Structural parameter design method for a fast-steering mirror based on a closed-loop bandwidth

Abstract

When a fast-steering mirror (FSM) system is designed, satisfying the performance requirements before fabrication and assembly is vital. This study proposes a structural parameter design approach for an FSM system based on the quantitative analysis of the required closed-loop bandwidth. First, the open-loop transfer function of the FSM system is derived. In accordance with the transfer function, the notch filter and proportional-integral (PI) feedback controller are designed as a closed-loop controller. The gains of the PI controller are determined by maximizing the closed-loop bandwidth while ensuring the robustness of the system. Then, the two unknown variables of rotational radius and stiffness in the open-loop transfer function are optimized, considering the bandwidth as a constraint condition. Finally, the structural parameters of the stage are determined on the basis of the optimized results of rotational radius and stiffness. Simulations are conducted to verify the theoretical analysis. A prototype of the FSM system is fabricated, and corresponding experimental tests are conducted. Experimental results indicate that the bandwidth of the proposed FSM system is 117.6 Hz, which satisfies the minimum bandwidth requirement of 100 Hz.

Keywords

fast-steering mirror / structural parameter / PI controller / bandwidth / notch filter

Introduction

A fast-steering mirror (FSM) is a rotational positioning stage that plays an important role in optical beam control systems to achieve pointing, tracking, and acquisition [¹]. FSMs have been widely used in many fields, such as imaging [²,³] and laser processing systems [⁴], free-space optical communications [⁵,⁶], and laser weaponry [⁷,⁸]. With a high bandwidth, FSMs are capable of fast response and good disturbance rejection [⁹]. However, a high bandwidth typically requires a rigid elastic support [¹⁰,¹¹]. A large actuator output force, which is costly and difficult to implement, is desirable for certain strokes. Moreover, the FSM is generally designed for a target application for which the bandwidth requirement is specific. If the FSM’s structural parameters are poorly designed, the final bandwidth may fail to satisfy the practical requirements of the target application. Therefore, developing a structural parameter design approach, which can ensure that the FSM can satisfy the bandwidth requirement for a given target application, is crucial.

Many design approaches have been developed to achieve this goal. Kluk et al. [⁹] and Csencsics et al. [¹²] designed FSM mechanical systems, ensuring low stiffness in the motion direction to enable a large travel range and placing structural mode well above the crossover frequency to maximize the bandwidth. Yuan et al. [¹³] determined the structural parameters of an FSM by setting the designed stroke slightly larger than what was required and making the resonant frequency coincide with the expectation. However, mechanical structures in these studies were designed on the basis of the qualitative analysis of the open-loop frequency characteristic. Whether or not resultant FSMs will satisfy the bandwidth requirement remains unknown until they are fabricated and assembled. In contrast to the aforementioned open-loop approach, Lu et al. [¹¹] developed an FSM system based on the desired closed-loop bandwidth. However, the suspension mode frequency was simply set to be 0.25–0.5 times of the closed-loop bandwidth. Given that the final bandwidth is affected by the structure, load, and closed-loop control strategy, this design approach is inaccurate. Thus, designing FSM systems based on the quantitative analysis of the required closed-loop bandwidth is rare.

As can be inferred from previous strategies aimed at addressing bandwidth challenges, the two major determinants of final bandwidth are the open-loop transfer function of the stage and the corresponding closed-loop control strategy. The open-loop transfer function is determined by the structural parameters and load of the stage. As the resonant modes substantially limit bandwidth, the control strategy is developed to eliminate vibration effects [¹⁴]. Effective methods to eliminate resonant modes mainly include the notch filter [¹⁵,¹⁶] and feedforward [¹⁷,¹⁸], input shaping [¹⁹], and integral resonant controllers [²⁰]. Feedback controllers are designed to compensate for modeling uncertainties and disturbance. The proportional-integral (PI) controller is widely used as a feedback controller owing to its robustness to modeling errors and simplicity of implementation [²¹]. Therefore, to conduct the quantitative analysis of the bandwidth, the open-loop transfer function, resonant modes’ damping methods, and feedback controller should all be considered.

In this study, a novel approach to the FSM system design is proposed. Structural parameters are designed on the basis of the quantitative analysis of the required closed-loop bandwidth. The bandwidth is derived from the theoretical analysis, with consideration of the open-loop transfer function and the closed-loop control strategy. With the bandwidth considered a constraint condition, structural parameters of the FSM are optimized to ensure that the final system can satisfy the bandwidth requirement. The rest of this paper is organized as follows: Section 2 presents the mechanical design of the FSM; Section 3 presents the open-loop transfer function of the stage and the closed-loop control strategy; Section 4 determines the structural parameters; Sections 5 and 6 detail the conducted simulations and experimental testing, respectively, finally, Section 7 concludes the study.

Mechanical design

In this study, the proposed FSM is applied in a laser weapon system to compensate for base vibration. The vibration of the base has low frequency (usually less than 100 Hz) and large magnitude. Therefore, the required bandwidth of the proposed FSM system is set at 100 Hz. The corresponding elliptical mirror has the dimensions Ø100 mm×Ø70.7 mm×15 mm (long axis×short axis×height). The mechanical structure is composed of a mirror, a monolithic compliant mechanism (MCM), and a base, as shown in Fig. 1. The connected link is designed to connect the MCM and the voice coil actuator (VCA). The stage consists of four VCAs, with two sets of two VCAs arranged in a push-pull configuration to rotate the stage. The MCM is composed of a rigid frame and four elastic kinematic chains, as shown in Fig. 2(a). The elastic kinematic chain is composed of four translational plates, a rotational plate, and a decoupled plate, as shown in Fig. 2(b). The translational plate is designed to achieve large translational motion along the z-axis. The rotational plate is designed to convert the translational motion of the VCA into the rotational motion of the FSM. The decoupled plate is used to minimize the transverse displacement of the coil and avoid any undesirable contact between the stator and the coil [²²].

Transfer function of the FSM

Given that the mirror is elliptical, the inertial moment along {\theta }_x-axis is larger than that along {\theta }_y-axis, as shown in Fig. 1. When the same control strategy is applied, the bandwidth of the {\theta }_x-axis is smaller than that of the {\theta }_y-axis. Therefore, a theoretical analysis is conducted to study the characteristics of the {\theta }_x-axis.

Open-loop transfer function of the stage

To achieve closed-loop control, deriving the transfer function of the FSM is necessary [²³]. The Laplace transform of the dynamic model of the FSM is written as follows:

(1)\mathrm{2}r{k}_{i}i(s)-K\theta (s)=s^{\mathrm{2}}J\theta (s),

where s denotes the Laplace operator, r and K denote the rotational radius and the rotational stiffness of the FSM, respectively, k_i and i denote the force constant of the VCA and the current of the coil, respectively, and J and q denote the inertial moment and rotational angle of the FSM, respectively.

According to Lagrange’s equation, the inertial moment of the stage is written as follows:

(2)J=\mathrm{2}{r}^{\mathrm{2}}(m_{\mathrm{1}}+m_{\mathrm{2}})+J_{\mathrm{1}},

where m_{\mathrm{1}} and m_{\mathrm{2}} denote the mass of the VCA and connected link, respectively, and J_{\mathrm{1}} denotes the inertial moment of the mirror.

The function of the electrical model of the VCA is derived from Kirchhoff’s law. The corresponding Laplace transform is written as follows:

(3)U(s)=(R+Ls)i(s)+k_{i}r\theta (s)s,

where U denotes the input voltage of the VCA, and R and L are the resistance and inductance of the VCA, respectively.

In accordance with Eqs. (1)–(3), the transfer function (G(s)) of the FSM is derived as follows:

(4)G(s)={\displaystyle \frac{\theta (s)}{U(s)}}={\displaystyle \frac{\mathrm{20.2}r}{B_{\mathrm{1}}{s}^{\mathrm{3}}+B_{\mathrm{2}}{s}^{\mathrm{2}}+B_{\mathrm{3}}s+\mathrm{10.1}K}},

where

\begin{array}{l}{B}_{\mathrm{1}}=\mathrm{4.26}\times {\mathrm{10}}^{-\mathrm{4}}{r}^{\mathrm{2}}+\mathrm{6.62}\times {\mathrm{10}}^{-\mathrm{7}},\\ B_{\mathrm{2}}=\mathrm{1.16}{r}^{\mathrm{2}}+\mathrm{1.81}\times {\mathrm{10}}^{-\mathrm{3}},\\ B_{\mathrm{3}}=\mathrm{204.02}{r}^{\mathrm{2}}+\mathrm{3.7}\times {\mathrm{10}}^{-\mathrm{3}}K.\end{array}

Closed-loop control strategy

As shown in Fig. 3, the closed-loop control system is composed of the notch filter, PI controller and the transfer function of the FSM.

As discussed in Section 3.1, the open-loop transfer function of the FSM is a third-order system, which contains a second-order oscillation link and a first-order inertial link. The resonant modes with low damping ratio result in low gain margins that limit the application of the high-gain feedback controllers [¹⁶]. Thus, a notch filter is generally used to suppress resonant behavior to improve the high-bandwidth performance of the closed-loop system. The transfer function of the notch filter (N(s)) is defined as follows [²¹]:

(5)N(s)={\displaystyle \frac{s^{\mathrm{2}}+\mathrm{2}\zeta {\omega }_{\mathrm{0}}s+{{\omega }_{\mathrm{0}}}^{\mathrm{2}}}{s^{\mathrm{2}}+\mathrm{2}{\omega }_{\mathrm{0}}s+{{\omega }_{\mathrm{0}}}^{\mathrm{2}}}},

where \zeta and {\omega }_{\mathrm{0}} denote the damping ratio and the resonance frequency of the second-order oscillation link, respectively.

A PI controller is widely used due to its robustness to modeling errors and simplicity of implementation. The PI controller transfer function (C(s)) is written as follows:

(6)C(s)=k_p+{\displaystyle \frac{k_i}{s}},

where k_p and k_i denote proportional gain and integral gain, respectively.

Calculation of the bandwidth

As the open-loop transfer function of the FSM and corresponding notch filter are determined, the bandwidth of the closed-loop control system depends on the design of the PI controller. Given that the bandwidth varies substantially according to the chosen parameters of the PI controller, obtaining the optimal parameters of the PI controller is necessary to maximize the bandwidth.

The optimization process is conducted under two constraint conditions. First, the gain and phase margins are set to be larger than 3 and 60°, respectively, to guarantee the robustness of the system [²⁴,²⁵]. Second, the settling time (2% tolerance) is set to be less than 20 ms to guarantee the time domain performance. Then, the maximum bandwidth of the closed-loop control system can be obtained using the iteration method. The method is explained as follows: k_p and k_i are increased incrementally until the stability margin of the closed-loop control system is out of scope. Various values for k_pand k_i, which satisfy the two constraint conditions, are obtained, and the corresponding closed-loop bandwidth is calculated using MATLAB software. Thus, the maximum bandwidth with optimal parameters for the PI controller can be obtained. The process is shown in Fig. 4.

Design of structural parameters

Determination of rotational radius and stiffness

As discussed in Section 3.1, two variables (stiffness K and rotational radius r) are unknown in the open-loop transfer function of the FSM. The effects of each variable on the bandwidth of the closed-loop control system are shown in Fig. 5.

The bandwidth increases with the increase in K (Fig. 5(a)), whereas the bandwidth decreases with the increase in r (Fig. 5(b)). Therefore, the bandwidth of the closed-loop control system can be improved by setting a relatively large K and relatively small r.

Parameters of variables r and K are obtained through an optimization process. The constrained objective function is listed as follows:

1) The objective of optimization is to guarantee that the bandwidth of the closed-loop control system will be larger than 100 Hz.

2) Parameter ranges: 25 mm≤r≤45 mm; 30 N·m/rad≤K≤60 N·m/rad.

The optimization results are shown as follows:

r=\mathrm{33}\mathrm{m}\mathrm{m},K=\mathrm{50}\mathrm{N}\cdot \mathrm{m}/\mathrm{r}\mathrm{a}\mathrm{d}.

Static modeling

Analytical static modeling is conducted using the compliance matrix method to obtain the relationship between the structural parameters and rotational stiffness of the FSM [²⁶,²⁷]. For a flexure hinge, the compliance matrices in the coordinate frames o_i-xyz and o_j-xyz are {\mathit{C}}_i and {\mathit{C}}_j, respectively. The coordinate transformation can be written as

(7){\mathit{C}}_j={\mathit{R}}_i(\theta )\mathit{P}({\mathit{r}}_i){\mathit{C}}_i\mathit{P}{({\mathit{r}}_i)}^{\mathrm{T}}{\mathit{R}}_i{(\theta )}^{\mathrm{T}},

where \mathit{P}(r_i) is the translational transformation matrix, {\mathit{r}}_i={[x,y,z]}^{\mathrm{T}}is the position coordinate of the point o_j in the reference frame o_i-xyz, {\mathit{R}}_i(\theta ) is the rotational transformation matrix, and \theta is the rotational angle about the i-axis.

As shown in Fig. 6, the compliance matrices of the translational and decoupled plates are defined as {\mathit{C}}_{\mathrm{1}} and {\mathit{C}}_{\mathrm{2}} in coordinate frames o_t-xyz and o_d-xyz, respectively.

(8){\mathit{C}}_{\mathrm{1}}=\left[\begin{array}{llllll}{c}_{\mathrm{11}}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& c_{\mathrm{12}}& \mathrm{0}& \mathrm{0}& \mathrm{0}& c_{\mathrm{13}}\\ \mathrm{0}& \mathrm{0}& c_{\mathrm{14}}& \mathrm{0}& -c_{\mathrm{15}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& c_{\mathrm{16}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -c_{\mathrm{15}}& \mathrm{0}& c_{\mathrm{17}}& \mathrm{0}\\ \mathrm{0}& c_{\mathrm{13}}& \mathrm{0}& \mathrm{0}& \mathrm{0}& c_{\mathrm{18}}\end{array}\right],

(9){\mathit{C}}_{\mathrm{2}}=\left[\begin{array}{llllll}{c}_{\mathrm{21}}& \mathrm{0}& \mathrm{0}& \mathrm{0}& c_{\mathrm{22}}& \mathrm{0}\\ \mathrm{0}& c_{\mathrm{23}}& \mathrm{0}& -c_{\mathrm{24}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& c_{\mathrm{25}}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -c_{\mathrm{24}}& \mathrm{0}& c_{\mathrm{26}}& \mathrm{0}& \mathrm{0}\\ c_{\mathrm{22}}& \mathrm{0}& \mathrm{0}& \mathrm{0}& c_{\mathrm{27}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& c_{\mathrm{28}}\end{array}\right],

where c_{\mathrm{11}}={\displaystyle \frac{L_{\mathrm{1}}}{E{t}_{\mathrm{1}}{b}_{\mathrm{1}}}}, c_{\mathrm{12}}={\displaystyle \frac{\mathrm{4}{L}_{\mathrm{1}}^{\mathrm{3}}}{E{b}_{\mathrm{1}}^{\mathrm{3}}{t}_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1.175}{L}_{\mathrm{1}}}{G{t}_{\mathrm{1}}{b}_{\mathrm{1}}}}, c_{\mathrm{13}}={\displaystyle \frac{\mathrm{6}{L}_{\mathrm{1}}^{\mathrm{2}}}{E{b}_{\mathrm{1}}^{\mathrm{3}}{t}_{\mathrm{1}}}}, c_{\mathrm{14}}={\displaystyle \frac{\mathrm{4}{L}_{\mathrm{1}}^{\mathrm{3}}}{E{b}_{\mathrm{1}}{t}_{\mathrm{1}}^{\mathrm{3}}}}+{\displaystyle \frac{\mathrm{1.175}{L}_{\mathrm{1}}}{G{t}_{\mathrm{1}}{b}_{\mathrm{1}}}}, c_{\mathrm{15}}={\displaystyle \frac{\mathrm{6}{L}_{\mathrm{1}}^{\mathrm{2}}}{E{b}_{\mathrm{1}}{t}_{\mathrm{1}}^{\mathrm{3}}}}, c_{\mathrm{16}}={\displaystyle \frac{L_{\mathrm{1}}}{Gk{b}_{\mathrm{1}}{t}_{\mathrm{1}}^{\mathrm{3}}}}, c_{\mathrm{17}}={\displaystyle \frac{\mathrm{12}{L}_{\mathrm{1}}}{E{b}_{\mathrm{1}}{t}_{\mathrm{1}}^{\mathrm{3}}}}, c_{\mathrm{18}}={\displaystyle \frac{\mathrm{12}{L}_{\mathrm{1}}}{E{b}_{\mathrm{1}}^{\mathrm{3}}{t}_{\mathrm{1}}}}, c_{\mathrm{21}}={\displaystyle \frac{\mathrm{4}{L}_{\mathrm{2}}^{\mathrm{3}}}{E{b}_{\mathrm{2}}^{\mathrm{3}}{t}_{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{1.175}{L}_{\mathrm{2}}}{G{t}_{\mathrm{2}}{b}_{\mathrm{2}}}}, c_{\mathrm{22}}={\displaystyle \frac{\mathrm{6}{L}_{\mathrm{2}}^{\mathrm{2}}}{E{b}_{\mathrm{2}}^{\mathrm{3}}{t}_{\mathrm{2}}}}, c_{\mathrm{23}}={\displaystyle \frac{\mathrm{4}{L}_{\mathrm{2}}^{\mathrm{3}}}{E{b}_{\mathrm{2}}{t}_{\mathrm{2}}^{\mathrm{3}}}}+{\displaystyle \frac{\mathrm{1.175}{L}_{\mathrm{2}}}{G{t}_{\mathrm{2}}{b}_{\mathrm{2}}}}, c_{\mathrm{24}}={\displaystyle \frac{\mathrm{6}{L}_{\mathrm{2}}^{\mathrm{2}}}{E{b}_{\mathrm{2}}{t}_{\mathrm{2}}^{\mathrm{3}}}}, c_{\mathrm{25}}={\displaystyle \frac{L_{\mathrm{2}}}{E{t}_{\mathrm{2}}{b}_{\mathrm{2}}}}, c_{\mathrm{26}}={\displaystyle \frac{\mathrm{12}{L}_{\mathrm{2}}}{E{b}_{\mathrm{2}}{t}_{\mathrm{2}}^{\mathrm{3}}}}, c_{\mathrm{27}}={\displaystyle \frac{\mathrm{12}{L}_{\mathrm{2}}}{E{b}_{\mathrm{2}}^{\mathrm{3}}{t}_{\mathrm{2}}}}, and c_{\mathrm{28}}={\displaystyle \frac{L_{\mathrm{2}}}{Gk{b}_{\mathrm{2}}{t}_{\mathrm{2}}^{\mathrm{3}}}}. Here, E is the Young’s modulus, and G denotes the shear modulus.

The compound basic parallelogram mechanism (CBPM) is composed of four translational plates, which are defined as flexure hinge 1, 2, 3, and 4, as shown in Fig. 7(a). The compliance matrices of the four translational plates in coordinate frame o_{\mathrm{5}}-xyz are derived as follows:

(10){\mathit{C}}_{o_{\mathrm{1}}}^{o_{\mathrm{5}}}=\mathit{P}({\mathit{r}}_{\mathrm{1}}){\mathit{C}}_{\mathrm{1}}\mathit{P}{({\mathit{r}}_{\mathrm{1}})}^{\mathrm{T}},

(11){\mathit{C}}_{o_{\mathrm{2}}}^{o_{\mathrm{5}}}={\mathit{R}}_x(\mathrm{\pi }){\mathit{C}}_{o_{\mathrm{1}}}^{o_{\mathrm{5}}}{\mathit{R}}_x{(\mathrm{\pi })}^{\mathrm{T}}={\mathit{R}}_x(\mathrm{\pi })\mathit{P}({\mathit{r}}_{\mathrm{1}}){\mathit{C}}_{\mathrm{1}}\mathit{P}{({\mathit{r}}_{\mathrm{1}})}^{\mathrm{T}}{\mathit{R}}_x{(\mathrm{\pi })}^{\mathrm{T}},

(12){\mathit{C}}_{o_{\mathrm{3}}}^{o_{\mathrm{5}}}={\mathit{R}}_z(\mathrm{\pi }){\mathit{C}}_{o_{\mathrm{1}}}^{o_{\mathrm{5}}}{\mathit{R}}_z{(\mathrm{\pi })}^{\mathrm{T}}={\mathit{R}}_z(\mathrm{\pi })\mathit{P}({\mathit{r}}_{\mathrm{1}}){\mathit{C}}_{\mathrm{1}}\mathit{P}{({\mathit{r}}_{\mathrm{1}})}^{\mathrm{T}}{\mathit{R}}_z{(\mathrm{\pi })}^{\mathrm{T}},

(13){\mathit{C}}_{o_{\mathrm{4}}}^{o_{\mathrm{5}}}={\mathit{R}}_x(\mathrm{\pi }){\mathit{C}}_{o_{\mathrm{3}}}^{o_{\mathrm{5}}}{\mathit{R}}_x{(\mathrm{\pi })}^{\mathrm{T}}={\mathit{R}}_x(\mathrm{\pi }){\mathit{R}}_z(\mathrm{\pi })\mathit{P}({\mathit{r}}_{\mathrm{1}}){\mathit{C}}_{\mathrm{1}}\mathit{P}{({\mathit{r}}_{\mathrm{1}})}^{\mathrm{T}}{\mathit{R}}_z{(\mathrm{\pi })}^T{\mathit{R}}_x{(\mathrm{\pi })}^{\mathrm{T}},

where {\mathit{r}}_{\mathrm{1}}={[m/\mathrm{2},\mathrm{0},-n/\mathrm{2}]}^{\mathrm{T}}.

The four translational plates are arranged in parallel. Thus, the compliance matrix of the CBPM in the coordinate frame o_{\mathrm{5}}-xyz is written as follows:

(14){\mathit{C}}_{o_{\mathrm{5}}}={({({\mathit{C}}_{o_{\mathrm{1}}}^{o_{\mathrm{5}}})}^{-\mathrm{1}}+{({\mathit{C}}_{o_{\mathrm{2}}}^{o_{\mathrm{5}}})}^{-\mathrm{1}}+{({\mathit{C}}_{o_{\mathrm{3}}}^{o_{\mathrm{5}}})}^{-\mathrm{1}}+{({\mathit{C}}_{o_{\mathrm{4}}}^{o_{\mathrm{5}}})}^{-\mathrm{1}})}^{-\mathrm{1}}.

The compliance matrix of the CBPM in the coordinate frame o-xyz is derived as follows:

(15){\mathit{C}}_{o_{\mathrm{5}}}^o=\mathit{P}({\mathit{r}}_{\mathrm{2}}){\mathit{C}}_{o_{\mathrm{5}}}\mathit{P}{({\mathit{r}}_{\mathrm{2}})}^{\mathrm{T}},

where {\mathit{r}}_{\mathrm{2}}={[\mathrm{0},(a+b)/\mathrm{2},h+c+d/\mathrm{2}]}^{\mathrm{T}}.

The compliance matrices of the rotational and decoupled plates in the coordinate frame o-xyz are derived as follows:

(16){\mathit{C}}_{o_{\mathrm{6}}}^o=\mathit{P}({\mathit{r}}_{\mathrm{3}}){\mathit{C}}_{\mathrm{2}}\mathit{P}{({\mathit{r}}_{\mathrm{3}})}^{\mathrm{T}},

(17){\mathit{C}}_{o_{\mathrm{7}}}^o={\mathit{R}}_z\left(-{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}\right)\mathit{P}({\mathit{r}}_{\mathrm{4}}){\mathit{C}}_{\mathrm{2}}\mathit{P}{({\mathit{r}}_{\mathrm{4}})}^{\mathrm{T}}{\mathit{R}}_z{\left(-{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}\right)}^{\mathrm{T}},

where {\mathit{r}}_{\mathrm{3}}={[\mathrm{0},(a+b)/\mathrm{2},c+h-L_{\mathrm{2}}]}^{\mathrm{T}} and {\mathit{r}}_{\mathrm{4}}={[-(a+b)/\mathrm{2},\mathrm{0},h]}^{\mathrm{T}}.

Given that the CBPM, rotational plate, and decoupled plate are connected in series, the compliance matrix of the elastic kinematic chain can be written as follows:

(18){\mathit{C}}_{o_a}^o={\mathit{C}}_{o_{\mathrm{5}}}^o+{\mathit{C}}_{o_{\mathrm{6}}}^o+{\mathit{C}}_{o_{\mathrm{7}}}^o.

By rotating {\mathit{C}}_{o_a}^o {\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}, π, and -{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}, the compliance matrices of the other three elastic kinematic chains can be written as follows:

(19){\mathit{C}}_{o_b}^o={\mathit{R}}_z\left({\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}\right){\mathit{C}}_{o_a}^o{\mathit{R}}_z{\left({\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}\right)}^{\mathrm{T}},

(20){\mathit{C}}_{o_c}^o={\mathit{R}}_z(\mathrm{\pi }){\mathit{C}}_{o_a}^o{\mathit{R}}_z{(\mathrm{\pi })}^{\mathrm{T}},

(21){\mathit{C}}_{o_d}^o={\mathit{R}}_z\left(-{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}\right){\mathit{C}}_{o_a}^o{\mathit{R}}_z{\left(-{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}\right)}^{\mathrm{T}}.

The four elastic kinematic chains are arranged in parallel, as shown in Fig. 8. Therefore, the compliance matrix of the FSM is derived as follows:

(22){\mathit{C}}_o={({{\mathit{C}}_{o_a}^o}^{-\mathrm{1}}+{{\mathit{C}}_{o_b}^o}^{-\mathrm{1}}+{{\mathit{C}}_{o_c}^o}^{-\mathrm{1}}+{{\mathit{C}}_{o_d}^o}^{-\mathrm{1}})}^{-\mathrm{1}}.

The rotational stiffness of the FSM is written as follows:

(23)K={\displaystyle \frac{\mathrm{1}}{C_o(\mathrm{5},\mathrm{5})}}.

Determination of structural parameters

In accordance with Section 4.1, the rotational radius and rotational stiffness of the FSM are set at 33 mm and 50 N·m/rad, respectively.

On the basis of Eq. (23), the structural parameters of the FSM are optimized based on three constraint conditions:

1) K is approximately equal to 50 N·m/rad;

2) {\displaystyle \frac{a+b}{\mathrm{2}}}=\mathrm{33} mm;

3) Rage of parameters (unit: mm): \mathrm{20}\le L_{\mathrm{1}}\le \mathrm{35}, \mathrm{4}\le L_{\mathrm{2}}\le \mathrm{10}, \mathrm{6}\le b_{\mathrm{1}}=b_{\mathrm{2}}\le \mathrm{12}, \mathrm{0.4}\le t_{\mathrm{1}}\le \mathrm{0.6}, and \mathrm{0.3}\le t_{\mathrm{2}}\le \mathrm{0.6}.

The structural parameters optimization results are L_{\mathrm{1}}=\mathrm{25} mm, L_{\mathrm{2}}=\mathrm{5} mm, b_{\mathrm{1}}=b_{\mathrm{2}}=\mathrm{8} mm, t_{\mathrm{1}}=\mathrm{0.52} mm, and t_{\mathrm{2}}=\mathrm{0.42} mm.

Simulation analyses

Stiffness of the FSM structure

The structural parameters of the FSM are derived from the theoretical calculation of rotational stiffness. Therefore, verifying the effectiveness of the static model is necessary. Finite element analysis is conducted using ANSYS Workbench software. The FSM structure is constructed from Al-7075. Forces are applied in a push-pull configuration. The deformation behavior is shown in Fig. 9. According to the finite element analysis, the rotational stiffness of the FSM is 49.58 N·m/rad. When this value is compared with the theoretical calculation (48.33 N·m/rad), the determined error is 2.52%, indicating the effectiveness of the static model.

Bandwidth of the closed-loop control system

According to Section 4.3, the structural parameters of the FSM is determined. According to the simulation results, the variables for rotational radius r and stiffness K are 33 mm and 49.58 N·m/rad, respectively.

In accordance with Section 3.1, the open-loop transfer function of the FSM on the {\theta }_x-axis is written as follows:

(24)G(s)={\displaystyle \frac{\mathrm{5.92}\times {\mathrm{10}}^{\mathrm{5}}}{s^{\mathrm{3}}+\mathrm{2.73}\times {\mathrm{10}}^{\mathrm{3}}{s}^{\mathrm{2}}+\mathrm{3.60}\times {\mathrm{10}}^{\mathrm{5}}s+\mathrm{4.45}\times {\mathrm{10}}^{\mathrm{8}}}}.

In accordance with Section 3.2 and Section 3.3, the notch filter and PI controller are written as follows:

(25)N(s)={\displaystyle \frac{s^{\mathrm{2}}+\mathrm{72.46}s+\mathrm{1.67}\times {\mathrm{10}}^{\mathrm{5}}}{s^{\mathrm{2}}+\mathrm{817.32}s+\mathrm{1.67}\times {\mathrm{10}}^{\mathrm{5}}}},

(26)C(s)=\mathrm{1480}+{\displaystyle \frac{\mathrm{221501}}{s}}.

Frequency response characteristic analyses of the {\theta }_x-axis are conducted using MATLAB. The open-loop frequency response characteristic is shown in Fig. 10(a). The magnitude and phase margins are 17.74 dB and 62°, respectively, indicating the robustness of the closed-loop system. The frequency response of the closed-loop system is shown in Fig. 10(b). The closed-loop bandwidth is 116 Hz, which exceeds the minimum bandwidth requirement of 100 Hz and thus more than satisfies the requirement.

Experimental tests

The compliant mechanism of the FSM is fabricated through a wire electrical discharge machining process. The VCAs (LVCA-038-038-01) are driven by two linear servo amplifiers (LA-210s-02-RA, VREDAN, Inc.). Two linear scales (CE300-40, MicroE, Inc.) are used to measure the displacement, and the resolution is 1 mm. A Dspace-DS1103 is used to control the FSM system. The experimental stage is shown in Fig. 11.

Rotational stiffness

Different VCA forces are applied to the FSM, and corresponding displacements are measured using a linear scale. Force displacement tests are repeated thrice to reduce random errors. Experimental stiffness is obtained by linear fitting according to experimental test data. As shown in Fig. 12, stiffness around the x- and y-axes are 50.41 and 52.93 N·m/rad, respectively. Compared with the theoretical calculation, the errors are 4.13% and 8.69%. These stiffness errors may be caused by a measurement error in the linear scale or a manufacturing error in the construction of the stage or a combination of both factors.

Closed-loop bandwidth

The frequency response of the stage is obtained by conducting system identification. A chirp signal is applied, and the output displacement of the stage is tested. The open-loop frequency response of the stage around the x-axis is shown in Fig. 13. The corresponding transfer function is calculated as follows:

(27)G(s)={\displaystyle \frac{\mathrm{1.379}\times {\mathrm{10}}^{\mathrm{9}}}{s^{\mathrm{3}}+\mathrm{3491}{s}^{\mathrm{2}}+\mathrm{3.61}\times {\mathrm{10}}^{\mathrm{5}}s+\mathrm{5.862}\times {\mathrm{10}}^{\mathrm{8}}}}.

The damping ratio and resonance frequency are 0.0671 and 413.07 rad/s, respectively.

In accordance with Section 3.2, the notch filter is designed as follows:

(28)N(s)={\displaystyle \frac{s^{\mathrm{2}}+\mathrm{55.43}s+\mathrm{1.71}\times {\mathrm{10}}^{\mathrm{5}}}{s^{\mathrm{2}}+\mathrm{826.15}s+\mathrm{1.71}\times {\mathrm{10}}^{\mathrm{5}}}}.

In accordance with Section 3.3, the gain parameters of the PI controller are obtained as follows:

{{\displaystyle k}}_p=\mathrm{0.8},{{\displaystyle k}}_i=\mathrm{131.}

In consideration of the PI controller and notch filter, the open-loop frequency response of the closed-loop control system in the {\theta }_x-axis is shown in Fig. 14(a). The magnitude and phase margins are 8.06 dB and 65.2°, respectively. The closed-loop frequency response of the {\theta }_x-axis are shown in Fig. 14(b). The bandwidth is 117.6 Hz. The experimental results are in good agreement with the theoretical analysis, verifying the effectiveness of the proposed structural parameter design method.

Similarly, the frequency response for the {\theta }_y-axis is shown in Figs. 14(c) and 14(d). The gain and phase margins are 6.82 dB and 64.9°, respectively, whereas the bandwidth is 131.9 Hz. As such, the designed FSM system clearly satisfies the bandwidth requirement.

Motion characteristic test

The travel range and motion resolution of the stage are tested. The peak force of the applied VCA is 40.4 N. The characteristic of the peak force is intermittent due to heat generated in the coils. Thus, the current signal applied to the VCAs is a sinusoidal wave with a magnitude of 4 A. The sinusoidal signal responses are shown in Fig. 15. The maximum rotational angles of the FSM around the x- and the y-axes are ±44.35 and ±43.88 mrad, respectively.

The closed-loop control system includes the notch filter and PI feedback controller, as illustrated in Fig. 3. The motion resolution of the stage is tested by applying a consecutive step signal. As shown in Fig. 16, the resolution of the FSM is ±0.03 mrad.

Conclusions

This study presents a structural parameter design method for an FSM system based on the required bandwidth. The closed-loop controller, including the notch filter and PI controller, is designed in accordance with the derived transfer function of the FSM. The two unknown variables (rotational radius r and stiffness K) in the transfer function are then optimized taking the bandwidth as a constraint condition. On the basis of the optimization results for r and K, the structural parameters of the FSM are determined. Finally, simulations and experimental tests are conducted to verify the theoretical analysis, thus proving the efficacy of the proposed method.

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The Author(s) 2019. This article is published with open access at link.springer.com and journal.hep.com.cn