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 EngineeringMeasured [radians]Measured [radians]Measured [radians]5002500-250-500500250x displacement-500 -250 0 250 500Command [radians]0-250-5005002500-250-500y displacement-500 -250 0 250 500Command [radians]z displacement-500 -250 0 250 500Command [radians]Error [m]Error [m]Error [m]1050-5-1010-5-10x - Parasitic errorsxyzTheta yTheta z-500 -250 0 250 500Command [microns]50y - Parasitic errors-500 -250 0 250 500Command [microns]1050-5-10xyzTheta xTheta zz - Parasitic errorsxyzTheta xTheta y-500 -250 0 250 500Command [microns]Figure 21: Orientation tests results which compare the measured vs. commanded HPF behavior (left)and the parasitic errors as a function of commanded behavior (right)Several linear relationships between the commanded displacement and the parasitic errors may bee seenwithin the parasitic error plots. Some of these relationships are denoted on the right sides of Figs. 20 and21 by line fits. The linear relationship between command and error is important as it shows thesesystematic errors are well-correlated to the actuation inputs. This opens the door to an elegant calibrationtechnique which will be described later in this section. The systematic errors are in-part due to:020100-10-202010-10-2020100-10-20Error [radian]Error [radian]Error [radian]18
Design of a Hybrid Positioner-Fixture for Six-axis Nanopositioning and Precision FixturingSubmitted to Precision Engineering(1) Geometry errors: The physical components which define the HPFs kinematic chains differ from theideal components which drive the kinematic model. For example there are errors in the reference andaligning features of each component which affect the assembly of each component. There are also errorsin the size of the components which define the HPFs kinematic chains, repeatable parasitic errors in theguide flexures which are due to imperfect geometry of the compliant parts, and actuator misalignmenterrors.(2) Material property errors: The model assumes isotropic material properties for Aluminum and aYoung’s Modulus of 30 ksi Young’s Modulus. It is not uncommon for the modulus of Aluminum alloyswhich have been formed into sheet to differ by 5-10% fro the generic 30ksi value.6.3. Removing systematic errors via calibrationThe calibrated position and orientation vector, PO C , may be related to the model’s predicted position andorientation vector, PO M , via the calibration matrix, C, given in Eqxn. 29. In this approach, we arecalibrating the HPF by modifying the model rather than modifying the hardware.POCXcYcZc C PO MxcyczcCalibratedC11C21C31C41C51C61C12C22C32C42C52C62C13C23C33C43C53C63C14C24C34C44C54C64C15C25C35C45C55C65C16C26C36C46C56C66XcYcZcxcyczcModelThe rows of the matrix may be populated using the results of the single-axis tests. For instance, during anX axis move, PO M , would be [Xc, 0, 0, 0, 0, 0] T and the pre-calibrated, experimentally determinedposition and orientation vector, PO E , would be given by Eqxn 30.(29)XcYcZcxcycFor POMzcExperimental [ Xc,0,0,0,0,0]TC Xc11C Xc21C Xc31C Xc41C Xc51C Xc61The slope of the lines which were fitted to the data plotted in Figs. 20 and 21 may be used as theconstants, C i1 , for the first column of the calibration matrix. The same procedure may be used to populatethe 2 nd , 3 rd , 4 th , 5 th , and 6 th , columns using the Yc, Zc, xc, yc, and zc tests respectively. Thecompleted matrix for the moving groove HPF prototype is provided in Eqxn. 31.(30)19
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