has been decoupled. As we concern the vertical three DOFs, the main object of this paper is to design control input uv=u4,u5,u6T to achieve active control of the vertical three DOFs of the system.


## 3. Desired Compensation Adaptive Robust Control (DCARC) Approach

### 3.1. Controller Design

Consider theZ-direction motion of the platform described as
(7)mz¨+3kfz−z0=u4.Defining the state variablesx1=z and x2=ż, the dynamic equation in (7) can be rewritten in the following state-space form
(8)ẋ1=x2,ẋ2=1θ1u−θ2x1+dx,t,where u=u4 is controller output, θ1=m and θ2=3kf are unknown model parameters, and dx,t is external disturbance and unmodeled dynamics including ground vibration. Define a vector as θ=θ1,θ2T.It is assumed that(9)θi∈Ωθ≜θi:θi,min<θi<θi,max,i=1,2,dx,t≤δ,where θmin=θ1min,θ2minT, θmax=θ1max,θ2maxT, and δ are known.Define an error operator as(10)s=ė+k1e,where e=x1–x1d,x1d is the reference trajectory, and k1 is a positive gain. Differentiating (10) and noting (8), derivative of error can be written as
(11)θ1ṡ=θ1x¨1−x¨1d+k1ė=u−θ2x1+dx,t−θ1x¨1d+θ1k1ė=u+φTθ+dx,t,where φ=k1ė−x¨1d,−x1T, and φd and φ~ are defined as φd=k1ė−x¨1d,−x1dT and φ~=φ−φd. There exists a certain function δφ satisfying
(12)φ~Tθ=φTθ−φdTθ≤δφs.The DCARC control law is designed as(13)u=ua+ur,ua=−φdTθ^,ur=ur1+ur2,ur1=−kss,where ua is the model compensation term, and θ^ is the estimate of θ. The parameter adaptation law is given by
(14)θ^̇=Projθ^Γφds,Projθ^i•i=0,ifθ^i=θimax,•i>0,0,ifθ^i=θimin,•i<0,•i,otherwise,where Γ is a diagonal symmetric positive definite matrix. The projection mapping is a simple output limitation treatment for θ^ and it guarantees
(15)iθ^∈Ωθ≜θ^:θmin≤θ^≤θmax,iiθ~TΓ−1Projθ^Γφds−φds≤0,where θ~=θ^−θ is the estimation error. Due to the reason that using the desired trajectory instead of the measurement output to calculate φ, the computing time and the influence of the measurement noise can be reduced effectively.ur is the deterministic robust control term. ur1 is a linear feedback control to stabilize the nominal system, and the feedback gain ks satisfies
(16)ks≥ks1+δφ,where ks and ks1 are positive; ur2 is a robust feedback term which can reduce the influence of parameter uncertainty and disturbance and should satisfy
(17)isur2≤0,iisur2−φdTθ~+dx,t≤ε,where ε is a small real positive constant which can be designed; ur2 can be chosen as
(18)ur2=−htanhκhεs,where κ=0.2785, h≥θMφd+δ, and θM=θmax–θmin.The DCARC control block diagram is shown in Figure3. The DRC can reduce the influence of disturbance, but it is difficult to realize zero tracking error. The AC can eliminate the influence of parametric uncertainties by using adaptive law to estimate model parameters and realize asymptotic tracing. Adaptive robust control (ARC) which combines DRC and AC features accurate trajectory tracking with parametric uncertainty and high robustness. Furthermore, due to the reason that using the desired trajectory instead of measurement output to calculate regressor, the computing time and the influence of the measurement noise can be reduced effectively.Figure 3
DCARC block diagram.


### 3.2. Stability Analysis

Theorem 1.
For the system in (8), with the DCARC control law (13) and the parameter adaptation law (14), the following results hold
(A)
In general, all signals in system (8) are bounded. Furthermore, the positive definite function Vs is defined as
(19)Vs=12θ1s2,which is bounded above by
(20)Vs≤Vs0exp−λt+ελ1−exp−λt,where λ=2ks1/θ1max.
(B)
If, after a finite time, the system is subjected to parametric uncertainties only (i.e., all the disturbances vanish after a finite time), in addition to results in (A), the tracking error will converge to zero asymptotically, that is,e→0 and s→0 as t→∞.Proof.
DifferentiatingVs results in
(21)V̇s=θ1sṡ=s−kss+ur2−φdTθ^+φTθ+dx,t=−kss2+sφTθ−φdTθ+sur2−φdTθ~+dx,t≤−kss2+δφs2+sur2−φdTθ~+dx,t≤−ks1s2+sur2−φdTθ~+dx,t.
Noting condition ii of (17) and selecting λ=2ks1/θ1max, we have
(22)V̇s≤−ks1s2+ε=−2ks1θ1Vs+ε≤−2ks1θ1maxVs+ε=−λVs+ε,which can further result in (20) and thus proves the results in (A) of Theorem 1.
If the disturbances vanish after a finite time, that is,dx,t=0,∀t≥t0, choose a positive definite function Va as
(23)Va=Vs+12θ~TΓ−1θ~.
Noting condition i of (17) and condition ii of (15), we can obtain
(24)V̇a=V̇s+θ~TΓ−1θ~̇=−kss2+sφTθ−φdTθ+sur2−φdTθ~+θ~TΓ−1θ^̇≤−kss2+δφs2+sur2−φdTθ~+θ~TΓ−1θ^̇≤−ks1s2+sur2+θ~TΓ−1θ^̇−θ~Tφds≤−ks1s2+θ~TΓ−1θ^̇−φds=−ks1s2+θ~TΓ−1Projθ^Γφds−φds≤−ks1s2≤0.
Therefore,Vat≤Va0, that is, Va has an upper bound. Integrating both sides of (24), we can obtain
(25)∫0ts2τdτ=−1ks1∫0tV̇aτdτ=−1ks1Vat−Va0≤1ks1Va0.
Therefore,s∈L2, that is, s is square integrable. Noting ṡ is bounded; s is uniformly continuous. By Barbalat’s lemma, s converges to zero as t→∞, and thus, e→0.


## 3.1. Controller Design

Consider theZ-direction motion of the platform described as
(7)mz¨+3kfz−z0=u4.Defining the state variablesx1=z and x2=ż, the dynamic equation in (7) can be rewritten in the following state-space form
(8)ẋ1=x2,ẋ2=1θ1u−θ2x1+dx,t,where u=u4 is controller output, θ1=m and θ2=3kf are unknown model parameters, and dx,t is external disturbance and unmodeled dynamics including ground vibration. Define a vector as θ=θ1,θ2T.It is assumed that(9)θi∈Ωθ≜θi:θi,min<θi<θi,max,i=1,2,dx,t≤δ,where θmin=θ1min,θ2minT, θmax=θ1max,θ2maxT, and δ are known.Define an error operator as(10)s=ė+k1e,where e=x1–x1d,x1d is the reference trajectory, and k1 is a positive gain. Differentiating (10) and noting (8), derivative of error can be written as
(11)θ1ṡ=θ1x¨1−x¨1d+k1ė=u−θ2x1+dx,t−θ1x¨1d+θ1k1ė=u+φTθ+dx,t,where φ=k1ė−x¨1d,−x1T, and φd and φ~ are defined as φd=k1ė−x¨1d,−x1dT and φ~=φ−φd. There exists a certain function δφ satisfying
(12)φ~Tθ=φTθ−φdTθ≤δφs.The DCARC control law is designed as(13)u=ua+ur,ua=−φdTθ^,ur=ur1+ur2,ur1=−kss,where ua is the model compensation term, and θ^ is the estimate of θ. The parameter adaptation law is given by
(14)θ^̇=Projθ^Γφds,Projθ^i•i=0,ifθ^i=θimax,•i>0,0,ifθ^i=θimin,•i<0,•i,otherwise,where Γ is a diagonal symmetric positive definite matrix. The projection mapping is a simple output limitation treatment for θ^ and it guarantees
(15)iθ^∈Ωθ≜θ^:θmin≤θ^≤θmax,iiθ~TΓ−1Projθ^Γφds−φds≤0,where θ~=θ^−θ is the estimation error. Due to the reason that using the desired trajectory instead of the measurement output to calculate φ, the computing time and the influence of the measurement noise can be reduced effectively.ur is the deterministic robust control term. ur1 is a linear feedback control to stabilize the nominal system, and the feedback gain ks satisfies
(16)ks≥ks1+δφ,where ks and ks1 are positive; ur2 is a robust feedback term which can reduce the influence of parameter uncertainty and disturbance and should satisfy
(17)isur2≤0,iisur2−φdTθ~+dx,t≤ε,where ε is a small real positive constant which can be designed; ur2 can be chosen as
(18)ur2=−htanhκhεs,where κ=0.2785, h≥θMφd+δ, and θM=θmax–θmin.The DCARC control block diagram is shown in Figure3. The DRC can reduce the influence of disturbance, but it is difficult to realize zero tracking error. The AC can eliminate the influence of parametric uncertainties by using adaptive law to estimate model parameters and realize asymptotic tracing. Adaptive robust control (ARC) which combines DRC and AC features accurate trajectory tracking with parametric uncertainty and high robustness. Furthermore, due to the reason that using the desired trajectory instead of measurement output to calculate regressor, the computing time and the influence of the measurement noise can be reduced effectively.Figure 3
DCARC block diagram.


## 3.2. Stability Analysis

Theorem 1.
For the system in (8), with the DCARC control law (13) and the parameter adaptation law (14), the following results hold
(A)
In general, all signals in system (8) are bounded. Furthermore, the positive definite function Vs is defined as
(19)Vs=12θ1s2,which is bounded above by
(20)Vs≤Vs0exp−λt+ελ1−exp−λt,where λ=2ks1/θ1max.
(B)
If, after a finite time, the system is subjected to parametric uncertainties only (i.e., all the disturbances vanish after a finite time), in addition to results in (A), the tracking error will converge to zero asymptotically, that is,e→0 and s→0 as t→∞.Proof.
DifferentiatingVs results in
(21)V̇s=θ1sṡ=s−kss+ur2−φdTθ^+φTθ+dx,t=−kss2+sφTθ−φdTθ+sur2−φdTθ~+dx,t≤−kss2+δφs2+sur2−φdTθ~+dx,t≤−ks1s2+sur2−φdTθ~+dx,t.
Noting condition ii of (17) and selecting λ=2ks1/θ1max, we have
(22)V̇s≤−ks1s2+ε=−2ks1θ1Vs+ε≤−2ks1θ1maxVs+ε=−λVs+ε,which can further result in (20) and thus proves the results in (A) of Theorem 1.
If the disturbances vanish after a finite time, that is,dx,t=0,∀t≥t0, choose a positive definite function Va as
(23)Va=Vs+12θ~TΓ−1θ~.
Noting condition i of (17) and condition ii of (15), we can obtain
(24)V̇a=V̇s+θ~TΓ−1θ~̇=−kss2+sφTθ−φdTθ+sur2−φdTθ~+θ~TΓ−1θ^̇≤−kss2+δφs2+sur2−φdTθ~+θ~TΓ−1θ^̇≤−ks1s2+sur2+θ~TΓ−1θ^̇−θ~Tφds≤−ks1s2+θ~TΓ−1θ^̇−φds=−ks1s2+θ~TΓ−1Projθ^Γφds−φds≤−ks1s2≤0.
Therefore,Vat≤Va0, that is, Va has an upper bound. Integrating both sides of (24), we can obtain
(25)∫0ts2τdτ=−1ks1∫0tV̇aτdτ=−1ks1Vat−Va0≤1ks1Va0.
Therefore,s∈L2, that is, s is square integrable. Noting ṡ is bounded; s is uniformly continuous. By Barbalat’s lemma, s converges to zero as t→∞, and thus, e→0.


## 4. Numerical Simulation

The proposed control strategy is validated by numerical simulation. The common performance indexes of AVIS which are transmissibility, position tracking error, and compliance are applied to evaluate the quality of the control strategy. Besides the DCARC controller, the DRC controller, the AC controller, and the passive system are all simulated for a comparison.The parameters of the AVIS used in the simulation are listed in Table1.Table 1
Parameters of the AVIS.
Parameter
Value
m
1300–1500 kg
Ix
6.2–7.2 kg m2
Iy
1.6–1.8 kg m
Iz
7.7-8.9 kg m2
kf
32–38 kN/m
L1x=L3x
0.935 m
L2x
0.905 m
L1y=L3y
0.305 m
Lh1x=Lh3x=Lh2y
0.291 m
Lh1z=Lh3z
0.519 m
Lh2z
1.119 m
Lv1x=Lv3x
1.148 m
Lv1y=Lv3y
0.305 m
Lv2x
0.905 m
Lv2y
0.213 mThe transmissibility of system (8) with the DCARC controller, DRC controller, and AC controller is simulated as shown in Figure 4. The natural frequency of the system is 1.4 Hz. With the AC controller, the transmissibility is larger than 0 dB when the frequency is lower than 0.7 Hz, which means the vibration from the ground with the frequency lower than 0.7 Hz will be amplified. With the DCARC controller or DRC controller, the transmissibility is less than −12 dB for the entire frequency range. So both the DCARC and DRC controllers can restrain ground vibration effectively.Figure 4
Transmissibility.Figure5 shows the ability of trajectory tracking with the DCARC controller, DRC controller, or AC controller. The desired trajectory is 1 mm, and there is no external disturbance on the platform. With the DRC controller, the tracking error is not zero due to parametric uncertainties. With the AC controller, the tracking error can be close to zero but transient time is long. DCARC can realize zero tracking error and short transient time in theory.Figure 5
Trajectory tracking.The external disturbing force shown in Figure6(a) is applied to the system (8) with the DCARC controller, DRC controller, or AC controller. The displacement response is shown in Figure 6(b). The maximum displacement is 8 μm with the DCARC controller or DRC controller while it is 50 μm with the AC controller. The external disturbance can be restrained effectively with the DCARC controller or DRC controller.Figure 6
Rejection to disturbing force: (a) disturbing force and (b) displacement response.
(a)
(b)
(c)


## 5. Experiment

### 5.1. Experiment Setup

The DCARC control strategy is tested on an AVIS in the laboratory as shown in Figure7. The platform is supported by three active vibration isolators. Each isolator has two VCMs, and each VCM needs one current amplifier which is TA115 in this paper. The gas tanks connected to the isolators are used to increase the volume of the main chamber for stabilizing the pressure. The 941B vibration sensors are used to measure the velocity of the platform and the velocity of ground. The velocity data is collected by INV 3020 data acquisition system and is transmitted to PXI controller for feedback control and also to PC for result display by data acquisition and signal processing (DASP) software. The vertical position of the platform is measured by three eddy current sensors which are fixed on the isolators. The resolution of the eddy current sensor is 1 μm, and the measuring range is 2 mm to 5 mm; the linearity is 0.12% FS. The analog voltage signal from the sensor can be collected and processed by the PXI controller. The vibration on the ground comes from the environment. The disturbance acted on the platform is generated by the interference source. The output force of the interference source is programmable.Figure 7
Experiment setup.


### 5.2. Experiment Result

Measured transmissibility is shown in Figure8. The blue curve shows that the natural frequency of the passive vibration isolation system is about 1.4 Hz, which is close to the simulation result and the magnitude crosses the 0 dB line at 2 Hz. With the AC controller, the vibrations below 0.6 Hz are amplified due to the transmissibility larger than 0 dB. With the DCARC controller or DRC controller, the transmissibility is less than −10 dB for the entire frequency range and the system achieves good vibration isolation performance. By using the proposed DCARC controller, the transmissibility is reduced obviously below 9 Hz.Figure 8
Measured transmissibility.To test the tracking performance, an experiment is carried out. The reference input is 0.1 mm step signal, and the position error is shown in Figure9. With the DRC controller, the tracking error is about 10 μm. With the AC controller, the tracking error is close to zero but transient time is as long as 3 s. With the DCARC controller, the tracking error is nearly zero and the transient time is less than 1 s.Figure 9
Measured position error of a 0.1 mm step reference input.With the external disturbing force, the displacement response and the velocity response of AVIS are tested and shown in Figures10 and 11. With the AC controller, the maximum displacement and velocity of the platform are about 60 μm and about 0.5 mm/s, respectively. In the case of the DCARC controller and DRC controller, the maximum displacement and velocity are 20 μm and less than 0.2 mm/s, respectively. Compared with passive vibration isolation system, the maximum displacement is reduced from 200 μm to 20 μm by using the proposed DCARC controller and the maximum velocity is reduced from 1.2 mm/s to less than 0.2 mm/s.Figure 10
Displacement response to external disturbing force: (a) passive vibration isolation system, (b) AVIS with DCARC controller, (c) AVIS with DRC controller, and (d) AVIS with AC controller.
(a)
(b)
(c)
(d)Figure 11
Velocity response to external disturbing force: (a) passive vibration isolation system, (b) AVIS with DCARC controller, (c) AVIS with DRC controller, and (d) AVIS with AC controller.
(a)
(b)
(c)
(d)


## 5.1. Experiment Setup

The DCARC control strategy is tested on an AVIS in the laboratory as shown in Figure7. The platform is supported by three active vibration isolators. Each isolator has two VCMs, and each VCM needs one current amplifier which is TA115 in this paper. The gas tanks connected to the isolators are used to increase the volume of the main chamber for stabilizing the pressure. The 941B vibration sensors are used to measure the velocity of the platform and the velocity of ground. The velocity data is collected by INV 3020 data acquisition system and is transmitted to PXI controller for feedback control and also to PC for result display by data acquisition and signal processing (DASP) software. The vertical position of the platform is measured by three eddy current sensors which are fixed on the isolators. The resolution of the eddy current sensor is 1 μm, and the measuring range is 2 mm to 5 mm; the linearity is 0.12% FS. The analog voltage signal from the sensor can be collected and processed by the PXI controller. The vibration on the ground comes from the environment. The disturbance acted on the platform is generated by the interference source. The output force of the interference source is programmable.Figure 7
Experiment setup.


## 5.2. Experiment Result

Measured transmissibility is shown in Figure8. The blue curve shows that the natural frequency of the passive vibration isolation system is about 1.4 Hz, which is close to the simulation result and the magnitude crosses the 0 dB line at 2 Hz. With the AC controller, the vibrations below 0.6 Hz are amplified due to the transmissibility larger than 0 dB. With the DCARC controller or DRC controller, the transmissibility is less than −10 dB for the entire frequency range and the system achieves good vibration isolation performance. By using the proposed DCARC controller, the transmissibility is reduced obviously below 9 Hz.Figure 8
Measured transmissibility.To test the tracking performance, an experiment is carried out. The reference input is 0.1 mm step signal, and the position error is shown in Figure9. With the DRC controller, the tracking error is about 10 μm. With the AC controller, the tracking error is close to zero but transient time is as long as 3 s. With the DCARC controller, the tracking error is nearly zero and the transient time is less than 1 s.Figure 9
Measured position error of a 0.1 mm step reference input.With the external disturbing force, the displacement response and the velocity response of AVIS are tested and shown in Figures10 and 11. With the AC controller, the maximum displacement and velocity of the platform are about 60 μm and about 0.5 mm/s, respectively. In the case of the DCARC controller and DRC controller, the maximum displacement and velocity are 20 μm and less than 0.2 mm/s, respectively. Compared with passive vibration isolation system, the maximum displacement is reduced from 200 μm to 20 μm by using the proposed DCARC controller and the maximum velocity is reduced from 1.2 mm/s to less than 0.2 mm/s.Figure 10
Displacement response to external disturbing force: (a) passive vibration isolation system, (b) AVIS with DCARC controller, (c) AVIS with DRC controller, and (d) AVIS with AC controller.
(a)
(b)
(c)
(d)Figure 11
Velocity response to external disturbing force: (a) passive vibration isolation system, (b) AVIS with DCARC controller, (c) AVIS with DRC controller, and (d) AVIS with AC controller.
(a)
(b)
(c)
(d)


## 6. Conclusion

In this paper, a DCARC control strategy is proposed for an AVIS to achieve high-performance vibration isolation as well as low trajectory tracking error for positioning. The following conclusions can be drawn.(1)
The system composition of the AVIS is introduced. The dynamic model of the AVIS is deduced in the vertical three DOFs and is simplified into three SISO systems.
(2)
The DCARC control scheme is proposed where the AC term is designed to estimate the unknown parameters and the DRC term is designed to improve the robustness of the system. Different from the traditional ARC control strategy, desired compensation is employed to reduce the computing time and the influence of the measurement noise.
(3)
The numerical simulation and comparative experiments are carried out under the conditions of using the DCARC, DRC, and AC controllers. The experimental results validate that the proposed DCARC control strategy outperforms other control method and possesses both high-performance vibration isolation and low tracking error.



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*Source: 4136972-2018-07-19.xml*