Design of a Hybrid Positioner-Fixture for Six-axis Nanopositioning ...

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Design of a Hybrid Positioner-Fixture for Six-axis Nanopositioning and Precision FixturingSubmitted to Precision EngineeringStaticCS GrooveDisplacedCSC r 1br 1b||Dyr x zBr 1ar 1cEr A zFigure 5: Detailed view of in-plane kinematic model in relationship to kinematic coupling geometry(top component removed for clarity). Note the static coordinate system (CS) and displaced CS. Thecoordinate systems lie at the coupling centroid as calculated from the layout of the ball positions whenall r ib = 0.For each loop i, Eqxn. 3 represents the two relationships shown in Eqxns. 4 and 5.LLiaiaiaLibcosiaLibsiniaLiccos icXccos || (4)iaLibsiniaLibcosiaLicsin icYcsin || (5)The variables L ij represent the length of vector r ij . In this model ic and z are linked by Eqxn. 6, and L iais equal to L ic .ic ia z(6)When Eqxn. 6 is substituted into Eqxns. 4 and 5, L ic is substituted for L ia , and we assume small angleapproximations (sin(z c )~ (z c ) and cos(z c )~1), we obtain the matrix equation provided in Eqxn. 7.101010010101 L L L L L L1c1c2c2c3c3c sin cos sin cos sin cos cos1a sin1a 0 cos2a 0 sin2a 00 cos3a 00 sin3a1a1a002a2a00003a3a Xc L1b sin 1a Yc L1b cos1a z c L2b sin2a L1b|| L2b cos2a L 2b||L3b sin3a L3b|| L3b cos 3aSmall angle approximations are appropriate as the second order effects on position due to angle couplingresult in errors on the order of 10 -3 – 10 -6 smaller than the displacements of interest. The right side ofEqxn. 7 consists of the actuation inputs, L ib , and geometry variables. This kinematic relationship isdifferent from previous approaches in that it enables the immediate calculation of forward and reversekinematic solutions without iteration.(7)6
Design of a Hybrid Positioner-Fixture for Six-axis Nanopositioning and Precision FixturingSubmitted to Precision Engineering3.3. Relationship between ball center displacements and the fixture’s out-ofplanedisplacementsThe out-of-plane displacement, Z c , and rotations, x c and y c , are captured using translations of theball centers perpendicular to part containing the grooves [01]. Figure 6 shows the planes formed by thehomed and displaced position of the ball centers. The motion of each ball relative to the homed plate isgiven by Eqxn. 2 as z i .Homedz 1z Ball 1Ball 3Figure 6: Out-of-plane vector model of HPFz 3Displacedz 2Ball 2Given the location of each ball center, one may form an equation for the plane containing the ball centersand then solve for (1) the location of the coupling centroid in z and (2) the orientation of the plane asdescribed by its normal vector. Changes in position and normal vector due to the out-of-plane balldisplacements, z i , are used to determine the change in Z c , x c and y c [01] The relationshipsbetween ball displacements and fixture displacements are provided in Eqxn. 8 – 10 [01]. Theabbreviations s[] = sin[] and c[] = cos[] have been used in Eqxns 8 – 10. L1.(s[1].z c[1]).(z2 z3) L2.(s[2].z c[2]).(z3 z1) L3.(s[3].z c[3]).(z1 z2)x c (8)L1.L2.s[21] L2.L3.s[32] L3.L1.s[13] L1.(s[1] c[1].z).( z2 z3) L2.(s[2] c[2]. z).( z3 z1) L3.(s[3] c[3].z).( z1 z2)yc (9)L1.L2.s[21] L2.L3.s[32] L3.L1.s[13]Zc1xc.s1 1 L ( z(10)1yc.c)3.4. Implementation of kinematic theory in a spreadsheet modelEquations 7 – 10 may be used to provide forward and inverse kinematic solutions for six axis motions.The theory was incorporated into an Excel spreadsheet (available for download atpsdam.mit.edu/tools/index.html) which solves the forward and reverse kinematics of a moving grooveHPF. The accuracy of the kinematic model was checked by comparing simulated displacements withdisplacements measured from a solid model of the HPF. The difference between predicted and measuredresults was less than 5 nanometers/1 radians in each of the following scenarios:(1) The planar displacement X c , Y c and z c examples portrayed in X(2) The non-planar displacements, x c , y c and Z c(3) Various combinations of planar and non-planar displacements4. Stiffness modeling approachIn this section we provide a brief overview of how the stiffness of the compliant components are modeledand then combined to form a model for the stiffness of the HPF.7
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Design of a Hybrid Positioner-Fixture for Six-axis Nanopositioning ...

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