Active vibration isolation using a six-axis orthogonal vibration isolation platform with piezoelectric actuators

1. Introduction

High resolution equipment and sensitive instruments taken by spacecraft or satellites require really quiet environment [1-4]. The performance of these facilities will be significantly reduced because of the perturbation generated by rotating machineries and compliant mechanisms, such as control moment gyroscope, reaction flywheel and solar wing [5-7]. A common way to deal with this problem is adding vibration isolation system between the vibration source and the precision devices [8-11]. Vibration isolation platform is usually applied because of compact structure and good capability of multi-degree-of-freedom vibration isolation. Nowadays, the passive, semi-active, active are generally applied in platform structure [9, 11-14].

Passive isolator is designed for certain system to achieve optimum isolation property, which owns the characteristic of high reliability, high stability and good high-frequency vibration isolation performance. However, the parameters of passive isolator are unchangeable after designed, which means it may not performs well when the property of control object alters. And furthermore, low-frequency vibration isolation effect is bad [15].

Semi-active vibration isolation technique using magnetorheological fluids, electrorheological fluids and some other smart materials can partly overcome the disadvantages of poor adaptability of passive vibration isolation. The damping coefficient or stiffness coefficient varies with the changing of motion state of control object. However, the low-frequency vibration isolation performance still needs to be improved [16].

Active vibration isolation systems own good low-frequency vibration isolation ability by using sensors and actuators based on different control methods [17]. So far, there have been many researches on actuators and control strategies, which are the core component of active vibration control platforms. Six-axis active vibration isolation platforms can be divided into soft platforms and stiff platforms according to the stiffness of actuators [18]. The resonant frequencies of soft platforms are generally very low, which significantly decrease the burden of control system in the low-frequency band. However, the output forces are usually small. So, they are not suitable for heavy payload vibration control [12, 19, 20]. Actuators used in stiff platforms have the properties of high stiffness and big output force, such as DC motors, magnetostrictive actuators and piezoelectric actuators (PEAs). In comparison, PEAs have been more widely investigated and applied in stiff platforms, because of the advantages of fast response, high output force, high stability, small size and light weight. Lots of platforms based on PEAs have been proposed, such as SEPTRA platform, MAIS platform and ULB piezoelectric Stewart platform [14, 21, 22]. The PEAs are set between the moving base and up-platform. Each PEA transfers the vibration energy from base to up-platform and generates control force under some certain control strategies to make the up-platform stable at the same time. Generally, the Stewart platform is a coupled system, which is difficult to establish controller. There are mainly two methods to deal with this problem [23-26]. One way is configuration design, which simplifies the dynamic behavior of Stewart platform by changing the layouts of actuators, such as the cubic Stewart platform. The other way is control method design, which decouples the dynamic equation of Stewart platform into sets of generalized single-degree-of-freedom subsystem and establishes a SISO controller for each subsystem. Generally, the algorism is very complicated and has to be changed if the configuration of Stewart platform altered.

In this paper, a Stewart platform employs PEAs with symmetric structure is designed to fit the demands of heavy load, large diameter and small installation size. The dynamic model platform with PEAs is established using Newton-Euler method and proven to be able to be decoupled under the proposed conditions. By applying modal superposition method, six SISO subsystems are obtained. Then LQR controller is designed for each SISO system. The verification experiment is conducted to evaluate the effectiveness of the control method. The experimental data indicates the decoupling control of the six-axis orthogonal vibration isolation platform with piezoelectric actuators is effective with the frequency of excitation during the frequency range of 20 Hz to 200 Hz.

2. Design and dynamic modeling of the platform

2.2. Dynamic modeling of the PEA

Assume the stiffness and damping coefficient of each supporting leg are equal and all the PEAs have the uniform property. The simplified dynamic system of the $i$th supporting leg is illustrated in Fig. 2.

Fig. 1Design process of platform and prototype of platform

a) Design process of platform

b) Prototype of platform

Fig. 2Simplified dynamic system of supporting leg based on PEA

The dynamic function of PEA in an active vibration isolation system is given by Eq. (1).

1

$\epsilon =\frac{\sigma }{E_H}+\delta -\frac{c_d}{E_H}\dot{\epsilon }-\frac{\rho l^2}{3E_H}\ddot{\epsilon ,}$

where$\epsilon$, $E_H$,$\delta$,$\sigma$,$c_d$, $\rho$ and $l$ are strain along the axial direction, Young’s modulus, strain caused by input voltage, stress, material damping, density and length. When the base is movable, the force act on the $i$th PEA ($F_i$) can be calculated by Eq. (2):

2

$F_i=\frac{E_H{A}_r}{l}\left(x_i-r_i-{\lambda }_i\right)+\frac{c_d{A}_r}{l}\left({\dot{x}}_i-{\dot{r}}_i\right)+\frac{\rho l{A}_r}{3}{\ddot{x}}_i.$

In which$A_r$, $x_i$, $r_i$ and${\lambda }_i$ are the cross-section area of PZT, the output displacement, the displacement of base and output displacement generated by input voltage. In this paper, the mass of PEA is much less than that of platform. Then Eq. (2) can be written as Eq. (3).

3

$F=\frac{E_H{A}_r}{l}\left(∆l_i-{\lambda }_i\right)+\frac{c_d{A}_r}{l}{\dot{l}}_i,$

where ${∆l}_i=x_i-r_i$ denotes the elongation of PEA, and${\dot{l}}_i={\dot{x}}_i-{\dot{r}}_i$ is the time rate of change of length.

The force acts on the payload ($F_p$) is given in Eq. (4):

4

$F_{pi}=-c{\dot{l}}_i-k{∆l}_i+F_{ri},$

where:

$c=c_l+\frac{c_d{A}_r}{l},k=k_l+\frac{E_H{A}_r}{l},F_{ri}=\frac{E_H{A}_r}{l}{\lambda }_i.$

In which $k$, $c$ and $F_{ri}$ are the equivalent damping, stiffness coefficient and generalized control force related to hysteresis displacement generated by input voltage. In the author’s former work, the nonlinear model of hysteresis displacement of$\lambda$ and feedforward-feedback control method is researched which will not be discussed here.

2.3. Dynamic modeling of platform

The coordinate system of the proposed platform is shown in Fig. 3. The moving coordinate frame $\left\{P\right\}$ is located at and attached to the up-platform’s center of mass. $Z$-axis is set along the centerline of cylinder shell structure. The direction of $X$-axis is along the symmetry of the platform. $\left\{B\right\}$ is a coordinate system attached to the moving base. The center of $\left\{B\right\}$ coincides with the mass center of base. $\left\{O\right\}$ is the global inertial coordinate system. In order to make the kinetic equation in a simple style, the direction of axis of $\left\{B\right\}$ and $\left\{U\right\}$ is set to be the same as $\left\{P\right\}$. The 6 supporting legs are numbered as illustrated in Fig. 3.

Fig. 3Coordinate system of the platform

Define$F={\left[\begin{array}{ccc}{F}_x& F_y& \begin{array}{ccc}{F}_z& M_x& \begin{array}{cc}{M}_y& M_Z\end{array}\end{array}\end{array}\right]}^T$ as the force acted on the payload through supporting legs and $F_p={\left[\begin{array}{ccc}{F}_{p1}& F_{p2}& \begin{array}{cc}\cdots & F_{p6}\end{array}\end{array}\right]}^T$ as the matrix of actuator forces then:

5

$F=J{F}_p,$

where $J$ is the Jacobi matrix of the platform as given in Eq. (6):

6

$J=\left[\begin{array}{lll}{\overrightarrow{\tau }}_1& {\overrightarrow{\tau }}_2& \cdots \\ {\overrightarrow{p}}_1\times {\overrightarrow{\tau }}_1& {\overrightarrow{p}}_2\times {\overrightarrow{\tau }}_2& \cdots \end{array}\begin{array}{l}{\overrightarrow{\tau }}_6\\ {\overrightarrow{p}}_6\times {\overrightarrow{\tau }}_6\end{array}\right].$

In which ${\overrightarrow{\tau }}_i$ is the unit vector of supporting leg vector ${\overrightarrow{l}}_i$ and ${\overrightarrow{p}}_i$ is the position vector described in $\left\{O\right\}$ ($i=$ 1, 2, …, 6). Substitute Eq. (5-6) into (4), we can get:

7

$F=-cJ\dot{L}-kJ∆L+J{F}_r,$

where $\dot{L}={\left[\begin{array}{ccc}{\dot{l}}_1& {\dot{l}}_2& \begin{array}{cc}\cdots & {\dot{l}}_6\end{array}\end{array}\right]}^T$ is the matrix of varying speed of lengths, $∆L={\left[\begin{array}{ccc}{∆l}_1& {∆l}_2& \begin{array}{cc}\cdots & {∆l}_6\end{array}\end{array}\right]}^T$ is the set of elongation of PEAs and $F_r={\left[\begin{array}{ccc}{F}_{r1}& F_{r2}& \begin{array}{cc}\cdots & F_{r6}\end{array}\end{array}\right]}^T$ denotes the general control forces of PEAs. According to Fig. 3 and theorem of vector operation, the following Eq. (8) and Eq. (9) can be obtained:

8

${\overrightarrow{l}}_i=\overrightarrow{OP}-\overrightarrow{OB}+{\overrightarrow{p}}_i-{\overrightarrow{b}}_i,$

9

$\dot{{\overrightarrow{l}}_i}=\dot{\overrightarrow{OP}}-\dot{\overrightarrow{OB}}+{\omega }_P\times {\overrightarrow{p}}_i-{\omega }_B\times {\overrightarrow{b}}_i.$

Let $X_P={\left[\begin{array}{ccc}{x}_P& y_P& \begin{array}{ccc}{z}_P& {\theta }_{xP}& \begin{array}{cc}{\theta }_{yP}& {\theta }_{zP}\end{array}\end{array}\end{array}\right]}^T$and$X_B={\left[\begin{array}{ccc}{x}_B& y_B& \begin{array}{ccc}{z}_B& {\theta }_{xB}& \begin{array}{cc}{\theta }_{yB}& {\theta }_{zB}\end{array}\end{array}\end{array}\right]}^T$ represent the attitude matrix of the payload and base, respectively, then Eq. (10) is deduced by ${\overrightarrow{\tau }}_i\bullet$Eq. (9):

10

$\dot{L}=J^T{\dot{X}}_P-J_B^T{\dot{X}}_B,$

where:

11

$J_B=\left[\begin{array}{lll}{\overrightarrow{\tau }}_1& {\overrightarrow{\tau }}_2& \cdots \\ {\overrightarrow{b}}_1\times {\overrightarrow{\tau }}_1& {\overrightarrow{b}}_2\times {\overrightarrow{\tau }}_2& \cdots \end{array}\begin{array}{l}{\overrightarrow{\tau }}_6\\ {\overrightarrow{b}}_6\times {\overrightarrow{\tau }}_6\end{array}\right],$

when the motion is in the case of small vibration condition, the elongation vector of actuators can be expressed by Eq. (12):

12

$∆L\approx J^T{X}_P-J_B^T{X}_B.$

Substituting Eq. (10) and Eq. (12) into Eq. (7), the force matrix can be written as Eq. (13):

13

$F=-cJ{J}^T{\dot{X}}_P+cJ{J}_B^T{\dot{X}}_B-kJ{J}^T{X}_P+kJ{J}_B^T{X}_B+J{F}_r.$

According to Newton-Euler formulation, the kinetic equation of the platform is:

14

$F=M{\ddot{X}}_P+\left[\begin{array}{l}{0}_{3\times 1}\\ {\omega }_P\times I_P{\omega }_P\end{array}\right],$

where $M=\left[\begin{array}{cc}m{I}_3& 0_{3\times 3}\\ 0_{3\times 3}& I_P\end{array}\right]$ denotes the mass matrix of payload.

As the center of $\left\{P\right\}$ coincides with the payload mass center and all the directions of axis are along the symmetries axis of cylinder structure, therefore $I_P$ is in the diagonal form and given as$I_p=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\mathrm{}\left\{\mathrm{}{I}_x{I}_y{I}_z\right\}$. Thus $M$ is a diagonal matrix. By assuming the small motion condition, the quadratic item of speed can be neglected. Let $J{F}_r=\widehat{F}={\left[\begin{array}{ccc}{f}_x& f_y& f_z\end{array}\begin{array}{ccc}{f}_{\alpha }& f_{\beta }& f_{\gamma }\end{array}\right]}^T$ to be the generalized control force. Then the actual desired control output is:

15

$F_r=J^{-}\widehat{F}.$

In which$J^{-}$is the inverse matrix of $J$. Thus, the kinetic equation is written as Eq. (16) by combining Eq. (13) and (14):

16

$M{\ddot{X}}_P+cJ{J}^T{\dot{X}}_P+kJ{J}^T{X}_P=cJ{J}_B^T{\dot{X}}_B+kJ{J}_B^T{X}_B+\widehat{F}.$

3. Coupled property analysis of platform

In order to simplify the proof procedure, the parameters angel $\alpha$,$\beta$ and radius $r$ and $R$ are defined as Fig. 4.

Fig. 4Schematic diagram of parameters

Define matrix $P_i$ and $B_i$. The $i$th column of $P_i$ and $B_i$ denote ${\overrightarrow{p}}_i$ and ${\overrightarrow{b}}_i$, respectively:

In which $h_C$ is the distance between the mass center of payload and the plane to which the up part of legs are attached.

It’s obvious that ${\overrightarrow{\tau }}_i={\left(\begin{array}{ccc}{\tau }_{1i}& {\tau }_{2i}& {\tau }_{3i}\end{array}\right)}^T$ equals to the $i$th column of P-B divided by the length of leg ($l$). As all the legs are assumed to be the same, they have the equal ${\tau }_{3i}$ ($i=$ 1, 2, …, 6). Let the angel between the shadow of the first leg in the $x$-$y$ plane and $x$-axis to be$\gamma$ and the length of the shadow is $\delta$. Define matrix $\tau$, in which the $i$th column denotes ${\overrightarrow{\tau }}_i$:

In which $l$ denotes the length of supporting leg and $h$ is the projection of $l$ in the $z$-axis direction.

4. Decoupling control of the platform

As mentioned before, $M$ is a diagonal matrix. With the derivation got from chapter 3, it can be concluded that the translation motion along $z$-axis and the rotation around $z$-axis are SISO dynamic systems, while the translation motion in the $x$-axis and the rotation around $y$-axis compose a MIMO dynamic system. The motion along the $y$-axis and the rotation around $x$-axis are in the same situation.

The controllers for two SISO systems can be designed separately. As to the MIMO system, the modal superposition method is adopted to make them in decoupling form [27].

The kinetic equation of the $x-{\theta }_y$ direction is:

where:

As $M_{x{\theta }_y}$ and $K_{x{\theta }_y}$ (or $C_{x{\theta }_y}$) are symmetric matrixes, there exists a two-dimensional matrix $\mathrm{\Phi }$ which satisfies:

In which, ${\lambda }_1$ and ${\lambda }_2$ are the generalized eigenvalues of $M_{x{\theta }_y}$ and $K_{x{\theta }_y}$ (or $C_{x{\theta }_y}$), and the column of$\mathrm{\Phi }$are the eigenvectors corresponding to ${\lambda }_1$ and ${\lambda }_2$. $I_2$ is a 2×2 unit matrix.

Let$X_{P1}=\mathrm{\Phi }\xi$, then substitute it into Eq. (37) and multiply both sides of Eq. (37) with ${\mathrm{\Phi }}^T$. Define ${\stackrel{̿}{F}}_{x{\theta }_y}={\mathrm{\Phi }}^T{\widehat{F}}_{x{\theta }_y}$ as the new generalized control inputs. Thus Eq. (39) is obtained:

Obviously, parameter matrixes are all diagonal. As a result, the system with ${\stackrel{̿}{F}}_{x{\theta }_y}$ as input and $\xi$ is decoupled. The processing method of $y-{\theta }_x$ subsystem is discussed in the same way.

With the decoupling algorithm proposed, the dynamic system is translated into six independent SISO systems. The vibration controller for each system is designed separately with LQR method to suppress the vibration response [28]. Let’s take the $z$-axis SISO system as example. Define $x\left(t\right)$ as the state variables and given as$x\left(t\right)={\left[\begin{array}{cc}{z}_P& \dot{z_P}\end{array}\right]}^T$, $u\left(t\right)$ as the desired control input that is equal to $f_z$. The state equation of vibration along z direction can be written as Eq. (40):

where $w\left(t\right)={\left[\begin{array}{cc}{z}_B& \dot{z_B}\end{array}\right]}^T$ is the perturbation vector. $y\left(t\right)$ is the output which is chosen as ${\ddot{z}}_P$. The matrixes in Eq. (40) are listed below:

The objective of control system is to minimize acceleration response of isolated object. The LQR coefficients are obtained by minimizing the cost function given in Eq. (41):

where $Q$ is the weighting matrix of state variable which is symmetric positive semi-definite, $R$ is the symmetric positive definite weighting matrix of input signal.

The control signal is in the form of Eq. (42):

where $K={\left[\begin{array}{cc}{K}_1& K_2\end{array}\right]}^T$ is a vector obtained by solving the Riccati equations given as Eq. (43) and Eq. (44):

During the parameters adjusting procedure of the LQR controller, $Q$ and $R$ are regulated separately according to the control performance. For the dynamic system established in this paper, $Q$ is a diagonal matrix and$Q=diag\left(\left[\begin{array}{cc}{Q}_1& Q_2\end{array}\right]\right)$. Usually, the regulation method of $Q$is $Q=Q_0+qB{B}^T$, where$Q_0$ is the initial real matrix and $q$ is the gain coefficient. $R$ is a real diagonal matrix. In this paper, $R$ is set as one.

If the base vibrates in the $z$ direction, the acceleration transmissibility of payload along $z$-axis is illustrated in Fig. 5, where the dashed line is the result for passive isolation and the solid line is the transmissibility when the active controller is applied. It can be observed that the active control method proposed significantly improves the dynamic characteristic of the platform. The vibration at the resonant frequency is attenuated by more than 30 dB. During the low-frequency band below 82 Hz, the controller effectively reduces the energy transfer from base to the payload, which is impossible for the traditional passive isolation. While in the high-frequency range, the vibration isolation effects of the two methods get close to each other. It’s worth noting that the vibration isolation effect is still improved by nearly 5 dB before 200 Hz.

Fig. 5Acceleration transmissibility of the platform in z-axis

When the excitement on the base is in the $x$-axis direction, as analyzed above, there is only $x$-axis direction vibration response and rotational response around $y$-axis. Fig. 6(a) is the $x$-axis acceleration transmissibility of the payload. The dashed line is the passive isolation result which indicates that the platform owns two resonant frequencies at 18 Hz and 95 Hz, and an anti-resonant frequency at 38 Hz. The acceleration response presents the attenuation characteristics except the frequency bands near resonant frequencies.

With the active controller working, the $x$-axis acceleration transmissibility is shown as solid line. It can be found that the control method makes the vibration isolation performance of the platform worse at the frequency nearby anti-resonant peak. But during the other frequency range, the controller can achieve a significant performance increase. Fig. 6(b) is the coupling vibration response around y-axis. By comparing the result, the controller proposed decreases acceleration transmissibility by average of more than 5dB when the excitation frequency is above 10 Hz.