From e753cb6e753914726b4bf125a92c1e44c3561afa Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Mon, 7 Apr 2025 16:41:25 +0200 Subject: [PATCH] Christophe's review --- figs/inkscape/ustation_stage_motion.svg | 21 ++++-- figs/ustation_cad_view.pdf | Bin 1129881 -> 1129885 bytes figs/ustation_combined_transformation.pdf | Bin 11485 -> 11487 bytes figs/ustation_compliance_meas.pdf | Bin 16014 -> 16015 bytes ...station_errors_dy_vertical_remove_mean.pdf | Bin 40333 -> 40912 bytes ...station_errors_dy_vertical_remove_mean.png | Bin 49051 -> 50432 bytes figs/ustation_errors_ty_setup.pdf | Bin 738462 -> 738464 bytes figs/ustation_hexapod_stage.pdf | Bin 1028862 -> 1028864 bytes figs/ustation_rotation.pdf | Bin 7062 -> 7064 bytes figs/ustation_ry_stage.pdf | Bin 661783 -> 661785 bytes figs/ustation_rz_stage.pdf | Bin 768176 -> 768178 bytes figs/ustation_simscape_stage_example.pdf | Bin 73840 -> 73844 bytes figs/ustation_stage_motion.pdf | Bin 11087 -> 14343 bytes figs/ustation_stage_motion.png | Bin 24608 -> 25820 bytes figs/ustation_transformation.pdf | Bin 10833 -> 10835 bytes figs/ustation_translation.pdf | Bin 12094 -> 12098 bytes figs/ustation_ty_stage.pdf | Bin 1242451 -> 1242454 bytes preamble.tex | 5 ++ simscape-micro-station.org | 59 +++++++---------- simscape-micro-station.pdf | Bin 8419662 -> 8425971 bytes simscape-micro-station.tex | 61 ++++++------------ 21 files changed, 61 insertions(+), 85 deletions(-) diff --git a/figs/inkscape/ustation_stage_motion.svg b/figs/inkscape/ustation_stage_motion.svg index 2686219..d54b830 100644 --- a/figs/inkscape/ustation_stage_motion.svg +++ b/figs/inkscape/ustation_stage_motion.svg @@ -6,7 +6,7 @@ version="1.2" id="svg251" sodipodi:docname="ustation_stage_motion.svg" - inkscape:version="1.4 (e7c3feb100, 2024-10-09)" + inkscape:version="1.4.1 (93de688d07, 2025-03-30)" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd" 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\setglossarystyle{long-name-desc} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{cases} +\usepackage{empheq} + \makeindex \makeglossaries diff --git a/simscape-micro-station.org b/simscape-micro-station.org index 940d0d1..1b4bf23 100644 --- a/simscape-micro-station.org +++ b/simscape-micro-station.org @@ -344,17 +344,6 @@ To do so, the effects of these disturbances were first measured experimental and To validate the accuracy of the micro-station model, "real world" experiments are simulated and compared with measurements in Section ref:sec:ustation_experiments. -# #+name: tab:ustation_section_matlab_code -# #+caption: Report sections and corresponding Matlab files -# #+attr_latex: :environment tabularx :width 0.6\linewidth :align lX -# #+attr_latex: :center t :booktabs t -# | *Sections* | *Matlab File* | -# |---------------------------------------+-----------------------------| -# | Section ref:sec:ustation_kinematics | =ustation_1_kinematics.m= | -# | Section ref:sec:ustation_modeling | =ustation_2_modeling.m= | -# | Section ref:sec:ustation_disturbances | =ustation_3_disturbances.m= | -# | Section ref:sec:ustation_experiments | =ustation_4_experiments.m= | - * Micro-Station Kinematics :PROPERTIES: :HEADER-ARGS:matlab+: :tangle matlab/ustation_1_kinematics.m @@ -364,10 +353,10 @@ To validate the accuracy of the micro-station model, "real world" experiments ar 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. #+name: fig:ustation_cad_view -#+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. +#+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). #+attr_latex: :width \linewidth [[file:figs/ustation_cad_view.png]] @@ -415,7 +404,7 @@ The "controlled" DoF of each stage (for instance the $D_y$ direction for the tra The translation stage is used to position and scan the sample laterally with respect to the X-ray beam. A linear motor was first used to enable fast and accurate scans. -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]. +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]. An optical linear encoder is used to measure the stage motion and for controlling the position. Four cylindrical bearings[fn:ustation_4] are used to guide the motion (i.e. minimize the parasitic motions) and have high stiffness. @@ -529,14 +518,10 @@ The /orientation/ of a rigid body is the same at all its points (by definition). 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). @@ -553,7 +538,7 @@ For rotations along $x$, $y$ or $z$ axis, the formulas of the corresponding rota \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} @@ -643,7 +628,7 @@ 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 \\ @@ -685,8 +670,9 @@ 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} @@ -845,7 +831,7 @@ Joints are used to impose kinematic constraints between solid bodies and to spec External forces can be used to model disturbances, and "sensors" can be used to measure the relative pose between two defined frames. #+name: fig:ustation_simscape_stage_example -#+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 +#+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 [[file:figs/ustation_simscape_stage_example.png]] Therefore, the micro-station is modeled by several solid bodies connected by joints. @@ -1095,7 +1081,7 @@ Similar to what is done for the accelerometers, a Jacobian matrix $\bm{J}_F$ is 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 $\{\mathcal{X}\}$. @@ -1348,8 +1334,8 @@ exportFig('figs/ustation_frf_compliance_Rxyz_model.pdf', 'width', 'half', 'heigh 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. @@ -1398,7 +1384,7 @@ Therefore, from a control perspective, they are not important. **** Ground Motion -The ground motion was measured by using a sensitive 3-axis geophone[fn:ustation_11] 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. @@ -1470,7 +1456,7 @@ A special optical element (called a "straightness interferometer"[fn:ustation_9] 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. #+name: fig:ustation_errors_ty_setup -#+caption: Experimental setup to measure the flatness (vertical deviation) of the translation stage +#+caption: Experimental setup to measure the straightness (vertical deviation) of the translation stage [[file:figs/ustation_errors_ty_setup.png]] Six scans were performed between $-4.5\,mm$ and $4.5\,mm$. @@ -1513,7 +1499,7 @@ exportFig('figs/ustation_errors_dy_vertical.pdf', 'width', 'half', 'height', 'no %% Measurement of the linear (vertical) deviation of the Translation stage - Remove best linear fit figure; plot(ty_errors.setpoint, ty_errors.ty_z - straight_line, 'k-') -xlabel('$D_y$ position [mm]'); ylabel('Vertical error [$\mu$m]'); +xlabel('$D_y$ position [mm]'); ylabel('Residual vertical error [$\mu$m]'); xlim([-5, 5]); ylim([-0.4, 0.4]); #+end_src @@ -1963,7 +1949,7 @@ exportFig('figs/ustation_dist_source_spindle.pdf', 'width', 'third', 'height', ' 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:&preumont94_random_vibrat_spect_analy chap. 12.11]]. +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]]. Examples of the obtained time-domain disturbance signals are shown in Figure ref:fig:ustation_dist_sources_time. #+begin_src matlab @@ -2192,7 +2178,7 @@ exportFig('figs/ustation_errors_model_spindle_axial.pdf', 'width', 'half', 'heig #+end_subfigure #+end_figure -** Raster Scans with the translation stage +** Scans with the translation stage <> A second experiment was performed in which the translation stage was scanned at constant velocity. @@ -4538,7 +4524,7 @@ Note that there is trade-off between: :END: **** Theory -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, $\bm{L} = [l_1, l_2, \dots, l_6]^T$. +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, $\bm{L} = [l_1, l_2, \dots, l_6]^{\intercal}$. From the geometry of the manipulator, the loop closure for each limb, $i = 1, 2, \dots, 6$ can be written as \begin{align*} @@ -4548,12 +4534,12 @@ From the geometry of the manipulator, the loop closure for each limb, $i = 1, 2, To obtain the length of each actuator and eliminate $\hat{\bm{s}}_i$, it is sufficient to dot multiply each side by itself: \begin{equation} - 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] + 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] \end{equation} Hence, for $i = 1, 2, \dots, 6$, each limb length can be uniquely determined by: \begin{equation} - 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} + 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} \end{equation} 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. @@ -4613,8 +4599,7 @@ Otherwise, when the limbs' lengths derived yield complex numbers, then the posit * Footnotes -[fn:ustation_12]It was probably caused by rust of the linear guides along its stroke. -[fn:ustation_11]A 3-Axis L4C geophone manufactured Sercel was used. +[fn:ustation_11]It was probably caused by rust of the linear guides along its stroke. [fn:ustation_10]Laser source is manufactured by Agilent (5519b). [fn:ustation_9]The special optics (straightness interferometer and reflector) are manufactured by Agilent (10774A). 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