applied sciences Article
Systematic Design Method and Experimental Validation of a 2-DOF Compliant Parallel Mechanism with Excellent Input and Output Decoupling Performances Yao Jiang 1,2 , Tiemin Li 2,3, *, Liping Wang 2,3 and Feifan Chen 1 1 2 3
*
Department of Precision Instrument, Institute of Instrument Science and Technology, Tsinghua University, Beijing 10084, China;
[email protected] (Y.J.);
[email protected] (F.C.) Beijing Key Lab of Precision/Ultra-Precision Manufacturing, Equipments and Control, Tsinghua University, Beijing 100084, China;
[email protected] Department of Mechanical Engineering, Institute of Manufacturing Engineering, Tsinghua University, Beijing 100084, China Correspondence:
[email protected]; Tel.: +86-010-6279-2792
Academic Editor: Giuseppe Lacidogna Received: 10 April 2017; Accepted: 2 June 2017; Published: 8 June 2017
Abstract: The output and input coupling characteristics of the compliant parallel mechanism (CPM) bring difficulty in the motion control and challenge its high performance and operational safety. This paper presents a systematic design method for a 2-degrees-of-freedom (DOFs) CPM with excellent decoupling performance. A symmetric kinematic structure can guarantee a CPM with a complete output decoupling characteristic; input coupling is reduced by resorting to a flexure-based decoupler. This work discusses the stiffness design requirement of the decoupler and proposes a compound flexure hinge as its basic structure. Analytical methods have been derived to assess the mechanical performances of the CPM in terms of input and output stiffness, motion stroke, input coupling degree, and natural frequency. The CPM’s geometric parameters were optimized to minimize the input coupling while ensuring key performance indicators at the same time. The optimized CPM’s performances were then evaluated by using a finite element analysis. Finally, a prototype was constructed and experimental validations were carried out to test the performance of the CPM and verify the effectiveness of the design method. The design procedure proposed in this paper is systematic and can be extended to design the CPMs with other types of motion. Keywords: compliant mechanism; input and output decoupling; design method
1. Introduction Manipulators with ultrahigh precision and resolution are urgently needed and play more and more important roles in modern technology, such as atomic force microscopy, ultra precision machining, biological cell manipulation, and chip assembly in the semiconductor industry [1–3]. However, conventional mechanisms make it difficult to meet these requirements due to their coarse accuracies caused by inevitable backlash, friction, and joint wear. By contrast, compliant mechanisms, which are composed of rigid links and flexure hinges, transmit motion solely through the deformation of materials. Therefore, they can provide high accuracy and resolution in micro-scale motion, benefitting from no friction, no backlash, no wear, and monolithic structure, while being easy to manufacture and assemble [4,5]. This kind of mechanism has already caught researcher’s attention.
Appl. Sci. 2017, 7, 591; doi:10.3390/app7060591
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A typical compliant mechanism mainly consists of a mechanical structure with flexure members for motion transmission, piezoelectric elements for actuation, ultra-precision sensors for displacement output measurement, and controllers for motion control. Among them, the mechanical structure determines the performance of the compliant mechanism to a great extent. Two kinds of kinematic structures are commonly adopted, i.e., serial and parallel types [6]. Serial compliant mechanisms generally adopt a stack structure with one 1-DOF positioning stage mounted on another 1-DOF stage. This is a simple structure that is easy to manufacture and control, which has already been successfully applied in commercial products. However, it has the disadvantage of different dynamic characteristics of each axis, a reduced natural frequency as a result of the stacked stage increasing the moving mass, and cumulative motion errors [7]. The serial compliant stage will be suitable for applications where the motion along one axis is considerably faster than the others. Kenton [8] utilized a serial-kinematic nanopositioning stage for raster-type scanning, in which one axis is chosen for high-speed scanning while the other is designed for low-speed motion. Compared with the serial type, the parallel structure has inherent advantages in terms of high stiffness, large load capacity, low inertia, high natural frequency, and an absence of accumulation errors [9]. However, there have been many problems during the development of the parallel manipulator, particularly regarding conventional joints and rigid links. First, there are passive joints in the parallel manipulator, which are difficult to manufacture with high precision, such as sphere and universal joints. Meanwhile, the kinematic chains of the parallel manipulator are generally composed of elongated rods, which are prone to deformation and vibrate under external loads [10,11]. Because the displacement output of the terminal platform cannot be measured in real time to realize a closed-loop control, the above adverse factors will significantly decrease the motion accuracy of the parallel manipulator. Moreover, the kinematic model of parallel manipulator is complicated, which creates difficulties in kinematic calibration and motion control. The performance indicators, such as stiffness, dexterity, acceleration ability, will all change when the pose of the manipulator varies. Therefore, conventional parallel manipulators have not achieved their expected success in industry applications. Fortunately, the above issues will cease to be problems when the parallel structure is utilized in the development of a compliant mechanism. Apparently, there are no joints in the compliant parallel mechanism (CPM) and its displacement output can be measured directly as it relates to its micro motion stroke. Moreover, the pose of the CPM can be considered as a constant because its motion stroke is very small when compared to the geometric dimensions. This guarantees that the CPM can have a linear force–displacement relationship, identical mechanical bandwidth for each axis, and even decoupled kinematic model. Therefore, parallel structure has a great potential to be utilized in the development of a compliant mechanism, and numerous researchers have already begun working on it [12–15]. Though the CPM has advantages over its rigid counterpart, there are specific challenges in its design and development, which include the realization of cross-axis decoupling and actuation isolation [16–18], also known as output and input decoupling. Output coupling refers to any motion along one axis is affected by the actuation/input force along another axis. It results in a complicated kinematic model and additional calibration should be performed in the absence of terminal feedback. This is a typical character of a parallel manipulator due to its parallel kinematic type. The input coupling means that the input force on one actuation stage will cause the primary motion of another motion stage and thus bring additional loads on the actuator. Its effects can be neglected in a conventional parallel manipulator because the adopted motors have large stiffness and can bear any type of reacting force. For a CPM, however, the input coupling will cause unwanted deformations and even clearance between an actuator and the actuation point, which will result in difficulties to the motion control. Moreover, CPM adopts a piezoelectric actuator as its actuation element due to the actuator’s high precision and resolution, large stiffness, high push force, and fast response. However, the piezoelectric actuator is made of multiple piezoelectric layers glued together and is thereby sensitive to pulling and bending forces that may damage the actuator. Therefore, the input coupling has adverse effects on the CPM and should be carefully considered in its design and development
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process. Awtar [16,19] first proposed the decoupling performance design requirements of the parallel XY compliant mechanism. In order to realize actuation isolation, the actuator was required to be connected to the actuation point by means of a decoupler, which transmitted the axial force and absorbed any transverse motion without generating transverse loads. Lai [20] employed two types of distributed-compliance compound flexure modules and a mirror symmetric structure to reduce the input and output coupling of a novel 2-DOF compliant stage. Qin [21] utilized statically indeterminate symmetric structures to attenuate the cross-axis coupling of a proposed XY CPM. Li [22] investigated the design and development of a decoupled compliant stage with displacement amplifiers. The output decoupling was achieved by resorting to compound parallelogram flexures; the double compound parallelogram flexure and the amplifier acted as the decoupler to ensure this stage had excellent input decoupling characteristics. There have already been several efficient and intuitive design approaches related to the compliant mechanisms, including the kinematic substitution method [20,23], constraint-based design method [24], and freedom and constraint topology [25,26]. However, these methods focus on the type synthesis and only guarantee the DOFs of the compliant mechanism instead of considering other performances. Additionally, they regard the degree and constrain of the compliant mechanism as those of the rigid mechanism. While the compliant mechanism moves through the deformations of materials, its DOFs are coupled with the stiffness property. Therefore, the type synthesis and structural design of the compliant mechanism are supposed to be integrated with the consideration of multiple performance indicators. The systematic design method for compliant mechanisms with excellent performances is still a great challenge. The practical CPMs are mostly directly provided in the present studies and there lacks a systematic process for guiding the design of a CPM with excellent decoupling performance. Considering the 2-DOF CPM [16–23,27–30] is the most practical compliant mechanism, this work’s motivation is to present a systematic design of a decoupled 2-DOF CPM. The requirement on the kinematic structure for realizing totally output decoupling performance has been discussed. A flexure-based decoupler was chosen and its detail structure design was studied to minimize the input coupling of the CPM. Unlike the previous works, which only provided qualitative analyses, the specific definition and analytical model of the input coupling degree (ICD) of the CPM have been provided. According to the established model, the geometric parameters of the 2-DOF were optimized to directly minimize its ICD and the other key performance indicators were also guaranteed through constraint conditions. Finally, a prototype of the 2-DOF CPM was fabricated based on the optimized result, and several experimental tests were performed. A novel method was proposed within the tests, in order to evaluate the input coupling of the 2-DOF CPM by detecting the variations of axial forces at each actuation point instead of measuring their transverse deformations, which is difficult to achieve experimentally. The results demonstrated the decoupled CPM’s excellent performances and also verified the effectiveness of the design method. 2. Structure Design of the 2-DOF CPM The structure design, compared with geometric parameter optimization, will be more crucial and effective in guaranteeing the CPM’s performance. As stated above, the input and output decoupling performances of the 2-DOF CPM are important and must be well guaranteed. Therefore, the structure of the CPM is determined according to the requirement of the decoupling characteristics in this section. The other performance indicators are ensured by resorting to the geometric parameter optimization in the following material. The schematic of the kinematic structure of the 2-DOF is depicted in Figure 1.
and effective in guaranteeing the CPM’s performance. As stated above, the input and output decoupling performances of the 2-DOF CPM are important and must be well guaranteed. Therefore, the structure of the CPM is determined according to the requirement of the decoupling characteristics in this section. The other performance indicators are ensured by resorting to the geometric parameter optimization is Appl. Sci. 2017, 7, in 591the following material. The schematic of the kinematic structure of the 2-DOF 4 of 28 depicted in Figure 1. xi
yi
θi
y
ri
x
ri = rix , riy , 0
Flexure Hinges
T
x
Moving Platform
Figure 1. Kinematic structure of a 2-DOF CPM. Figure 1. Kinematic structure of a 2-DOF CPM.
2.1. Realization of Output Decopuling Performance The requirement of the output decoupling performance on the structure design of the 2-DOF CPM is studied first. It is expected that translational motion of the terminal platform along the xand y-axes are independent, thereby ensuring the CPM with a total output decoupling characteristic. Therefore, the output stiffness matrix of the CPM should satisfy the form as
Kx− Fx K= 0 0
0 Ky− Fy 0
0 0
Kα− Mz
(1)
The output stiffness of the CPM can be calculated through the summation of the stiffness of each limb as [31] n
K=
∑ Tif−1 Kil Tid−1
(2)
i =1
where Kil is the local stiffness matrix of the ith limb, and T if and T id are the force and displacement transformation matrices, respectively, which take on the form
Tif = TTid
− sin θi cos θi −rix
cos θi = sin θi riy
0 0 1
(3)
The output stiffness matrix of the CPM should satisfy the form expressed in Equation (1). However, it is difficult to obtain a satisfied structure by solving the equations directly. To simplify this problem, the structure of the CPM is divided into two parts and the local coordinate systems of each subsystem are established, as shown in Figure 2. There are two ways to build the local coordinate systems, i.e., mirror symmetry and rotation symmetry. The stiffness matrices of the two subsystems shown in Figure 2a under their corresponding local coordinate systems are expressed as Kim− Fx m = Kiy− Fx m Kiα − Fx
Km i
m Kix − Fy m K1y− Fy m K1α − Fy
m Kix − Mz m Kiy − Mz m Kiα − Mz
(i = 1, 2)
(4)
Then, the stiffness matrix of the CPM under the globe coordinate system can be obtained according to Equation (2) as Km =
2
∑ (Tmif )−1 Kmi (Tmid )−1
i =1 m where Tm 1f = T1d
T
m = diag([1, 1, 1]), and Tm 2f = T2d
T
= diag([−1, −1, 1]).
(5)
iy
ix
The output stiffness matrix of the CPM should satisfy the form expressed in Equation (1). However, it is difficult to obtain a satisfied structure by solving the equations directly. To simplify this problem, the structure of the CPM is divided into two parts and the local coordinate systems of Appl. 2017, 7, 591are established, as shown in Figure 2. There are two ways to build the local coordinate 5 of 28 eachSci. subsystem systems, i.e., mirror symmetry and rotation symmetry. y2
y1
z2
x2
x1 y
z1
z
x
(a)
y1
y
x2
x
z
z2
x1
z1 y2 (b)
Figure 2. Division of a 2-DOF CPM: (a) mirror symmetry coordinate systems; and (b) rotation Figure 2. Division of a 2-DOF CPM: (a) mirror symmetry coordinate systems; and (b) rotation symmetry symmetry coordinate coordinate systems. systems.
The stiffness matrices of the two subsystems shown in Figure 2a under their corresponding local If the structure of the CPM is mirror symmetry, the local stiffness matrix of the two subsystems coordinatemsystems are expressed as satisfies K1 = Km 2 . Then Equation (5) can be simplified as m m m K i − Fx K ix − Fy K ix − Mz mm m m m 0 2K K im = K2K 1, 2 K K i = 1x − Fx 1 y − Fy (4) iy − Mz (1x − Mz ) iy − Fx m m 0 Km = K m 0 K m 2K1y (6) − Fy K iα − Mz 1α − Fy iα −mFx m 2K1α− Fx 0 2K1α− Mz Then, the stiffness matrix of the CPM under the globe coordinate system can be obtained This shows that the motion along the y-axis will not affect the other two axes, while according to Equation (2)translational as the translational motion along the x-axis and the rotational motion about the z-axis are coupled. Similarly, when the structure of the CPM is rotation symmetry, its stiffness matrix can be obtained as s s 2K1x 0 − Fx 2K1x − Fy s s Ks = 2K1y (7) 0 − Fx 2K1y− Fy 0
0
s K1α − Mz
This indicates that the rotational motion about the z-axis has no influence on the other two axes, while the translational motions along the x- and y-axes are coupled. It is verifiable that, when the structure of the CPM both satisfies the mirror and rotation symmetry, the translational and rotational motions of the terminal platform are independent. It is clear that the CPM is symmetric around the x- and y-axes when it satisfies both the mirror and the rotation symmetry. In order to obtain a total output decoupling characteristic, the actuation forces must be applied along the two symmetric axes at the same time. Therefore, linear actuators are needed, such as the piezoelectric actuator [32] and voice coil motor [33], which should be mounted along the symmetric axes. The above discussion shows that a symmetric kinematic structure can sufficiently ensure the 2-DOF CPM with total output decoupling performance. 2.2. Realization of Input Decopuling Performance Next, the requirement of the input decoupling performance on the structure design of the 2-DOF CPM is investigated. The effective way of reducing the input coupling of the CPM is to adopt a flexure-based decoupler between an actuator and the terminal platform. There are two types of the decouplers, and their kinematic structures are shown in Figure 3.
2.2. Realization of Input Decopuling Performance Next, the requirement of the input decoupling performance on the structure design of the 2-DOF CPM is investigated. The effective way of reducing the input coupling of the CPM is to adopt a flexure-based decoupler between an actuator and the terminal platform. There are two types of the Appl. Sci. 2017, 7, 591 6 of 28 decouplers, and their kinematic structures are shown in Figure 3.
y x
y x (a)
(b)
Figure 3. 3. Kinematic Kinematic Figure Kinematic structures structuresof offlexure-based flexure-baseddecoupler. decoupler.(a) (a)Kinematic KinematicStructure Structure1.1.(b) (b) Kinematic Structure 2. Structure 2. Appl. Sci. 7, 591 of 26 The2017, difference between the two decouplers is the actuator’s mounting type. The actuator is 6fixed
The difference between the two decouplers is the actuator’s mounting type. The actuator is fixed on the ground in the first decoupler, while the other installs the actuator on the moving part. on the ground inbe theinserted first decoupler, while the other installs the second actuator on the moving The The actuator can into a displacement amplifier in the decoupler, whichpart. increases actuator can stroke be inserted into a displacement amplifier in the second decoupler, which increases the the motion of the CPM and gives the actuator double protection from input coupling. motion stroke theof CPM and gives the actuator protection from input coupling. However, this However, thisof type decoupler results in a moredouble complex structure. Moreover, the actuator increases type of decoupler in a more structure. Moreover, the actuator increases the moving the moving mass results of the CPM and complex thus reduces its natural frequency. Therefore, the first type of mass of theisCPM and thus its natural Therefore, the first decoupler to is decoupler preferred for reduces this paper, and itsfrequency. detailed structure design willtype be of investigated preferred paper, and its detailed structure design be investigated to guarantee the of input guaranteefor thethis input decoupling characteristic of the CPM.will Because of the symmetric structure the decoupling of the Because of the the y-axis symmetric of the 2-DOF CPM, we can 2-DOF CPM,characteristic we can choose the CPM. decoupler along as thestructure study object and its stiffness model choose theindecoupler is shown Figure 4. along the y-axis as the study object and its stiffness model is shown in Figure 4.
y2 x2
O
y x
z2
K2
z K1−1
A
y1
x1
K1−2
z1 (a)
(b)
Figure decoupler. (a) (a) Kinematic Kinematic structure. structure. (b) (b) Stiffness Stiffness model. model. Figure 4. 4. Stiffness Stiffness model model of of flexure-based flexure-based decoupler.
For the 2-DOF CPM, excellent input decoupling performance requires that the transverse and For the 2-DOF CPM, excellent input decoupling performance requires that the transverse and rotational deformations at one actuation point are small enough that the terminal platform is actuated rotational deformations at one actuation point are small enough that the terminal platform is actuated by the other actuator. According to Figure 4, utilizing a force equilibrium condition of point A1 allows by the other actuator. According to Figure 4, utilizing a force equilibrium condition of point A1 allows us to obtain the relationship between the displacement of the terminal platform and the deformation us to obtain the relationship between the displacement of the terminal platform and the deformation at the actuation point as at the actuation point as −1 K 1 δa + K 2 ( δa − δo ) = 0 δa = ( K 1 + K 2 ) K 2 δo (8) K1 δa + K2 (δa − δo ) = 0 ⇒ δa = (K1 + K2 )−1 K2 δo (8) where δo is the is the displacement of the terminal platform, δa is the deformation of actuation point A1, and andis K 2 are the stiffness matrices ofplatform, Parts I and ofdeformation the decoupler expressed in the where δo K is1 the the displacement of the terminal δa isIIthe of actuation point A1 , y1z1}, respectively. coordinate and K1 andframe K2 are{A-x the 1stiffness matrices of Parts I and II of the decoupler expressed in the coordinate bending and rotational deformations at the actuation point caused by the motion of the frameThe {A-x 1 y1 z1 }, respectively. terminal obtaineddeformations through the components of δa;point they are expected to motion be as small as The platform bending can and be rotational at the actuation caused by the of the possible. platform Through can a qualitative analysis of Equation (8), the stiffness requirement of the decoupler terminal be obtained through the components of δa ; they are expected to be as small is: as the bending stiffness along the analysis x-axis and rotation about the z-axis of II should be possible. Through a qualitative of the Equation (8),stiffness the stiffness requirement ofPart the decoupler is: much smaller than in Part I. With the exception of protecting the actuator from suffering transverse load, the decoupler also transmits the motion of the actuator to the terminal platform. Therefore, the axial stiffness along the y-axis of Part I should not be too large, in order to avoid the motion stroke of the actuator losing too much because of its deformation. In addition, the axial stiffness of Part II should be large in order to reduce the motion lost between the actuator and the terminal platform,
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the bending stiffness along the x-axis and the rotation stiffness about the z-axis of Part II should be much smaller than in Part I. With the exception of protecting the actuator from suffering transverse load, the decoupler also transmits the motion of the actuator to the terminal platform. Therefore, the axial stiffness along the y-axis of Part I should not be too large, in order to avoid the motion stroke of the actuator losing too much because of its deformation. In addition, the axial stiffness of Part II should be large in order to reduce the motion lost between the actuator and the terminal platform, thereby improving the motion transmission efficiency. The design requirements on the decoupler will be utilized to guide its detailed structure design. The kinematic structure of Part I of the decoupler adopts a parallel type, as shown in Figure 5. In order to obtain a compact structure, the limb is constructed from a single leaf spring due to its advantages of compact size, lightweight, and easy manufacturing [34,35]. Appl. Sci. 2017, 7, 591 7 of 26
yi
xi oi θ i x
y1 r T i ri = rix , riy , 0 z z1 A x1 i
Figure Figure 5. 5. Kinematic Kinematic structure structure of of Part Part II of of flexure-based flexure-based decoupler. decoupler.
The stiffness of Part I is the summation of the stiffness of each limb. Therefore, the influence of stiffness of Partparameters I is the summation of the stiffness limb. Therefore, influence of each The limb’s mechanical on the stiffness of Part Iof is each investigated first. Thethe stiffness of the each limb’s mechanical parameters on the stiffness of Part I is investigated first. The stiffness of the ith ith limb can be expressed in the coordinate system {A-x1y1z1} as limb can be expressed in the coordinate system {A-x1 y1 z1 } as kio = Ti −f 1kil Ti −d1 (9) o −1 l −1 ki = Tif ki Tid (9) in its local coordinate frame {o-xiyizi}, which where kil is the stiffness matrix of the ith limb expressed takes the form where kil is the stiffness matrix of the ith limb expressed in its local coordinate frame {o-xi yi zi }, which takes the form 00 00 − Fx kixk−ix Fx l = 0 k − Mz (10) − Fy i kil =k (10) kkiyiy− 0 Fy kiyiy − Mz 00 k k k iαiα− − Fy − Mz Fy kiαiα − Mz The axial axial and and bending bending stiffness The stiffness of of the the limb limb can can be be obtained obtained from from Equation Equation (9) (9) as as 2 kixo −= − sin ) −Fy koix− Fx k ixk−ix −Fx sin2θθi i⋅ (· kixk−ixFx−−Fxk iy−− Fyk iy Fx = Fx − o 2 + sin ) −Fy Fy = ix −Fy Fy + koiy−kFyiy − = k ixk− sin2θθi i⋅ (· k ixk−ixFx−−Fxk iy−− Fyk iy
(11) (11) (12) (12)
kiy kiy−Fy Considering the the stiffness stiffness components componentsof ofthe theleaf leafspring springsatisfy satisfykixkix−Fx the value value of of angle −Fx −Fy,,the θ large bending bending stiffness stiffness and and small small axial axial stiffness stiffness of of Part PartI.I. θ ii should be zero to ensure the large Then the rotation stiffness of the limb can be written written as as kioα −rMz2=· kriy 2 ⋅ kix −+ + rix22·⋅kkiy − Fy ++ kiαk− Mz + 2r+ kiy − Mz· k Fx r ix ⋅2r koiα− Mz = iy ix − Fx ix iy− Fy iα− Mz ix iy− Mz
(13) (13)
Because each component of the stiffness matrix kl il is greater than zero, the increment of rix and Because each component of the stiffness matrix ki is greater than zero, the increment of rix and riy riy will both increase the rotation stiffness. will both increase the rotation stiffness. According to the discussion above, in order to meet the stiffness requirement of Part I of the According to the discussion above, in order to meet the stiffness requirement of Part I of the decoupler, each limb should be parallel to the x-axis and their positions should be far from the centric decoupler, each limb should be parallel to the x-axis and their positions should be far from the centric of the actuation platform if possible. of the actuation platform if possible. Next, the influence of the leaf spring’s geometric parameters is studied. The stiffness matrix of a single leaf spring can be expressed as EDt L kL = 0
0 EDt 3 L3
0 EDt 3 2 L2
(14)
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Next, the influence of the leaf spring’s geometric parameters is studied. The stiffness matrix of a single leaf spring can be expressed as
EDt L
0
kL =
0
0
0
EDt3 L3 EDt3 2L2
EDt3 2L2 EDt3 3L
(14)
where E is the elastic modulus, and L, t, and D are the length, thick and height of the leaf spring, respectively. According to the stiffness requirement of Part I and the above analysis, the bending stiffness of the leaf spring should be small, while the axial and rotation stiffness should be large. According to Equation (13), however, it is impossible to reduce the bending stiffness of the single leaf spring Appl. Sci. 2017, 7, 591 that the axial and rotation stiffness remain unchanged or increased by adjusting 8 ofits 26 while guaranteeing geometric parameters. Therefore, another type of hinge should be proposed to construct a satisfied geometric parameters. Therefore, another type of hinge should be proposed to construct a satisfied limb. As opposed to utilizing a single flexure hinge, a compound hinge composed of n identical limb. As opposed to utilizing a single flexure hinge, a compound hinge composed of n identical parallel leaf springs is chosen, as shown in Figure 6. parallel leaf springs is chosen, as shown in Figure 6.
L
D
L
D d
(a)
t n
t
(b)
Figure 6. Single and compound leaf springs. (a)(a) Single leafleaf spring. (b)(b) Compound leafleaf spring. Figure 6. Single and compound leaf springs. Single spring. Compound spring.
The length, height and total mass of the compound hinge are the same as the single leaf spring, and its expressed Thestiffness length, matrix height can andbe total mass ofas the compound hinge are the same as the single leaf spring, and its stiffness matrix can be expressed as k 0 0 1x − Fx 2 2 kK 1xn−= Fx 0 10n ) ⋅ k1 y − Fy 1 n ) ⋅ k10y − Mz ( ( (15) 0 (1/n)2 · k1y 1/n)22 · k1y− Mz ( 2 − Fy 2 Kn = (15) 0 (1 n ) ⋅ k1α − Fy ( n + 1) ( n − 1) ⋅ γ +2(1 n ) ⋅ k12α − Mz 0 (1/n)2 · k1α− Fy (n + 1)/(n − 1) · γ + (1/n) · k1α− Mz
(
)
where γ =d / 2t , and k1m−n is the stiffness component of the single leaf spring. where γ = d/2t, and k1m−n is the stiffness component of the single leaf spring. Equation (14) shows that the axial stiffness of the compound hinge is equal to the single leaf Equation (14) shows that the axial stiffness of the compound hinge is equal to the single leaf spring, spring, and the bending stiffness is always smaller and decreases rapidly as number n grows. and the bending stiffness is always smaller and decreases rapidly as number n grows. The rotation The rotation stiffness depends on the number n and parameter γ. When γ > 1, the rotation stiffness is stiffness depends on the number n and parameter γ. When γ > 1, the rotation stiffness is always larger always larger than that of a single leaf spring. Therefore, the compound hinge will meet that than that of a single leaf spring. Therefore, the compound hinge will meet that requirement quite well. requirement quite well. Because the stiffness requirement of Part II of the decoupler is contrary to Part I, its limbs should Because the stiffness requirement of Part II of the decoupler is contrary to Part I, its limbs should be parallel to the y-axis according to the investigation above. The leaf spring is still utilized to construct be parallel to the y-axis according to the investigation above. The leaf spring is still utilized to the limbs in Part II and its stiffness requirement is illustrated as: the axial stiffness should be large construct the limbs in Part II and its stiffness requirement is illustrated as: the axial stiffness should while the bending and rotation stiffness should be small. The heights of the flexure hinges adopted in be large while the bending and rotation stiffness should be small. The heights of the flexure hinges Parts I and II are the same due to the monolithic structure of the CPM. In this case, no matter what, adopted in Parts I and II are the same due to the monolithic structure of the CPM. In this case, no the single leaf spring and the compound hinge cannot meet the stiffness requirement. Considering matter what, the single leaf spring and the compound hinge cannot meet the stiffness requirement. the terminal platform of the 2-DOF CPM only has the translation motion, minimizing the bending Considering the terminal platform of the 2-DOF CPM only has the translation motion, minimizing stiffness and shear stiffness ka −Fy can ensure the decoupler with an excellent decoupling character the bending stiffness and shear stiffness ka−Fy can ensure the decoupler with an excellent decoupling according to Equation (8). The bending and shear stiffness of the compound hinge all decrease sharply character according to Equation (8). The bending and shear stiffness of the compound hinge all decrease sharply as the number n grows. Therefore, the compound hinge is still the best choice when constructing the limbs of Part II of the decoupler. As mentioned above, the excellent input decoupling character requires that the bending and shear stiffness of the compound hinges of Part II are small, and the axial and rotation stiffness of the compound hinge of Part I are large. Except for the axial stiffness, the other stiffness components of
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as the number n grows. Therefore, the compound hinge is still the best choice when constructing the limbs of Part II of the decoupler. As mentioned above, the excellent input decoupling character requires that the bending and shear stiffness of the compound hinges of Part II are small, and the axial and rotation stiffness of the compound hinge of Part I are large. Except for the axial stiffness, the other stiffness components of the compound hinge will all decrease as number n increases. Therefore, the number n of the compound hinge of Part I would be greater than that of Part II. In addition, the number n of the compound hinge should not be overly large, due to difficulties in manufacturing and because it is then prone to buckling under axial load. Taking all these factors into consideration, the number n of the compound hinges of Parts I and II are chosen to be two and three respectively in this paper. 2.3. Detailed Structure of the 2-DOF CPM According to the discussion above, the detailed structure of the 2-DOF CPM is given in Figure 7. The output decoupling performance of the CPM is realized by the symmetric kinematic structure, and the input coupling is reduced by the flexure-based decoupler composed of the compound hinges. Appl. Sci. 2017, 7, 591 9 of 26
2d
L
t2 A1
2d 2W1
t2
L
t1 2d
y
2W2
A2
x Figure 7. Detailed structure of 2-DOF CPM. Figure 7. Detailed structure of 2-DOF CPM.
3. Analytical Modeling of the 2-DOF CPM 3. Analytical Modeling of the 2-DOF CPM The detailed structure of the 2-DOF CPM has already been determined to guarantee the The detailed structure of the 2-DOF CPM has already been determined to guarantee the decoupling performances, and in this section its geometric parameters will be optimized to obtain an decoupling performances, and in this section its geometric parameters will be optimized to obtain ideal mechanism. With the exception of decoupling performance, the concerned performance an ideal mechanism. With the exception of decoupling performance, the concerned performance indicators also include the input and output stiffness, motion stroke, and natural frequency. Their indicators also include the input and output stiffness, motion stroke, and natural frequency. Their analytical models should be established before optimization. analytical models should be established before optimization. 3.1. Input and Output Stiffness 3.1. Input and Output Stiffness The stiffness model of the 2-DOF CPM is given in Figure 8a, and the input and output stiffness The stiffness model of the 2-DOF CPM is given in Figure 8a, and the input and output stiffness matrices will be derived first. Because the stiffness modeling is related to specific coordinate frames, matrices will be derived first. Because the stiffness modeling is related to specific coordinate frames, the globe coordinate frame is defined at the center of the terminal platform and the local coordinate the globe coordinate frame is defined at the center of the terminal platform and the local coordinate systems of each chain are shown in Figure 8b. systems of each chain are shown in Figure 8b. K 4l −1b
K 4l −1a K 4l −2
K1l−1a l
K il−1b
K 3l −1b
Kl
y
yi −1b
Part I-b xi −1b
3.1. Input and Output Stiffness The stiffness model of the 2-DOF CPM is given in Figure 8a, and the input and output stiffness matrices will be derived first. Because the stiffness modeling is related to specific coordinate frames, the globe coordinate frame is defined at the center of the terminal platform and the local coordinate Appl. Sci. 2017, 7, 591 10 of 28 systems of each chain are shown in Figure 8b.
K 4l −1b K
K 4l −1a K 4l −2
l 1−1a
K1l− 2
A1
K
K il−1b
l 3−1b
O
yi −1b
yi
K 3l − 2
K
l 1−1b
K
K 2l −2 K
l 2 −1a
K
Part II
zi
l 3−1a
zi −1a
l xi K i−2
Oi
y
l 2 −1b
Bi
yi −1a
x
O
K
z
A2
(a)
Part I-b xi −1b
l i −1a
zi −1a
xi −1a Part I-a
(b)
Figure 8. Stiffness model and coordinate frames of 2-DOF CPM. (a) Stiffness model of the CPM. Figure 8. Stiffness model and coordinate frames of 2-DOF CPM. (a) Stiffness model of the CPM. (b) Stiffness Stiffness model model of of aa chain. chain. (b)
The matrix method [36,37], which has the advantages of terse form and high modeling accuracy, The matrix method [36,37], which has the advantages of terse form and high modeling accuracy, has already been widely utilized in the stiffness modeling of the compliant mechanism. Therefore, Appl.already Sci. 2017, 7, 591 widely utilized in the stiffness modeling of the compliant mechanism. Therefore, 10 of 26 has been it is adopted to establish the analytical model of the 2-DOF CPM. To unify the description of the P K 2-DOF it is adopted to establish the analytical of the CPM. To unify the description of the stiffness matrices in this paper, we utilize model Ki−m l and i−m to represent the stiffness of Part m in the ith stiffness matrices in this paper, we utilize K i−ml and PKi−m to represent the stiffness of Part m in the ith chain expressed in its local coordinate frame and the coordinate frame {P}. The stiffness expressed in chain expressed in its local coordinate frame and the coordinate frameexpression {P}. The stiffness expressed in different coordinate frames can be transformed through the following as different coordinate frames can be transformed through the following expression as Q 1P K = T− K T−1 (16) Q K = Tff−1 P K Tdd−1 (16) where thethe force andand displacement transformation matrices between the coordinate frames andTdTdare are force displacement transformation matrices between the coordinate where TTf fand {P} and {Q}. frames {P} and {Q}. The of Parts The stiffness stiffness matrices matrices of Parts II and and II II of of each each chain chain under under their their local local coordinate coordinate frames frames are are calculated first. These two parts consist of compound hinges; the geometric parameters of calculated first. These two parts consist of compound hinges; the geometric parameters of the the hinges hinges and and their their local local coordinate coordinate frames frames are are shown shown in in Figure Figure 9. 9. yi − 2
yi −1 zi −1
zi − 2
xi −1
2d
xi − 2
d
t2
t1
(a)
(b)
Figure 9. Geometric parameters andand local coordinate frames of Parts I and II. II. (a)(a) Part I. (b) Part II. II. Figure 9. Geometric parameters local coordinate frames of Parts I and Part I. (b) Part
According to Equation (14), the stiffness of the two parts can be obtained, respectively, as According to Equation (14), the stiffness of the two parts can be obtained, respectively, as 22EDt EDt11 0 0 0 L 3 3 L 2EDt EDt 1 1 3 3 3 2 Kil−1a = Kil−1b = 0 (17) L L EDt 2 EDt l l 3 1 1 3 2 EDt EDt 0 K i −1a = K i −1b = (17) d 1 2 2 0 6 t · 3L1 L23 L+ L 1 2 EDt 3 EDt13 d 1 6 + 2⋅ 0 2 3 L t L 1
K il− 2
3EDt2 L = 0
0
0
3EDt23 L3
3EDt23 2 L2
(18)
Appl. Sci. 2017, 7, 591
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3EDt2 L
Kil−2 =
0 0
0
0
3EDt32 L3 3EDt32 2L2
3EDt32
2L2 2 6 td2 + 3 ·
EDt32 3L
(18)
The chain of the CPM is composed of Parts I and II connected in series; therefore, its compliance matrix under its local coordinate frames {Oi } can be calculated as Oi
Ci =
Oi
Ki−1a + Oi Ki−1b
−1
+ Oi Ki−−12
(19)
The CPM is composed of the four identical chains connected in parallel type, and its output stiffness can be obtained as 4 Koutput = ∑ O Ti−1 Oi Ki O Ti−T (20) i =1
−1 0 0 0 1 0 1 0 0 where O T1 = 0 −1 0 , O T2 = −1 0 0 , O T3 = 0 1 0 , and 0 W1 1 −W1 0 1 0 −W1 1 0 −1 0 OT = 1 0 0 . 4 Appl. Sci. 2017, 11 of 26 W7,1 591 0 1 The input stiffness of the 2-DOF CPM along its two axes is identical. The input stiffness model of The in input stiffness of the 2-DOF CPM10, along itsOtwo identical. The model the CPM the x-axis is depicted in Figure where KT1 axes is theistotal stiffness ofinput three stiffness chains from the OKT1 is the total stiffness of three chains from of the CPM in the x-axis is depicted in Figure 10, where second to the fourth chain and can be obtained as the second to the fourth chain and can be obtained as 4 O KOT1 = ∑4 OOTi−−11 OOi Ki OO T− Ti−T (21) K T 1 =i= (21) 2 Ti ( K i ) Ti i
i=2
y1
Actuator I
K1l−1a K1l−2
A1 x1
O
O
KT 1
z1
K1l−1b
Figure in the the x-axis. x-axis. Figure 10. 10. Input Input stiffness stiffness model model of of CPM CPM in
The input stiffness of the CPM can be obtained according to Figure 11 as The input stiffness of the CPM can be obtained according to Figure 11 as
(
)
K Input = A1 K1−1a + A1 K1−1b + A1 K1−−12 + A1T T ( O KT−11 ) A1T −1 1 A1 KInput = A1 K1−1a + A1 K1−1b + A1 K1−−12 + A1 TT O K− T T1 0 0 1 where A1T = 0 1 1 0 0 . 0 where A1 T = 1 2 1 0 . 0 0L + W1 + 2W 0 L + W1 + 2W2 1 3.2. Motion Stroke −1
(22) (22)
The motion strokes along the x- and y-axes of the 2-DOF CPM are the same. Considering the stiffness of the actuator, the input stiffness of the CPM in the x-axis is given in Figure 11. δ n1
kn1
δ a1 A1
K xIn− Fx
0 L + W1 + 2W2
1
3.2. Motion Stroke The motion strokes along the x- and y-axes of the 2-DOF CPM are the same. Considering the Appl. Sci. 2017, 7, 591 12 of 28 stiffness of the actuator, the input stiffness of the CPM in the x-axis is given in Figure 11. δ n1
kn1
δ a1
K xIn− Fx
A1 Figure Figure 11. 11. Input Input stiffness stiffness model model of of CPM CPM considering considering the stiffness of the actuator.
The relationship between the actual and nominal displacement output of the actuator can be 3.2. Motion Stroke obtained according to Figure 11 as The motion strokes along the x- and y-axes of the 2-DOF CPM are the same. Considering the knof 1 x-axis is given in Figure 11. stiffness of the actuator, the inputδ stiffness 1 the CPM in the ⋅ δ n1 = ⋅ δ n1 (23) a1 = In In kn1 + K K x − Fx kn 1 +displacement The relationship between the actual and output of the actuator can be x − Fxnominal obtained according to Figure 11 as where kn1 is the axial stiffness of the actuator, and K xIn− Fx is the input axial stiffness of the CPM along k n1 1 the x-axis. δa1 = · δn1 = · δn1 (23) k n1 + KxIn− Fx 1 + KxIn− Fxpoint /k n to the terminal platform should The efficiency of the motion transferred from the actuation also be calculated to obtain the motion stroke of the CPM. According to Figure 10, utilizing a force where kn1 is condition the axial stiffness and KxIn− Fx is the input axial stiffness of the CPM of along equilibrium of pointof O the canactuator, obtain the relationship between the displacement output the the x-axis.platform and the deformation at the actuation point A1 as terminal The efficiency of the motion transferred from the actuation point to the terminal platform should −1 O O O O K1− 2motion K1− 2 δa10, theδoCPM. = ( O K ( δo − δa1stroke ) = 0 of also be calculated toKobtain According utilizing a force (24) T 1 δo + the 1− 2 + K T 1 )to Figure 1 equilibrium condition of point O can obtain the relationship between the displacement output of the The motion of the CPM along can be calculated terminal platformstroke and the deformation atthe thex-axis actuation point A1 as according to Equations (22) and (23) as −1 O O O O O KT1 δo + K1−2 (δo − δa1 ) = 0 ⇒ δo = K1−2 + KT1 K1−2 δa1 (24) The motion stroke of the CPM along the x-axis can be calculated according to Equations (22) and (23) as h iT −1 O OK O K1−2 e1 e1 1−2 + KT1 δox = · δn (25) 1 + KxIn− Fx /k n where e1 = [1, 0, 0]T is a unit vector. Additionally, the maximum stress generated inside the flexure hinges should be kept within the allowable value when the CPM moves to the maximum displacement. The maximum stress of Parts I and II can be calculated as [22] σ1max = α
σy σy 12δamax Et1 12δomax Et2 ≤ , σ2max = α ≤ 2 n n L L2
(26)
where α is the stress concentration factor chosen as 2, δamax and δomax are the maximum displacements of the actuation platform and terminal platform, σy is the yield stress of the material, and n is the safety factor chosen as 1.5. According to Equation (23), the maximum displacement of the terminal platform is always smaller than that of the actuation platform. Therefore, the motion stroke of the CPM should satisfy (
δomax
σy L2 σy L2 ≤ min , 12nαEt1 12nαEt2
) (27)
3.3. Decoupling Character The symmetric structure of the 2-DOF CPM has already ensured it with a total output decoupling performance. Therefore, only the analytical model for the input decoupling characteristic is investigated and evaluated using the ICD. Without loss of generality, the ICD of the 2-DOF CPM can
3.3. Decoupling Character The symmetric structure of the 2-DOF CPM has already ensured it with a total output decoupling performance. Therefore, only the analytical model for the input decoupling characteristic is investigated using the ICD. Without loss of generality, the ICD of the13 2-DOF CPM Appl. Sci. and 2017, 7,evaluated 591 of 28 can be defined using the ratio of the bending deformation at the actuation point A2 along the x-axis to the axial deformation atratio the ofactuation point A1 along theactuation x-axis when Actuator I is actuated. be defined using the the bending deformation at the point Aonly 2 along the x-axis to the The axial relationship between the displacement output of when the terminal platform and the deformation deformation at the actuation point A1 along the x-axis only Actuator I is actuated. The relationship between the displacement outputin of Equation the terminal(23). platform andthe the deformation at the actuation point A1 has already been obtained Then deformation at the at the actuation point A1 has already been obtained in Equation (23). Then the deformation at the actuation point A2 caused by the motion of the terminal platform must be calculated to obtain the actuation point A2 caused by the motion of the terminal platform must be calculated to obtain the ICD ICD of the CPM. The stiffness model of the CPM along the y-axis is given in Figure 12, where °KT2 is of the CPM. The stiffness model of the CPM along the y-axis is given in Figure 12, where O KT2 is the the total stiffness of the first, andfourth fourth chains. total stiffness of the first,third third and chains.
O
KT 2
O K 2l −2
K 2l −1a
K 2l −1b
y2
x2
A2
z2
Figure Stiffnessmodel model of the the y-axis. Figure 12. 12. Stiffness ofCPM CPMalong along y-axis.
Similarly, the relationship between thedeformation deformation atat the actuation pointpoint A2 andAthe displacement Similarly, the relationship between the the actuation 2 and the displacement of the terminal platform can be obtained by resorting to a force equilibrium condition of point A2 as of the terminal platform can be obtained by resorting to a force equilibrium condition of point A2 as
−1
A2 + AA22 K2−1b + AA22 K2−2 −1 + K 2 −1b + K 2 − 2 ) δa 2 = ( A2 KK22−−1a 1 a
δa2 =
A2 A2
K2−2 δo
(28)
K 2 − 2 δo
(28)
Combining Equations (23) and (27), the relationship between the deformations at two actuation
Combining Equations (23) Iand (27),can theberelationship points when only Actuator is driven obtained as between the deformations at two actuation points when only Actuator I is driven can be obtained as δa2 =
A2
K + A2 2−1a 2 −1a
δa 2 = ( K
A2
+
K + A2 2−1b 2 −1b
K
−1
A2
+
A2
K K A2 2−2 −1 A2 2−2 2− 2 2− 2
K
)
K
(
O
O
−1
O
K + KT1 −1 K1−2 δa1 O 1−2 O O 1− 2 1− 2 a1 T1
K
+ K
)
(29)
K δ
Therefore, the analytical model of the ICD of the 2-DOF CPM can be defined as ICD =
A2
K2−1a + A2 K2−1b + A2 K2−2
−1
A2
K2 − 2
O
K1−2 + O KT1
−1
O
T K1 − 2 e 1
e1
(30)
3.4. Natural Frequency In order to ensure the excellent dynamic property of the CPM, its mechanical natural frequency should be large. The first two natural frequencies of the 2-DOF CPM are identical, and the corresponding vibration motions should be the translational modes along the two axes. Owing to the adopted decoupler, the coupling effects between the actuators of the CPM can be ignored. Therefore, the translational mode along the x-axis can be depicted as shown in Figure 13, where ma and mo are the masses of the actuation platform and terminal platform, respectively, and m1 and m2 are the masses of the single leaf spring in Parts I and II, respectively.
(29)
should large. firstthe twoexcellent natural frequencies of the 2-DOF areitsidentical, and natural the corresponding In be order to The ensure dynamic property of theCPM CPM, mechanical frequency vibration motions the translational modes alongCPM the are twoidentical, axes. Owing the adopted should be large. Theshould first twobenatural frequencies of the 2-DOF and thetocorresponding decoupler, the coupling the actuators of the can be ignored. vibration motions shouldeffects be the between translational modes along the CPM two axes. Owing to theTherefore, adopted the translational mode along the x-axis can be depicted as shown in Figure 13, where m a and mo are decoupler, the coupling effects between the actuators of the CPM can be ignored. Therefore, the masses themode actuation platform, respectively, andwhere m1 and m2 are the Appl.translational Sci. 2017,of 7, 591 14o of 28 the along platform the x-axisand can terminal be depicted as shown in Figure 13, ma and m are masses of the single leaf spring in Parts I and II, respectively. the masses of the actuation platform and terminal platform, respectively, and m1 and m2 are the masses of the single leaf spring in Parts I and II, respectively.
m1
m2
m1
m2
ma mo
ma
mo
Figure 13. Translational mode along the x-axis of 2-DOF CPM. Figure 13. Translational mode along the x-axis of 2-DOF CPM. Figure 13. Translational mode along the x-axis of 2-DOF CPM.
Due to the continuum structure of the CPM, its dynamic response analysis should utilize the Due to the continuum structure of theare CPM, its dynamic analysis should utilize the partial differential equation structure models which difficult to solve.response Considering that should the axial stiffness Due to the continuum of the CPM, its dynamic response analysis utilize the partial differential equation models which are difficult to solve. Considering that the axial stiffness of of Part differential II of each chain is large, the displacements of thetoactuation platform, terminal platform and partial equation models which areof difficult solve. Considering that the axial stiffness Part II of each chain is large, the displacements the actuation platform, terminal platform and Part II Part II of the first and third chains can be considered to beactuation almost identical in the translational mode of Part II of each chain is large, the displacements of the platform, terminal platform and of the the firstx-axis. and third chains the canCPM be considered to be identical in the translational mode along along equivalent toalmost atogeneralized single degree-of-freedom (SDOF) Part II of the firstTherefore, andthe third chains canisbe considered be almost identical in the translational mode the x-axis. Therefore, CPM is equivalent to a generalized single degree-of-freedom (SDOF) system system in x-axis. this vibration mode, shown in Figure to 14.a generalized single degree-of-freedom (SDOF) along Therefore, the as CPM is equivalent in thisthe vibration mode, as shown in Figure 14. system in this vibration mode, as shown in Figure 14. Fixed
Fixed
m1
m2
m1
m2 M T = 2ma + mo + 6m2 M T = 2ma + mo + 6m2
Fixed Fixed
Figure 14. Translational mode of an equivalent generalized SDOF system. Figure SDOF system. system. Figure 14. 14. Translational Translational mode mode of of an an equivalent equivalent generalized generalized SDOF
The Rayleigh method can be utilized to calculate the natural frequency of the equivalent SDOF system as s Keq 1 f = (31) 2π Meq where Keq and Meq are the equivalent stiffness and mass of the system. The equivalent stiffness of the SDOF can be obtained as Keq = 8
Z L 0
EI1 ( x )[ψ00 ( x )]2 dx + 6
Z L 0
EI2 ( x )[ψ00 ( x )]2 dx
(32)
f =
1 2π
K eq
(31)
M eq
where Keq and Meq are the equivalent stiffness and mass of the system. stiffness of the SDOF can be obtained as
The2017, equivalent Appl. Sci. 7, 591
L
2
L
15 of 28 2
Keq = 8 EI1 ( x ) ψ ′′ ( x ) dx + 6 EI 2 ( x ) ψ ′′ ( x ) dx (32) 0 0 where I1 (x) and I2 (x) are the moment of inertia of the single leaf spring in Parts I and II, respectively, where I2(x) are the moment of inertia of the single leaf spring in Parts I and II, respectively, and ψ( xI1)(x) is and the shape function. ψ ( xequivalent and The function. mass of the SDOF system can be obtained as ) is the shape The equivalent mass of the SDOF system can be obtained as Meq = MT + M f (33) M eq = M T + M f (33) where MT = 2ma + mo + 6m2 is the mass of the rigid moving part, and Mf is the equivalent mass of the where MT = 2ma + mo + 6m2 is the mass of the rigid moving part, and Mf is the equivalent mass of the flexure members, which flexure members, which Z L L
Z L L
00
00
22 MM 8 m1 ( x )[ψψ(( xx)] dx )[ψ( (xx) )]2 2dx f = dx + + 66 m2m(2x()xψ ) dx f = 8 m1 ( x )
(34) (34)
The calculation calculationaccuracy accuracy of natural the natural frequency of thesystem SDOFdepends systemondepends on the The of the frequency of the SDOF the authenticity authenticity of the estimated shape function. According tocharacteristics, the constraint the characteristics, the flexure of the estimated shape function. According to the constraint flexure members in the members in the translational vibration mode be treated as fixed-fixed beams. Therefore, thecan shape translational vibration mode can be treated ascan fixed-fixed beams. Therefore, the shape function be function canasbeshown determined as shown in Figure 15, where the of rotational angles ofthe both ends under determined in Figure 15, where the rotational angles both ends under applied loads the applied loads equal to zero. equal to zero. v ( x ) = Z 0ψ ( x ) Z0
x L
M F
Figure 15. 15. Determination Determination of of the the shape shape function. function. Figure
According to Figure 15, the shape function of the flexure members can be determined as According to Figure 15, the shape function of the flexure members can be determined as v ( x ) 3x 2 2 x3 ψ ( x) = = 22 − 3 3 (35) v( x ) 2x 3x (35) ψ( x ) = Z 0 = L2 − L 3 Z0 L L Substituting Equation (34) in Equations (31) and (32) can obtain the equivalent stiffness and mass Substituting Equation (34) in Equations (31) and (32) can obtain the equivalent stiffness and mass of the SDOF system as of the SDOF system as 8 EDt313 66EDt EDt233 8EDt (36) 1 K = + 3 Keq eq= + L33 2 (36) LL3 L 104 288 104 288 MeqM= 2m a + moo + m11++ mm (37) + m (37) eq = 2m a +m 2 2 35 35 3535 4. Geometric Parameters Optimization and Performance Evaluation 4. Geometric Parameters Optimization and Performance Evaluation 4.1. Geometric Parameters Optimization 4.1. Geometric Parameters Optimization To ensure the excellent performances of the 2-DOF CPM, it is necessary to optimize its geometric parameters. The is to guarantee the excellent of geometric the CPM. To ensure theoptimum excellentgoal performances of the 2-DOF CPM, itdecoupling is necessaryperformance to optimize its Because the symmetric structure of the CPM already possesses a total output decoupling character, parameters. The optimum goal is to guarantee the excellent decoupling performance of the CPM. the minimization of the structure input coupling selected as the optimization objective, i.e., min (ICD). Because the symmetric of the will CPMbealready possesses a total output decoupling character, The CPM is fabricated from a single block with a specific thickness of 15 mm, and the independent geometric parameters to be optimized are L, t1 , t2 , and d. The other performances of the CPM, including the workspace, output rotation stiffness, and natural frequency, are guaranteed by the constraint conditions described as follows. (1)
Workspace: δx × δy ≥ 47.5 µm × 47.5 µm. According to the analytical models, the workspace of the CPM is influenced by the stiffness of the actuator and the mechanism, and is also limited by the maximum stress of the materials. The nominal motion stroke of the piezoelectric actuator
Appl. Sci. 2017, 7, 591
(2)
(3)
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adopted in this CPM is 60 µm, and its static axial stiffness is about 15 N/µm. Additionally, alloy Al-7075 with a yield stress of 525 MPa is chosen to fabricate the CPM. According to the given workspace, Equations (24) and (26) can be utilized to determine the satisfied ranges of the geometric parameters. Output rotation stiffness: Kα ≥ 3.5 × 104 Nm/rad. The translational deformations of the terminal platform caused by external loads along the x- and y-axes can be compensated through the closed-loop control. However, the rotational deformation about the z-axis cannot be compensated, and thus will affect the motion precision of the CPM. Therefore, the output rotation stiffness is expected to be large enough. Natural frequency: f ≥ 800 Hz. The first two natural frequencies of the 2-DOF CPM are identical and correspond to the translational vibration modes along the two axes. Therefore, they should be as high as possible to guarantee the CPM with a rapid dynamic response and robustness against the disturbances.
The optimization process of the CPM’s geometric parameters is performed with the “fmincon” function. It is an optimization function in MATLAB used to find a constrained minimum of a scalar function of several variables and its grammar is given as
[x, fval] = fmincon(fun, x0, A, b, Aeq, beq, lb, ub, nonlcon)
(38)
where x is the optimized solution, fval is the value of the objective function at the solution x, fun is the objective function, x0 is the initial point for x, A is the matrix for linear inequality constraints, b is the vector for linear inequality constraints, Aeq is the matrix for linear equality constraints, beq is the vector for linear equality constraints, lb is the vector of lower bounds, ub is the vector of upper bounds, and nonlcon is the nonlinear constraint function. The minimization of the ICD is adopted as the objective function and the requirement of the workspace, output rotation stiffness, and the natural frequency of the CPM are used as the nonlinear constraint conditions in the “fimincon” function. There are no linear inequality and linear equality constraints in the geometric parameters optimization, and therefore set A = [], b = [], Aeq = [], beq = []. Additionally, the ranges of the geometric parameters are given as 50 mm ≤ L ≤ 75 mm, 1 mm ≤ t1 ≤ 5 mm, 1 mm ≤ t2 ≤ 5 mm, 5 mm ≤ d ≤ 10 mm, and their initial values are given as L = 60 mm, t1 = 3 mm, t2 = 3 mm, and d = 7.5 mm. Finally, the optimized parameters are obtained as L = 50 mm, t1 = 3.56 mm, t2 = 2.57 mm, and d = 10 mm. The ICD of the optimized CPM is as low as 0.279%, which indicates that the 2-DOF CPM possesses an excellent input decoupling character. The analytical results of other performance indicators are also obtained. The results show that the CPM has a workspace of 47.54 µm × 47.54 µm. The input and output stiffness are 3.68 N/µm and 3.50 × 104 Nm/rad, respectively. In addition, the natural frequency is 876 Hz. The analytical results show that the key performances of the optimized CPM all satisfy the design requirements. 4.2. Performance Evaluation with Finite Element Analysis The optimized mechanism’s performance will be evaluated through FEA, performed with the ANSYS WorkBench software (Workbench 14.5, ANSYS Inc., Cannonburg, PA, USA). The physical and mechanical parameters of the adopted material alloy Al-7075 are: Young’s modulus = 71.7 GPa, Poisson’s ratio = 0.33, Density = 2770 kg/m3 , and yield stress = 525 MPa. A good finite element model with a reasonably high number of DOFs is first required to guarantee the accuracy of FEA results. The finite element model of the CPM is presented in Figure 16, wherein the mesh is generated by using the hex dominant method. The maximum element size is set as 2 mm and the numbers of the nodes and elements are 139,471 and 32,016, respectively.
Appl. Sci. 2017, 7, 591
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Appl. Sci. 2017, 7, 591
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A good finite element model with a reasonably high number of DOFs is first required to guarantee thefinite accuracy of FEA results. Theafinite elementhigh model of the CPM is presented Figure 16, A good element model with reasonably number of DOFs is first in required to Appl. Sci. 2017, 7, 591 17 of 28 wherein the mesh is generated by using the hex dominant method. The maximum element size is set guarantee the accuracy of FEA results. The finite element model of the CPM is presented in Figure 16, as 2 mm the andmesh the numbers of the elements are 139,471 and 32,016, respectively. wherein is generated bynodes usingand the hex dominant method. The maximum element size is set as 2 mm and the numbers of the nodes and elements are 139,471 and 32,016, respectively.
Figure 16. 16. Finite Finite element element model model of of the the CPM. CPM. Figure Figure 16. Finite element model of the CPM.
The input axial stiffness of the CPM is calculated by applying a force at the actuation point A1, The input axial stiffness of the CPM is calculated by applying a force at the actuation point A1 , as shown in Figure The relationship the applied forceaand theatdeformation pointAA1,1 The input axial17a. stiffness of the CPM between is calculated by applying force the actuationatpoint as shown in Figure 17a. The relationship between the applied force and the deformation at point is obtained and is given in Figure 17b. The force–displacement curve fits into a line very well, as shown in Figure 17a. The relationship between the applied force and the deformation at point A1 A1 is obtained and is given in Figure 17b. The force–displacement curve fits into a line very well, which indicates linearity the CPM. axial stiffness is curve obtained calculating the slope is obtained and the is given in of Figure 17b. The Theinput force–displacement fitsby into a line very well, which indicates the linearity of the CPM. The input axial stiffness is obtained by calculating the of the indicates fitted linethe and resorting to CPM. the least method, which is 3.36 by N/μm. In actual which linearity of the The squares input axial stiffness is obtained calculating the CPM, slope slope of the fitted line and resorting to the least squares method, which is 3.36 N/µm. In actual preloading forces will cause initial displacements along is each axis, which influence of the fittedand lineactuation and resorting to the least squares method, which 3.36 N/μm. Inmay actual CPM, CPM, preloading and actuation forces will cause initial displacements along each axis, which may the CPM’s and stiffness. Taking a will small initial displacement (lessalong thaneach 100 axis, μm) which along may one influence axis into preloading actuation forces cause initial displacements influence the CPM’s stiffness. Taking a small initial displacement (less than 100 µm) along one axis consideration, both the theoretical and finite element models have demonstrated that the CPM’s the CPM’s stiffness. Taking a small initial displacement (less than 100 μm) along one axis into into consideration, both the theoretical and finite element models have demonstrated that the CPM’s stiffness remains unchanged. consideration, both the theoretical and finite element models have demonstrated that the CPM’s stiffness remains unchanged. stiffness remains unchanged.
(a)
(b)
(a) model and results of input axial stiffness. (b) (a) Finite element model. Figure 17. Finite element (b) Force-displacement curve. Figure 17. Finite axial stiffness. stiffness. (a) Figure 17. Finite element element model model and and results results of of input input axial (a) Finite Finite element element model. model. (b) Force-displacement curve. (b) Force-displacement curve.
Similarly, the output rotation stiffness of the CPM can be analyzed through applying bending moment at the terminal platform, shown in 18a.can Thebe relationship betweenapplying the rotation angle Similarly, the output rotationasstiffness ofFigure the CPM analyzed through bending Similarly, the output rotation stiffness of the CPM can be analyzed through applying bending and applied given in as Figure 18b, and the output rotation stiffness canthe berotation calculated as moment at themoment terminalisplatform, shown in Figure 18a. The relationship between angle moment the terminal platform, as shown in Figure 18a. The relationship between the rotation angle 4at 3.33 × 10 Nm/rad. and applied moment is given in Figure 18b, and the output rotation stiffness can be calculated as and applied is given in input Figureaxial 18b, stiffness and the and output stiffness stiffness can be calculated as 4 Nm/rad. bemoment observed that the therotation output rotation of the CPM 3.33 ×It10can 4 Nm/rad. 3.33 × 10 calculated FEA that are 8.7% and 4.9% than thethe theoretical values. The reason because It can through be observed the input axialsmaller stiffness and output rotation stiffness of isthe CPM It can beplatform observedand that the input axial stiffness and the output rotation stiffness of themodel, CPM the actuation terminal platform are considered to be rigid parts in the analytical calculated through FEA are 8.7% and 4.9% smaller than the theoretical values. The reason is because calculated through are 8.7% under and 4.9% than the theoretical values. Thestress reasondistribution is because butactuation they also sufferFEA deformation the smaller applied loads. This from the platform and terminal platform are considered to is beevident rigid parts inthe the analytical model, the actuation platform and terminal platform are considered to be rigid parts in the analytical model, diagrams shown indeformation Figure 19. Therefore, the compliance of the platforms thedistribution stiffness of but they also suffer under the applied loads. This is evident fromreduces the stress but they also suffer deformation under the applied loads. This is evident from the stress distribution the CPM. diagrams shown in Figure 19. Therefore, the compliance of the platforms reduces the stiffness of diagrams shown in Figure 19. Therefore, the compliance of the platforms reduces the stiffness of the CPM. the CPM.
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(a) (b) (a) (b) Figure 18. Finite element model and results of output rotation stiffness. (a) Finite element model. Figure Finite element element model model and and results results of of output (a) Finite Finite element element model. model. Figure 18. 18. Finite output rotation rotation stiffness. stiffness. (a) (b) Force-displacement curve. (b) Force-displacement curve. (b) Force-displacement curve.
(a) (a)
(b) (b)
Figure 19. Stress diagrams of the CPM under applied force and moment. (a) Stress diagram related to Figure 19. Stress diagrams of the CPM under applied force and moment. (a) Stress diagram related to Figure Stress of the CPM under applied force and moment. (a) Stress diagram related to the axial19. force. (b)diagrams Stress diagram related to the moment. the axial force. (b) Stress diagram related to the moment. the axial force. (b) Stress diagram related to the moment.
The finite element model shown in Figure 17a can be utilized to calculate the ICD of the CPM. The finite element model shown in Figure 17a can be utilized to calculate the ICD of the CPM. WhenThe the driving forcemodel 100 N is applied at the actuation pointtoAcalculate 1 along the x-axis, the axial element shown in Figure 17a can be utilized the ICD of the When thefinite driving force 100 N is applied at the actuation point A1 along the x-axis, the CPM. axial deformation at point A 1 and the bending deformation at point A2 are 29.8 μm and 8.66 × 10−2 μm, When the driving force 100 N is applied at the actuation point A along the x-axis, the axial deformation 1 −2 deformation at point A1 and the bending deformation at point A2 are 29.8 μm and 8.66 × 10 μm, respectively. Thethe ICD can bedeformation calculated as 0.291%, is close to the theoretical value. This not at point A1 and bending point Awhich µm and 8.66 × 10−2 µm, respectively. 2 are 29.8 respectively. The ICD can be calculated asat0.291%, which is close to the theoretical value. This not only verifies the validity of the analytical model but also demonstrates the excellent input decoupling The can the be calculated 0.291%, which is close thedemonstrates theoretical value. This notinput only decoupling verifies the onlyICD verifies validity ofas the analytical model but to also the excellent characteristic the CPM.model but also demonstrates the excellent input decoupling characteristic of validity of theof analytical characteristic of the CPM. Furthermore, the resonant frequencies and corresponding vibration modes of the CPM are also the CPM. Furthermore, the resonant frequencies and corresponding vibration modes of the CPM are also simulated. The first the six natural frequencies areand 808 Hz, 808 Hz, 1802 Hz, 3153modes Hz, 3153 Hz,CPM and 3430 Hz, Furthermore, frequencies vibration of the are also simulated. The first six resonant natural frequencies are 808corresponding Hz, 808 Hz, 1802 Hz, 3153 Hz, 3153 Hz, and 3430 Hz, and the first three vibration modes are shown in Figure 20. The simulation results show that the first simulated. first six natural frequencies 808 Hz, 1802 Hz,results 3153 Hz, and and the firstThe three vibration modes are shownare in Figure 20.808 TheHz, simulation show3153 thatHz, the first two natural arevibration identical, and their exist along the x- and y-axes, which 3430 Hz, andfrequencies the first three are vibration shown in modes Figure The simulation results show that two natural frequencies are identical,modes and their vibration modes 20. exist along the x- and y-axes, which correspond to the structural characteristic of the CPM. The third vibration mode is out-of-plane the first two to natural frequencies are identical,ofand vibration modes exist along and y-axes, correspond the structural characteristic thetheir CPM. The third vibration modetheisx-out-of-plane motion and its frequency is more than twice as large as the first two natural frequencies. Additionally, which correspond to the structural characteristic of the CPM. The third vibration mode is out-of-plane motion and its frequency is more than twice as large as the first two natural frequencies. Additionally, the rotational mode corresponds the eighth vibration mode with a frequencyAdditionally, of 4636 Hz. motion and itsvibration frequency is more than twiceto large as the first two natural frequencies. the rotational vibration mode corresponds toasthe eighth vibration mode with a frequency of 4636 Hz. Therefore, thevibration vibration modes with higher frequencies have less of an effect on the translation the rotational corresponds to the eighth vibration mode a frequency 4636 Hz. Therefore, the vibrationmode modes with higher frequencies have less of with an effect on the of translation motion of the CPM. Compared with the simulation results, the analytical model overestimates the Therefore, the vibration modes with higher frequencies have less of an effect on the translation motion motion of the CPM. Compared with the simulation results, the analytical model overestimates the first two natural frequencies with a derivation of 8.4%. This may arise from an overestimation oftwo the of the CPM. Compared with the simulation results, the analytical model overestimates the first first two natural frequencies with a derivation of 8.4%. This may arise from an overestimation of the equivalent stiffness with in Equation (30).of The more accurate equivalent can be replaced by the natural frequencies a derivation 8.4%. This may arise from an stiffness overestimation the equivalent equivalent stiffness in Equation (30). The more accurate equivalent stiffness can beofreplaced by the simulated input axial stiffness, and the first two natural frequencies can be calculated as 805 Hz, simulated input axial stiffness, and the first two natural frequencies can be calculated as 805 Hz, which is close to the simulation result. which is close to the simulation result.
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stiffness in Equation (30). The more accurate equivalent stiffness can be replaced by the simulated input axial stiffness, and the first two natural frequencies can be calculated as 805 Hz, which is close to the simulation result. Appl. Sci. 2017, 7, 591 18 of 26
(a)
(b)
(c)
Figure of the the CPM. CPM. (a) (a) First First mode. mode. (b) mode. (c) Figure 20. 20. First First three three vibration vibration modes modes of (b) Second Second mode. (c) Third Third mode. mode.
5. Experimental Test 5. Experimental Test In this section, a prototype of the 2-DOF CPM was fabricated and several experimental tests In this section, a prototype of the 2-DOF CPM was fabricated and several experimental tests were carried out to evaluate its performance and demonstrate the effectiveness of the proposed were carried out to evaluate its performance and demonstrate the effectiveness of the proposed design method. Referring to the optimized results, the dimensions of the prototype were selected as: design method. Referring to the optimized results, the dimensions of the prototype were selected as: L = 50 mm, t1 = 3.5 mm, t2 = 2.5 mm, and d = 10 mm. The 3-D model and the experimental table of the L = 50 mm, t1 = 3.5 mm, t2 = 2.5 mm, and d = 10 mm. The 3-D model and the experimental table of 2-DOF CPM are shown in Figure 21. The main body of the CPM was fabricated using the milling the 2-DOF CPM are shown in Figure 21. The main body of the CPM was fabricated using the milling process with a single block of Al-7075. From the stress diagrams shown in Figure 19, we can see there process with a single block of Al-7075. From the stress diagrams shown in Figure 19, we can see there is is a concentration of stress where the leaf springs connect to the platforms and the fixed base. a concentration of stress where the leaf springs connect to the platforms and the fixed base. Therefore, Therefore, fillets with a 2 mm radius were machined at both ends of the flexure members to reduce fillets with a 2 mm radius were machined at both ends of the flexure members to reduce the effects the effects of stress concentration. Two 60-μm-stroke piezoelectric actuators (P-843.40 from Physik of stress concentration. Two 60-µm-stroke piezoelectric actuators (P-843.40 from Physik Instrument) Instrument) were adopted to drive the stage. They were driven by a modular piezo controller & were adopted to drive the stage. They were driven by a modular piezo controller & driver (E-621 from driver (E-621 from Physik Instrument), which generates voltage up to 120 V. Two force sensors Physik Instrument), which generates voltage up to 120 V. Two force sensors (U93-1kN from HBM) (U93-1kN from HBM) were mounted at the end of each actuator to detect their actuation forces. were mounted at the end of each actuator to detect their actuation forces. Compared to mounting them Compared to mounting them in front of the actuators, this installation type reduces the moving mass in front of the actuators, this installation type reduces the moving mass of the stage and thus improves of the stage and thus improves its dynamic response. A four-channel high dynamic signal processor its dynamic response. A four-channel high dynamic signal processor (MX410 from HBM) was utilized (MX410 from HBM) was utilized to amplify and record the analog voltage output of the force sensors. to amplify and record the analog voltage output of the force sensors. The displacement outputs of the The displacement outputs of the terminal platform along the two axes were measured using two high terminal platform along the two axes were measured using two high precision length gauges (MT-1281 precision length gauges (MT-1281 from Heidenhain). A DSPACE processor board DS1007 equipped from Heidenhain). A DSPACE processor board DS1007 equipped with a 16-bit ADC card (DS2004), a with a 16-bit ADC card (DS2004), a 16-bit DAC card (DS2101), and a six-channel incremental encoder 16-bit DAC card (DS2101), and a six-channel incremental encoder interface card (DS3002) was utilized interface card (DS3002) was utilized to output the excitation voltage for the piezo servo controller to output the excitation voltage for the piezo servo controller and capture the data from the force and capture the data from the force sensor signal processor and length gauges. Preloads were applied sensor signal processor and length gauges. Preloads were applied along the two axes through preload along the two axes through preload baffles to prevent the piezoelectric actuators from suffering baffles to prevent the piezoelectric actuators from suffering pulling forces during the motion process pulling forces during the motion process and to eliminate the clearances between the actuators and and to eliminate the clearances between the actuators and the mechanism. The preloading forces were the mechanism. The preloading forces were detectable by the force sensors and they were set to about detectable by the force sensors and they were set to about 300 N initially. 300 N initially. The input axial stiffness along the two axes was tested first. The actuation forces were detected using the force sensors and the displacements were measured using the length gauges. Figure 22 shows the relationship between the tested actuation forces and the corresponding displacements along the two axes. The tested displacement–force curves fit into lines well and indicate the stiffness of the CPM is linear, which will be easy to control. The least squares method was utilized to fit the results and the input axial stiffness of the CPM was calculated as 3.87 N/µm and 3.93 N/µm, respectively. The axial stiffness measurements were greater than the theoretical and simulated results. The differences occur for two reasons. First, the fillets machined at both ends of the leaf springs increase their stiffness. Second, the displacement output of the terminal platform is larger than that measured at the actuation point, which results in a larger measured stiffness.
(a)
(b)
interface card (DS3002) was utilized to output the excitation voltage for the piezo servo controller and capture the data from the force sensor signal processor and length gauges. Preloads were applied along the two axes through preload baffles to prevent the piezoelectric actuators from suffering pulling forces during the motion process and to eliminate the clearances between the actuators and Appl.mechanism. Sci. 2017, 7, 591The preloading forces were detectable by the force sensors and they were set to 20 of 28 the about 300 N initially.
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Appl. Sci. 2017,axial 7, 591 stiffness along the two axes was tested first. The actuation forces were 19 of 26 The input detected using the force sensors and the displacements were measured using the length gauges. Figure 22 The input axial stiffness along the two axes was tested first. The actuation forces were detected shows the relationship between the tested actuation forces and the corresponding displacements using the force sensors and the displacements were measured using the length gauges. Figure 22 alongshows the two The tested displacement–force curves fit and intothe lines well and indicate the stiffness theaxes. relationship between the tested actuation forces corresponding displacements of thealong CPMtheistwo linear, will displacement–force be easy to control.curves The least squares method was utilized to fit the axes.which The tested fit into lines well and indicate the stiffness results andCPM the isinput of the CPM The wasleast calculated as 3.87 was N/μm andto3.93 N/μm, of the linear,axial whichstiffness will be easy to control. squares method utilized fit the respectively. The measurements were than the theoretical andN/μm, simulated results and theaxial inputstiffness axial stiffness of the CPM was greater calculated as 3.87 N/μm and 3.93 respectively. The axial stiffness measurements were the greater than the theoretical and simulated results. The differences occur for two reasons. First, fillets machined at both ends of the leaf (a) for two reasons. First, the fillets machined at both (b) results. The differences occur of the leaf springs increase their stiffness. Second, the displacement output of the terminalends platform is larger springs increase their stiffness. Second, the displacement output of the terminal platform is larger than that measured at the which resultsCPM. in a (a) larger stiffness. Figure 21. 3-D actuation model andpoint, prototype of 2-DOF 3-D measured model. (b) Prototype. Figure 21. 3-D model and prototype of 2-DOF CPM. (a) 3-D model. (b) Prototype.
than that measured at the actuation point, which results in a larger measured stiffness.
500
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X x Axis x-Axis X x Axis x-Axis Y Axis yY y-Axis y Axis y-Axis
450 450 400 400 350 350 300
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Figure 22. Experimental results of input axial stiffness.
Figure 22. Experimental results of input axial stiffness. Figure 22. Experimental results of input axial stiffness. The workspace of the CPM was tested through running a square trajectory in the open-loop trajectory open-loop control. The experimental resultwas andtested estimated workspace are agiven in Figure 23. The measured The workspace of the CPM through running square in the workspace was 45.21 μm × 45.63 μm, which is a little smaller than the estimated result. The decreased control. The experimental result and estimated workspace are given in Figure 23. The measured The experimental result and estimated workspace 23. motion of the CPM lies in the deformations of the joint surfaces between the parts connected by bolts. workspace was 45.21 µm × 45.63 µm, which is a little smaller than the estimated result. The decreased workspace was 45.21 μm × 45.63 μm, which is a little smaller than the estimated result. The decreased
in the the deformations deformations of of the the joint joint surfaces surfaces between between the the parts parts connected connected by bybolts. bolts. motion of the CPM lies in 50 50
40
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4030
Measured Estimated
3020 2010 10 0 0
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0 Figure 23. Workspace of 2-DOF CPM. 0 10 20 30 40
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Then the natural frequency of the CPM was tested using the hammering method with the piezoelectric actuators dismantled. A modal hammer was utilized Figure CPM.to apply impulse forces at the Figure 23. 23. Workspace Workspace of of 2-DOF 2-DOF CPM. actuation points along the two axes, respectively. The dynamic response of the terminal platform was captured the length gauges and on the experimental results have been given Then thebynatural frequency of the thespectral CPM analyses was tested using the hammering method with the in Figure 24. The first two natural frequencies of the CPM were obtained as 793 Hz and 799 Hz, piezoelectric actuators dismantled. A modal hammer was utilized to apply impulse forces at the which are close to the simulated values.
actuation points along the two axes, respectively. The dynamic response of the terminal platform was captured by the length gauges and the spectral analyses on the experimental results have been given in Figure 24. The first two natural frequencies of the CPM were obtained as 793 Hz and 799 Hz,
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Then the natural frequency of the CPM was tested using the hammering method with the piezoelectric actuators dismantled. A modal hammer was utilized to apply impulse forces at the actuation points along the two axes, respectively. The dynamic response of the terminal platform was captured by the length gauges and the spectral analyses on the experimental results have been given in Figure 24. The first two natural frequencies of the CPM were obtained as 793 Hz and 799 Hz, which Appl. Sci. 2017, 7, 591 20 of 26 are close to the simulated values.
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Figure Figure 24. 24. Natural Natural frequency frequency test test of of the the CPM. CPM. Figure 24. Natural frequency test of the CPM.
The step response of the CPM was tested where only the piezoelectric actuators adopted the The step step response response of of the CPM was tested where only the piezoelectric piezoelectric actuators actuators adopted adopted the the The closed-loop control. The result of the 1 μm step response test of the x-axis is shown in Figure 25. closed-loop control. The result test of of thethe x-axis is shown in Figure 25. The closed-loop result of ofthe the11µm μmstep stepresponse response test x-axis is shown in Figure 25. The rising time was 8 ms without overshoot. Similar results in the y-axis were obtained, but are not rising timetime was 8was ms8without overshoot. Similar resultsresults in thein y-axis were obtained, but arebut notare shown The rising ms without overshoot. Similar the y-axis were obtained, not shown herein. The results show that the CPM has a rapid response performance. herein. herein. The results show that thethat CPM a rapid performance. shown The results show thehas CPM has aresponse rapid response performance.
Displacement (μm) Displacement (μm)
4 4 3 3 2 2 1 1 0 0
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1.0 1.0 0.5 0.5 0 02.53 2.53
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Figure 25. Step response test of the x-axis of the stage. Figure Figure 25. 25. Step Step response response test test of of the the x-axis x-axis of of the the stage. stage.
The positioning resolution of the CPM was tested in the closed-loop control. A consecutive-step The positioning resolution of the CPM was tested in the closed-loop control. A consecutive-step positioning with a step size of 2ofnm out for each axis; results are shown in Figure 26. The positioning resolution thewas CPMcarried was tested in the closed-loop control. A consecutive-step positioning with a step size of 2 nm was carried out for each axis; results are shown in Figure 26. The steps can be clearly identified, which indicates that the positioning resolution of the CPM better positioning with a step size of 2 nm was carried out for each axis; results are shown in Figureis26. The The steps can be clearly identified, which indicates that the positioning resolution of the CPM is better than nm.beMoreover, the resolution be improved by filtering resolution the measurement noise. steps2can clearly identified, which can indicates that the positioning of the CPM is better than than 2 nm. Moreover, the resolution can be improved by filtering the measurement noise. 2 nm.10Moreover, the resolution can be improved by filtering the measurement noise. 10 10 10 The output decoupling performance of the 2-DOF 8 CPM was evaluated by actuating one axis 8 8 platform along the other axis. The CPM’s 8 and measuring the displacement output of the terminal 6 6 maximum output motion ranges along the two axes were both set as 40 µm. Figure 27 shows the 6 6 4 4 coupled 4 motions along the two axes when the preload 4force along the x-axis changed from 240 N to 2 unchanged. It can be seen that the coupling 320 N2 while the preload force along the y-axis remained 2 2 0 motions along the y-axis varied significantly as the preload force changed. However, the variation of 0 0 0 the coupling motions along the x-axis was slight. -2 -2 0 0
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(a) (b) (a) (b) Figure 26. Positioning resolution test of the stage. (a) Positioning resolution of the x-axis. (b) Positioning Figure 26. Positioning resolution test of the stage. (a) Positioning resolution of the x-axis. (b) Positioning resolution of the y-axis. resolution of the y-axis.
Figure 25. Step response test of the x-axis of the stage.
The positioning resolution of the CPM was tested in the closed-loop control. A consecutive-step positioning with a step size of 2 nm was carried out for each axis; results are shown in Figure 26. Appl. Sci. 2017, 7, 591 of 28 The steps can be clearly identified, which indicates that the positioning resolution of the CPM is 22 better than 2 nm. Moreover, the resolution can be improved by filtering the measurement noise. 10
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21 of 26 force along thethe y-axis remained unchanged. It can be seen that the coupling Figure 26. Positioning resolution test of stage. (a) Positioning resolution of the x-axis. (b) Positioning Figure 26. Positioning resolution test of the stage. (a) Positioning resolution of the x-axis. (b) Positioning motions alongthe they-axis. y-axis varied significantly as the preload force changed. However, the variation of resolution 320 N whileof the preload force the y-axis remained unchanged. It can be seen that the coupling resolution of the y-axis. the coupling motions along thealong x-axis was slight. motions along the y-axis varied significantly as the preload force changed. However, the variation of 0.8 0.3 was evaluated by actuating one axis and The outputmotions decoupling ofslight. the 2-DOF CPM the coupling alongperformance the x-axis was
y-Axis y-Axis Displacement (μm) (μm) Displacement
Appl. Sci.while 2017, 7,the 591preload 320 N
0.6 the displacement output of the terminal platform along the other axis. The CPM’s measuring 0.8 0.3 0.2 maximum 0.4 output motion ranges along the two axes were both set as 40 μm. Figure 27 shows the 0.6 coupled 0.2 motions along the two axes when the preload force 0.2 0.1 along the x-axis changed from 240 N to 0.4 0 0.2 -0.2 0 0 -0.2
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Figure 27. Coupling motions along the two axes when the preload force along the x-axis changes.
(a) (b) Figure 27. Coupling motions along the two axes when the preload force along the x-axis changes. (a) Coupling motion along the x-axis. (b) Coupling motion along the y-axis. (a) Coupling motion along the x-axis. (b) Coupling motion along the y-axis. Figure 27. Coupling motions along the two axes when the preload force along the x-axis changes. (a) motion along the x-axis. (b) axes Coupling motion along the y-axis. TheCoupling coupling motions along the two when the preload force along the y-axis changed from
The motions the 28, twoand axes when conclusion the preloadcan force along the y-axis changed from 240 N coupling to 320 N are shownalong in Figure a similar be obtained. The coupling motions along the two axes when the preload force along the y-axis changed from 240 N to 320 N are shown in Figure 28, and a similar conclusion can be obtained. 0.6 y-Axis y-Axis Displacement (μm) (μm) Displacement
0.4 240 N to 320 N are shown in Figure 28, and a similar conclusion can be obtained.
0.6 0.4
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Figure 28. Coupling motions along the two axes when the preload force along (a) (b) the y-axis changes. (a) Coupling motion along the x-axis. (b) Coupling motion along the y-axis. Figure 28. Coupling motions along the two axes when the preload force along the y-axis changes. Figure 28. Coupling motions along the two axes when the preload force along the y-axis changes. (a) motion the x-axis. motion the y-axis. errors in the flatness of the TheCoupling cases shown inalong Figures 27 and(b) 28Coupling may come fromalong manufacturing
(a) Coupling motion along the x-axis. (b) Coupling motion along the y-axis. measuring datum planes. Figure 29 provides an explanation for the tested results of the coupling The cases in Figures and 28 may come from manufacturing flatness of of the motions whenshown the preload force27along the x-axis changes. It can be seenerrors that in thethe variation the The cases shown in Figures 27 and 28 may come from manufacturing errors in the flatness measuring datum planes. Figure 29 provides an explanation for the tested results of the coupling preload load force along the x-axis will alter the initial measuring point on the terminal platform of when thecoupling preload force along thethe x-axis It can bethe seen that the variation of the themotions measuring datum planes.motions Figure 29 provides anchanges. explanation for tested results of the coupling while testing the along y-axis. The initial measuring point, however, remains preload load force along the x-axis will alter the initial measuring point on the terminal platform motions when thetesting preload force along the x-axis changes. It can be the seen that the variation of the the same when the coupling motions along the x-axis. Therefore, flatness of the measuring while testing the coupling motions along the y-axis. The initial measuring point, however, remains preload load force along the x-axis will alter the initial measuring point on the terminal platform while datum planes will influence the test of the coupling motions along the y-axis when the preload force the same when testing the coupling motions along the x-axis. Therefore, the flatness of the measuring testing the coupling motions along the y-axis. The initial measuring point, however, remains the same along the x-axis varies. datum planes will influence the test of the coupling motions along the y-axis when the preload force when testing the coupling motions along the x-axis. Therefore, the flatness of the measuring datum along the x-axis varies.
planes will influence the test of the coupling motions along the y-axis when the preload force along the x-axis varies.
preload load force along the x-axis will alter the initial measuring point on the terminal platform while testing the coupling motions along the y-axis. The initial measuring point, however, remains the same when testing the coupling motions along the x-axis. Therefore, the flatness of the measuring datum planes influence the test of the coupling motions along the y-axis when the 23 preload force Appl. Sci. will 2017, 7, 591 of 28 along the x-axis varies.
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Figure 29. Coupling motion tests the preload along x-axis (a) Coupling Figure 29. Coupling motion tests when when the preload force force along the x-axisthe changes. (a) changes. Coupling motion along the (b)(b) Coupling motion along the x-axis. motion along the y-axis. Coupling motion along the x-axis. Appl. Sci. 2017, 7, y-axis. 591 22 of 26 Because Because the the preload preload forces forces affect the coupling coupling motions motions of of the CPM, we can adjust the preload forces to minimize the cross-axis outputs. Through Through trial and and error, error, the preload forces along the x- and y-axes, respectively, were chosen as 250 N and 270 N. The test for the two axes’ output coupling was performed and the results are shown in Figure 30. It can be seen that the maximum maximum output couplings between the two axes are less than 0.5%. 0.05
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Figure Figure 30. 30. Output Output coupling coupling evaluation evaluation of of CPM CPM through through adjusting adjusting the the preload preload forces. forces.
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Since the flatness of the measuring datum planes has a negative effect on the output decoupling Since the flatness of the measuring datum planes has a negative effect on the output decoupling performance of the CPM, the direct way of reducing its output coupling is to improve the performance of the CPM, the direct way of reducing its output coupling is to improve the manufacturing accuracy of the measuring base. The flatness of the measuring datum planes was manufacturing accuracy of the measuring base. The flatness of the measuring datum planes was improved to 1 μm. The preload forces along the two axes were set to 300 N. Then the output coupling improved to 1 µm. The preload forces along the two axes were set to 300 N. Then the output coupling of the CPM was re-evaluated and the results are given in Figure 31. It shows that the cross-axis of the CPM was re-evaluated and the results are given in Figure 31. It shows that the cross-axis outputs outputs of the CPM are significantly reduced and the maximum output couplings between the two of the CPM are significantly reduced and the maximum output couplings between the two axes are axes are only 0.105% and 0.045%, respectively. only 0.105% and 0.045%, respectively. The actuation point deformation should be measured to evaluate the ICD of the CPM, which is X-axis X-Axis 0.02 impossible to be achieved experimentally in this prototype. To conquer Y-axis Y-Axisthis problem, we proposed utilizing the ratio of the axial deformations at the two actuation points to reflect the input coupling 0.01 performance of the 2-DOF CPM. Because of the linear displacement–force relation as shown in Figure 22, the changes of axial 0forces along the two axes were detected to indirectly evaluate the input coupling of the CPM. Tests were run on the variations of axial forces along the two axes when the x- or -0.01 32. The results show that when the CPM moves through the whole y-axis is actuated, shown in Figure range along the x-axis, the changes of the axial forces along the x- and y-axes are 169.4 N and 0.41 N, -0.02 0 through 10 the whole 20 range along 30 the x-axis, 40 the changes of the axial respectively. When the CPM moves forces along the x- and y-axes are 0.34 N and 170.2 N, respectively. The corresponding coupling ratios Figure 31. Output coupling evaluation of CPM through improving the manufacturing accuracy of the measuring datum.
The actuation point deformation should be measured to evaluate the ICD of the CPM, which is impossible to be achieved experimentally in this prototype. To conquer this problem, we proposed
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Figure 30. Output coupling evaluation of CPM through adjusting the preload forces.
Since the flatness of the measuring datum planes has a negative effect on the output decoupling Appl. Sci. 2017, 7, 591 24 of 28 performance of the CPM, the direct way of reducing its output coupling is to improve the manufacturing accuracy of the measuring base. The flatness of the measuring datum planes was can be calculated 0.24% andforces 0.20%,along where theoretical calculated (28) are improved to 1 μm.as The preload thethe two axes werevalues set to 300 N. Thenby theEquation output coupling all 0.084%. The difference may lie in manufacturing and assembling errors. Regardless, this method of the CPM was re-evaluated and the results are given in Figure 31. It shows that the cross-axis providesofa the feasible evaluate the input coupling of the CPMoutput and demonstrates this prototype’s outputs CPMway are to significantly reduced and the maximum couplings between the two excellent input decoupling performance. axes are only 0.105% and 0.045%, respectively. X-axis X-Axis Y-axis Y-Axis
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The actuation point deformation should be measured to evaluate the ICD of the CPM, which is x x-Axis 0 y-Axis y impossible to be achieved experimentally in this prototype. To conquer this problem, we proposed utilizing the ratio of the axial-0.1 deformations at the two actuation points to reflect the input coupling performance of the 2-DOF CPM. Because of the linear displacement–force relation as shown in Figure 22, -0.2 the two axes were detected to indirectly evaluate the input coupling the changes of axial forces along of the CPM. Tests were run on the variations of axial forces along the two axes when the x- or y-axis is actuated, shown-0.3 in Figure 32. The results show that when the CPM moves through the whole range along the x-axis,-0.4 the changes of the axial forces along the x- and y-axes are 169.4 N and 0 100whole range 150 along the x-axis, the changes of 0.41 N, respectively. When the CPM moves50 through the the axial forces along the x- and y-axes are 0.34 N and 170.2 N, respectively. The corresponding Figure Input coupling evaluation CPM. coupling ratios can be calculated 0.24% 0.20%, where of the theoretical values calculated by Figureas32. 32. Input and coupling evaluation of CPM. Equation (28) are all 0.084%. The difference may lie in manufacturing and assembling errors. This study evaluated the acontour the CPM 2-Dcoupling trajectories, to reflect the Regardless, this also method provides feasibletracking way to of evaluate the for input of the CPM and This study also evaluated the contour tracking of the CPM for 2-D trajectories, to reflect the two-axis cooperative tracking performance. To adequately verify the decoupling performances of demonstrates this prototype’s excellent input decoupling performance. two-axis cooperative tracking performance. To adequately verify the decoupling performances the CPM, the actuators in the CPM were controlled independently without any additional of the CPM, the actuators in the CPM were controlled independently without any additional kinematic calibration or compensation. A Proportion-Integration algorithm was utilized to realize the kinematic calibration or compensation. A Proportion-Integration algorithm was utilized to realize the closed-loop control. closed-loop control. The 2-D tracking results for ◦30°, ◦45°, and◦ 60° lines with a contouring velocity of 40 μm/s are The 2-D tracking results for 30 , 45 , and 60 lines with a contouring velocity of 40 µm/s are shown shown in Figure 33a, and the corresponding contouring errors have been plotted in Figure 33b. in Figure 33a, and the corresponding contouring errors have been plotted in Figure 33b. The contouring The contouring error here is defined as the minimum distance between the actual position and error here is defined as the minimum distance between the actual position and desired trajectory along desired trajectory along an orthogonal direction to the trajectory. The results show that the maximum an orthogonal direction to the trajectory. The results show that the maximum contouring error is contouring error is smaller than 25 nm. smaller than 25 nm. The experimental tests of circular contouring at different frequencies are shown in Figure 34, 20 25 where the radius of the circles is 10 µm. The measured circular trajectories are coincident with the 10 ideal trajectory without elliptical distortion, which contributes to the identical dynamic characteristics 20 decoupling performance of the two axes. 0To obtain a more intuitive comparison, the and excellent 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 maximum and average contouring errors of the circular contour with different frequencies has been illustrated in15Figure 35. The results show that the contouring errors grow up as the frequency increases, 20 and the maximum error is less than 150 nm. The contouring accuracy can be improved with the help 10 of an advanced control strategy, which will be investigated in our future work. 10 0
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kinematic calibration or compensation. A Proportion-Integration algorithm was utilized to realize the closed-loop control. The 2-D tracking results for 30°, 45°, and 60° lines with a contouring velocity of 40 μm/s are shown in Figure 33a, and the corresponding contouring errors have been plotted in Figure 33b. The contouring error here is defined as the minimum distance between the actual position and Appl. Sci. 2017,trajectory 7, 591 desired along an orthogonal direction to the trajectory. The results show that the maximum25 of 28 contouring error is smaller than 25 nm. 20
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Figure 33. Experimental results of 2-D linear contouring: (a) linear contours; and (b) contour accuracy.
Figure 33. Experimental results of 2-D linear contouring: (a) linear contours; and (b) contour accuracy. Appl. Sci. 2017, 7, 591 24 of 26
The experimental tests of circular contouring at different frequencies are shown in Figure 34, Y 10 circles is 10 μm. The measured circular trajectories where the radius of the are coincident with the ideal trajectory without elliptical distortion, which contributes to the identical dynamic characteristics 5 and excellent decoupling performanceActual of the two axes. To obtain a more intuitive comparison, the maximum and average contouringIdeal errors of the circular contour with different frequencies has 0 X as the frequency been illustrated in Figure 35. The results show that the contouring errors grow up increases, and the maximum error is less than 150 nm. The contouring accuracy can be improved -5 with the help of an advanced control strategy, which will be investigated in our future work. -10 -10
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Figureand 34.(c) Circular contouring performance at different frequencies: (a) f = 0.5 Hz; (b) f = 1 Hz; and f = 5 Hz. (c) f = 5 Hz. Contouring Error (μm)
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6. 6. Conclusions Conclusions A output decoupling A systematic systematic design design method method of of aa 2-DOF 2-DOF CPM CPM with with excellent excellent input input and and output decoupling performances performances was was presented presented in in this this paper. paper. The The symmetric symmetric kinematic kinematic structure structure of of the the CPM CPM can can ensure ensure the mechanism with a total output decoupling character. The input coupling was reduced by resorting to a flexure-based decoupler and geometric parameter optimization. The ICD of the optimized mechanism calculated using the analytical model was as low as 0.279%. A prototype was fabricated according to the optimized result with a measured workspace of 45.21 µm × 45.63 µm; the first two natural frequencies were 793 Hz and 799 Hz. The consecutive-step tests showed that this stage has a high resolution that was better than 2 nm and the rising time of 1 µm step test was only 8 ms without overshoot. The cross-axis outputs of the CPM were minimized by adjusting the preload forces and improving the manufacturing accuracy of the measuring base, and the maximum output coupling between the two axes was only about 0.1%. The input coupling was indirectly evaluated by detecting the axial force variations along the two axes and the coupling ratios were 0.24% and 0.20%. Moreover, the experimental studies show that high tracking accuracy was obtained in the biaxial linear and circular contouring tests, in which the two actuators were controlled independently without any additional kinematic calibration or compensation. Considering they performed well, the developed CPM can be suitable for applications in which high precision and resolution are needed, such as the atomic force scanner and biological manipulation; this will be the goal of our future work. Acknowledgments: This research was supported by the National Natural Science Foundation of China (Project No. 51675952), the National Science and Technology Major Project (Project No. 2015ZX04001002), and the Tsinghua University Initiative Scientific Research Program (Project No. 2014z22068). Author Contributions: Yao Jiang and Tiemin Li proposed the design idea; Yao Jiang performed the design method and developed the prototype of the 2-DOF compliant parallel mechanism; Liping Wang and Feifan Chen performed the experiments and analyzed the data; and Tiemin Li and Yao Jiang wrote the paper together. Conflicts of Interest: The authors declare no conflict of interest.
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