Christophe's review
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\setabbreviationstyle[acronym]{long-short}
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\setglossarystyle{long-name-desc}
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\usepackage{amssymb}
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\usepackage{amsmath}
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\usepackage{cases}
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\usepackage{empheq}
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\makeindex
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\makeglossaries
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@ -344,17 +344,6 @@ To do so, the effects of these disturbances were first measured experimental and
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To validate the accuracy of the micro-station model, "real world" experiments are simulated and compared with measurements in Section ref:sec:ustation_experiments.
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# #+name: tab:ustation_section_matlab_code
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# #+caption: Report sections and corresponding Matlab files
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# #+attr_latex: :environment tabularx :width 0.6\linewidth :align lX
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# #+attr_latex: :center t :booktabs t
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# | *Sections* | *Matlab File* |
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# |---------------------------------------+-----------------------------|
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# | Section ref:sec:ustation_kinematics | =ustation_1_kinematics.m= |
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# | Section ref:sec:ustation_modeling | =ustation_2_modeling.m= |
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# | Section ref:sec:ustation_disturbances | =ustation_3_disturbances.m= |
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# | Section ref:sec:ustation_experiments | =ustation_4_experiments.m= |
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* Micro-Station Kinematics
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:PROPERTIES:
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:HEADER-ARGS:matlab+: :tangle matlab/ustation_1_kinematics.m
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@ -364,10 +353,10 @@ To validate the accuracy of the micro-station model, "real world" experiments ar
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The micro-station consists of 4 stacked positioning stages (Figure ref:fig:ustation_cad_view).
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From bottom to top, the stacked stages are the translation stage $D_y$, the tilt stage $R_y$, the rotation stage (Spindle) $R_z$ and the positioning hexapod.
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Such a stacked architecture allows high mobility, but the overall stiffness is reduced, and the dynamics is very complex. complex dynamics.
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Such a stacked architecture allows high mobility, but the overall stiffness is reduced, and the dynamics is very complex.
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#+name: fig:ustation_cad_view
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#+caption: CAD view of the micro-station with the translation stage (in blue), tilt stage (in red), rotation stage (in yellow) and positioning hexapod (in purple). On top of these four stages, a solid part (shown in green) will be replaced by the stabilization stage.
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#+caption: CAD view of the micro-station with the translation stage (in blue), tilt stage (in red), rotation stage (in yellow) and positioning hexapod (in purple).
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#+attr_latex: :width \linewidth
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[[file:figs/ustation_cad_view.png]]
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@ -415,7 +404,7 @@ The "controlled" DoF of each stage (for instance the $D_y$ direction for the tra
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The translation stage is used to position and scan the sample laterally with respect to the X-ray beam.
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A linear motor was first used to enable fast and accurate scans.
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It was later replaced with a stepper motor and lead-screw, as the feedback control used for the linear motor was unreliable[fn:ustation_12].
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It was later replaced with a stepper motor and lead-screw, as the feedback control used for the linear motor was unreliable[fn:ustation_11].
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An optical linear encoder is used to measure the stage motion and for controlling the position.
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Four cylindrical bearings[fn:ustation_4] are used to guide the motion (i.e. minimize the parasitic motions) and have high stiffness.
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@ -529,14 +518,10 @@ The /orientation/ of a rigid body is the same at all its points (by definition).
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Hence, the orientation of a rigid body can be viewed as that of a moving frame attached to the rigid body.
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It can be represented in several different ways: the rotation matrix, the screw axis representation, and the Euler angles are common descriptions.
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The rotation matrix ${}^A\bm{R}_B$ is a $3 \times 3$ matrix containing the Cartesian unit vectors of frame $\{\bm{B}\}$ represented in frame $\{\bm{A}\}$ eqref:eq:ustation_rotation_matrix.
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The rotation matrix ${}^A\bm{R}_B$ is a $3 \times 3$ matrix containing the Cartesian unit vectors $[{}^A\hat{\bm{x}}_B,\ {}^A\hat{\bm{y}}_B,\ {}^A\hat{\bm{z}}_B]$ of frame $\{\bm{B}\}$ represented in frame $\{\bm{A}\}$ eqref:eq:ustation_rotation_matrix.
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\begin{equation}\label{eq:ustation_rotation_matrix}
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{}^A\bm{R}_B = \left[ {}^A\hat{\bm{x}}_B | {}^A\hat{\bm{y}}_B | {}^A\hat{\bm{z}}_B \right] = \begin{bmatrix}
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u_{x} & v_{x} & z_{x} \\
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u_{y} & v_{y} & z_{y} \\
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u_{z} & v_{z} & z_{z}
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\end{bmatrix}
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{}^A\bm{R}_B = \left[ {}^A\hat{\bm{x}}_B | {}^A\hat{\bm{y}}_B | {}^A\hat{\bm{z}}_B \right]
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\end{equation}
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Consider a pure rotation of a rigid body ($\{\bm{A}\}$ and $\{\bm{B}\}$ are coincident at their origins, as shown in Figure ref:fig:ustation_rotation).
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@ -553,7 +538,7 @@ For rotations along $x$, $y$ or $z$ axis, the formulas of the corresponding rota
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\begin{align}
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\bm{R}_x(\theta_x) &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta_x) & -\sin(\theta_x) \\ 0 & \sin(\theta_x) & \cos(\theta_x) \end{bmatrix} \\
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\bm{R}_y(\theta_y) &= \begin{bmatrix} \cos(\theta_y) & 0 & \sin(\theta_y) \\ 0 & 1 & 0 \\ -\sin(\theta_y) & 0 & \cos(\theta_y) \end{bmatrix} \\
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\bm{R}_z(\theta_z) &= \begin{bmatrix} \cos(\theta_z) & -\sin(\theta_z) & 0 \\ \sin(\theta_z) & \cos(\theta_x) & 0 \\ 0 & 0 & 1 \end{bmatrix}
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\bm{R}_z(\theta_z) &= \begin{bmatrix} \cos(\theta_z) & -\sin(\theta_z) & 0 \\ \sin(\theta_z) & \cos(\theta_z) & 0 \\ 0 & 0 & 1 \end{bmatrix}
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\end{align}
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\end{subequations}
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@ -643,7 +628,7 @@ Another key advantage of homogeneous transformation is the easy inverse transfor
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{}^B\bm{T}_A = {}^A\bm{T}_B^{-1} =
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\left[ \begin{array}{ccc|c}
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& & & \\
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& {}^A\bm{R}_B^T & & -{}^A \bm{R}_B^T {}^AP_{O_B} \\
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& {}^A\bm{R}_B^{\intercal} & & -{}^A \bm{R}_B^{\intercal} {}^AP_{O_B} \\
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& & & \cr
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\hline
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0 & 0 & 0 & 1 \\
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@ -685,8 +670,9 @@ The homogeneous transformation matrix corresponding to the micro-station $\bm{T}
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$\bm{T}_{\mu\text{-station}}$ represents the pose of the sample (supposed to be rigidly fixed on top of the positioning-hexapod) with respect to the granite.
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If the transformation matrices of the individual stages are each representing a perfect motion (i.e. the stages are supposed to have no parasitic motion), $\bm{T}_{\mu\text{-station}}$ then represent the pose setpoint of the sample with respect to the granite.
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If the transformation matrices of the individual stages are each representing a perfect motion (i.e. the stages are supposed to have no parasitic motion), $\bm{T}_{\mu\text{-station}}$ then represents the pose setpoint of the sample with respect to the granite.
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The transformation matrices for the translation stage, tilt stage, spindle, and positioning hexapod can be written as shown in Equation eqref:eq:ustation_transformation_matrices_stages.
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The setpoints are $D_y$ for the translation stage, $\theta_y$ for the tilt-stage, $\theta_z$ for the spindle, $[D_{\mu x},\ D_{\mu y}, D_{\mu z}]$ for the micro-hexapod translations and $[\theta_{\mu x},\ \theta_{\mu y}, \theta_{\mu z}]$ for the micro-hexapod rotations.
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\begin{equation}\label{eq:ustation_transformation_matrices_stages}
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\begin{align}
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@ -845,7 +831,7 @@ Joints are used to impose kinematic constraints between solid bodies and to spec
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External forces can be used to model disturbances, and "sensors" can be used to measure the relative pose between two defined frames.
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#+name: fig:ustation_simscape_stage_example
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#+caption: Example of a stage (here the tilt-stage) represented in the multi-body model software (Simscape). It is composed of two solid bodies connected by a 6-DoF joint. One joint DoF (here the tilt angle) can be imposed, the other DoFs are represented by springs and dampers. Additional disturbances forces for all DoF can be included
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#+caption: Example of a stage (here the tilt-stage) represented in the multi-body model software (Simscape). It is composed of two solid bodies connected by a 6-DoF joint. One joint DoF (here the tilt angle) can be imposed, the other DoFs are represented by springs and dampers. Additional disturbing forces for all DoF can be included
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[[file:figs/ustation_simscape_stage_example.png]]
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Therefore, the micro-station is modeled by several solid bodies connected by joints.
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@ -1095,7 +1081,7 @@ Similar to what is done for the accelerometers, a Jacobian matrix $\bm{J}_F$ is
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The equivalent forces and torques applied at center of $\{\mathcal{X}\}$ are then computed using eqref:eq:ustation_compute_cart_force.
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\begin{equation}\label{eq:ustation_compute_cart_force}
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F_{\mathcal{X}} = \bm{J}_F^t \cdot F_{\mathcal{L}}
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F_{\mathcal{X}} = \bm{J}_F^{\intercal} \cdot F_{\mathcal{L}}
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\end{equation}
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Using the two Jacobian matrices, the FRF from the 10 hammer impacts to the 12 accelerometer outputs can be converted to the FRF from 6 forces/torques applied at the origin of frame $\{\mathcal{X}\}$ to the 6 linear/angular accelerations of the top platform expressed with respect to $\{\mathcal{X}\}$.
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@ -1348,8 +1334,8 @@ exportFig('figs/ustation_frf_compliance_Rxyz_model.pdf', 'width', 'half', 'heigh
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The goal of this section is to obtain a realistic representation of disturbances affecting the micro-station.
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These disturbance sources are then used during time domain simulations to accurately model the micro-station behavior.
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The focus on stochastic disturbances because, in principle, it is possible to calibrate the repeatable part of disturbances.
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Such disturbances include ground motions and vibrations induces by scanning the translation stage and the spindle.
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The focus is on stochastic disturbances because, in principle, it is possible to calibrate the repeatable part of disturbances.
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Such disturbances include ground motions and vibrations induce by scanning the translation stage and the spindle.
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In the multi-body model, stage vibrations are modeled as internal forces applied in the stage joint.
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In practice, disturbance forces cannot be directly measured.
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@ -1398,7 +1384,7 @@ Therefore, from a control perspective, they are not important.
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**** Ground Motion
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The ground motion was measured by using a sensitive 3-axis geophone[fn:ustation_11] placed on the ground.
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The ground motion was measured by using a sensitive 3-axis geophone shown in Figure ref:fig:ustation_geophone_picture placed on the ground.
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The generated voltages were recorded with a high resolution DAC, and converted to displacement using the Geophone sensitivity transfer function.
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The obtained ground motion displacement is shown in Figure ref:fig:ustation_ground_disturbance.
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@ -1470,7 +1456,7 @@ A special optical element (called a "straightness interferometer"[fn:ustation_9]
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A similar setup was used to measure the horizontal deviation (i.e. in the $x$ direction), as well as the pitch and yaw errors of the translation stage.
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#+name: fig:ustation_errors_ty_setup
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#+caption: Experimental setup to measure the flatness (vertical deviation) of the translation stage
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#+caption: Experimental setup to measure the straightness (vertical deviation) of the translation stage
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[[file:figs/ustation_errors_ty_setup.png]]
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Six scans were performed between $-4.5\,mm$ and $4.5\,mm$.
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@ -1513,7 +1499,7 @@ exportFig('figs/ustation_errors_dy_vertical.pdf', 'width', 'half', 'height', 'no
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%% Measurement of the linear (vertical) deviation of the Translation stage - Remove best linear fit
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figure;
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plot(ty_errors.setpoint, ty_errors.ty_z - straight_line, 'k-')
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xlabel('$D_y$ position [mm]'); ylabel('Vertical error [$\mu$m]');
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xlabel('$D_y$ position [mm]'); ylabel('Residual vertical error [$\mu$m]');
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xlim([-5, 5]); ylim([-0.4, 0.4]);
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#+end_src
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@ -1963,7 +1949,7 @@ exportFig('figs/ustation_dist_source_spindle.pdf', 'width', 'third', 'height', '
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The disturbances are characterized by their power spectral densities, as shown in Figure ref:fig:ustation_dist_sources.
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However, to perform time domain simulations, disturbances must be represented by a time domain signal.
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To generate stochastic time-domain signals with a specific power spectral densities, the discrete inverse Fourier transform is used, as explained in [[cite:&preumont94_random_vibrat_spect_analy chap. 12.11]].
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To generate stochastic time-domain signals with a specific power spectral density, the discrete inverse Fourier transform is used, as explained in [[cite:&preumont94_random_vibrat_spect_analy chap. 12.11]].
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Examples of the obtained time-domain disturbance signals are shown in Figure ref:fig:ustation_dist_sources_time.
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#+begin_src matlab
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@ -2192,7 +2178,7 @@ exportFig('figs/ustation_errors_model_spindle_axial.pdf', 'width', 'half', 'heig
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#+end_subfigure
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#+end_figure
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** Raster Scans with the translation stage
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** Scans with the translation stage
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<<sec:ustation_experiments_ty_scans>>
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A second experiment was performed in which the translation stage was scanned at constant velocity.
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@ -4538,7 +4524,7 @@ Note that there is trade-off between:
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:END:
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**** Theory
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For inverse kinematic analysis, it is assumed that the position ${}^A\bm{P}$ and orientation of the moving platform ${}^A\bm{R}_B$ are given and the problem is to obtain the joint variables, namely, $\bm{L} = [l_1, l_2, \dots, l_6]^T$.
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For inverse kinematic analysis, it is assumed that the position ${}^A\bm{P}$ and orientation of the moving platform ${}^A\bm{R}_B$ are given and the problem is to obtain the joint variables, namely, $\bm{L} = [l_1, l_2, \dots, l_6]^{\intercal}$.
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From the geometry of the manipulator, the loop closure for each limb, $i = 1, 2, \dots, 6$ can be written as
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\begin{align*}
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@ -4548,12 +4534,12 @@ From the geometry of the manipulator, the loop closure for each limb, $i = 1, 2,
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To obtain the length of each actuator and eliminate $\hat{\bm{s}}_i$, it is sufficient to dot multiply each side by itself:
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\begin{equation}
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l_i^2 \left[ {}^A\hat{\bm{s}}_i^T {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^T \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]
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l_i^2 \left[ {}^A\hat{\bm{s}}_i^{\intercal} {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^{\intercal} \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]
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\end{equation}
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Hence, for $i = 1, 2, \dots, 6$, each limb length can be uniquely determined by:
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\begin{equation}
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l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i}
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l_i = \sqrt{{}^A\bm{P}^{\intercal} {}^A\bm{P} + {}^B\bm{b}_i^{\intercal} {}^B\bm{b}_i + {}^A\bm{a}_i^{\intercal} {}^A\bm{a}_i - 2 {}^A\bm{P}^{\intercal} {}^A\bm{a}_i + 2 {}^A\bm{P}^{\intercal} \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^{\intercal} {}^A\bm{a}_i}
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\end{equation}
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If the position and orientation of the moving platform lie in the feasible workspace of the manipulator, one unique solution to the limb length is determined by the above equation.
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@ -4613,8 +4599,7 @@ Otherwise, when the limbs' lengths derived yield complex numbers, then the posit
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* Footnotes
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[fn:ustation_12]It was probably caused by rust of the linear guides along its stroke.
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[fn:ustation_11]A 3-Axis L4C geophone manufactured Sercel was used.
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[fn:ustation_11]It was probably caused by rust of the linear guides along its stroke.
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[fn:ustation_10]Laser source is manufactured by Agilent (5519b).
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[fn:ustation_9]The special optics (straightness interferometer and reflector) are manufactured by Agilent (10774A).
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[fn:ustation_8]C8 capacitive sensors and CPL290 capacitive driver electronics from Lion Precision.
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% Created 2025-03-25 Tue 22:22
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% Created 2025-04-07 Mon 16:40
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% Intended LaTeX compiler: pdflatex
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\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
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@ -13,7 +13,7 @@
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pdftitle={Multi-Body Model - Micro Station},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Emacs 29.4 (Org mode 9.6)},
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pdfcreator={Emacs 30.1 (Org mode 9.7.26)},
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pdflang={English}}
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\usepackage{biblatex}
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@ -23,7 +23,6 @@
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\tableofcontents
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\clearpage
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From the start of this work, it became increasingly clear that an accurate micro-station model was necessary.
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First, during the uniaxial study, it became clear that the micro-station dynamics affects the nano-hexapod dynamics.
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@ -43,18 +42,16 @@ Disturbances affecting the positioning accuracy also need to be modeled properly
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To do so, the effects of these disturbances were first measured experimental and then injected into the multi-body model (Section \ref{sec:ustation_disturbances}).
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To validate the accuracy of the micro-station model, ``real world'' experiments are simulated and compared with measurements in Section \ref{sec:ustation_experiments}.
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\chapter{Micro-Station Kinematics}
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\label{sec:ustation_kinematics}
|
||||
The micro-station consists of 4 stacked positioning stages (Figure \ref{fig:ustation_cad_view}).
|
||||
From bottom to top, the stacked stages are the translation stage \(D_y\), the tilt stage \(R_y\), the rotation stage (Spindle) \(R_z\) and the positioning hexapod.
|
||||
Such a stacked architecture allows high mobility, but the overall stiffness is reduced, and the dynamics is very complex. complex dynamics.
|
||||
Such a stacked architecture allows high mobility, but the overall stiffness is reduced, and the dynamics is very complex.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,width=\linewidth]{figs/ustation_cad_view.png}
|
||||
\caption{\label{fig:ustation_cad_view}CAD view of the micro-station with the translation stage (in blue), tilt stage (in red), rotation stage (in yellow) and positioning hexapod (in purple). On top of these four stages, a solid part (shown in green) will be replaced by the stabilization stage.}
|
||||
\caption{\label{fig:ustation_cad_view}CAD view of the micro-station with the translation stage (in blue), tilt stage (in red), rotation stage (in yellow) and positioning hexapod (in purple).}
|
||||
\end{figure}
|
||||
|
||||
There are different ways of modeling the stage dynamics in a multi-body model.
|
||||
@ -65,7 +62,6 @@ s can be tuned separately for each DoF.
|
||||
The ``controlled'' DoF of each stage (for instance the \(D_y\) direction for the translation stage) is modeled as infinitely rigid (i.e. its motion is imposed by a ``setpoint'') while the other DoFs have limited stiffness to model the different micro-station modes.
|
||||
\section{Motion Stages}
|
||||
\label{ssec:ustation_stages}
|
||||
|
||||
\paragraph{Translation Stage}
|
||||
|
||||
The translation stage is used to position and scan the sample laterally with respect to the X-ray beam.
|
||||
@ -75,7 +71,6 @@ It was later replaced with a stepper motor and lead-screw, as the feedback contr
|
||||
An optical linear encoder is used to measure the stage motion and for controlling the position.
|
||||
|
||||
Four cylindrical bearings\footnote{Ball cage (N501) and guide bush (N550) from Mahr are used.} are used to guide the motion (i.e. minimize the parasitic motions) and have high stiffness.
|
||||
|
||||
\paragraph{Tilt Stage}
|
||||
|
||||
The tilt stage is guided by four linear motion guides\footnote{HCR 35 A C1, from THK.} which are placed such that the center of rotation coincide with the X-ray beam.
|
||||
@ -99,14 +94,12 @@ To precisely control the \(R_y\) angle, a stepper motor and two optical encoders
|
||||
\captionof{figure}{\label{fig:ustation_ry_stage}Tilt Stage}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
|
||||
\paragraph{Spindle}
|
||||
|
||||
Then, a rotation stage is used for tomography experiments.
|
||||
It is composed of an air bearing spindle\footnote{Made by LAB Motion Systems.}, whose angular position is controlled with a 3 phase synchronous motor based on the reading of 4 optical encoders.
|
||||
|
||||
Additional rotary unions and slip-rings are used to be able to pass electrical signals, fluids and gazes through the rotation stage.
|
||||
|
||||
\paragraph{Micro-Hexapod}
|
||||
|
||||
Finally, a Stewart platform\footnote{Modified Zonda Hexapod by Symetrie.} is used to position the sample.
|
||||
@ -128,7 +121,6 @@ It can also be used to precisely position the PoI vertically with respect to the
|
||||
\captionof{figure}{\label{fig:ustation_hexapod_stage}Micro Hexapod}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
|
||||
\section{Mathematical description of a rigid body motion}
|
||||
\label{ssec:ustation_motion_description}
|
||||
In this section, mathematical tools\footnote{The tools presented here are largely taken from \cite{taghirad13_paral}.} that are used to describe the motion of positioning stages are introduced.
|
||||
@ -176,14 +168,10 @@ The \emph{orientation} of a rigid body is the same at all its points (by definit
|
||||
Hence, the orientation of a rigid body can be viewed as that of a moving frame attached to the rigid body.
|
||||
It can be represented in several different ways: the rotation matrix, the screw axis representation, and the Euler angles are common descriptions.
|
||||
|
||||
The rotation matrix \({}^A\bm{R}_B\) is a \(3 \times 3\) matrix containing the Cartesian unit vectors of frame \(\{\bm{B}\}\) represented in frame \(\{\bm{A}\}\) \eqref{eq:ustation_rotation_matrix}.
|
||||
The rotation matrix \({}^A\bm{R}_B\) is a \(3 \times 3\) matrix containing the Cartesian unit vectors \([{}^A\hat{\bm{x}}_B,\ {}^A\hat{\bm{y}}_B,\ {}^A\hat{\bm{z}}_B]\) of frame \(\{\bm{B}\}\) represented in frame \(\{\bm{A}\}\) \eqref{eq:ustation_rotation_matrix}.
|
||||
|
||||
\begin{equation}\label{eq:ustation_rotation_matrix}
|
||||
{}^A\bm{R}_B = \left[ {}^A\hat{\bm{x}}_B | {}^A\hat{\bm{y}}_B | {}^A\hat{\bm{z}}_B \right] = \begin{bmatrix}
|
||||
u_{x} & v_{x} & z_{x} \\
|
||||
u_{y} & v_{y} & z_{y} \\
|
||||
u_{z} & v_{z} & z_{z}
|
||||
\end{bmatrix}
|
||||
{}^A\bm{R}_B = \left[ {}^A\hat{\bm{x}}_B | {}^A\hat{\bm{y}}_B | {}^A\hat{\bm{z}}_B \right]
|
||||
\end{equation}
|
||||
|
||||
Consider a pure rotation of a rigid body (\(\{\bm{A}\}\) and \(\{\bm{B}\}\) are coincident at their origins, as shown in Figure \ref{fig:ustation_rotation}).
|
||||
@ -200,7 +188,7 @@ For rotations along \(x\), \(y\) or \(z\) axis, the formulas of the correspondin
|
||||
\begin{align}
|
||||
\bm{R}_x(\theta_x) &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta_x) & -\sin(\theta_x) \\ 0 & \sin(\theta_x) & \cos(\theta_x) \end{bmatrix} \\
|
||||
\bm{R}_y(\theta_y) &= \begin{bmatrix} \cos(\theta_y) & 0 & \sin(\theta_y) \\ 0 & 1 & 0 \\ -\sin(\theta_y) & 0 & \cos(\theta_y) \end{bmatrix} \\
|
||||
\bm{R}_z(\theta_z) &= \begin{bmatrix} \cos(\theta_z) & -\sin(\theta_z) & 0 \\ \sin(\theta_z) & \cos(\theta_x) & 0 \\ 0 & 0 & 1 \end{bmatrix}
|
||||
\bm{R}_z(\theta_z) &= \begin{bmatrix} \cos(\theta_z) & -\sin(\theta_z) & 0 \\ \sin(\theta_z) & \cos(\theta_z) & 0 \\ 0 & 0 & 1 \end{bmatrix}
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
@ -220,7 +208,6 @@ Such rotation can be parameterized by three Euler angles \((\alpha,\ \beta,\ \ga
|
||||
\gamma &= \text{atan2}(-R_{12}/\cos(\beta),\ R_{11}/\cos(\beta))
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
\paragraph{Motion of a Rigid Body}
|
||||
|
||||
Since the relative positions of a rigid body with respect to a moving frame \(\{B\}\) attached to it are fixed for all time, it is sufficient to know the position of the origin of the frame \(O_B\) and the orientation of the frame \(\{B\}\) with respect to the fixed frame \(\{A\}\), to represent the position of any point \(P\) in the space.
|
||||
@ -294,13 +281,12 @@ Another key advantage of homogeneous transformation is the easy inverse transfor
|
||||
{}^B\bm{T}_A = {}^A\bm{T}_B^{-1} =
|
||||
\left[ \begin{array}{ccc|c}
|
||||
& & & \\
|
||||
& {}^A\bm{R}_B^T & & -{}^A \bm{R}_B^T {}^AP_{O_B} \\
|
||||
& {}^A\bm{R}_B^{\intercal} & & -{}^A \bm{R}_B^{\intercal} {}^AP_{O_B} \\
|
||||
& & & \cr
|
||||
\hline
|
||||
0 & 0 & 0 & 1 \\
|
||||
\end{array} \right]
|
||||
\end{equation}
|
||||
|
||||
\section{Micro-Station Kinematics}
|
||||
\label{ssec:ustation_kinematics}
|
||||
|
||||
@ -338,8 +324,9 @@ The homogeneous transformation matrix corresponding to the micro-station \(\bm{T
|
||||
|
||||
\(\bm{T}_{\mu\text{-station}}\) represents the pose of the sample (supposed to be rigidly fixed on top of the positioning-hexapod) with respect to the granite.
|
||||
|
||||
If the transformation matrices of the individual stages are each representing a perfect motion (i.e. the stages are supposed to have no parasitic motion), \(\bm{T}_{\mu\text{-station}}\) then represent the pose setpoint of the sample with respect to the granite.
|
||||
If the transformation matrices of the individual stages are each representing a perfect motion (i.e. the stages are supposed to have no parasitic motion), \(\bm{T}_{\mu\text{-station}}\) then represents the pose setpoint of the sample with respect to the granite.
|
||||
The transformation matrices for the translation stage, tilt stage, spindle, and positioning hexapod can be written as shown in Equation \eqref{eq:ustation_transformation_matrices_stages}.
|
||||
The setpoints are \(D_y\) for the translation stage, \(\theta_y\) for the tilt-stage, \(\theta_z\) for the spindle, \([D_{\mu x},\ D_{\mu y}, D_{\mu z}]\) for the micro-hexapod translations and \([\theta_{\mu x},\ \theta_{\mu y}, \theta_{\mu z}]\) for the micro-hexapod rotations.
|
||||
|
||||
\begin{equation}\label{eq:ustation_transformation_matrices_stages}
|
||||
\begin{align}
|
||||
@ -371,7 +358,6 @@ The transformation matrices for the translation stage, tilt stage, spindle, and
|
||||
\end{bmatrix}
|
||||
\end{align}
|
||||
\end{equation}
|
||||
|
||||
\chapter{Micro-Station Dynamics}
|
||||
\label{sec:ustation_modeling}
|
||||
In this section, the multi-body model of the micro-station is presented.
|
||||
@ -396,7 +382,7 @@ External forces can be used to model disturbances, and ``sensors'' can be used t
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/ustation_simscape_stage_example.png}
|
||||
\caption{\label{fig:ustation_simscape_stage_example}Example of a stage (here the tilt-stage) represented in the multi-body model software (Simscape). It is composed of two solid bodies connected by a 6-DoF joint. One joint DoF (here the tilt angle) can be imposed, the other DoFs are represented by springs and dampers. Additional disturbances forces for all DoF can be included}
|
||||
\caption{\label{fig:ustation_simscape_stage_example}Example of a stage (here the tilt-stage) represented in the multi-body model software (Simscape). It is composed of two solid bodies connected by a 6-DoF joint. One joint DoF (here the tilt angle) can be imposed, the other DoFs are represented by springs and dampers. Additional disturbing forces for all DoF can be included}
|
||||
\end{figure}
|
||||
|
||||
Therefore, the micro-station is modeled by several solid bodies connected by joints.
|
||||
@ -436,7 +422,6 @@ Hexapod & \(10\,N/\mu m\) & \(10\,N/\mu m\) & \(100\,N/\mu m\) & \(1.5\,Nm/rad\)
|
||||
\bottomrule
|
||||
\end{tabularx}
|
||||
\end{table}
|
||||
|
||||
\section{Comparison with the measured dynamics}
|
||||
\label{ssec:ustation_model_comp_dynamics}
|
||||
|
||||
@ -469,7 +454,6 @@ Tuning the numerous model parameters to better match the measurements is a highl
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_comp_com_response}FRFs between the hammer impacts on the translation stage and the measured stage acceleration expressed at its CoM. Comparison of the measured and extracted FRFs from the multi-body model. Different directions are computed for different stages.}
|
||||
\end{figure}
|
||||
|
||||
\section{Micro-station compliance}
|
||||
\label{ssec:ustation_model_compliance}
|
||||
|
||||
@ -534,7 +518,7 @@ Similar to what is done for the accelerometers, a Jacobian matrix \(\bm{J}_F\) i
|
||||
The equivalent forces and torques applied at center of \(\{\mathcal{X}\}\) are then computed using \eqref{eq:ustation_compute_cart_force}.
|
||||
|
||||
\begin{equation}\label{eq:ustation_compute_cart_force}
|
||||
F_{\mathcal{X}} = \bm{J}_F^t \cdot F_{\mathcal{L}}
|
||||
F_{\mathcal{X}} = \bm{J}_F^{\intercal} \cdot F_{\mathcal{L}}
|
||||
\end{equation}
|
||||
|
||||
Using the two Jacobian matrices, the FRF from the 10 hammer impacts to the 12 accelerometer outputs can be converted to the FRF from 6 forces/torques applied at the origin of frame \(\{\mathcal{X}\}\) to the 6 linear/angular accelerations of the top platform expressed with respect to \(\{\mathcal{X}\}\).
|
||||
@ -559,13 +543,12 @@ Considering the complexity of the micro-station compliance dynamics, the model c
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_frf_compliance_model}Compliance of the micro-station expressed in frame \(\{\mathcal{X}\}\). The measured FRFs are display by translucent lines, while the FRFs extracted from the multi-body models are shown by opaque lines. Both translation terms (\subref{fig:ustation_frf_compliance_xyz_model}) and rotational terms (\subref{fig:ustation_frf_compliance_Rxyz_model}) are displayed.}
|
||||
\end{figure}
|
||||
|
||||
\chapter{Estimation of Disturbances}
|
||||
\label{sec:ustation_disturbances}
|
||||
The goal of this section is to obtain a realistic representation of disturbances affecting the micro-station.
|
||||
These disturbance sources are then used during time domain simulations to accurately model the micro-station behavior.
|
||||
The focus on stochastic disturbances because, in principle, it is possible to calibrate the repeatable part of disturbances.
|
||||
Such disturbances include ground motions and vibrations induces by scanning the translation stage and the spindle.
|
||||
The focus is on stochastic disturbances because, in principle, it is possible to calibrate the repeatable part of disturbances.
|
||||
Such disturbances include ground motions and vibrations induce by scanning the translation stage and the spindle.
|
||||
|
||||
In the multi-body model, stage vibrations are modeled as internal forces applied in the stage joint.
|
||||
In practice, disturbance forces cannot be directly measured.
|
||||
@ -582,7 +565,7 @@ The tilt stage and the micro-hexapod also have positioning errors; however, they
|
||||
Therefore, from a control perspective, they are not important.
|
||||
\paragraph{Ground Motion}
|
||||
|
||||
The ground motion was measured by using a sensitive 3-axis geophone\footnote{A 3-Axis L4C geophone manufactured Sercel was used.} placed on the ground.
|
||||
The ground motion was measured by using a sensitive 3-axis geophone shown in Figure \ref{fig:ustation_geophone_picture} placed on the ground.
|
||||
The generated voltages were recorded with a high resolution DAC, and converted to displacement using the Geophone sensitivity transfer function.
|
||||
The obtained ground motion displacement is shown in Figure \ref{fig:ustation_ground_disturbance}.
|
||||
|
||||
@ -599,7 +582,6 @@ The obtained ground motion displacement is shown in Figure \ref{fig:ustation_gro
|
||||
\captionof{figure}{\label{fig:ustation_geophone_picture}(3D) L-4C geophone}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
|
||||
\paragraph{Ty Stage}
|
||||
|
||||
To measure the positioning errors of the translation stage, the setup shown in Figure \ref{fig:ustation_errors_ty_setup} is used.
|
||||
@ -609,7 +591,7 @@ A similar setup was used to measure the horizontal deviation (i.e. in the \(x\)
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/ustation_errors_ty_setup.png}
|
||||
\caption{\label{fig:ustation_errors_ty_setup}Experimental setup to measure the flatness (vertical deviation) of the translation stage}
|
||||
\caption{\label{fig:ustation_errors_ty_setup}Experimental setup to measure the straightness (vertical deviation) of the translation stage}
|
||||
\end{figure}
|
||||
|
||||
Six scans were performed between \(-4.5\,mm\) and \(4.5\,mm\).
|
||||
@ -634,7 +616,6 @@ Similar result is obtained for the \(x\) lateral direction.
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_errors_dy}Measurement of the linear (vertical) deviation of the Translation stage (\subref{fig:ustation_errors_dy_vertical}). A linear fit is then removed from the data (\subref{fig:ustation_errors_dy_vertical_remove_mean}).}
|
||||
\end{figure}
|
||||
|
||||
\paragraph{Spindle}
|
||||
|
||||
To measure the positioning errors induced by the Spindle, a ``Spindle error analyzer''\footnote{The Spindle Error Analyzer is made by Lion Precision.} is used as shown in Figure \ref{fig:ustation_rz_meas_lion_setup}.
|
||||
@ -687,7 +668,6 @@ The vertical motion induced by scanning the spindle is in the order of \(\pm 30\
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_errors_spindle}Measurement of the radial (\subref{fig:ustation_errors_spindle_radial}), axial (\subref{fig:ustation_errors_spindle_axial}) and tilt (\subref{fig:ustation_errors_spindle_tilt}) Spindle errors during a 60rpm spindle rotation. The circular best fit is shown by the dashed circle. It represents the misalignment of the spheres with the rotation axis.}
|
||||
\end{figure}
|
||||
|
||||
\section{Sensitivity to disturbances}
|
||||
\label{ssec:ustation_disturbances_sensitivity}
|
||||
|
||||
@ -716,7 +696,6 @@ The obtained transfer functions are shown in Figure \ref{fig:ustation_model_sens
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_model_sensitivity}Extracted transfer functions from disturbances to relative motion between the micro-station's top platform and the granite. The considered disturbances are the ground motion (\subref{fig:ustation_model_sensitivity_ground_motion}), the translation stage vibrations (\subref{fig:ustation_model_sensitivity_ty}), and the spindle vibrations (\subref{fig:ustation_model_sensitivity_rz}).}
|
||||
\end{figure}
|
||||
|
||||
\section{Obtained disturbance sources}
|
||||
\label{ssec:ustation_disturbances_results}
|
||||
|
||||
@ -747,7 +726,7 @@ The obtained power spectral density of the disturbances are shown in Figure \ref
|
||||
|
||||
The disturbances are characterized by their power spectral densities, as shown in Figure \ref{fig:ustation_dist_sources}.
|
||||
However, to perform time domain simulations, disturbances must be represented by a time domain signal.
|
||||
To generate stochastic time-domain signals with a specific power spectral densities, the discrete inverse Fourier transform is used, as explained in \cite[chap. 12.11]{preumont94_random_vibrat_spect_analy}.
|
||||
To generate stochastic time-domain signals with a specific power spectral density, the discrete inverse Fourier transform is used, as explained in \cite[chap. 12.11]{preumont94_random_vibrat_spect_analy}.
|
||||
Examples of the obtained time-domain disturbance signals are shown in Figure \ref{fig:ustation_dist_sources_time}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
@ -771,7 +750,6 @@ Examples of the obtained time-domain disturbance signals are shown in Figure \re
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_dist_sources_time}Generated time domain disturbance signals. Ground motion (\subref{fig:ustation_dist_source_ground_motion_time}), translation stage (\subref{fig:ustation_dist_source_translation_stage_time}) and spindle (\subref{fig:ustation_dist_source_spindle_time}).}
|
||||
\end{figure}
|
||||
|
||||
\chapter{Simulation of Scientific Experiments}
|
||||
\label{sec:ustation_experiments}
|
||||
To fully validate the micro-station multi-body model, two time-domain simulations corresponding to typical use cases were performed.
|
||||
@ -803,8 +781,7 @@ A good correlation with the measurements is observed both for radial errors (Fig
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:ustation_errors_model_spindle}Simulation results for a tomography experiment at constant velocity of 60rpm. The comparison is made with measurements for both radial (\subref{fig:ustation_errors_model_spindle_radial}) and axial errors (\subref{fig:ustation_errors_model_spindle_axial}).}
|
||||
\end{figure}
|
||||
|
||||
\section{Raster Scans with the translation stage}
|
||||
\section{Scans with the translation stage}
|
||||
\label{sec:ustation_experiments_ty_scans}
|
||||
|
||||
A second experiment was performed in which the translation stage was scanned at constant velocity.
|
||||
@ -818,7 +795,6 @@ A similar error amplitude was observed, thus indicating that the multi-body mode
|
||||
\includegraphics[scale=1]{figs/ustation_errors_model_dy_vertical.png}
|
||||
\caption{\label{fig:ustation_errors_model_dy_vertical}Vertical errors during a constant-velocity scan of the translation stage. Comparison of the measurements and simulated errors.}
|
||||
\end{figure}
|
||||
|
||||
\chapter*{Conclusion}
|
||||
\label{sec:ustation_conclusion}
|
||||
|
||||
@ -829,6 +805,5 @@ After tuning the model parameters, a good match with the measured compliance was
|
||||
|
||||
The disturbances affecting the sample position should also be well modeled.
|
||||
After experimentally estimating the disturbances (Section \ref{sec:ustation_disturbances}), the multi-body model was finally validated by performing a tomography simulation (Figure \ref{fig:ustation_errors_model_spindle}) as well as a simulation in which the translation stage was scanned (Figure \ref{fig:ustation_errors_model_dy_vertical}).
|
||||
|
||||
\printbibliography[heading=bibintoc,title={Bibliography}]
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user