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Rigid 6-DOF parallel platform for precision 3-D micromanipulation
International Journal of Machine Tools & Manufacture 41 (2001) 1229–1250
Rigid 6-DOF parallel platform for precision 3-D micromanipulation Vladimir T. Portman*, Ben-Zion Sandler, Eliahu Zahavi Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105, Israel Received 1 November 2000; accepted 31 January 2001
Abstract A new type of 6-DOF platform mechanism characterized by very high stiffness and a high degree of accuracy is described. Welded joints connecting the changeable links with the platform and base provide the high stiffness. Hydraulically controlled microactuators use longitudinal elastic deformations for elongation of the legs. Infinitesimal kinematics of a conventional platform, the finite-elements method, and an identification procedure are used for the analysis of the mechanism. Experimental investigations of a pilot set-up of the proposed platform confirmed the predictions concerning the accuracy and stiffness of the proposed device. A practical application of the platform as a secondary table for precise machining on a CNC milling center is described. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Parallel mechanism; Micromanipulation; Secondary table; Milling
1. Introduction Parallel platform constitutes a very effective tool for accomplishing a variety of tasks in 3-D space [1] providing six degrees of freedom (DOF) manipulations. Under consideration here is the development of a specific type of the platform designed for high precision manipulation in microdomain. The term ‘micro’ relates to spatial displacements in the domain of about 10⫺6–10⫺4 m, while the required accuracy of the device is of the order of 10⫺9–10⫺7 m, i.e. up to and including the nanometer range. The high accuracy of such mechanisms stems from high structural stiffness and non-serial accumulation of the joint and link errors inherent in these mechanisms. The maximal accuracy is achieved by designing the platform mechanism as an ideal axially symmetric * Corresponding author. Tel.: +972-7-647-7093; fax: +972-7-647-2813. E-mail address: [email protected] (V.T. Portman).
0890-6955/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 0 1 ) 0 0 0 2 7 - X
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structure. This structure has been successfully applied for the development of precision devices with six DOF for a variety of applications. Some examples of these are a six-component force/torque transducer [1–3], an ultra-precision wafer-positioning table for lithography systems [4], a micro-positioning stage with six degrees of freedom for a fine–coarse positioning system [5], etc. The features of the structures of these platform-type micromanipulators are essentially defined by the type of connection of the changeable links (hereafter referred to as ‘legs’) with the base and the upper platform. To enhance the accuracy of platform-type systems for micromanipulations, elastic hinges are used instead of traditional Hooke’s joints. A variety of types of elastic hinges— revolute, prismatic, and spherical—have been applied for parallel micromanipulators [6]. Different types of drives are used within precision manipulators. Most in use are piezoelectric actuators [7,8] as, for example, piezoelectric actuators possessing 3-DOF [9] and 6-DOF [10]. Another type of high-stiffness precision actuator was given in Ref. 11. The latter constitutes an elastic cylinder deformed by controlled internal hydraulic pressure and is used for the so-called ‘rigid platform’ for micropositioning [12]. In this paper, the utilization of the rigid platform as a ‘secondary micrometric coordinate table’ for precision machining is discussed. The laboratory arrangement of a ‘fine–coarse feed’ system mounted on a milling center is shown in Fig. 1. As shown, the XYZ moving units of the machine tool represent a coarse feed subsystem, while the platform fastened on the machine table represents a fine feed subsystem. This platform mechanism has a simple and reliable structure and is relatively inexpensive to produce. It may be used for accurate positioning of workpieces, cutting tools, or end-effectors under relatively high loads in machine tool technology and in robotics. The theoretical and experimental results concerning this platform, its kinematics, control, and preliminary practical applications are discussed. A description of the structure of the mechanism (Section 2) is followed by a consideration of infinitesimal kinematics in Section 3. The statics of the mechanism, including an application of the finite-elements method, is discussed in Section 4, and the dynamics problem is considered in
Fig. 1. Laboratory arrangement of a ‘fine-course feed’ system on the milling center: 1, CNC machine tool; 2, 6-DOF platform; 3, hydraulic control arrangement; 4, computer for the platform control; 5, display of the measuring device.
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Fig. 2. Layout of the platform for micropositioning.
Section 5. Section 6 presents an experimental investigation of the platform, and an application example is given in Section 7. 2. Structure of the platform for micropositioning The mechanism under consideration (Fig. 2) consists of an upper platform connected to a base by six legs M1N1,…,M6N6. The ‘leg-to-platform’ and ‘leg-to-base’ joints are elastic hinges with three degrees of freedom. The platform’s position and its slope in space depend on the lengths of the legs. Six measurement devices, marked in Fig. 2 by arrows 1–6, give displacement data of six points of the upper platform in relation to the base. The mechanism comprises a 6×6 platform of the greatest possible stiffness, achieved by an introduction of the welded ‘leg-to-platform’ and ‘leg-to-base’ joints in the structure [11]. Six independent built-in microactuators, responsible for elongation of the legs, provide the spatial micromanipulation of the platform. An experimental design of the leg connections with the platform and the base is shown in Fig. 3. An elastic deformable steel cylinder is used as a microactu-
Fig. 3. General view of the platform (a) and the design of the leg connections (b): 1, upper platform; 2, leg connection to the upper platform; 3, leg; 4, base; 5, leg connection to the base; LVDT, device for measurement of the leg elongation.
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ator controlling displacements in the micron and submicron ranges [11]. The deformations are caused by hydraulic pressure inside the cylinder. This microactuator, which facilitates delicate and accurate displacements, provides high stiffness and high carrying capacity at a relatively low cost compared with conventional microactuators. The design of the microactuator for displacements in the range 0–40 µm is shown in Fig. 4. The microactuator consists of a base and a deformable hollow cylinder, which create a closed volume filled with a hydraulic liquid under controlled pressure p. This pressure causes axial deformation ⌬l of the relatively thin walls of the cylinder: ⌬l⫽
plpD2in ⫽k p (D2out−D2in)E p
(1)
where Dout and Din are the external and internal diameters of the leg (Fig. 4); lp is the length of the leg to be elongated; E is the Young’s modulus of the leg material; and kp is a transfer factor for leg elongation, kp=⌬l/p. Thus, the nominal dimensions are: E=2.1×105 MPa, lp=70 mm, Dout=35 mm, Din=30 mm, and the transfer factor kp=0.09187 µm/bar. Based on the fact that the last member of the kinematic chain is a deformable stiff metal cylinder, this microactuator will exhibit small hysteresis, good linearity, and good resolution in the domain of possible elastic deformations. 3. Infinitesimal kinematics Infinitesimal kinematics of the conventional platform (i.e. the platform with kinematic hinges between the legs and both plates) is generally described by the Jacobian matrix. In our mechanism, however, substitution of kinematic hinges with elastic or ‘rigid’ hinges results in enhanced stiffness of the mechanism and, hence, in increasing deviations between the computational models of the corresponding structures. In accordance with Fig. 2, the following formal attributes for the ith leg (i=1,…,6) may be introduced: a vector ri of the leg; a unit vector ai of the leg direction; a moment mi of the vector ai relative to the origin of the coordinate system; a six-order vector Ri of Plucker’s coordinates, and the 6×6 Jacobian matrix Rd composed of these vectors Ri:
Fig. 4.
Microactuator layout.
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ri⫽rNi⫺rMi;
(2)
ai⫽(axi,ayi,azi)T⫽ri/li;
(3)
mi⫽(mxi,myi,mzi)T⫽rNi⫻ai;
(4)
Ri⫽(axi,ayi,azi,mxi,myi,mzi)T;
(5)
冢
ax1 ay1 az1 mx1 my1 mz1
冣
R d⫽ % % % % % % , ax6 ay6 az6 mx6 my6 mz6
(6)
where rNi=(xNi,yNi,zNi)T and rMi=(xMi,yMi,zMi)T are the position vectors of the upper and lower ends of the ith leg, respectively; li is the length of the ith leg, li=|ri|; axi, ayi and azi are the direction cosines of the ith leg; and mxi, myi, and mzi are the moments of vector ai relative to the X, Y, and Z axes, respectively. Since the Jacobian Rd is non-singular (det Rd⫽0), vector ⌬ of the small displacements of the platform and vector d of the elongation of the legs are related to each other, accurate to the first order, as follows: ⌬⫽(Rd)−1 d and d⫽Rd⌬,
(7)
d⫽(⌬l1,⌬l2,…,⌬l6)T,
(8)
⌬⫽(dx,dy,dz,a,b,g)T,
(9)
with
where ⌬l1, ⌬l2,…,⌬l6 are the small elongations of six legs; dx, dy, and dz are the small translations of the platform along the X, Y, and Z axes, respectively, or (which is the same) small linear displacements of the origin O⬘ of the coordinate system associated with the platform (see Fig. 2); and a, b, and g are the small angular rotations of the platform relative to the same axes. Expressions (7) may be used to solve the direct and inverse kinematics problems for small displacements of the platform. In order to obtain sufficient data concerning small displacements of the upper platform, considered as a rigid body, it is necessary to measure the relative displacements of six predetermined
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points of the platform. The interrelationships between these displacements and the vector ⌬, Eq. (9) may be presented in the form: dm⫽Rm⌬ and ⌬⫽(Rm)−1 dm
(10)
dm⫽(dm1,dm2,dm3,dm4,dm5,dm6)T
(11)
冢
Ax1 Ay1 Az1 Mx1 My1 Mz1
Rm⫽ % % % %
%
%
冣
(12)
Ax6 Ay6 Az6 Mx6 My6 Mz6
where dm is the 6×1 vector of the displacements dmi (i=1,2,…,6) measured by the ith device; Axi, Ayi, and Azi are the direction cosines of the ith measurement direction (i.e. direction of the positive displacement of the ith measuring probe); and Mxi, Myi, and Mzi are the moments of vector Ai relative to the X, Y, and Z axes, respectively. Using expressions (7) and (10), we obtain the following relationships: dm⫽Rtr d, or d⫽(Rtr)−1 dm,
(13)
Rtr⫽Rm(Rd)−1
(14)
with
4. Statics of the ‘rigid platform’ 4.1. Rigidity-based modification of the Jacobian Eqs. (7) and (13) are valid when: (a) the platform, base and legs are absolutely rigid bodies; and (b) the joints between the legs and the platform are ideal kinematic hinges. These hypotheses are only partly valid for the mechanism under consideration, where the bodies are elastic and the joints are welded. These facts result in the following: 앫 an additional interrelationship between the leg elongations is not taken into consideration by the above-discussed theory; 앫 non-homogeneity of the legs’ stiffness; 앫 deflections of the platform and base themselves; and, 앫 difficulties in accurate definition of the position vectors rNi and rMi.
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Therefore, the infinitesimal kinematics has to be reconsidered for improving the estimation of the displacement. The deviations between the Jacobian-based expression (7) and the computational model may be expressed in terms of undesired additional (superfluous) displacements of the platform points caused by rigid connections. These displacements are taken into account through influence factors kij, which relate the relative elongation of the ith leg to the elongation of the jth leg (kij =1, if j=i and kij 1 if j⫽i):
冘 6
di,real⫽
kij dj , with i⫽1,…,6,
(15)
j⫽1
where dj (j=1,…,6) and di,real (i=1,…,6) are the given and real displacements of the legs, respectively. Eq. (15) may be written in matrix form as follows: dreal⫽K d
(16)
where d is a 6×1 vector composed of six given elongations di (i=1,…,6) of the legs, d=(d1, d2,…,d6)T; dreal is a similar vector of the real displacements of the leg ends, dreal=(d1,real,d2,real,…,d6,real)T; K is the 6×6 influence matrix composed of the influence factors kij; the structure of this matrix is defined by the circular symmetry of the platform. This symmetry (Fig. 5) results in the following interrelationships of elongations between leg pairs: kii⫽1, for i⫽1,…,6; kij ⫽kji, for i,j⫽1,…,6; kij ⫽a, for i⫽1, 3, 5; j⫽i⫹1; kij ⫽d, for i ⫽2, 4, 6; j⫽i⫹1; kij ⫽b, for i⫽1,…,6; j⫽i⫹2; kij ⫽c, for i⫽1, 3, 5; j⫽i⫹3;
Fig. 5.
Display for explanation of the circular symmetry of the platform.
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where a, b, c, d are small real numbers. Therefore, matrix K has the following symmetrical form:
冢 冣 1 a b c b d
a 1 d b c b
K⫽
b d 1 a b c
c b a 1 d b
.
(17)
b c b d 1 a d b c b a 1
The dependence of the factors kij on the structural parameters was investigated by the finiteelements method (see below) and then verified experimentally. The results of the numerical experiment with leg No. 5, after it was elongated (Fig. 6), show that: 앫 There is good accord between the hypothesis about the equality of the factors k51=k53=b; and the experimental results. It is evident from the fact that the curves No. 1 and No. 3 practically coincide. 앫 The minimal interdependency between the elongations of neighboring legs, i.e. the minimum of factors k12=k34=k56=a, corresponds to the height of the platform h=53 mm. In this condition, the angle a between the directions of the axes of neighboring legs is about 100°. Note that this value corresponds to an accepted platform structure, as defined by the minimum value of the determinant of Jacobian matrix (10). For our design, the values of a, b, c, and d calculated by the finite-elements method are:
Fig. 6. Undesired elongations of legs No. 1, 2, 3, 4, and 6 for the specified elongation of leg No. 5 vs. the height of the platform.
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a=⫺0.02688, b=0.0114, c=⫺0.00706, and d=0.00681. It is important to note that the maximal correction factor a forced by the rigidity of the device (as can be seen from this matrix) is about a⬵0.027; all other deviations are smaller. Matrix K is used for modification of the Jacobian matrix Rd of the platform mechanism. For this purpose, a real small displacement ⌬real of the upper platform is considered, and the actual Jacobian Rd,real of the rigid platform is calculated through the theoretical Jacobian Rd and the influence matrix K as follows: ⌬real⫽(Rd)−1 dreal⫽(Rd)−1K d⫽(Rd,real)−1 d, i.e. Rd,real⫽K −1 Rd
(18)
4.2. Finite-elements model To solve the elastic-deformable problem for the given design of the whole mechanical system of the platform, the finite-elements method was applied. Computations of the linear and angular displacements of the platform points caused by elastic deformations of the entire elastic system were carried out with the ANSYS 5.5 finite-elements method program [13]. A Silicon Graphics Challenge L computer with an Irix 6.5 operating system was used. A layout of the finite-elements model built from three-dimensional elements is shown in Fig. 7: upper platform 1 and base 4 are considered as deformable plates, and legs 3 (six in total) are seen as restrained beams with specified tension-compression, bending, and torque properties. The legs are rigidly bound to base 4 and upper platform 1 by prismatic elements 2 and 5. The finiteelements model is built from three-dimensional elements. The upper platform 1 and the base 4
Fig. 7. Isometric view of the FEM model.
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are constructed from elements in the form of eight-node hexahedrals. The connecting parts, i.e. the six legs, comprising three parts each, are made up as follows: the middle comprises a pipe element 3, while the connectors, at each end, are made of prismatic elements 2 and 5. The pipe element of the middle, which is seen as a straight line 3, is subject to bending, torsion and axial forces, i.e. it has finite tension, bending, and torsion rigidity. The prismatic elements, shown as six-node prisms 2 and 5, represent the rigid connection of the legs to the upper and lower platforms. The mechanical properties of the elements correspond to those of standard steel. To simulate the conditions of the actual experiments, the finite-elements model rests on a fixed surface, similar to the positioning of the experimental device. The finite-elements model is supported by a contact between the base and the fixed surface as node-to-node interface elements, taking into account the deformation of the base. The penalty parameters of the interface elements, simulating the contact, equal 2 000 000 N/mm. In computation, loading is applied to one leg only, in the form of axial tension, causing it to stretch. For the solution, based on the nonlinear behavior caused by the contact, a Newton–Raphson iterative procedure was applied. The finite-elements method is used to reveal real deflections of the platform itself caused by the rigid welded joints between the links of this mechanism. The vertical displacements along the Z-axis of the platform points situated along the X and Y axes are shown (Fig. 8a). Deflections from linearity of the points lying along the X and Y axes in the upper plane of the platform are shown in Fig. 8b. The graphs in Fig. 8 were calculated for the condition that the pressure in leg No. 5 equals 200 bars and the pressures in the other legs equal zero. These deformations, of course, hinder the use of the device, but two actions can be taken to counter their effect: to increase the stiffness of the platform’s body or to create reference points on the working surface of the platform so as to counter the influence of deformation on the workpiece. 4.3. Identification of platform transfer factors This procedure is aimed at calculating the transfer matrix Rtr, Eq. (14), by applying results of certain specified measurements. For this purpose, the measurements defined by Eq. (14) must be carried out six times at least. The interrelationships between elongations of the legs and results of measurements may be represented as follows:
Fig. 8. Vertical displacements of the platform points: (a) absolute displacements; (b) deviations from linearity in two axial sections of the platform.
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Dm⫽RtrD
1239
(19)
where D and Dm are the n×6 matrices (n=6) composed of six vectors d by expression (8) and six vectors dm by expression (11), respectively: D⫽(d1,d2,…,d6) and Dm⫽(dm1,dm2,…,dm6)
(20)
If n=6 (i.e. D and Dm are 6×6 matrices), we can calculate the transformation matrix Rtr using six experiments, designed so that det D⫽0: Rtr⫽DmD−1
(21)
It is the best to compose the experiments for identification as follows: d1⫽(d1,0,0,0,0,0)T⫽d1(1,0,0,0,0,0)T, ……………., d6⫽(0,0,0,0,0,d6)T⫽d6(0,0,0,0,0,1)T
(22)
where di (i=1,…,6) is the elongation of the ith leg in the ith experiment. In this case, the transformation matrix is: Rtr=Dm=(dm1/d1,…,dm6/d6). 5. Dynamics and control problems The automatic control system used in our experiments is shown in Fig. 9. The layout of the open and closed control systems (CS) for the secondary table (ST), i.e. for the 6-DOF parallel platform, is shown in Fig. 10. When the open-loop approach is applied, the displacement is controlled with varying the pressure (as measured by a pressure gauge) according to control algorithm of Eq. (18). With the one-directional displacement control, the accuracy achieved was no better than 0.2 µm. The application of the closed-loop system reduced considerably the effect of pressure errors on the accuracy of the platform’s displacement. The relationships of the elements of the CS are performed using the LABVIEW software package [14]. To discuss the behavior of the system under automatic control conditions, we begin with the one-dimensional control structures (i.e. the control system for a single microactuator). The trans-
Fig. 9. Layout of the automatic control system for programmable elongation of the legs.
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Fig. 10. Control loop layout for six microactuators driving the upper platform: (a) open system; (b) closed system: CS, control system on the base of the PC and LABVIEW software package; ST, secondary table (6-DOF platform).
formations of the input signals ⌿(s) into the output signals ⌽0(s) and ⌽1(s) in the plane s for the open and closed control loop, respectively, have the following form [15]: ⌽0(s)⫽⌿(s)W0(s)W1(s)W2(s)W3(s);
⌽1(s)⫽⌿(s)
W0(s)W1(s)W2(s)W3(s) , 1+W0(s)W1(s)W2(s)W3(s)W4(s)
(23)
(24)
where ⌿(s) is the Laplace transform for input influence, which is the given scalar function of time d=d(t) for elongation of the microactuator; ⌽0(s) and ⌽1(s) are the Laplace transforms for the output function, which is the real elongation ⌬l=⌬l1(t) of the microactuator for the open and closed control loop, respectively; W0(s), W1(s), W2(s), W3(s), and W4(s) are the transfer functions of the digital-to-analog transformer, the proportional pressure-limiting hydraulic valve, the actuator, the electronic driver of the valve, and the measuring device, respectively. In our arrangement, the system comprises the following devices: 앫 앫 앫 앫
the proportional pressure-limiting hydraulic valve of the type PMV42; the actuator produced in our laboratory; the electronic driver of the valve of the type E-MI-AC, Plug format [16]; and the measuring device LVDT (inductive measuring probe 1301 with digital device Militron 1202D, Mahr GmbH). Their transfer functions are: W0(s)⫽(1⫺e−sT0)/s⬵T0/(1⫹0.5s T0);
(25)
W1(s)⫽(T 21s2⫹T2s⫹1)−1;
(26)
W2(s)⫽(7.8)10−14 m3/N; W3(s)⫽10 A/V; W4(s)⫽(3.0)105 V/m.
(27)
where T0 is the given sampling period of time; T1 and T2 are the time constants for the control valve considered to be an oscillating link [15]. These constants are experimentally determined. For this purpose, a single actuator was investigated by applying step-wise inputs.
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In contrast to the known electro-hydraulic servo systems for devices of a similar nature [17– 19], our device has no moving masses (pistons, piston rods, etc.). In addition, there are no liquid flows in the device actuators, which work in a hydrostatic regime. The input for the multi-channel system is the spatial position of the platform described by the 6×1 vector ⌿⌺=(⌿1,…,⌿6)T, where ⌿i (i=1,…,6), is a specified spatial position of the platform, given as a function of time ⌿i=⌿i(t), transformed in the LABVIEW package (block CS in Fig. 10) by the matrix Rd,real, Eq. (18), as follows:
冘
冘
6
d1,r⫽
6
⌿ib1i,…,d6,r⫽
i⫽1
⌿ib6i
i⫽1
where bki (k,i=1,…,6) are corresponding elements of the matrix Rd,real. The output response is the vector ⌽⌺=(⌽1,…,⌽6)T, where ⌽1, ⌽2, and ⌽3 are small linear displacements along the X, Y and Z axes; and ⌽4, ⌽5, and ⌽6 are the small rotations relative to the same axes. These provide the required posture of the platform as a function of time. The rectangle designated as ST represents the upper platform in the layout given in Fig. 10, and in expanded form in Fig. 11. The six components ⌬li (i=1,…,6), that are the actual elongations of the actuators, undergo transformation by matrix (18) composed of 36 elements aij (i,j=1,…,6).
Fig. 11.
Layout of transformations of the vector dr into the vector ⌽⌺.
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Fig. 12. Measured responses of the microactuator (displacement in relative units vs. time in seconds) to a step-like input produced by an open loop control. Polygonal and smooth lines correspond to experimental and approximated data, respectively.
These elements serve as amplifiers, the outputs of which are afterwards summed to create six components ⌽1⫽⌺6i=1⌬lia6i,…,⌽6,r⫽⌺6i=1⌬lia6i. The averaged results of the experimental responses were approximated by the following expression: ⌽0(t)⫽1⫺exp[⫺at]⫹a exp[⫺bt]sin[wt] where a=0.6; a=20 s⫺1; b=4 s⫺1 and w=6.28 s⫺1. The experimental and approximated results are shown in Fig. 12. As seen in this figure, there is an overshooting of about 20% and the transient time is about 1 s. Fig. 13 presents the Bode diagrams computed by the MathLab software system. These diagrams
Fig. 13. Bode diagram: magnitude (in dB) and phase (in degrees) of the control response vs. frequency.
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indicate the stability of the control dynamics, which follows from the relationship between the values of the magnitude (0 dB) and the phase (⬵50°) of the response at frequency ⬵11 s⫺1. Since the mechanical part of the platform has very high stiffness, its natural frequencies are relatively high. To estimate the natural frequencies of the platform, a scheme of a rigid body with six degrees of freedom, which bears up against six elastic supports (the legs of the platform), is used (Fig. 14). The corresponding calculations show that the natural frequencies of the mechanical part lie in the range between 4×103 and 14×103 rad/s, i.e. they are two orders higher than the natural frequencies of the control system. 6. Experimental investigations of the platform’s accuracy The nominal location and orientation of the legs of the experimental platform shown in Fig. 2 are given in Table 1. Displacements of the platform points are measured by six devices (Fig. 2). The location and orientation of the probes for measurement of the displacements of the upper platform points are presented in Table 2. The elongations of the legs are measured by the probes, which are located on the leg parallel to its axis (Fig. 3b). The measurement results are presented in form of matrices Rd, Eq. (6), and Rm, Eq. (12), as follows:
冢
R d⫽
−0.77242
0.20598 0.600775
90.9745 −38.5354
0.77242
0.20598 0.600775
90.9745
0.19774 −0.77377 0.60182
−11.9504
−0.56672
0.56672 0.598046
−78.757
0.56672
0.56672 0.598046
−78.757
−0.19774 −0.77377 0.60182
−11.9504
130.179
冣
38.5354 −130.179 99.128
131.377
59.691 −131.196 −59.691
131.196
−99.128 −131.377
Fig. 14. Scheme for the calculation of the platform natural frequencies.
(28)
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Table 1 Nominal coordinates of the legs’ ends, mm Leg
Upper ends
Lower ends
no.
xNi
yNi
zNi
xMi
yMi
zMi
1 2 3 4 5 6
32 ⫺32 ⫺156.5 ⫺123.5 123.5 156.5
160 160 ⫺52 ⫺108 ⫺108 ⫺52
25 25 25 25 25 25
167 ⫺167 ⫺191 ⫺24 24 191
124 214 83 ⫺207.5 ⫺207.5 83
⫺80 ⫺80 ⫺80 ⫺80 ⫺80 ⫺80
Table 2 Location of the measuring devices I
Coordinates of the contact points Xi
Yi
⫺47.148 63.311 55.060 122.622 1.220 ⫺125.842
1 2 3 4 5 6
Rm⫽
冢
⫺80.337 ⫺39.658 30.634 ⫺72.387 143.887 ⫺68.036
Direction of measurement Zi
Axi
Ayi
Azi
55 55 55 25 25 25
0.5 0.5 0.866 0 0 0
⫺0.866 ⫺0.866 0.5 0 0 0
0 0 0 1 1 1
0.5
−0.8660 0 47.6313
27.5
81
0.5
−0.8660 0 47.6313
27.5
−35 1
0.8660 0.5
0 −27.5
47.6313
0
0
1 47.6313
−122.622 0
0
0
1 143.887
−1.21985 0
0
0
1 −68.0359 125.841
0
冣
(29)
Elongations ⌬i are measured as a function of the pressure p in the cylinder built in the ith leg over a pressure range of 0–200 bars. The results show that elongations were obtained with reasonable linearity (correlation coefficients in the range of 0.997 and higher) and with hysteresis less than 2.5%. For example, elongation ⌬1 (in µm) of leg No. 1 depends on pressure p as follows: 앫 for growing pressure ⌬1=0.1430p⫺0.82 (correlation factor 0.9982);
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앫 for decreasing pressure ⌬1=0.1398p⫺0.80 (correlation factor 0.9977); 앫 hysteresis is 2.24%. The repeatability of the results was tested by a set of repeated measurements for two modes. When the elongation of the leg is repeated from 0 to 11 µm, the deviations from linearity were in the range of 40 nm. When the pressure p=100 bar in the leg is repeated, elongation of the leg deviates for about 60 nm. The real values of the coordinates of the legs’ fixed points deviate from the nominal values. An identification procedure based on experimentation with an existing device was applied to obtain the actual platform characteristics. Thirty-six components of the vectors Rd were obtained as the regression factors resulting from the experiments according to conditions (22), i.e. the experiments in which only the ith leg was under pressure pi, and the other pressures were zero. An example of these dependencies—the experimental points, regression lines, and regression equations—are shown in Fig. 15. The plot shows the readings of six measuring devices vs. elongation of leg no. 5 when p5=0–250 bar and p1=p2=p3=p4=p6=0. As a result, we obtained matrix Rtr,real by expression (21) in the form:
冢
Rtr,real⫽
0.0094 0.0923 0.3061 −0.3276 −0.1182 −0.0325 −0.1035 0.2148 0.2053 −0.1967 −0.2265 −0.2355 0.1995 0.065
0.0454
冣
0.0845
0.1789 −0.2402
0.0055 −0.0037
0.3869
0.4804
0.3734 0.4613 0.0304 −0.0037
0.0127
0.002
0.0097 0.0084 0.3655
0.0469 −0.0095
0.0566 0
0.4999
(30)
Fig. 15. Experimental results for identification of transfer factors for elongation of leg no. 5. The numbers of the measurement devices correspond to those in Fig. 2.
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In this matrix, the fifth column is composed of six coefficients of regression given in Fig. 15 for corresponding measuring devices; for example the element a15 is the regression coefficient for the readings of the device no. 1, i.e. a15=⫺0.1182 (see Fig. 15), and so on. The other columns are composed of coefficients of regression, which are obtained by similar measurements made for leg nos. 1, 2, 3, 4, and 6. A set of legs’ elongations d providing the desired position ⌬ of the platform is d=(Rtr,real)−1Rm⌬. Thus, a scalar displacement d of the platform point in the specified direction given by unit vector a=(ax,ay,az)T is: d⫽RT⌬⫽RT(Rm)−1Rtr,reald
(31)
where R is the 6×1 vector (5) of Plucker’s coordinates of this point. By applying the identification matrix (30) one may accurately predict the displacements of the platform points. The prediction was tested experimentally. During the experiments, different combinations of hydraulic pressures in the legs’ cylinders were used: p⫽(p,0,p,0,p,0)T, p⫽(0,p,0,p,0,p)T, and p⫽(p,p,p,p,p,p)T, where p is the vector of hydraulic pressures in the legs; the components p of the vector are controlled in the range p=0–250 bar. The values calculated using Eq. (31) agrees well with experimental measurements (Fig. 16). A relative deviation between the theoretical results obtained by application of expression (31) and experimental measurements lies in the range 0.68–2.79%. Note that the main part of the error is caused by deviations from the given values of the six actual values of the hydraulic pressure within the legs. Thus, the analysis shows that neither the conventional or modified Jacobian model, nor the
Fig. 16. Comparison of the theoretical (lines) and actual (points) displacements for the readings of measurement device nos. 1, 5 and 6 vs. pressure in the legs’ cylinders p2=p4=p6=0 and p1=p3=p5=40–200 bar. The numbers of the devices correspond to those in Fig. 2.
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finite-elements method can satisfy the requirements for the high precision device in the accuracy range of 10⫺7–10⫺8 m. Thus, the identification-based model is the model of choice. 7. Application example The results of the dynamic investigation may be applied for calculating the minimal segment length smin of the machined profile when the platform is used as a secondary table in the ‘coarse– fine’ systems. To compensate the cumulative and cyclic errors of the feed system of the machine tool, this length must be more than the resolution ts of the feed system, i.e. smin⫽(p/30)f/wmax⬎ts where f is cutting feed along the machined profile in mm/min, f=√f 2x +f 2y ; fx and fy are the feeds along the X and Y axes, respectively; and wmax is the maximal allowed forced frequency, rad/s. For example, when wmax=8 rad/s and f=100 mm/min, this expression yields smin=1.3 mm. This feature can be improved by modifying the hydraulic system. Experiments were carried out, in which disturbances were added to nominal feeds. The nominal feeds are performed with the X–Y motions of the table of the milling center, while the additional feeds are made by the platform mechanism. The said feeds provided the appearance of a predesigned distortion of the machined profile. For example, a specified elliptical shape is expressed by an expression: R⫽R0[1⫹d sin(2⍀t⫹j0)], where R is the current polar radius of the specified workpiece profile; R0 is the initial polar radius of the circle; d is the specified programmed relative deviation of the profile; w is the circular frequency in rad/s corresponding to the profile feed f, i.e. ⍀=pf/(30R0); t is time; and j0 is the initial phase angle. The profile according to this expression is shown in Fig. 17a. Simulation of this profile performed with LABVIEW based control system is presented in Fig. 17b. The profile of the workpiece machined on the milling center and measured on the Talyrond device is shown in Fig. 17c. Fig. 18 illustrates roundness polar plots of a cylindrical part machined. The out-of-roundness of a cylinder milled without correction (Fig. 18a) is 14.25 µm, and a milled cylinder after correction by means of the device has out-of-roundness 8.55 µm (Fig. 18b). 8. Conclusions 1. The 6-DOF platform-type mechanism for manipulations in the range of small displacements possessing rigid joints between the legs and the platform as well as between the legs and the base is developed and investigated. The operating range and displacement accuracy are established. The accepted accuracy of the device under discussion is better than 0.1 µm when a closed-loop automatic control system is used. 2. Analysis of infinitesimal kinematics is performed using the models of classical and modified Stewart platform kinematics, the finite-elements method, and an experimentally based identifi-
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Fig. 17. Profile with elliptical-type distortion (a), simulation performed with LABVIEW based control system (b) and out-of-roundness polar plot of the machined profile measured with Talyrond device (c).
cation method. The last-listed method is the most suitable for calculations of the considered platform. 3. Hydraulically driven microactuators are suitable for driving the platform and providing the necessary stiffness, manipulation linearity, and precision. The dynamic restrictions of the investigated device depend on the hydraulic system. 4. The described device can serve as a secondary table for high-precision milling machines to improve the accuracy of those machines and for some other tasks.
Acknowledgements The authors express their gratitude to the Paul Ivanier Robotics Research and Production Management Center for its help and encouragement.
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Fig. 18. Out-of-roundness polar plots of a cylindrical part machined on a CNC milling machine: (a) without correction; and (b) with acting correction. An initial orientation label is denoted by x.
References [1] C.C. Nguyen, S.S. Antrazi, Z.-L. Zhou, C.E. Campbell Jr, Analysis and experimentation of a Stewart platformbased force/torque sensor, Int. J. Robotics Automation 7 (3) (1992) 133–140. [2] D.R. Kerr, Analysis, properties, and design of Stewart-platform transducers, Trans. ASME, J. Mech. Transmissions, Automation Design 111 (1989) 24–28. [3] A. Gaillet, C. Reboulet, An isostatic six component force and torque sensor, in: Proceedings of the 13th International Symposium on Industrial Robots in Chicago, 18/102-18/111, 1983. [4] M. Taniguchi, M. Ikeda, A. Inagaki, R. Funatsu, Ultra precision wafer positioning by six-axis micro-motion mechanism, Int. J. Jpn. Soc. Prec. Eng. 26 (1) (1992) 35–40. [5] S. Asano, T. Goto, Micro-positioning stage of 6 degrees-of-freedom with nanometer resolution, in: P. Seyfried, H. Kunzmann, P. McKeown, M. Weck (Eds.), Progress in Precision Engineering, Proceeedings of the Sixth International Precision Engineering Seminar Springer-Verlag, Berlin, 1991, p. 362. [6] A. Hara, K. Sugimoto, Synthesis of parallel micromanipulators, Trans. ASME, J. Mech. Transmissions, Automations Design 111 (1989) 34–39. [7] A. Slocum, Precision Machine Design, MIT Press, 1992. [8] Bi Zhang, Z. Zhu, Bolin Zhang, W.-E. Fu, J. Wang, A linear piezomotor of high stiffness for nanometer resolution, Annals CIRP 46 (1) (1997) 305–308. [9] K.-M. Lee, S. Arjunan, A three-degrees-of-freedom micromotion in-parallel actuated manipulator, IEEE Trans. Robotics Automation 7 (1999) 238–247. [10] P. Gao, S.M. Swei, A six-degree-of-freedom micro-manipulator based on piezoelectric translator, Nanotechnology 10 (1999) 447–452. [11] V. Portman, B.Z. Sandler, High stiffness precision actuator for small displacements, Int. J. Machine Tools Manufacture 39 (5) (1999) 823–837. [12] V. Portman, B.-Z. Sandler, E. Zahavi, Rigid 6×6 parallel platform for precision 3-d micromanipulation, in: Proceedings of the Tenth World Congress on the Theory of Machines and Mechanisms, vol. 2, 1999, pp. 828–834. [13] ANSYS 5.2, User Manual, Theory, vol. II, Swanson Analysis System, 1996.
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[14] LABVIEW, User Manual, National Instrument, parts NN 320997C-01, 320999B-01, 321296B-01, 321526B-01, January 1998. [15] R.C. Dorf, R.H. Bishop, Modern control systems (7th ed), Addison-Wesley, New York, 1998. [16] Proportional pressure limiting valves type PMV and PDV, HAWE Hydraulic. Electronic drivers type E-MI-ACPlug format, ATOS spa, Italy. [17] W.-S. Yu, T.-S. Kuo, Continuous-time indirect adaptive control of the electro-hydraulic servo systems, IEEE Trans. Control Syst. Technol. 5 (1997) 163–177. [18] G.A. Sohl, J.E. Bobrow, Experiments and simulations on the nonlinear control of a hydraulic servosystem, IEEE Trans. Control Technol. 7 (1999) 238–247. [19] G. Vossoughi, M. Donath, Dynamic feedback linearization for electro-hydraulic actuated control system, J. Dynamic Syst. Meas., Control 117 (1995) 468–477.
Rigid 6-DOF parallel platform for precision 3-D micromanipulation Vladimir T. Portman*, Ben-Zion Sandler, Eliahu Zahavi Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105, Israel Received 1 November 2000; accepted 31 January 2001
Abstract A new type of 6-DOF platform mechanism characterized by very high stiffness and a high degree of accuracy is described. Welded joints connecting the changeable links with the platform and base provide the high stiffness. Hydraulically controlled microactuators use longitudinal elastic deformations for elongation of the legs. Infinitesimal kinematics of a conventional platform, the finite-elements method, and an identification procedure are used for the analysis of the mechanism. Experimental investigations of a pilot set-up of the proposed platform confirmed the predictions concerning the accuracy and stiffness of the proposed device. A practical application of the platform as a secondary table for precise machining on a CNC milling center is described. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Parallel mechanism; Micromanipulation; Secondary table; Milling
1. Introduction Parallel platform constitutes a very effective tool for accomplishing a variety of tasks in 3-D space [1] providing six degrees of freedom (DOF) manipulations. Under consideration here is the development of a specific type of the platform designed for high precision manipulation in microdomain. The term ‘micro’ relates to spatial displacements in the domain of about 10⫺6–10⫺4 m, while the required accuracy of the device is of the order of 10⫺9–10⫺7 m, i.e. up to and including the nanometer range. The high accuracy of such mechanisms stems from high structural stiffness and non-serial accumulation of the joint and link errors inherent in these mechanisms. The maximal accuracy is achieved by designing the platform mechanism as an ideal axially symmetric * Corresponding author. Tel.: +972-7-647-7093; fax: +972-7-647-2813. E-mail address: [email protected] (V.T. Portman).
0890-6955/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 0 1 ) 0 0 0 2 7 - X
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structure. This structure has been successfully applied for the development of precision devices with six DOF for a variety of applications. Some examples of these are a six-component force/torque transducer [1–3], an ultra-precision wafer-positioning table for lithography systems [4], a micro-positioning stage with six degrees of freedom for a fine–coarse positioning system [5], etc. The features of the structures of these platform-type micromanipulators are essentially defined by the type of connection of the changeable links (hereafter referred to as ‘legs’) with the base and the upper platform. To enhance the accuracy of platform-type systems for micromanipulations, elastic hinges are used instead of traditional Hooke’s joints. A variety of types of elastic hinges— revolute, prismatic, and spherical—have been applied for parallel micromanipulators [6]. Different types of drives are used within precision manipulators. Most in use are piezoelectric actuators [7,8] as, for example, piezoelectric actuators possessing 3-DOF [9] and 6-DOF [10]. Another type of high-stiffness precision actuator was given in Ref. 11. The latter constitutes an elastic cylinder deformed by controlled internal hydraulic pressure and is used for the so-called ‘rigid platform’ for micropositioning [12]. In this paper, the utilization of the rigid platform as a ‘secondary micrometric coordinate table’ for precision machining is discussed. The laboratory arrangement of a ‘fine–coarse feed’ system mounted on a milling center is shown in Fig. 1. As shown, the XYZ moving units of the machine tool represent a coarse feed subsystem, while the platform fastened on the machine table represents a fine feed subsystem. This platform mechanism has a simple and reliable structure and is relatively inexpensive to produce. It may be used for accurate positioning of workpieces, cutting tools, or end-effectors under relatively high loads in machine tool technology and in robotics. The theoretical and experimental results concerning this platform, its kinematics, control, and preliminary practical applications are discussed. A description of the structure of the mechanism (Section 2) is followed by a consideration of infinitesimal kinematics in Section 3. The statics of the mechanism, including an application of the finite-elements method, is discussed in Section 4, and the dynamics problem is considered in
Fig. 1. Laboratory arrangement of a ‘fine-course feed’ system on the milling center: 1, CNC machine tool; 2, 6-DOF platform; 3, hydraulic control arrangement; 4, computer for the platform control; 5, display of the measuring device.
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Fig. 2. Layout of the platform for micropositioning.
Section 5. Section 6 presents an experimental investigation of the platform, and an application example is given in Section 7. 2. Structure of the platform for micropositioning The mechanism under consideration (Fig. 2) consists of an upper platform connected to a base by six legs M1N1,…,M6N6. The ‘leg-to-platform’ and ‘leg-to-base’ joints are elastic hinges with three degrees of freedom. The platform’s position and its slope in space depend on the lengths of the legs. Six measurement devices, marked in Fig. 2 by arrows 1–6, give displacement data of six points of the upper platform in relation to the base. The mechanism comprises a 6×6 platform of the greatest possible stiffness, achieved by an introduction of the welded ‘leg-to-platform’ and ‘leg-to-base’ joints in the structure [11]. Six independent built-in microactuators, responsible for elongation of the legs, provide the spatial micromanipulation of the platform. An experimental design of the leg connections with the platform and the base is shown in Fig. 3. An elastic deformable steel cylinder is used as a microactu-
Fig. 3. General view of the platform (a) and the design of the leg connections (b): 1, upper platform; 2, leg connection to the upper platform; 3, leg; 4, base; 5, leg connection to the base; LVDT, device for measurement of the leg elongation.
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ator controlling displacements in the micron and submicron ranges [11]. The deformations are caused by hydraulic pressure inside the cylinder. This microactuator, which facilitates delicate and accurate displacements, provides high stiffness and high carrying capacity at a relatively low cost compared with conventional microactuators. The design of the microactuator for displacements in the range 0–40 µm is shown in Fig. 4. The microactuator consists of a base and a deformable hollow cylinder, which create a closed volume filled with a hydraulic liquid under controlled pressure p. This pressure causes axial deformation ⌬l of the relatively thin walls of the cylinder: ⌬l⫽
plpD2in ⫽k p (D2out−D2in)E p
(1)
where Dout and Din are the external and internal diameters of the leg (Fig. 4); lp is the length of the leg to be elongated; E is the Young’s modulus of the leg material; and kp is a transfer factor for leg elongation, kp=⌬l/p. Thus, the nominal dimensions are: E=2.1×105 MPa, lp=70 mm, Dout=35 mm, Din=30 mm, and the transfer factor kp=0.09187 µm/bar. Based on the fact that the last member of the kinematic chain is a deformable stiff metal cylinder, this microactuator will exhibit small hysteresis, good linearity, and good resolution in the domain of possible elastic deformations. 3. Infinitesimal kinematics Infinitesimal kinematics of the conventional platform (i.e. the platform with kinematic hinges between the legs and both plates) is generally described by the Jacobian matrix. In our mechanism, however, substitution of kinematic hinges with elastic or ‘rigid’ hinges results in enhanced stiffness of the mechanism and, hence, in increasing deviations between the computational models of the corresponding structures. In accordance with Fig. 2, the following formal attributes for the ith leg (i=1,…,6) may be introduced: a vector ri of the leg; a unit vector ai of the leg direction; a moment mi of the vector ai relative to the origin of the coordinate system; a six-order vector Ri of Plucker’s coordinates, and the 6×6 Jacobian matrix Rd composed of these vectors Ri:
Fig. 4.
Microactuator layout.
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ri⫽rNi⫺rMi;
(2)
ai⫽(axi,ayi,azi)T⫽ri/li;
(3)
mi⫽(mxi,myi,mzi)T⫽rNi⫻ai;
(4)
Ri⫽(axi,ayi,azi,mxi,myi,mzi)T;
(5)
冢
ax1 ay1 az1 mx1 my1 mz1
冣
R d⫽ % % % % % % , ax6 ay6 az6 mx6 my6 mz6
(6)
where rNi=(xNi,yNi,zNi)T and rMi=(xMi,yMi,zMi)T are the position vectors of the upper and lower ends of the ith leg, respectively; li is the length of the ith leg, li=|ri|; axi, ayi and azi are the direction cosines of the ith leg; and mxi, myi, and mzi are the moments of vector ai relative to the X, Y, and Z axes, respectively. Since the Jacobian Rd is non-singular (det Rd⫽0), vector ⌬ of the small displacements of the platform and vector d of the elongation of the legs are related to each other, accurate to the first order, as follows: ⌬⫽(Rd)−1 d and d⫽Rd⌬,
(7)
d⫽(⌬l1,⌬l2,…,⌬l6)T,
(8)
⌬⫽(dx,dy,dz,a,b,g)T,
(9)
with
where ⌬l1, ⌬l2,…,⌬l6 are the small elongations of six legs; dx, dy, and dz are the small translations of the platform along the X, Y, and Z axes, respectively, or (which is the same) small linear displacements of the origin O⬘ of the coordinate system associated with the platform (see Fig. 2); and a, b, and g are the small angular rotations of the platform relative to the same axes. Expressions (7) may be used to solve the direct and inverse kinematics problems for small displacements of the platform. In order to obtain sufficient data concerning small displacements of the upper platform, considered as a rigid body, it is necessary to measure the relative displacements of six predetermined
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points of the platform. The interrelationships between these displacements and the vector ⌬, Eq. (9) may be presented in the form: dm⫽Rm⌬ and ⌬⫽(Rm)−1 dm
(10)
dm⫽(dm1,dm2,dm3,dm4,dm5,dm6)T
(11)
冢
Ax1 Ay1 Az1 Mx1 My1 Mz1
Rm⫽ % % % %
%
%
冣
(12)
Ax6 Ay6 Az6 Mx6 My6 Mz6
where dm is the 6×1 vector of the displacements dmi (i=1,2,…,6) measured by the ith device; Axi, Ayi, and Azi are the direction cosines of the ith measurement direction (i.e. direction of the positive displacement of the ith measuring probe); and Mxi, Myi, and Mzi are the moments of vector Ai relative to the X, Y, and Z axes, respectively. Using expressions (7) and (10), we obtain the following relationships: dm⫽Rtr d, or d⫽(Rtr)−1 dm,
(13)
Rtr⫽Rm(Rd)−1
(14)
with
4. Statics of the ‘rigid platform’ 4.1. Rigidity-based modification of the Jacobian Eqs. (7) and (13) are valid when: (a) the platform, base and legs are absolutely rigid bodies; and (b) the joints between the legs and the platform are ideal kinematic hinges. These hypotheses are only partly valid for the mechanism under consideration, where the bodies are elastic and the joints are welded. These facts result in the following: 앫 an additional interrelationship between the leg elongations is not taken into consideration by the above-discussed theory; 앫 non-homogeneity of the legs’ stiffness; 앫 deflections of the platform and base themselves; and, 앫 difficulties in accurate definition of the position vectors rNi and rMi.
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Therefore, the infinitesimal kinematics has to be reconsidered for improving the estimation of the displacement. The deviations between the Jacobian-based expression (7) and the computational model may be expressed in terms of undesired additional (superfluous) displacements of the platform points caused by rigid connections. These displacements are taken into account through influence factors kij, which relate the relative elongation of the ith leg to the elongation of the jth leg (kij =1, if j=i and kij 1 if j⫽i):
冘 6
di,real⫽
kij dj , with i⫽1,…,6,
(15)
j⫽1
where dj (j=1,…,6) and di,real (i=1,…,6) are the given and real displacements of the legs, respectively. Eq. (15) may be written in matrix form as follows: dreal⫽K d
(16)
where d is a 6×1 vector composed of six given elongations di (i=1,…,6) of the legs, d=(d1, d2,…,d6)T; dreal is a similar vector of the real displacements of the leg ends, dreal=(d1,real,d2,real,…,d6,real)T; K is the 6×6 influence matrix composed of the influence factors kij; the structure of this matrix is defined by the circular symmetry of the platform. This symmetry (Fig. 5) results in the following interrelationships of elongations between leg pairs: kii⫽1, for i⫽1,…,6; kij ⫽kji, for i,j⫽1,…,6; kij ⫽a, for i⫽1, 3, 5; j⫽i⫹1; kij ⫽d, for i ⫽2, 4, 6; j⫽i⫹1; kij ⫽b, for i⫽1,…,6; j⫽i⫹2; kij ⫽c, for i⫽1, 3, 5; j⫽i⫹3;
Fig. 5.
Display for explanation of the circular symmetry of the platform.
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where a, b, c, d are small real numbers. Therefore, matrix K has the following symmetrical form:
冢 冣 1 a b c b d
a 1 d b c b
K⫽
b d 1 a b c
c b a 1 d b
.
(17)
b c b d 1 a d b c b a 1
The dependence of the factors kij on the structural parameters was investigated by the finiteelements method (see below) and then verified experimentally. The results of the numerical experiment with leg No. 5, after it was elongated (Fig. 6), show that: 앫 There is good accord between the hypothesis about the equality of the factors k51=k53=b; and the experimental results. It is evident from the fact that the curves No. 1 and No. 3 practically coincide. 앫 The minimal interdependency between the elongations of neighboring legs, i.e. the minimum of factors k12=k34=k56=a, corresponds to the height of the platform h=53 mm. In this condition, the angle a between the directions of the axes of neighboring legs is about 100°. Note that this value corresponds to an accepted platform structure, as defined by the minimum value of the determinant of Jacobian matrix (10). For our design, the values of a, b, c, and d calculated by the finite-elements method are:
Fig. 6. Undesired elongations of legs No. 1, 2, 3, 4, and 6 for the specified elongation of leg No. 5 vs. the height of the platform.
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a=⫺0.02688, b=0.0114, c=⫺0.00706, and d=0.00681. It is important to note that the maximal correction factor a forced by the rigidity of the device (as can be seen from this matrix) is about a⬵0.027; all other deviations are smaller. Matrix K is used for modification of the Jacobian matrix Rd of the platform mechanism. For this purpose, a real small displacement ⌬real of the upper platform is considered, and the actual Jacobian Rd,real of the rigid platform is calculated through the theoretical Jacobian Rd and the influence matrix K as follows: ⌬real⫽(Rd)−1 dreal⫽(Rd)−1K d⫽(Rd,real)−1 d, i.e. Rd,real⫽K −1 Rd
(18)
4.2. Finite-elements model To solve the elastic-deformable problem for the given design of the whole mechanical system of the platform, the finite-elements method was applied. Computations of the linear and angular displacements of the platform points caused by elastic deformations of the entire elastic system were carried out with the ANSYS 5.5 finite-elements method program [13]. A Silicon Graphics Challenge L computer with an Irix 6.5 operating system was used. A layout of the finite-elements model built from three-dimensional elements is shown in Fig. 7: upper platform 1 and base 4 are considered as deformable plates, and legs 3 (six in total) are seen as restrained beams with specified tension-compression, bending, and torque properties. The legs are rigidly bound to base 4 and upper platform 1 by prismatic elements 2 and 5. The finiteelements model is built from three-dimensional elements. The upper platform 1 and the base 4
Fig. 7. Isometric view of the FEM model.
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are constructed from elements in the form of eight-node hexahedrals. The connecting parts, i.e. the six legs, comprising three parts each, are made up as follows: the middle comprises a pipe element 3, while the connectors, at each end, are made of prismatic elements 2 and 5. The pipe element of the middle, which is seen as a straight line 3, is subject to bending, torsion and axial forces, i.e. it has finite tension, bending, and torsion rigidity. The prismatic elements, shown as six-node prisms 2 and 5, represent the rigid connection of the legs to the upper and lower platforms. The mechanical properties of the elements correspond to those of standard steel. To simulate the conditions of the actual experiments, the finite-elements model rests on a fixed surface, similar to the positioning of the experimental device. The finite-elements model is supported by a contact between the base and the fixed surface as node-to-node interface elements, taking into account the deformation of the base. The penalty parameters of the interface elements, simulating the contact, equal 2 000 000 N/mm. In computation, loading is applied to one leg only, in the form of axial tension, causing it to stretch. For the solution, based on the nonlinear behavior caused by the contact, a Newton–Raphson iterative procedure was applied. The finite-elements method is used to reveal real deflections of the platform itself caused by the rigid welded joints between the links of this mechanism. The vertical displacements along the Z-axis of the platform points situated along the X and Y axes are shown (Fig. 8a). Deflections from linearity of the points lying along the X and Y axes in the upper plane of the platform are shown in Fig. 8b. The graphs in Fig. 8 were calculated for the condition that the pressure in leg No. 5 equals 200 bars and the pressures in the other legs equal zero. These deformations, of course, hinder the use of the device, but two actions can be taken to counter their effect: to increase the stiffness of the platform’s body or to create reference points on the working surface of the platform so as to counter the influence of deformation on the workpiece. 4.3. Identification of platform transfer factors This procedure is aimed at calculating the transfer matrix Rtr, Eq. (14), by applying results of certain specified measurements. For this purpose, the measurements defined by Eq. (14) must be carried out six times at least. The interrelationships between elongations of the legs and results of measurements may be represented as follows:
Fig. 8. Vertical displacements of the platform points: (a) absolute displacements; (b) deviations from linearity in two axial sections of the platform.
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Dm⫽RtrD
1239
(19)
where D and Dm are the n×6 matrices (n=6) composed of six vectors d by expression (8) and six vectors dm by expression (11), respectively: D⫽(d1,d2,…,d6) and Dm⫽(dm1,dm2,…,dm6)
(20)
If n=6 (i.e. D and Dm are 6×6 matrices), we can calculate the transformation matrix Rtr using six experiments, designed so that det D⫽0: Rtr⫽DmD−1
(21)
It is the best to compose the experiments for identification as follows: d1⫽(d1,0,0,0,0,0)T⫽d1(1,0,0,0,0,0)T, ……………., d6⫽(0,0,0,0,0,d6)T⫽d6(0,0,0,0,0,1)T
(22)
where di (i=1,…,6) is the elongation of the ith leg in the ith experiment. In this case, the transformation matrix is: Rtr=Dm=(dm1/d1,…,dm6/d6). 5. Dynamics and control problems The automatic control system used in our experiments is shown in Fig. 9. The layout of the open and closed control systems (CS) for the secondary table (ST), i.e. for the 6-DOF parallel platform, is shown in Fig. 10. When the open-loop approach is applied, the displacement is controlled with varying the pressure (as measured by a pressure gauge) according to control algorithm of Eq. (18). With the one-directional displacement control, the accuracy achieved was no better than 0.2 µm. The application of the closed-loop system reduced considerably the effect of pressure errors on the accuracy of the platform’s displacement. The relationships of the elements of the CS are performed using the LABVIEW software package [14]. To discuss the behavior of the system under automatic control conditions, we begin with the one-dimensional control structures (i.e. the control system for a single microactuator). The trans-
Fig. 9. Layout of the automatic control system for programmable elongation of the legs.
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Fig. 10. Control loop layout for six microactuators driving the upper platform: (a) open system; (b) closed system: CS, control system on the base of the PC and LABVIEW software package; ST, secondary table (6-DOF platform).
formations of the input signals ⌿(s) into the output signals ⌽0(s) and ⌽1(s) in the plane s for the open and closed control loop, respectively, have the following form [15]: ⌽0(s)⫽⌿(s)W0(s)W1(s)W2(s)W3(s);
⌽1(s)⫽⌿(s)
W0(s)W1(s)W2(s)W3(s) , 1+W0(s)W1(s)W2(s)W3(s)W4(s)
(23)
(24)
where ⌿(s) is the Laplace transform for input influence, which is the given scalar function of time d=d(t) for elongation of the microactuator; ⌽0(s) and ⌽1(s) are the Laplace transforms for the output function, which is the real elongation ⌬l=⌬l1(t) of the microactuator for the open and closed control loop, respectively; W0(s), W1(s), W2(s), W3(s), and W4(s) are the transfer functions of the digital-to-analog transformer, the proportional pressure-limiting hydraulic valve, the actuator, the electronic driver of the valve, and the measuring device, respectively. In our arrangement, the system comprises the following devices: 앫 앫 앫 앫
the proportional pressure-limiting hydraulic valve of the type PMV42; the actuator produced in our laboratory; the electronic driver of the valve of the type E-MI-AC, Plug format [16]; and the measuring device LVDT (inductive measuring probe 1301 with digital device Militron 1202D, Mahr GmbH). Their transfer functions are: W0(s)⫽(1⫺e−sT0)/s⬵T0/(1⫹0.5s T0);
(25)
W1(s)⫽(T 21s2⫹T2s⫹1)−1;
(26)
W2(s)⫽(7.8)10−14 m3/N; W3(s)⫽10 A/V; W4(s)⫽(3.0)105 V/m.
(27)
where T0 is the given sampling period of time; T1 and T2 are the time constants for the control valve considered to be an oscillating link [15]. These constants are experimentally determined. For this purpose, a single actuator was investigated by applying step-wise inputs.
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In contrast to the known electro-hydraulic servo systems for devices of a similar nature [17– 19], our device has no moving masses (pistons, piston rods, etc.). In addition, there are no liquid flows in the device actuators, which work in a hydrostatic regime. The input for the multi-channel system is the spatial position of the platform described by the 6×1 vector ⌿⌺=(⌿1,…,⌿6)T, where ⌿i (i=1,…,6), is a specified spatial position of the platform, given as a function of time ⌿i=⌿i(t), transformed in the LABVIEW package (block CS in Fig. 10) by the matrix Rd,real, Eq. (18), as follows:
冘
冘
6
d1,r⫽
6
⌿ib1i,…,d6,r⫽
i⫽1
⌿ib6i
i⫽1
where bki (k,i=1,…,6) are corresponding elements of the matrix Rd,real. The output response is the vector ⌽⌺=(⌽1,…,⌽6)T, where ⌽1, ⌽2, and ⌽3 are small linear displacements along the X, Y and Z axes; and ⌽4, ⌽5, and ⌽6 are the small rotations relative to the same axes. These provide the required posture of the platform as a function of time. The rectangle designated as ST represents the upper platform in the layout given in Fig. 10, and in expanded form in Fig. 11. The six components ⌬li (i=1,…,6), that are the actual elongations of the actuators, undergo transformation by matrix (18) composed of 36 elements aij (i,j=1,…,6).
Fig. 11.
Layout of transformations of the vector dr into the vector ⌽⌺.
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Fig. 12. Measured responses of the microactuator (displacement in relative units vs. time in seconds) to a step-like input produced by an open loop control. Polygonal and smooth lines correspond to experimental and approximated data, respectively.
These elements serve as amplifiers, the outputs of which are afterwards summed to create six components ⌽1⫽⌺6i=1⌬lia6i,…,⌽6,r⫽⌺6i=1⌬lia6i. The averaged results of the experimental responses were approximated by the following expression: ⌽0(t)⫽1⫺exp[⫺at]⫹a exp[⫺bt]sin[wt] where a=0.6; a=20 s⫺1; b=4 s⫺1 and w=6.28 s⫺1. The experimental and approximated results are shown in Fig. 12. As seen in this figure, there is an overshooting of about 20% and the transient time is about 1 s. Fig. 13 presents the Bode diagrams computed by the MathLab software system. These diagrams
Fig. 13. Bode diagram: magnitude (in dB) and phase (in degrees) of the control response vs. frequency.
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indicate the stability of the control dynamics, which follows from the relationship between the values of the magnitude (0 dB) and the phase (⬵50°) of the response at frequency ⬵11 s⫺1. Since the mechanical part of the platform has very high stiffness, its natural frequencies are relatively high. To estimate the natural frequencies of the platform, a scheme of a rigid body with six degrees of freedom, which bears up against six elastic supports (the legs of the platform), is used (Fig. 14). The corresponding calculations show that the natural frequencies of the mechanical part lie in the range between 4×103 and 14×103 rad/s, i.e. they are two orders higher than the natural frequencies of the control system. 6. Experimental investigations of the platform’s accuracy The nominal location and orientation of the legs of the experimental platform shown in Fig. 2 are given in Table 1. Displacements of the platform points are measured by six devices (Fig. 2). The location and orientation of the probes for measurement of the displacements of the upper platform points are presented in Table 2. The elongations of the legs are measured by the probes, which are located on the leg parallel to its axis (Fig. 3b). The measurement results are presented in form of matrices Rd, Eq. (6), and Rm, Eq. (12), as follows:
冢
R d⫽
−0.77242
0.20598 0.600775
90.9745 −38.5354
0.77242
0.20598 0.600775
90.9745
0.19774 −0.77377 0.60182
−11.9504
−0.56672
0.56672 0.598046
−78.757
0.56672
0.56672 0.598046
−78.757
−0.19774 −0.77377 0.60182
−11.9504
130.179
冣
38.5354 −130.179 99.128
131.377
59.691 −131.196 −59.691
131.196
−99.128 −131.377
Fig. 14. Scheme for the calculation of the platform natural frequencies.
(28)
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Table 1 Nominal coordinates of the legs’ ends, mm Leg
Upper ends
Lower ends
no.
xNi
yNi
zNi
xMi
yMi
zMi
1 2 3 4 5 6
32 ⫺32 ⫺156.5 ⫺123.5 123.5 156.5
160 160 ⫺52 ⫺108 ⫺108 ⫺52
25 25 25 25 25 25
167 ⫺167 ⫺191 ⫺24 24 191
124 214 83 ⫺207.5 ⫺207.5 83
⫺80 ⫺80 ⫺80 ⫺80 ⫺80 ⫺80
Table 2 Location of the measuring devices I
Coordinates of the contact points Xi
Yi
⫺47.148 63.311 55.060 122.622 1.220 ⫺125.842
1 2 3 4 5 6
Rm⫽
冢
⫺80.337 ⫺39.658 30.634 ⫺72.387 143.887 ⫺68.036
Direction of measurement Zi
Axi
Ayi
Azi
55 55 55 25 25 25
0.5 0.5 0.866 0 0 0
⫺0.866 ⫺0.866 0.5 0 0 0
0 0 0 1 1 1
0.5
−0.8660 0 47.6313
27.5
81
0.5
−0.8660 0 47.6313
27.5
−35 1
0.8660 0.5
0 −27.5
47.6313
0
0
1 47.6313
−122.622 0
0
0
1 143.887
−1.21985 0
0
0
1 −68.0359 125.841
0
冣
(29)
Elongations ⌬i are measured as a function of the pressure p in the cylinder built in the ith leg over a pressure range of 0–200 bars. The results show that elongations were obtained with reasonable linearity (correlation coefficients in the range of 0.997 and higher) and with hysteresis less than 2.5%. For example, elongation ⌬1 (in µm) of leg No. 1 depends on pressure p as follows: 앫 for growing pressure ⌬1=0.1430p⫺0.82 (correlation factor 0.9982);
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앫 for decreasing pressure ⌬1=0.1398p⫺0.80 (correlation factor 0.9977); 앫 hysteresis is 2.24%. The repeatability of the results was tested by a set of repeated measurements for two modes. When the elongation of the leg is repeated from 0 to 11 µm, the deviations from linearity were in the range of 40 nm. When the pressure p=100 bar in the leg is repeated, elongation of the leg deviates for about 60 nm. The real values of the coordinates of the legs’ fixed points deviate from the nominal values. An identification procedure based on experimentation with an existing device was applied to obtain the actual platform characteristics. Thirty-six components of the vectors Rd were obtained as the regression factors resulting from the experiments according to conditions (22), i.e. the experiments in which only the ith leg was under pressure pi, and the other pressures were zero. An example of these dependencies—the experimental points, regression lines, and regression equations—are shown in Fig. 15. The plot shows the readings of six measuring devices vs. elongation of leg no. 5 when p5=0–250 bar and p1=p2=p3=p4=p6=0. As a result, we obtained matrix Rtr,real by expression (21) in the form:
冢
Rtr,real⫽
0.0094 0.0923 0.3061 −0.3276 −0.1182 −0.0325 −0.1035 0.2148 0.2053 −0.1967 −0.2265 −0.2355 0.1995 0.065
0.0454
冣
0.0845
0.1789 −0.2402
0.0055 −0.0037
0.3869
0.4804
0.3734 0.4613 0.0304 −0.0037
0.0127
0.002
0.0097 0.0084 0.3655
0.0469 −0.0095
0.0566 0
0.4999
(30)
Fig. 15. Experimental results for identification of transfer factors for elongation of leg no. 5. The numbers of the measurement devices correspond to those in Fig. 2.
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In this matrix, the fifth column is composed of six coefficients of regression given in Fig. 15 for corresponding measuring devices; for example the element a15 is the regression coefficient for the readings of the device no. 1, i.e. a15=⫺0.1182 (see Fig. 15), and so on. The other columns are composed of coefficients of regression, which are obtained by similar measurements made for leg nos. 1, 2, 3, 4, and 6. A set of legs’ elongations d providing the desired position ⌬ of the platform is d=(Rtr,real)−1Rm⌬. Thus, a scalar displacement d of the platform point in the specified direction given by unit vector a=(ax,ay,az)T is: d⫽RT⌬⫽RT(Rm)−1Rtr,reald
(31)
where R is the 6×1 vector (5) of Plucker’s coordinates of this point. By applying the identification matrix (30) one may accurately predict the displacements of the platform points. The prediction was tested experimentally. During the experiments, different combinations of hydraulic pressures in the legs’ cylinders were used: p⫽(p,0,p,0,p,0)T, p⫽(0,p,0,p,0,p)T, and p⫽(p,p,p,p,p,p)T, where p is the vector of hydraulic pressures in the legs; the components p of the vector are controlled in the range p=0–250 bar. The values calculated using Eq. (31) agrees well with experimental measurements (Fig. 16). A relative deviation between the theoretical results obtained by application of expression (31) and experimental measurements lies in the range 0.68–2.79%. Note that the main part of the error is caused by deviations from the given values of the six actual values of the hydraulic pressure within the legs. Thus, the analysis shows that neither the conventional or modified Jacobian model, nor the
Fig. 16. Comparison of the theoretical (lines) and actual (points) displacements for the readings of measurement device nos. 1, 5 and 6 vs. pressure in the legs’ cylinders p2=p4=p6=0 and p1=p3=p5=40–200 bar. The numbers of the devices correspond to those in Fig. 2.
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finite-elements method can satisfy the requirements for the high precision device in the accuracy range of 10⫺7–10⫺8 m. Thus, the identification-based model is the model of choice. 7. Application example The results of the dynamic investigation may be applied for calculating the minimal segment length smin of the machined profile when the platform is used as a secondary table in the ‘coarse– fine’ systems. To compensate the cumulative and cyclic errors of the feed system of the machine tool, this length must be more than the resolution ts of the feed system, i.e. smin⫽(p/30)f/wmax⬎ts where f is cutting feed along the machined profile in mm/min, f=√f 2x +f 2y ; fx and fy are the feeds along the X and Y axes, respectively; and wmax is the maximal allowed forced frequency, rad/s. For example, when wmax=8 rad/s and f=100 mm/min, this expression yields smin=1.3 mm. This feature can be improved by modifying the hydraulic system. Experiments were carried out, in which disturbances were added to nominal feeds. The nominal feeds are performed with the X–Y motions of the table of the milling center, while the additional feeds are made by the platform mechanism. The said feeds provided the appearance of a predesigned distortion of the machined profile. For example, a specified elliptical shape is expressed by an expression: R⫽R0[1⫹d sin(2⍀t⫹j0)], where R is the current polar radius of the specified workpiece profile; R0 is the initial polar radius of the circle; d is the specified programmed relative deviation of the profile; w is the circular frequency in rad/s corresponding to the profile feed f, i.e. ⍀=pf/(30R0); t is time; and j0 is the initial phase angle. The profile according to this expression is shown in Fig. 17a. Simulation of this profile performed with LABVIEW based control system is presented in Fig. 17b. The profile of the workpiece machined on the milling center and measured on the Talyrond device is shown in Fig. 17c. Fig. 18 illustrates roundness polar plots of a cylindrical part machined. The out-of-roundness of a cylinder milled without correction (Fig. 18a) is 14.25 µm, and a milled cylinder after correction by means of the device has out-of-roundness 8.55 µm (Fig. 18b). 8. Conclusions 1. The 6-DOF platform-type mechanism for manipulations in the range of small displacements possessing rigid joints between the legs and the platform as well as between the legs and the base is developed and investigated. The operating range and displacement accuracy are established. The accepted accuracy of the device under discussion is better than 0.1 µm when a closed-loop automatic control system is used. 2. Analysis of infinitesimal kinematics is performed using the models of classical and modified Stewart platform kinematics, the finite-elements method, and an experimentally based identifi-
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Fig. 17. Profile with elliptical-type distortion (a), simulation performed with LABVIEW based control system (b) and out-of-roundness polar plot of the machined profile measured with Talyrond device (c).
cation method. The last-listed method is the most suitable for calculations of the considered platform. 3. Hydraulically driven microactuators are suitable for driving the platform and providing the necessary stiffness, manipulation linearity, and precision. The dynamic restrictions of the investigated device depend on the hydraulic system. 4. The described device can serve as a secondary table for high-precision milling machines to improve the accuracy of those machines and for some other tasks.
Acknowledgements The authors express their gratitude to the Paul Ivanier Robotics Research and Production Management Center for its help and encouragement.
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Fig. 18. Out-of-roundness polar plots of a cylindrical part machined on a CNC milling machine: (a) without correction; and (b) with acting correction. An initial orientation label is denoted by x.
References [1] C.C. Nguyen, S.S. Antrazi, Z.-L. Zhou, C.E. Campbell Jr, Analysis and experimentation of a Stewart platformbased force/torque sensor, Int. J. Robotics Automation 7 (3) (1992) 133–140. [2] D.R. Kerr, Analysis, properties, and design of Stewart-platform transducers, Trans. ASME, J. Mech. Transmissions, Automation Design 111 (1989) 24–28. [3] A. Gaillet, C. Reboulet, An isostatic six component force and torque sensor, in: Proceedings of the 13th International Symposium on Industrial Robots in Chicago, 18/102-18/111, 1983. [4] M. Taniguchi, M. Ikeda, A. Inagaki, R. Funatsu, Ultra precision wafer positioning by six-axis micro-motion mechanism, Int. J. Jpn. Soc. Prec. Eng. 26 (1) (1992) 35–40. [5] S. Asano, T. Goto, Micro-positioning stage of 6 degrees-of-freedom with nanometer resolution, in: P. Seyfried, H. Kunzmann, P. McKeown, M. Weck (Eds.), Progress in Precision Engineering, Proceeedings of the Sixth International Precision Engineering Seminar Springer-Verlag, Berlin, 1991, p. 362. [6] A. Hara, K. Sugimoto, Synthesis of parallel micromanipulators, Trans. ASME, J. Mech. Transmissions, Automations Design 111 (1989) 34–39. [7] A. Slocum, Precision Machine Design, MIT Press, 1992. [8] Bi Zhang, Z. Zhu, Bolin Zhang, W.-E. Fu, J. Wang, A linear piezomotor of high stiffness for nanometer resolution, Annals CIRP 46 (1) (1997) 305–308. [9] K.-M. Lee, S. Arjunan, A three-degrees-of-freedom micromotion in-parallel actuated manipulator, IEEE Trans. Robotics Automation 7 (1999) 238–247. [10] P. Gao, S.M. Swei, A six-degree-of-freedom micro-manipulator based on piezoelectric translator, Nanotechnology 10 (1999) 447–452. [11] V. Portman, B.Z. Sandler, High stiffness precision actuator for small displacements, Int. J. Machine Tools Manufacture 39 (5) (1999) 823–837. [12] V. Portman, B.-Z. Sandler, E. Zahavi, Rigid 6×6 parallel platform for precision 3-d micromanipulation, in: Proceedings of the Tenth World Congress on the Theory of Machines and Mechanisms, vol. 2, 1999, pp. 828–834. [13] ANSYS 5.2, User Manual, Theory, vol. II, Swanson Analysis System, 1996.
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[14] LABVIEW, User Manual, National Instrument, parts NN 320997C-01, 320999B-01, 321296B-01, 321526B-01, January 1998. [15] R.C. Dorf, R.H. Bishop, Modern control systems (7th ed), Addison-Wesley, New York, 1998. [16] Proportional pressure limiting valves type PMV and PDV, HAWE Hydraulic. Electronic drivers type E-MI-ACPlug format, ATOS spa, Italy. [17] W.-S. Yu, T.-S. Kuo, Continuous-time indirect adaptive control of the electro-hydraulic servo systems, IEEE Trans. Control Syst. Technol. 5 (1997) 163–177. [18] G.A. Sohl, J.E. Bobrow, Experiments and simulations on the nonlinear control of a hydraulic servosystem, IEEE Trans. Control Technol. 7 (1999) 238–247. [19] G. Vossoughi, M. Donath, Dynamic feedback linearization for electro-hydraulic actuated control system, J. Dynamic Syst. Meas., Control 117 (1995) 468–477.