From c8b9ee19cee3cb58562fedcfb2c0a1d08473f681 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 23 Apr 2025 10:02:08 +0200 Subject: [PATCH] Rework all heading names --- phd-thesis.org | 312 ++++++++++++++++++++++++------------------------- 1 file changed, 156 insertions(+), 156 deletions(-) diff --git a/phd-thesis.org b/phd-thesis.org index 61cb715..d9f3794 100644 --- a/phd-thesis.org +++ b/phd-thesis.org @@ -990,7 +990,7 @@ Two key effects that may limit that positioning performances are then considered In this section, a uniaxial model of the micro-station is tuned to match measurements made on the micro-station. -***** Measured dynamics +***** Measured Dynamics The measurement setup is schematically shown in Figure\nbsp{}ref:fig:uniaxial_ustation_meas_dynamics_schematic where two vertical hammer hits are performed, one on the Granite (force $F_{g}$) and the other on the positioning hexapod's top platform (force $F_{h}$). The vertical inertial motion of the granite $x_{g}$ and the top platform of the positioning hexapod $x_{h}$ are measured using geophones[fn:uniaxial_1]. @@ -1040,7 +1040,7 @@ The parameters obtained are summarized in Table\nbsp{}ref:tab:uniaxial_ustation_ Two disturbances are considered which are shown in red: the floor motion $x_f$ and the stage vibrations represented by $f_t$. The hammer impacts $F_{h}, F_{g}$ are shown in blue, whereas the measured inertial motions $x_{h}, x_{g}$ are shown in black. -***** Comparison of model and measurements +***** Comparison of Model and Measurements The transfer functions from the forces injected by the hammers to the measured inertial motion of the positioning hexapod and granite are extracted from the uniaxial model and compared to the measurements in Figure\nbsp{}ref:fig:uniaxial_comp_frf_meas_model. Because the uniaxial model has three acrshortpl:dof, only three modes with frequencies at $70\,\text{Hz}$, $140\,\text{Hz}$ and $320\,\text{Hz}$ are modeled. @@ -1118,7 +1118,7 @@ For further analysis, 9 "configurations" of the uniaxial NASS model of Figure\nb #+end_subfigure #+end_figure -*** Disturbance Identification +*** Identification of Disturbances <> ***** Introduction :ignore: To quantify disturbances (red signals in Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass), three geophones[fn:uniaxial_2] are used. @@ -1217,7 +1217,7 @@ Then, the transfer functions from disturbances to the performance metric (here t Finally, these two types of information are combined to estimate the corresponding spectral density of the performance metric. This is very useful to identify what is limiting the performance of the system, and to compare the achievable performance with different system parameters. -***** Sensitivity to disturbances +***** Sensitivity to Disturbances From the uniaxial model of the acrshort:nass (Figure\nbsp{}ref:fig:uniaxial_model_micro_station_nass), the transfer function from the disturbances ($f_s$, $x_f$ and $f_t$) to the displacement $d$ are computed. This is done for two extreme sample masses $m_s = 1\,\text{kg}$ and $m_s = 50\,\text{kg}$ and three active platform stiffnesses: @@ -1522,7 +1522,7 @@ All three active damping techniques yielded similar damped plants. #+end_subfigure #+end_figure -***** Sensitivity to disturbances and Noise Budgeting +***** Sensitivity to Disturbances and Noise Budgeting Reasonable gains are chosen for the three active damping strategies such that the active platform suspension mode is well damped. The sensitivity to disturbances (direct forces $f_s$, stage vibrations $f_t$ and floor motion $x_f$) for all three active damping techniques are compared in Figure\nbsp{}ref:fig:uniaxial_sensitivity_dist_active_damping. The comparison is done with the active platform having a stiffness $k_n = 1\,\text{N}/\mu\text{m}$. @@ -1856,7 +1856,7 @@ The attainable vibration control performances were estimated for the three activ However, the stiff active platform ($k_n = 100\,\text{N}/\mu\text{m}$) is requiring the largest feedback bandwidth, which is difficult to achieve while being robust to the change of payload mass. A slight advantage can be given to the soft active platform as it requires less feedback bandwidth while providing better stability results. -*** Effect of limited micro-station compliance +*** Effect of Limited Support Compliance <> ***** Introduction :ignore: @@ -2029,7 +2029,7 @@ To study the effect of the sample dynamics, the models shown in Figure\nbsp{}ref #+end_subfigure #+end_figure -***** Impact on plant dynamics +***** Impact on Plant Dynamics To study the impact of the flexibility between the active platform and the payload, a first (reference) model with a rigid payload, as shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_rigid_schematic is used. Then "flexible" payload whose model is shown in Figure\nbsp{}ref:fig:uniaxial_paylaod_dynamics_schematic are considered. @@ -2081,7 +2081,7 @@ Even though the added sample's flexibility still shifts the high frequency mass #+end_subfigure #+end_figure -***** Impact on close loop performances +***** Impact on Close Loop Performances Having a flexibility between the measured position (i.e., the top platform of the active platform) and the acrshort:poi to be positioned relative to the x-ray may also impact the closed-loop performance (i.e., the remaining sample's vibration). @@ -2198,7 +2198,7 @@ After the dynamics of this system is studied, the objective will be to dampen th #+attr_latex: :scale 0.8 [[file:figs/rotating_3dof_model_schematic.png]] -***** Equations of motion and transfer functions +***** Equations of Motion and Transfer Functions To obtain the equations of motion for the system represented in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic, the Lagrangian equation\nbsp{}eqref:eq:rotating_lagrangian_equations is used. $L = T - V$ is the Lagrangian, $T$ the kinetic coenergy, $V$ the potential energy, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable $\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}$. These terms are derived in\nbsp{}eqref:eq:rotating_energy_functions_lagrange. @@ -2414,7 +2414,7 @@ This small displacement then increases the centrifugal force $m\Omega^2d_u = \fr \end{bmatrix} \end{equation} -***** Effect of rotation speed on IFF plant dynamics +***** Effect of Rotation Speed on IFF Plant Dynamics The transfer functions from actuator forces $[F_u,\ F_v]$ to the measured force sensors $[f_u,\ f_v]$ are identified for several rotating velocities and are shown in Figure\nbsp{}ref:fig:rotating_iff_bode_plot_effect_rot. As expected from the derived equations of motion: - when $\Omega < \omega_0$: the low-frequency gain is no longer zero and two (non-minimum phase) real zeros appear at low-frequencies. @@ -2558,7 +2558,7 @@ The same trade-off can be seen between achievable damping and loss of compliance #+end_subfigure #+end_figure -*** IFF with a stiffness in parallel with the force sensor +*** IFF with a Stiffness in Parallel with the Force Sensor <> ***** Introduction :ignore: @@ -2609,7 +2609,7 @@ Thus, if the added /parallel stiffness/ $k_p$ is higher than the /negative stiff \boxed{\alpha > \frac{\Omega^2}{{\omega_0}^2} \quad \Leftrightarrow \quad k_p > m \Omega^2} \end{equation} -***** Effect of parallel stiffness on the IFF plant +***** Effect of Parallel Stiffness on the IFF plant The IFF plant (transfer function from $[F_u, F_v]$ to $[f_u, f_v]$) is identified without parallel stiffness $k_p = 0$, with a small parallel stiffness $k_p < m \Omega^2$ and with a large parallel stiffness $k_p > m \Omega^2$. Bode plots of the obtained dynamics are shown in Figure\nbsp{}ref:fig:rotating_iff_effect_kp. The two real zeros for $k_p < m \Omega^2$ are transformed into two complex conjugate zeros for $k_p > m \Omega^2$. @@ -2636,7 +2636,7 @@ It is shown that if the added stiffness is higher than the maximum negative stif #+end_subfigure #+end_figure -***** Effect of $k_p$ on the attainable damping +***** Effect of $k_p$ on the Attainable Damping Even though the parallel stiffness $k_p$ has no impact on the open-loop poles (as the overall stiffness $k$ is kept constant), it has a large impact on the transmission zeros. Moreover, as the attainable damping is generally proportional to the distance between poles and zeros\nbsp{}[[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition]], the parallel stiffness $k_p$ is expected to have some impact on the attainable damping. To study this effect, Root Locus plots for several parallel stiffnesses $k_p > m \Omega^2$ are shown in Figure\nbsp{}ref:fig:rotating_iff_kp_root_locus_effect_kp. @@ -2662,7 +2662,7 @@ This is confirmed by the Figure\nbsp{}ref:fig:rotating_iff_kp_optimal_gain where #+end_subfigure #+end_figure -***** Damped plant +***** Damped Plant The parallel stiffness are chosen to be $k_p = 2 m \Omega^2$ and the damped plant is computed. The damped and undamped transfer functions from $F_u$ to $d_u$ are compared in Figure\nbsp{}ref:fig:rotating_iff_kp_added_hpf_damped_plant. Even though the two resonances are well damped, the IFF changes the low-frequency behavior of the plant, which is usually not desired. @@ -2717,7 +2717,7 @@ K_d(s) = g \cdot \frac{s}{s + \omega_d} #+attr_latex: :scale 0.8 [[file:figs/rotating_3dof_model_schematic_rdc.png]] -***** Equations of motion +***** Equations of Motion Let's note $\bm{G}_d$ the transfer function between actuator forces and measured relative motion in parallel with the actuators\nbsp{}eqref:eq:rotating_rdc_plant_matrix. The elements of $\bm{G}_d$ were derived in Section\nbsp{}ref:sec:rotating_system_description are shown in\nbsp{}eqref:eq:rotating_rdc_plant_elements. @@ -2982,7 +2982,7 @@ The gain is chosen such that 99% of modal damping is obtained (obtained gains ar #+latex: \captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the RDC} #+end_minipage -***** Comparison of the obtained damped plants +***** Comparison of the Obtained Damped Plants Now that the optimal parameters for the three considered active damping techniques have been determined, the obtained damped plants are computed and compared in Figure\nbsp{}ref:fig:rotating_nass_damped_plant_comp. Similar to what was concluded in the previous analysis: @@ -3014,14 +3014,14 @@ Similar to what was concluded in the previous analysis: #+end_subfigure #+end_figure -*** Nano-Active-Stabilization-System with rotation +*** Nano Active Stabilization System with Rotation <> ***** Introduction :ignore: Until now, the model used to study gyroscopic effects consisted of an infinitely stiff rotating stage with a X-Y suspended stage on top. While quite simplistic, this allowed us to study the effects of rotation and the associated limitations when active damping is to be applied. In this section, the limited compliance of the micro-station is considered as well as the rotation of the spindle. -***** Nano Active Stabilization System model +***** Nano Active Stabilization System Model To have a more realistic dynamics model of the NASS, the 2-DoF active platform (modeled as shown in Figure\nbsp{}ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure\nbsp{}ref:fig:rotating_nass_model for a 3D view): - the floor whose motion is imposed - a 2-DoF granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}$, $m_g = \SI{2500}{\kg}$) @@ -3036,7 +3036,7 @@ A payload is rigidly fixed to the active platform and the $x,y$ motion of the pa #+attr_latex: :scale 0.7 [[file:figs/rotating_nass_model.png]] -***** System dynamics +***** System Dynamics The dynamics of the undamped and damped plants are identified using the optimal parameters found in Section\nbsp{}ref:sec:rotating_nano_hexapod. The obtained dynamics are compared in Figure\nbsp{}ref:fig:rotating_nass_plant_comp_stiffness in which the direct terms are shown by the solid curves and the coupling terms are shown by the shaded ones. @@ -3070,7 +3070,7 @@ It can be observed that: #+end_subfigure #+end_figure -***** Effect of disturbances +***** Effect of Disturbances The effect of three disturbances are considered (as for the uniaxial model), floor motion $[x_{f,x},\ x_{f,y}]$ (Figure\nbsp{}ref:fig:rotating_nass_effect_floor_motion), micro-Station vibrations $[f_{t,x},\ f_{t,y}]$ (Figure\nbsp{}ref:fig:rotating_nass_effect_stage_vibration) and direct forces applied on the sample $[f_{s,x},\ f_{s,y}]$ (Figure\nbsp{}ref:fig:rotating_nass_effect_direct_forces). Note that only the transfer functions from the disturbances in the $x$ direction to the relative position $d_x$ between the sample and the granite in the $x$ direction are displayed because the transfer functions in the $y$ direction are the same due to the system symmetry. @@ -3464,7 +3464,7 @@ The $69 \times 3 \times 801$ frequency response matrix is then reduced to a $36 To validate this reduction of acrshort:dof and the solid body assumption, the frequency response function at the accelerometer location are "reconstructed" from the reduced frequency response matrix and are compared with the initial measurements in Section\nbsp{}ref:ssec:modal_solid_body_assumption. -**** From accelerometer DOFs to solid body DOFs +**** From Accelerometer DOFs to Solid Body DOFs <> Let us consider the schematic shown in Figure\nbsp{}ref:fig:modal_local_to_global_coordinates where the motion of a solid body is measured at 4 distinct locations (in $x$, $y$ and $z$ directions). @@ -3546,7 +3546,7 @@ Using\nbsp{}eqref:eq:modal_cart_to_acc, the frequency response matrix $\bm{H}_\t \end{bmatrix} \end{equation} -**** Verification of solid body assumption +**** Verification of the Solid Body Assumption <> From the response of one solid body expressed by its 6 acrshortpl:dof (i.e. from $\bm{H}_{\text{CoM}}$), and using equation\nbsp{}eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any considered location. @@ -3578,7 +3578,7 @@ The graphical display of the mode shapes can be computed from the modal model, w To validate the quality of the modal model, the full acrshort:frf matrix is computed from the modal model and compared to the initial measured acrshort:frf (section\nbsp{}ref:ssec:modal_model_validity). -**** Number of modes determination +**** Determination of the Number of Modes <> The acrshort:mif is applied to the $n\times p$ acrshort:frf matrix where $n$ is a relatively large number of measurement DOFs (here $n=69$) and $p$ is the number of excitation DOFs (here $p=3$). @@ -3632,7 +3632,7 @@ The obtained natural frequencies and associated modal damping are summarized in #+latex: \captionof{table}{\label{tab:modal_obtained_modes_freqs_damps}Identified modes} #+end_minipage -**** Modal parameter extraction +**** Modal Parameter Extraction <> Generally, modal identification is using curve-fitting a theoretical expression to the actual measured acrshort:frf data. @@ -3691,7 +3691,7 @@ The eigenvalues $s_r$ and $s_r^*$ can then be computed from equation\nbsp{}eqref s_r = \omega_r (-\xi_r + i \sqrt{1 - \xi_r^2}), \quad s_r^* = \omega_r (-\xi_r - i \sqrt{1 - \xi_r^2}) \end{equation} -**** Verification of the modal model validity +**** Verification of the Modal Model Validity <> To check the validity of the modal model, the complete $n \times n$ acrshort:frf matrix $\bm{H}_{\text{syn}}$ is first synthesized from the modal parameters. @@ -3887,7 +3887,7 @@ First, the tools to describe the pose of a solid body (i.e. it's position and or The motion induced by a positioning stage is described by transformation matrices. Finally, the motions of all stacked stages are combined, and the sample's motion is computed from each stage motion. -***** Spatial motion representation +***** Spatial Representation of Motion The /pose/ of a solid body relative to a specific frame can be described by six independent parameters. Three parameters are typically used to describe its position, and three other parameters describe its orientation. @@ -4178,7 +4178,7 @@ The spring values are summarized in Table\nbsp{}ref:tab:ustation_6dof_stiffness_ | Spindle | $700\,\text{N}/\mu\text{m}$ | $700\,\text{N}/\mu\text{m}$ | $2\,\text{kN}/\mu\text{m}$ | $10\,\text{Nm}/\mu\text{rad}$ | $10\,\text{Nm}/\mu\text{rad}$ | - | | Hexapod | $10\,\text{N}/\mu\text{m}$ | $10\,\text{N}/\mu\text{m}$ | $100\,\text{N}/\mu\text{m}$ | $1.5\,\text{Nm/rad}$ | $1.5\,\text{Nm/rad}$ | $0.27\,\text{Nm/rad}$ | -**** Comparison with the measured dynamics +**** Comparison with the Measured Dynamics <> The dynamics of the micro-station was measured by placing accelerometers on each stage and by impacting the translation stage with an instrumented hammer in three directions. @@ -4213,7 +4213,7 @@ Tuning the numerous model parameters to better match the measurements is a highl #+end_subfigure #+end_figure -**** Micro-station compliance +**** Micro-station Compliance <> As discussed in the previous section, the dynamics of the micro-station is complex, and tuning the multi-body model parameters to obtain a perfect match is difficult. @@ -4319,7 +4319,7 @@ Instead, the vibrations of the micro-station's top platform induced by the distu To estimate the equivalent disturbance force that induces such vibration, the transfer functions from disturbance sources (i.e. forces applied in the stages' joint) to the displacements of the micro-station's top platform with respect to the granite are extracted from the multi-body model (Section\nbsp{}ref:ssec:ustation_disturbances_sensitivity). Finally, the obtained disturbance sources are compared in Section\nbsp{}ref:ssec:ustation_disturbances_results. -**** Disturbance measurements +**** Disturbance Measurements <> ***** Introduction :ignore: In this section, ground motion is directly measured using geophones. @@ -4350,7 +4350,7 @@ The obtained ground motion displacement is shown in Figure\nbsp{}ref:fig:ustatio [[file:figs/ustation_geophone_picture.jpg]] #+end_minipage -***** Ty Stage +***** Translation Stage To measure the positioning errors of the translation stage, the setup shown in Figure\nbsp{}ref:fig:ustation_errors_ty_setup is used. A special optical element (called a "straightness interferometer"[fn:ustation_9]) is fixed on top of the micro-station, while a laser source[fn:ustation_10] and a straightness reflector are fixed on the ground. @@ -4442,7 +4442,7 @@ The vertical motion induced by scanning the spindle is in the order of $\pm 30\, #+end_subfigure #+end_figure -**** Sensitivity to disturbances +**** Sensitivity to Disturbances <> To compute the disturbance source (i.e. forces) that induced the measured vibrations in Section\nbsp{}ref:ssec:ustation_disturbances_meas, the transfer function from the disturbance sources to the stage vibration (i.e. the "sensitivity to disturbances") needs to be estimated. @@ -4473,7 +4473,7 @@ The obtained transfer functions are shown in Figure\nbsp{}ref:fig:ustation_model #+end_subfigure #+end_figure -**** Obtained disturbance sources +**** Obtained Disturbance Sources <> From the measured effect of disturbances in Section\nbsp{}ref:ssec:ustation_disturbances_meas and the sensitivity to disturbances extracted from the multi-body model in Section\nbsp{}ref:ssec:ustation_disturbances_sensitivity, the power spectral density of the disturbance sources (i.e. forces applied in the stage's joint) can be estimated. @@ -4569,7 +4569,7 @@ A good correlation with the measurements is observed both for radial errors (Fig #+end_subfigure #+end_figure -**** 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. @@ -4614,7 +4614,7 @@ Section\nbsp{}ref:sec:nhexa_control explores how the acrfull:haclac strategy, pr This adaptation requires fundamental decisions regarding both the control architecture (centralized versus decentralized) and the control frame (Cartesian versus strut space). Through careful analysis of system interactions and plant characteristics in different frames, a control architecture combining decentralized Integral Force Feedback for active damping with a centralized high authority controller for positioning was developed, with both controllers implemented in the frame of the struts. -*** Active Vibration Platforms +*** Review of Active Vibration Platforms <> **** Introduction :ignore: @@ -4752,7 +4752,7 @@ The primary control requirements focus on $[D_y,\ D_z,\ R_y]$ motions; however, <> The choice of the active platform architecture for the NASS requires careful consideration of several critical specifications. -The platform must provide control over five acrshortpl:dof ($D_x$, $D_y$, $D_z$, $R_x$, and $R_y$), with strokes exceeding $100\,\mu\text{m}$ to correct for micro-station positioning errors, while fitting within a cylindrical envelope of 300 mm diameter and 95 mm height. +The platform must provide control over five acrshortpl:dof ($D_x$, $D_y$, $D_z$, $R_x$, and $R_y$), with strokes exceeding $100\,\mu\text{m}$ to correct for micro-station positioning errors, while fitting within a cylindrical envelope of $300\,\text{mm}$ diameter and $95\,\text{mm}$ height. It must accommodate payloads up to $50\,\text{kg}$ while maintaining high dynamical performance. For light samples, the typical design strategy of maximizing actuator stiffness works well because resonance frequencies in the kilohertz range can be achieved, enabling control bandwidths up to $100\,\text{Hz}$. However, achieving such resonance frequencies with a $50\,\text{kg}$ payload would require unrealistic stiffness values of approximately $2000\,\text{N}/\mu\text{m}$. @@ -4814,7 +4814,7 @@ Furthermore, the successful implementation of Integral Force Feedback (IFF) cont #+end_subfigure #+end_figure -*** The Stewart platform +*** The Stewart Platform <> **** Introduction :ignore: @@ -4961,7 +4961,7 @@ The matrix $\bm{J}$ is called the Jacobian matrix and is defined by\nbsp{}eqref: Therefore, the Jacobian matrix $\bm{J}$ links the rate of change of the strut length to the velocity and angular velocity of the top platform with respect to the fixed base through a set of linear equations. However, $\bm{J}$ needs to be recomputed for every Stewart platform pose because it depends on the actual pose of the manipulator. -***** Approximate solution to the Forward and Inverse Kinematic problems +***** Approximate Solution to the Forward and Inverse Kinematic Problems For small displacements $\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^{\intercal}$ around an operating point $\bm{\mathcal{X}}_0$ (for which the Jacobian was computed), the associated joint displacement $\delta\bm{\mathcal{L}} = [\delta l_1,\,\delta l_2,\,\delta l_3,\,\delta l_4,\,\delta l_5,\,\delta l_6]^{\intercal}$ can be computed using the Jacobian\nbsp{}eqref:eq:nhexa_inverse_kinematics_approximate. @@ -4978,7 +4978,7 @@ Similarly, for small joint displacements $\delta\bm{\mathcal{L}}$, it is possibl These two relations solve the forward and inverse kinematic problems for small displacement in a /approximate/ way. While this approximation offers limited value for inverse kinematics, which can be solved analytically, it proves particularly useful for the forward kinematic problem where exact analytical solutions are difficult to obtain. -***** Range validity of the approximate inverse kinematics +***** Range Validity of the Approximate Inverse Kinematics The accuracy of the Jacobian-based forward kinematics solution was estimated by a simple analysis. For a series of platform positions, the exact strut lengths are computed using the analytical inverse kinematics equation\nbsp{}eqref:eq:nhexa_inverse_kinematics. @@ -5120,7 +5120,7 @@ This will be performed in the next section using a multi-body model. All these characteristics (maneuverability, stiffness, dynamics, etc.) are fundamentally determined by the platform's geometry. While a reasonable geometric configuration will be used to validate the NASS during the conceptual phase, the optimization of these geometric parameters will be explored during the detailed design phase. -*** Multi-Body Model +*** Multi-Body Model of Stewart Platforms <> **** Introduction :ignore: @@ -5226,7 +5226,7 @@ This modular approach to actuator modeling allows for future refinements as the #+latex: \captionof{table}{\label{tab:nhexa_actuator_parameters}Actuator parameters} #+end_minipage -**** Validation of the multi-body model +**** Validation of the Multi-body Model <> The developed multi-body model of the Stewart platform is represented schematically in Figure\nbsp{}ref:fig:nhexa_stewart_model_input_outputs, highlighting the key inputs and outputs: actuator forces $\bm{f}$, force sensor measurements $\bm{f}_n$, and relative displacement measurements $\bm{\mathcal{L}}$. @@ -5482,7 +5482,7 @@ This high gain, combined with the bounded phase, enables effective damping of th #+end_subfigure #+end_figure -**** MIMO High-Authority Control - Low-Authority Control +**** High Authority Control / Low Authority Control <> The design of the High Authority Control positioning loop is now examined. @@ -5592,7 +5592,7 @@ This approach combines decentralized Integral Force Feedback for active damping This study establishes the theoretical framework necessary for the subsequent development and validation of the NASS. -** Validation of the Concept +** Validation of the NASS Concept <> *** Introduction :ignore: @@ -5684,7 +5684,7 @@ Using these reference signals, the desired sample position relative to the fixed \end{align} \end{equation} -**** Computation of the sample's pose error +**** Computation of the Sample's Pose Error <> The external metrology system measures the sample position relative to the fixed granite. @@ -5706,7 +5706,7 @@ The measured sample pose is represented by the homogeneous transformation matrix \end{array} \right] \end{equation} -**** Position error in the frame of the struts +**** Position Error in the Frame of the Struts <> The homogeneous transformation formalism enables straightforward computation of the sample position error. @@ -5763,7 +5763,7 @@ Then, the high authority controller uses the computed errors in the frame of the Building on the uniaxial model study, this section implements decentralized Integral Force Feedback (IFF) as the first component of the acrshort:haclac strategy. The springs in parallel to the force sensors were used to guarantee the control robustness, as observed with the 3DoF rotating model. The objective here is to design a decentralized IFF controller that provides good damping of the active platform modes across payload masses ranging from $1$ to $50\,\text{kg}$ and rotational velocity up to $360\,\text{deg/s}$. -The payloads used for validation have a cylindrical shape with 250 mm height and with masses of $1\,\text{kg}$, $25\,\text{kg}$, and $50\,\text{kg}$. +The payloads used for validation have a cylindrical shape with $250\,\text{mm}$ height and with masses of $1\,\text{kg}$, $25\,\text{kg}$, and $50\,\text{kg}$. **** IFF Plant <> @@ -5999,7 +5999,7 @@ The current approach of controlling the position in the strut frame is inadequat #+end_subfigure #+end_figure -**** Controller design +**** Controller Design <> A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure\nbsp{}ref:fig:nass_hac_plants), and achievement of sufficient bandwidth (targeted at $10\,\text{Hz}$) for high performance operation. @@ -6031,7 +6031,7 @@ Second, the characteristic loci analysis presented in Figure\nbsp{}ref:fig:nass_ #+end_subfigure #+end_figure -**** Tomography experiment +**** Tomography Experiment <> The Nano Active Stabilization System concept was validated through time-domain simulations of scientific experiments, with a particular focus on tomography scanning because of its demanding performance requirements. @@ -6169,7 +6169,7 @@ The selected instrumentation is then experimentally characterized to verify comp The chapter concludes with a concise presentation of the obtained optimized active platform design, called the "nano-hexapod", in Section\nbsp{}ref:sec:detail_design, summarizing how the various optimizations contribute to a system that balances the competing requirements of precision positioning, vibration isolation, and practical implementation constraints. With the detailed design completed and components procured, the project advances to the experimental validation phase, which will be addressed in the subsequent chapter. -** Optimal Geometry +** Optimal Active Platform Geometry <> *** Introduction :ignore: @@ -6184,7 +6184,7 @@ Section\nbsp{}ref:sec:detail_kinematics_geometry develops the analytical framewo Section\nbsp{}ref:sec:detail_kinematics_cubic examines the cubic configuration, a specific architecture that has gathered significant attention, to evaluate its suitability for the NASS applications. Finally, Section\nbsp{}ref:sec:detail_kinematics_nano_hexapod presents the optimized active platform geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS. -*** Review of Stewart platforms +*** Review of Stewart Platforms <> The first parallel platform similar to the Stewart platform was built in 1954 by Gough\nbsp{}[[cite:&gough62_univer_tyre_test_machin]], for a tyre test machine (shown in Figure\nbsp{}ref:fig:detail_geometry_gough_paper). @@ -6306,7 +6306,7 @@ The influence of strut orientation and joint positioning on Stewart platform pro #+end_subfigure #+end_figure -*** Effect of geometry on Stewart platform properties +*** Kinematic Study of Stewart Platforms <> **** Introduction :ignore: @@ -6344,7 +6344,7 @@ The analysis is significantly simplified when considering small motions, as the Therefore, the mobility of the Stewart platform (defined as the set of achievable $[\delta x\ \delta y\ \delta z\ \delta \theta_x\ \delta \theta_y\ \delta \theta_z]$) depends on two key factors: the stroke of each strut and the geometry of the Stewart platform (embodied in the Jacobian matrix). More specifically, the XYZ mobility only depends on the $\hat{\bm{s}}_i$ (orientation of struts), while the mobility in rotation also depends on $\bm{b}_i$ (position of top joints). -***** Mobility in translation +***** Mobility in Translation For simplicity, only translations are first considered (i.e., the Stewart platform is considered to have fixed orientation). In the general case, the translational mobility can be represented by a 3D shape having 12 faces, where each actuator limits the stroke along its axis in positive and negative directions. @@ -6404,7 +6404,7 @@ The amplification factor increases when the struts have a high angle with the di #+end_subfigure #+end_figure -***** Mobility in rotation +***** Mobility in Rotation As shown by equation\nbsp{}eqref:eq:detail_kinematics_jacobian, the rotational mobility depends both on the orientation of the struts and on the location of the top joints. Similarly to the translational case, to increase the rotational mobility in one direction, it is advantageous to have the struts more perpendicular to the rotational direction. @@ -6439,7 +6439,7 @@ Having struts further apart decreases the "lever arm" and therefore reduces the #+end_subfigure #+end_figure -***** Combined translations and rotations +***** Combined Translations and Rotations It is possible to consider combined translations and rotations, although displaying such mobility becomes more complex. For a fixed geometry and a desired mobility (combined translations and rotations), it is possible to estimate the required minimum actuator stroke. @@ -6496,7 +6496,7 @@ This property depends on both the geometry and the chosen $\{A\}$ frame. For specific geometry and choice of $\{A\}$ frame, it is possible to achieve a diagonal $K$ matrix. This is discussed in Section\nbsp{}ref:ssec:detail_kinematics_cubic_static. -**** Dynamical properties +**** Dynamical Properties <> The dynamical equations (both in the Cartesian frame and in the frame of the struts) for the Stewart platform were derived during the conceptual phase with simplifying assumptions (massless struts and perfect joints). @@ -6582,7 +6582,7 @@ The ultimate objective is to determine the suitability of the cubic architecture **** Static Properties <> -***** Stiffness matrix for the Cubic architecture +***** Stiffness Matrix for the Cubic Architecture Consider the cubic architecture shown in Figure\nbsp{}ref:fig:detail_kinematics_cubic_schematic_full. The unit vectors corresponding to the edges of the cube are described by equation\nbsp{}eqref:eq:detail_kinematics_cubic_s. @@ -6647,7 +6647,7 @@ The stiffness matrix is therefore diagonal when the considered $\{B\}$ frame is This means that static forces (resp torques) applied at the cube's center will induce pure translations (resp rotations around the cube's center). This specific location where the stiffness matrix is diagonal is referred to as the acrfull:cok, analogous to the acrfull:com where the mass matrix is diagonal. -***** Effect of having frame $\{B\}$ off-centered +***** Effect of Having Frame $\{B\}$ Off-centered When the reference frames $\{A\}$ and $\{B\}$ are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix. @@ -6714,7 +6714,7 @@ When relative motion sensors are integrated in each strut (measuring $\bm{\mathc #+caption: Typical control architecture in the cartesian frame [[file:figs/detail_kinematics_centralized_control.png]] -***** Low frequency and High frequency coupling +***** Low Frequency and High Frequency Coupling As derived during the conceptual design phase, the dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ is described by Equation\nbsp{}eqref:eq:detail_kinematics_transfer_function_cart. At low frequency, the behavior of the platform depends on the stiffness matrix\nbsp{}eqref:eq:detail_kinematics_transfer_function_cart_low_freq. @@ -6761,7 +6761,7 @@ Conversely, when positioned at the acrlong:cok, coupling occurred at high freque #+end_subfigure #+end_figure -***** Payload's CoM at the cube's center +***** Payload's CoM at the Cube's Center An effective strategy for improving dynamical performances involves aligning the cube's center (acrlong:cok) with the acrlong:com of the moving components\nbsp{}[[cite:&li01_simul_fault_vibrat_isolat_point]]. This can be achieved by positioning the payload below the top platform, such that the acrlong:com of the moving body coincides with the cube's center (Figure\nbsp{}ref:fig:detail_kinematics_cubic_centered_payload). @@ -6888,7 +6888,7 @@ Three key parameters define the geometry of the cubic Stewart platform: $H$, the Depending on the cube's size $H_c$ in relation to $H$ and $H_{CoM}$, different designs emerge. In the following examples, $H = 100\,\text{mm}$ and $H_{CoM} = 20\,\text{mm}$. -***** Small cube +***** Small Cube When the cube size $H_c$ is smaller than twice the height of the CoM $H_{CoM}$ eqref:eq:detail_kinematics_cube_small, the resulting design is shown in Figure\nbsp{}ref:fig:detail_kinematics_cubic_above_small. @@ -6903,7 +6903,7 @@ Adjacent struts are parallel to each other, differing from the typical architect This approach yields a compact architecture, but the small cube size may result in insufficient rotational stiffness. #+name: fig:detail_kinematics_cubic_above_small -#+caption: Cubic architecture with cube's center above the top platform. A cube height of 40mm is used. +#+caption: Cubic architecture with cube's center above the top platform. A cube height of $40\,\text{mm}$ is used. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_above_small_iso}Isometric view} @@ -6926,7 +6926,7 @@ This approach yields a compact architecture, but the small cube size may result #+end_subfigure #+end_figure -***** Medium sized cube +***** Medium Sized Cube Increasing the cube's size such that\nbsp{}eqref:eq:detail_kinematics_cube_medium is verified produces an architecture with intersecting struts (Figure\nbsp{}ref:fig:detail_kinematics_cubic_above_medium). @@ -6937,7 +6937,7 @@ Increasing the cube's size such that\nbsp{}eqref:eq:detail_kinematics_cube_mediu This configuration resembles the design proposed in\nbsp{}[[cite:&yang19_dynam_model_decoup_contr_flexib]] (Figure\nbsp{}ref:fig:detail_kinematics_yang19), although their design is not strictly cubic. #+name: fig:detail_kinematics_cubic_above_medium -#+caption: Cubic architecture with cube's center above the top platform. A cube height of 140mm is used. +#+caption: Cubic architecture with cube's center above the top platform. A cube height of $140\,\text{mm}$ is used. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_above_medium_iso}Isometric view} @@ -6960,7 +6960,7 @@ This configuration resembles the design proposed in\nbsp{}[[cite:&yang19_dynam_m #+end_subfigure #+end_figure -***** Large cube +***** Large Cube When the cube's height exceeds twice the sum of the platform height and CoM height\nbsp{}eqref:eq:detail_kinematics_cube_large, the architecture shown in Figure\nbsp{}ref:fig:detail_kinematics_cubic_above_large is obtained. @@ -6969,7 +6969,7 @@ When the cube's height exceeds twice the sum of the platform height and CoM heig \end{equation} #+name: fig:detail_kinematics_cubic_above_large -#+caption: Cubic architecture with cube's center above the top platform. A cube height of 240mm is used. +#+caption: Cubic architecture with cube's center above the top platform. A cube height of $240\,\text{mm}$ is used. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_above_large_iso}Isometric view} @@ -6992,7 +6992,7 @@ When the cube's height exceeds twice the sum of the platform height and CoM heig #+end_subfigure #+end_figure -***** Platform size +***** Platform Size For the proposed configuration, the top joints $\bm{b}_i$ (resp. the bottom joints $\bm{a}_i$) and are positioned on a circle with radius $R_{b_i}$ (resp. $R_{a_i}$) described by Equation\nbsp{}eqref:eq:detail_kinematics_cube_joints. @@ -7026,7 +7026,7 @@ To address this limitation, modified cubic architectures have been proposed with Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform. This structural modification enables the alignment of the moving body's acrlong:com with the acrlong:cok, resulting in beneficial decoupling properties in the Cartesian frame. -*** Active Platform for the NASS +*** Kinematics of the Active Platform <> **** Introduction :ignore: @@ -7060,7 +7060,7 @@ Regarding dynamical properties, particularly for control in the frame of the str Consequently, the geometry was selected according to practical constraints. The height between the two plates is maximized and set at $95\,\text{mm}$. Both platforms take the maximum available size, with joints offset by $15\,\text{mm}$ from the plate surfaces and positioned along circles with radii of $120\,\text{mm}$ for the fixed joints and $110\,\text{mm}$ for the mobile joints. -The positioning angles, as shown in Figure\nbsp{}ref:fig:detail_kinematics_nano_hexapod_top, are $[255,\ 285,\ 15,\ 45,\ 135,\ 165]$ degrees for the top joints and $[220,\ 320,\ 340,\ 80,\ 100,\ 200]$ degrees for the bottom joints. +The positioning angles, as shown in Figure\nbsp{}ref:fig:detail_kinematics_nano_hexapod_top, are [255, 285, 15, 45, 135, 165] degrees for the top joints and [220, 320, 340, 80, 100, 200] degrees for the bottom joints. #+name: fig:detail_kinematics_nano_hexapod #+caption: Obtained architecture for the active platform @@ -7090,7 +7090,7 @@ However, positioning the cube's center $150\,\text{mm}$ above the top platform w Additionally, to benefit from the cubic configuration's dynamical properties, each payload would require careful calibration of inertia before placement on the active platform, ensuring that its acrlong:com coincides with the cube's center. Given the impracticality of consistently aligning the acrlong:com with the cube's center, the cubic architecture was deemed unsuitable for the NASS. -**** Required Actuator stroke +**** Required Actuator Stroke <> With the geometry established, the actuator stroke necessary to achieve the desired mobility can be determined. @@ -7109,7 +7109,7 @@ The diagram confirms that the required workspace fits within the system's capabi #+attr_latex: :scale 0.8 [[file:figs/detail_kinematics_nano_hexapod_mobility.png]] -**** Required Joint angular stroke +**** Required Joint Angular Stroke <> With the active platform geometry and mobility requirements established, the flexible joint angular stroke necessary to avoid limiting the achievable workspace can be determined. @@ -7138,7 +7138,7 @@ Modified cubic architectures with the cube's center positioned above the top pla For the active platform design, a key challenge was addressing the wide range of potential payloads (1 to $50\,\text{kg}$), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios. This led to a practical design approach where struts were oriented more vertically than in cubic configurations to address several application-specific needs: achieving higher resolution in the vertical direction by reducing amplification factors and better matching the micro-station's modal characteristics with higher vertical resonance frequencies. -** Component Optimization +** Hybrid Modelling for Component Optimization <> *** Introduction :ignore: @@ -7149,7 +7149,7 @@ The theoretical foundations and implementation are presented in Section\nbsp{}re The framework was then applied to optimize two critical active platform elements: the actuators (Section\nbsp{}ref:sec:detail_fem_actuator) and the flexible joints (Section\nbsp{}ref:sec:detail_fem_joint). Through this approach, system-level dynamic behavior under closed-loop control conditions could be successfully predicted while detailed component-level optimization was facilitated. -*** Reduced order flexible bodies +*** Reduced Order Flexible Bodies <> **** Introduction :ignore: @@ -7211,9 +7211,9 @@ The specific design of the acrshort:apa allows for the simultaneous modeling of #+latex: \centering #+attr_latex: :environment tabularx :width 0.55\linewidth :placement [b] :align Xc #+attr_latex: :booktabs t :float nil :center nil :font \footnotesize\sf -| *Parameter* | *Value* | -|----------------+---------------------| -| Nominal Stroke | $100\,\mu\text{m}$ | +| *Parameter* | *Value* | +|----------------+----------------------------| +| Nominal Stroke | $100\,\mu\text{m}$ | | Blocked force | $2100\,\text{N}$ | | Stiffness | $21\,\text{N}/\mu\text{m}$ | #+latex: \captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications} @@ -7359,7 +7359,7 @@ Measurement of the sensor stack voltage $V_s$ was performed using an acrshort:ad #+attr_latex: :width \linewidth [[file:figs/detail_fem_apa95ml_bench_schematic.png]] -***** Comparison of the dynamics +***** Comparison of the Dynamics Frequency domain system identification techniques were used to characterize the dynamic behavior of the APA95ML. The identification procedure required careful choice of the excitation signal\nbsp{}[[cite:&pintelon12_system_ident, chap. 5]]. @@ -7449,7 +7449,7 @@ These competing requirements suggest an optimal stiffness of approximately $1\,\ Additional specifications arise from the control strategy and physical constraints. The implementation of the decentralized Integral Force Feedback (IFF) architecture necessitates force sensors to be collocated with each actuator. -The system's geometric constraints limit the actuator height to 50mm, given the active platform's maximum height of 95mm and the presence of flexible joints at each strut extremity. +The system's geometric constraints limit the actuator height to $50\,\text{mm}$, given the active platform's maximum height of $95\,\text{mm}$ and the presence of flexible joints at each strut extremity. Furthermore, the actuator stroke must exceed the micro-station positioning errors while providing additional margin for mounting adjustments and operational flexibility. An actuator stroke of $\approx 200\,\mu\text{m}$ is therefore required. @@ -7818,7 +7818,7 @@ Based on this analysis, an axial stiffness specification of $100\,\text{N}/\mu\t #+end_subfigure #+end_figure -**** Specifications and Design flexible joints +**** Specifications and Design of Flexible Joints <> The design of flexible joints for precision applications requires careful consideration of multiple mechanical characteristics. @@ -7844,7 +7844,7 @@ Third, the architecture inherently provides high axial stiffness while maintaini The joint geometry was optimized through parametric acrshort:fea. The optimization process revealed an inherent trade-off between maximizing axial stiffness and achieving sufficiently low bending/torsional stiffness, while maintaining material stresses within acceptable limits. -The final design, featuring a neck dimension of 0.25mm, achieves mechanical properties closely matching the target specifications, as verified through acrshort:fea and summarized in Table\nbsp{}ref:tab:detail_fem_joints_specs. +The final design, featuring a neck dimension of $0.25\,\text{mm}$, achieves mechanical properties closely matching the target specifications, as verified through acrshort:fea and summarized in Table\nbsp{}ref:tab:detail_fem_joints_specs. #+name: fig:detail_fem_joints_design #+caption: Designed flexible joints. @@ -8312,7 +8312,7 @@ The typical magnitude response of a weighting function generated using\nbsp{}eqr \end{equation} #+end_minipage -***** Validation of the proposed synthesis method +***** Validation of the Proposed Synthesis Method The proposed methodology for designing complementary filters is now applied to a simple example. Consider the design of two complementary filters $H_1(s)$ and $H_2(s)$ with the following requirements: @@ -8353,7 +8353,7 @@ The resulting $\mathcal{H}_\infty$ norm is found to be close to unity, indicatin This is confirmed by the Bode plots of the obtained complementary filters in Figure\nbsp{}ref:fig:detail_control_sensor_hinf_filters_results. This straightforward example demonstrates that the proposed methodology for shaping complementary filters is both simple and effective. -**** Synthesis of a set of three complementary filters +**** Synthesis of a set of Three Complementary Filters <> Certain applications necessitate the fusion of more than two sensors\nbsp{}[[cite:&stoten01_fusion_kinet_data_using_compos_filter;&carreira15_compl_filter_desig_three_frequen_bands]]. @@ -8471,7 +8471,7 @@ Consequently, typical sensor fusion objectives can be effectively translated int For the NASS, the acrshort:haclac strategy was found to perform well and to offer the advantages of being both intuitive to understand and straightforward to tune. Looking forward, it would be interesting to investigate how sensor fusion (particularly between the force sensors and external metrology) compares to the HAC-IFF approach in terms of performance and robustness. -*** Decoupling +*** Decoupling Strategies for Parallel Manipulators <> **** Introduction :ignore: @@ -8507,7 +8507,7 @@ The decentralized plant (transfer functions from actuators to sensors integrated Three approaches are investigated across subsequent sections: Jacobian matrix decoupling (Section\nbsp{}ref:ssec:detail_control_decoupling_jacobian), modal decoupling (Section\nbsp{}ref:ssec:detail_control_decoupling_modal), and Singular Value Decomposition (SVD) decoupling (Section\nbsp{}ref:ssec:detail_control_decoupling_svd). Finally, a comparative analysis with concluding observations is provided in Section\nbsp{}ref:ssec:detail_control_decoupling_comp. -**** Test Model +**** 3-DoF Test Model <> Instead of using the Stewart platform for comparing decoupling strategies, a simplified parallel manipulator is employed to facilitate a more straightforward analysis. @@ -8594,7 +8594,7 @@ The matrices representing the payload inertia, actuator stiffness, and damping a The parameters employed for the subsequent analysis are summarized in Table\nbsp{}ref:tab:detail_control_decoupling_test_model_params, which includes values for geometric parameters ($l_a$, $h_a$), mechanical properties (actuator stiffness $k$ and damping $c$), and inertial characteristics (payload mass $m$ and rotational inertia $I$). -**** Control in the frame of the struts +**** Control in the Frame of the Struts <> The dynamics in the frame of the struts are first examined. @@ -8651,7 +8651,7 @@ The transfer function from $\bm{\mathcal{F}}_{\{O\}$ to $\bm{\mathcal{X}}_{\{O\} The frame $\{O\}$ can be selected according to specific requirements, but the decoupling properties are significantly influenced by this choice. Two natural reference frames are particularly relevant: the acrlong:com and the acrlong:cok. -***** Center Of Mass +***** Center of Mass When the decoupling frame is located at the acrlong:com (frame $\{M\}$ in Figure\nbsp{}ref:fig:detail_control_decoupling_model_test), the Jacobian matrix and its inverse are expressed as in\nbsp{}eqref:eq:detail_control_decoupling_jacobian_CoM_inverse. @@ -8709,7 +8709,7 @@ This phenomenon is illustrated in Figure\nbsp{}ref:fig:detail_control_decoupling #+end_subfigure #+end_figure -***** Center Of Stiffness +***** Center of Stiffness When the decoupling frame is located at the acrlong:cok, the Jacobian matrix and its inverse are expressed as in\nbsp{}eqref:eq:detail_control_decoupling_jacobian_CoK_inverse. @@ -8888,7 +8888,7 @@ This information can be obtained either experimentally or derived from a model. While this approach ensures effective decoupling near the chosen frequency, it provides no guarantees regarding decoupling performance away from this frequency. Furthermore, the quality of decoupling depends significantly on the accuracy of the real approximation, potentially limiting its effectiveness for plants with high damping. -***** Example +***** Test on the 3-DoF model Plant decoupling using the Singular Value Decomposition was then applied on the test model. A decoupling frequency of $\SI{100}{Hz}$ was used. @@ -8954,7 +8954,7 @@ The exceptional performance of SVD decoupling on the plant with collocated senso This effectiveness may be attributed to the symmetrical properties of the plant, as evidenced in the Bode plots of the decentralized plant shown in Figure\nbsp{}ref:fig:detail_control_decoupling_coupled_plant_bode. The phenomenon potentially relates to previous research on SVD controllers applied to systems with specific symmetrical characteristics\nbsp{}[[cite:&hovd97_svd_contr_contr]]. -**** Comparison of decoupling strategies +**** Comparison of Decoupling Strategies <> While the three proposed decoupling methods may appear similar in their mathematical implementation (each involving pre-multiplication and post-multiplication of the plant with constant matrices), they differ significantly in their underlying approaches and practical implications, as summarized in Table\nbsp{}ref:tab:detail_control_decoupling_strategies_comp. @@ -9061,7 +9061,7 @@ In this arrangement, the physical plant is controlled at low frequencies, while Although the control architecture shown in Figure\nbsp{}ref:fig:detail_control_cf_arch appears to be a multi-loop system, it should be noted that no non-linear saturation-type elements are present in the inner loop (containing $k$, $G$, and $H_H$, all numerically implemented). Consequently, this structure is mathematically equivalent to the single-loop architecture illustrated in Figure\nbsp{}ref:fig:detail_control_cf_arch_eq. -***** Asymptotic behavior +***** Asymptotic Behavior When considering the extreme case of very high values for $k$, the effective controller $K(s)$ converges to the inverse of the plant model multiplied by the inverse of the high-pass filter, as expressed in\nbsp{}eqref:eq:detail_control_cf_high_k. @@ -9098,7 +9098,7 @@ At frequencies where the model accurately represents the physical plant ($G^{-1} The sensitivity transfer function equals the high-pass filter $S = \frac{y}{dy} = H_H$, and the complementary sensitivity transfer function equals the low-pass filter $T = \frac{y}{n} = H_L$. Hence, when the plant model closely approximates the actual dynamics, the closed-loop transfer functions converge to the designed complementary filters, allowing direct translation of performance requirements into the design of the complementary. -**** Translating the performance requirements into the shape of the complementary filters +**** Translating the Performance Requirements into the Shape of the Complementary Filters <> ***** Introduction :ignore: Performance specifications in a feedback system can usually be expressed as upper bounds on the magnitudes of closed-loop transfer functions such as the sensitivity and complementary sensitivity transfer functions\nbsp{}[[cite:&bibel92_guidel_h]]. @@ -9202,7 +9202,7 @@ The acrfull:rp condition effectively combines both nominal performance\nbsp{}eqr If both NP and RS conditions are satisfied, robust performance will be achieved within a factor of 2\nbsp{}[[cite:&skogestad07_multiv_feedb_contr, chapt. 7.6]]. Therefore, for acrshort:siso systems, ensuring robust stability and nominal performance is typically sufficient. -**** Complementary filter design +**** Complementary Filter Design <> As proposed in Section\nbsp{}ref:sec:detail_control_sensor, complementary filters can be shaped using standard $\mathcal{H}_{\infty}\text{-synthesis}$ techniques. @@ -9324,7 +9324,7 @@ Figure\nbsp{}ref:fig:detail_control_cf_bode_plot_mech_sys illustrates both the n #+end_subfigure #+end_figure -***** Requirements and choice of complementary filters +***** Requirements and Choice of Complementary Filters As discussed in Section\nbsp{}ref:ssec:detail_control_cf_trans_perf, nominal performance requirements can be expressed as upper bounds on the shape of the complementary filters. For this example, the requirements are: @@ -9360,7 +9360,7 @@ There magnitudes are displayed in Figure\nbsp{}ref:fig:detail_control_cf_specs_S #+end_subfigure #+end_figure -***** Controller analysis +***** Controller Analysis The controller to be implemented takes the form $K(s) = \tilde{G}^{-1}(s) H_H^{-1}(s)$, where $\tilde{G}^{-1}(s)$ represents the plant inverse, which must be both stable and proper. To ensure properness, low-pass filters with high corner frequencies are added as shown in Equation\nbsp{}eqref:eq:detail_control_cf_test_plant_inverse. @@ -9375,7 +9375,7 @@ The loop gain reveals several important characteristics: - A notch at the plant resonance frequency (arising from the plant inverse) - A lead component near the control bandwidth of approximately $20\,\text{Hz}$, enhancing stability margins -***** Robustness and Performance analysis +***** Robustness and Performance Analysis Robust stability is assessed using the Nyquist plot shown in Figure\nbsp{}ref:fig:detail_control_cf_nyquist_robustness. Even when considering all possible plants within the uncertainty set, the Nyquist plot remains sufficiently distant from the critical point $(-1,0)$, indicating robust stability with adequate margins. @@ -9475,7 +9475,7 @@ Only the vertical direction is considered in this analysis as it presents the mo From these transfer functions, the maximum acceptable acrfull:asd of the noise sources is derived (Section\nbsp{}ref:ssec:detail_instrumentation_max_noise_specs). Since the voltage amplifier gain affects the amplification of acrshort:dac noise, an assumption of an amplifier gain of 20 was made. -**** Closed-Loop Sensitivity to Instrumentation Disturbances +**** Closed-Loop Sensitivity to Instrumentation Noise <> Several key noise sources are considered in the analysis (Figure\nbsp{}ref:fig:detail_instrumentation_plant). @@ -9490,7 +9490,7 @@ The transfer functions from these three noise sources (for one strut) to the ver #+attr_latex: :scale 0.8 [[file:figs/detail_instrumentation_noise_sensitivities.png]] -**** Estimation of maximum instrumentation noise +**** Estimation of Maximum Acceptable Instrumentation Noise <> The most stringent requirement for the system is maintaining vertical vibrations below the smallest expected beam size of $100\,\text{nm}$, which corresponds to a maximum allowed vibration of $15\,\text{nm RMS}$. @@ -9512,7 +9512,7 @@ In terms of RMS noise, these translate to less than $1\,\text{mV RMS}$ for the a If the Amplitude Spectral Density of the noise of the acrshort:adc, acrshort:dac, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below $15\,\text{nm RMS}$. -*** Choice of Instrumentation +*** Selection of Instrumentation <> **** Piezoelectric Voltage Amplifier ***** Introduction :ignore: @@ -9775,7 +9775,7 @@ This approach is effective because the noise approximates white noise and its am #+attr_latex: :scale 0.8 [[file:figs/detail_instrumentation_adc_noise_measured.png]] -***** Reading of piezoelectric force sensor +***** Reading of Piezoelectric Force Sensor To further validate the ADC's capability to effectively measure voltage generated by a piezoelectric stack, a test was conducted using the APA95ML. The setup is illustrated in Figure\nbsp{}ref:fig:detail_instrumentation_force_sensor_adc_setup, where two stacks are used as actuators (connected in parallel) and one stack serves as a sensor. @@ -9990,7 +9990,7 @@ The noise profile exhibits characteristics of white noise with an amplitude of a [[file:figs/detail_instrumentation_vionic_asd.png]] #+end_minipage -**** Noise budgeting from measured instrumentation noise +**** Noise Budgeting from Measured Instrumentation Noise After characterizing all instrumentation components individually, their combined effect on the sample's vibration was assessed using the multi-body model developed earlier. The vertical motion induced by the noise sources, specifically the acrshort:adc noise, acrshort:dac noise, and voltage amplifier noise, is presented in Figure\nbsp{}ref:fig:detail_instrumentation_cl_noise_budget. @@ -10025,7 +10025,7 @@ Finally, the measured noise characteristics of all instrumentation components we The combined effect of all noise sources was estimated to induce vertical sample vibrations of only $1.5\,\text{nm RMS}$, which is substantially below the $15\,\text{nm RMS}$ requirement. This rigorous methodology spanning requirement formulation, component selection, and experimental characterization validates the instrumentation's ability to fulfill the nano active stabilization system's demanding performance specifications. -** Obtained Design +** Obtained Design: the "Nano-Hexapod" <> *** Introduction :ignore: @@ -10294,7 +10294,7 @@ An important consequence of this measurement principle is that a relative rotati #+end_subfigure #+end_figure -***** Validation of the designed active platform +***** Validation of the Designed Active Platform The refined multi-body model of the nano-hexapod was integrated into the multi-body micro-station model. Dynamical analysis was performed, confirming that the platform's behavior closely approximates the dynamics of the "idealized" model used during the conceptual design phase. @@ -10371,7 +10371,7 @@ The acrshort:haclac control architecture is implemented and tested under various #+attr_latex: :width \linewidth [[file:figs/chapter3_overview.png]] -** Amplified Piezoelectric Actuator +** Amplified Piezoelectric Actuators <> *** Introduction :ignore: @@ -10396,7 +10396,7 @@ This more complex model also captures well capture the axial dynamics of the APA #+caption: Picture of 5 out of the 7 received APA300ML [[file:figs/test_apa_received.jpg]] -*** First Basic Measurements +*** Static Measurements <> **** Introduction :ignore: @@ -10585,7 +10585,7 @@ Another explanation is the shape difference between the manufactured APA300ML an #+attr_latex: :scale 0.8 [[file:figs/test_apa_meas_freq_compare.png]] -*** Dynamical measurements +*** Dynamical Measurements <> **** Introduction :ignore: After the measurements on the acrshort:apa were performed in Section\nbsp{}ref:sec:test_apa_basic_meas, a new test bench was used to better characterize the dynamics of the APA300ML. @@ -10843,7 +10843,7 @@ The two obtained root loci are compared in Figure\nbsp{}ref:fig:test_apa_iff_roo #+end_subfigure #+end_figure -*** APA300ML - 2 degrees-of-freedom Model +*** Two degrees-of-freedom Model <> ***** Introduction :ignore: @@ -10953,7 +10953,7 @@ This indicates that this model represents well the axial dynamics of the APA300M #+end_subfigure #+end_figure -*** APA300ML - Super Element +*** Reduced Order Flexible Model <> ***** Introduction :ignore: @@ -10972,7 +10972,7 @@ Finally, two /remote points/ (=4= and =5=) are located across the third piezoele #+caption: Finite Element Model of the APA300ML with "remotes points" on the left. Multi-Body model with included "Reduced Order Flexible Solid" on the right (here in Simulink-Simscape software). [[file:figs/test_apa_super_element_simscape.png]] -***** Identification of the Actuator and Sensor constants +***** Identification of the Actuator and Sensor "Constants" Once the APA300ML /super element/ is included in the multi-body model, the transfer function from $F_a$ to $d_L$ and $d_e$ can be extracted. The gains $g_a$ and $g_s$ are then tuned such that the gains of the transfer functions match the identified ones. @@ -11009,7 +11009,7 @@ From these parameters, $g_s = 5.1\,\text{V}/\mu\text{m}$ and $g_a = 26\,\text{N/ | $A$ | $10^{-4}\,\text{m}^2$ | Area of the piezoelectric stack | | $n$ | $160$ per stack | Number of layers in the piezoelectric stack | -***** Comparison of the obtained dynamics +***** Comparison of the Obtained Dynamics The obtained dynamics using the /super element/ with the tuned "sensor sensitivity" and "actuator sensitivity" are compared with the experimentally identified acrshortpl:frf in Figure\nbsp{}ref:fig:test_apa_super_element_comp_frf. A good match between the model and the experimental results was observed. @@ -11187,7 +11187,7 @@ However, what is more important than the true value of the thickness is the cons #+attr_latex: :scale 0.8 [[file:figs/test_joints_size_hist.png]] -**** Bad flexible joints +**** Defects in Flexible Joints Using this profilometer allowed to detect flexible joints with manufacturing defects such as non-symmetrical shapes (see Figure\nbsp{}ref:fig:test_joints_bad_shape) or flexible joints with machining chips stuck in the gap (see Figure\nbsp{}ref:fig:test_joints_bad_chips). @@ -11209,7 +11209,7 @@ Using this profilometer allowed to detect flexible joints with manufacturing def #+end_subfigure #+end_figure -*** Compliance Measurement Test Bench +*** Characterization Test Bench <> **** Introduction :ignore: The most important characteristic of the flexible joint to be measured is its bending stiffness $k_{R_x} \approx k_{R_y}$. @@ -11221,9 +11221,9 @@ The bending stiffness can then be computed from equation\nbsp{}eqref:eq:test_joi \boxed{k_{R_x} = \frac{T_x}{\theta_x}, \quad k_{R_y} = \frac{T_y}{\theta_y}} \end{equation} -**** Measurement principle +**** Measurement Principle <> -***** Torque and Rotation measurement +***** Torque and Rotation Measurement To apply torque $T_{y}$ between the two mobile parts of the flexible joint, a known "linear" force $F_{x}$ can be applied instead at a certain distance $h$ with respect to the rotation point. In this case, the equivalent applied torque can be estimated from equation\nbsp{}eqref:eq:test_joints_force_torque_distance. Note that the application point of the force should be sufficiently far from the rotation axis such that the resulting bending motion is much larger than the displacement due to shear. @@ -11257,7 +11257,7 @@ The deflection of the joint $d_x$ is measured using a displacement sensor. #+caption: Working principle of the test bench used to estimate the bending stiffness $k_{R_y}$ of the flexible joints by measuring $F_x$, $d_x$ and $h$ [[file:figs/test_joints_bench_working_principle.png]] -***** Required external applied force +***** Required External Applied Force The bending stiffness is foreseen to be $k_{R_y} \approx k_{R_x} \approx 5\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx \theta_{x,\text{max}}\approx 25\,\text{mrad}$. The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 22.5\,\text{mm}$ (see Figure\nbsp{}ref:fig:test_joints_bench_working_principle). @@ -11274,7 +11274,7 @@ The measurement range of the force sensor should then be higher than $5.5\,\text F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h} \approx 5.5\,\text{N} \end{equation} -***** Required actuator stroke and sensors range +***** Required Actuator Stroke and Sensors Range The flexible joint is designed to allow a bending motion of $\pm 25\,\text{mrad}$. The corresponding stroke at the location of the force sensor is given by\nbsp{}eqref:eq:test_joints_max_stroke. To test the full range of the flexible joint, the means of applying a force (explained in the next section) should allow a motion of at least $0.5\,\text{mm}$. @@ -11284,13 +11284,13 @@ Similarly, the measurement range of the displacement sensor should also be highe d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \approx 0.5\,\text{mm} \end{equation} -***** Force and Displacement measurements +***** Force and Displacement Measurements To determine the applied force, a load cell will be used in series with the mechanism that applied the force. The measured deflection of the flexible joint will be indirectly estimated from the displacement of the force sensor itself (see Section\nbsp{}ref:ssec:test_joints_test_bench). Indirectly measuring the deflection of the flexible joint induces some errors because of the limited stiffness between the force sensor and the displacement sensor. Such an effect will be estimated in the error budget (Section\nbsp{}ref:ssec:test_joints_error_budget) -**** Error budget +**** Error Budget <> ***** Introduction :ignore: To estimate the accuracy of the measured bending stiffness that can be obtained using this measurement principle, an error budget is performed. @@ -11326,7 +11326,7 @@ The estimated bending stiffness $k_{\text{est}}$ then depends on the shear stiff With an estimated shear stiffness $k_s = 13\,\text{N}/\mu\text{m}$ from the acrshort:fem and an height $h=25\,\text{mm}$, the estimation errors of the bending stiffness due to shear is $\epsilon_s < 0.1\,\%$ -***** Effect of load cell limited stiffness +***** Effect of Load Cell Limited Stiffness As explained in the previous section, because the measurement of the flexible joint deflection is indirectly performed with the encoder, errors will be made if the load cell experiences some compression. Suppose the load cell has an internal stiffness $k_f$, the same reasoning that was made for the effect of shear can be applied here. @@ -11338,7 +11338,7 @@ The estimation error of the bending stiffness due to the limited stiffness of th With an estimated load cell stiffness of $k_f \approx 1\,\text{N}/\mu\text{m}$ (from the documentation), the errors due to the load cell limited stiffness is around $\epsilon_f = 1\,\%$. -***** Estimation error due to height estimation error +***** Height Estimation Error Now consider an error $\delta h$ in the estimation of the height $h$ as described by\nbsp{}eqref:eq:test_joints_est_h_error. \begin{equation}\label{eq:test_joints_est_h_error} @@ -11353,7 +11353,7 @@ The computed bending stiffness will be\nbsp{}eqref:eq:test_joints_stiffness_heig The height estimation is foreseen to be accurate to within $|\delta h| < 0.4\,\text{mm}$ which corresponds to a stiffness error $\epsilon_h < 3.5\,\%$. -***** Estimation error due to force and displacement sensors accuracy +***** Force and Displacement Sensors Accuracy An optical encoder is used to measure the displacement (see Section\nbsp{}ref:ssec:test_joints_test_bench) whose maximum non-linearity is $40\,\text{nm}$. As the measured displacement is foreseen to be $0.5\,\text{mm}$, the error $\epsilon_d$ due to the encoder non-linearity is negligible $\epsilon_d < 0.01\,\%$. @@ -11489,7 +11489,7 @@ The load cell stiffness can then be estimated by computing a linear fit and is f #+end_subfigure #+end_figure -**** Bending Stiffness estimation +**** Bending Stiffness Estimation The actual stiffness is now estimated by manually moving the translation stage from a start position where the force sensor is not yet in contact with the flexible joint to a position where the flexible joint is on its mechanical stop. The measured force and displacement as a function of time are shown in Figure\nbsp{}ref:fig:test_joints_meas_bend_time. @@ -11517,7 +11517,7 @@ The bending stroke can also be estimated as shown in Figure\nbsp{}ref:fig:test_j #+end_subfigure #+end_figure -**** Measured flexible joint stiffness +**** Measured Flexible Joints' Stiffnesses The same measurement was performed for all the 16 flexible joints, both in the $x$ and $y$ directions. The measured angular motion as a function of the applied torque is shown in Figure\nbsp{}ref:fig:test_joints_meas_bending_all_raw_data for the 16 flexible joints. @@ -11586,7 +11586,7 @@ The model dynamics from the acrshort:dac voltage to the axial motion of the stru However, this is not the case for the dynamics from acrshort:dac voltage to the encoder displacement. It is found that the complex dynamics is due to a misalignment between the flexible joints and the acrshort:apa. -*** Mounting Procedure +*** Assembly Procedure <> A mounting bench was developed to ensure: @@ -11710,7 +11710,7 @@ Thanks to this mounting procedure, the coaxiality and length between the two fle #+end_subfigure #+end_figure -*** Measurement of flexible modes +*** Measurement of Flexible Modes <> A Finite Element Model[fn:test_struts_3] of the struts is developed and is used to estimate the flexible modes. @@ -11809,7 +11809,7 @@ This validates the quality of the acrshort:fem. | Y-Bending | $285\,\text{Hz}$ | $293\,\text{Hz}$ | $337\,\text{Hz}$ | | Z-Torsion | $400\,\text{Hz}$ | $381\,\text{Hz}$ | $398\,\text{Hz}$ | -*** Dynamical measurements +*** Dynamical Measurements <> **** Introduction :ignore: @@ -11841,7 +11841,7 @@ First, the effect of the encoder on the measured dynamics is investigated in Sec The dynamics observed by the encoder and interferometers are compared in Section\nbsp{}ref:ssec:test_struts_comp_enc_int. Finally, all measured struts are compared in terms of dynamics in Section\nbsp{}ref:ssec:test_struts_comp_all_struts. -**** Effect of the Encoder on the measured dynamics +**** Effect of the Encoder on the Measured Dynamics <> System identification was performed without the encoder being fixed to the strut (Figure\nbsp{}ref:fig:test_struts_bench_leg_front) and with one encoder being fixed to the strut (Figure\nbsp{}ref:fig:test_struts_bench_leg_coder). @@ -11894,7 +11894,7 @@ This means that the encoder should have little effect on the effectiveness of th #+end_subfigure #+end_figure -**** Comparison of the encoder and interferometer +**** Comparison of the Encoder and Interferometer <> The dynamics measured by the encoder (i.e. $d_e/u$) and interferometers (i.e. $d_a/u$) are compared in Figure\nbsp{}ref:fig:test_struts_comp_enc_int. @@ -11959,7 +11959,7 @@ The struts were then disassembled and reassemble a second time to optimize align #+attr_latex: :width 0.65\linewidth [[file:figs/test_struts_simscape_model.png]] -**** Model dynamics +**** Model Dynamics <> Two models of the APA300ML are used here: a simple two-degrees-of-freedom model and a model using a super-element extracted from a acrlong:fem. @@ -11997,7 +11997,7 @@ For the flexible model, it will be shown in the next section that by adding some #+end_subfigure #+end_figure -**** Effect of strut misalignment +**** Effect of Strut Misalignment <> As shown in Figure\nbsp{}ref:fig:test_struts_comp_enc_plants, the identified dynamics from DAC voltage $u$ to encoder measured displacement $d_e$ are very different from one strut to the other. @@ -12044,7 +12044,7 @@ This similarity suggests that the identified differences in dynamics are caused #+end_subfigure #+end_figure -**** Measured strut misalignment +**** Measured Strut Misalignment <> During the initial mounting of the struts, as presented in Section\nbsp{}ref:sec:test_struts_mounting, the positioning pins that were used to position the acrshort:apa with respect to the flexible joints in the $y$ directions were not used (not received at the time). @@ -12086,7 +12086,7 @@ With a better alignment, the amplitude of the spurious resonances is expected to #+attr_latex: :scale 0.8 [[file:figs/test_struts_comp_dy_tuned_model_frf_enc.png]] -**** Proper struts alignment +**** Better Struts Alignment <> After receiving the positioning pins, the struts were mounted again with the positioning pins. @@ -12148,7 +12148,7 @@ The Nano-Hexapod was then mounted on top of the suspended table such that its dy The obtained nano-hexapod dynamics is analyzed in Section\nbsp{}ref:sec:test_nhexa_dynamics, and compared with the multi-body model in Section\nbsp{}ref:sec:test_nhexa_model. -*** Nano-Hexapod Assembly Procedure +*** Assembly Procedure <> The assembly of the nano-hexapod is critical for both avoiding additional stress in the flexible joints (that would result in a loss of stroke) and for precisely determining the Jacobian matrix. The goal was to fix the six struts to the two nano-hexapod plates (shown in Figure\nbsp{}ref:fig:test_nhexa_nano_hexapod_plates) while the two plates were parallel and aligned vertically so that all the flexible joints did not experience any stress. @@ -12281,7 +12281,7 @@ Finally, some interface elements were designed, and mechanical lateral mechanica #+attr_latex: :width 0.7\linewidth [[file:figs/test_nhexa_suspended_table_cad.jpg]] -**** Modal analysis of the suspended table +**** Modal Analysis of the Suspended Table <> In order to perform a modal analysis of the suspended table, a total of 15 3-axis accelerometers[fn:test_nhexa_7] were fixed to the breadboard. @@ -12339,7 +12339,7 @@ The next modes are the flexible modes of the breadboard as shown in Figure\nbsp{ #+end_subfigure #+end_figure -**** Multi-body Model of the suspended table +**** Multi-body Model of the Suspended Table <> The multi-body model of the suspended table consists simply of two solid bodies connected by 4 springs. @@ -12360,7 +12360,7 @@ The obtained suspension modes of the multi-body model are compared with the meas | Multi-body | $1.3\,\text{Hz}$ | $1.8\,\text{Hz}$ | $6.8\,\text{Hz}$ | $9.5\,\text{Hz}$ | | Experimental | $1.3\,\text{Hz}$ | $2.0\,\text{Hz}$ | $6.9\,\text{Hz}$ | $9.5\,\text{Hz}$ | -*** Nano-Hexapod Measured Dynamics +*** Measured Active Platform Dynamics <> **** Introduction :ignore: @@ -12385,7 +12385,7 @@ The effect of the payload mass on the dynamics is discussed in Section\nbsp{}ref #+attr_latex: :width 0.9\linewidth [[file:figs/test_nhexa_nano_hexapod_signals.png]] -**** Modal analysis +**** Modal Analysis <> To facilitate the future analysis of the measured plant dynamics, a basic modal analysis of the nano-hexapod is performed. @@ -12433,7 +12433,7 @@ These modes are summarized in Table\nbsp{}ref:tab:test_nhexa_hexa_modal_modes_li #+end_subfigure #+end_figure -**** Identification of the dynamics +**** Identification of the Dynamics <> The dynamics of the nano-hexapod from the six command signals ($u_1$ to $u_6$) to the six measured displacement by the encoders ($d_{e1}$ to $d_{e6}$) and to the six force sensors ($V_{s1}$ to $V_{s6}$) were identified by generating low-pass filtered white noise for each command signal, one by one. @@ -12469,7 +12469,7 @@ The first flexible mode of the struts as $235\,\text{Hz}$ has large amplitude, a #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_identified_frf_Vs.png]] -**** Effect of payload mass on the dynamics +**** Effect of Payload Mass on the Dynamics <> One major challenge for controlling the NASS is the wanted robustness to a variation of payload mass; therefore, it is necessary to understand how the dynamics of the nano-hexapod changes with a change in payload mass. @@ -12516,7 +12516,7 @@ For all tested payloads, the measured acrshort:frf always have alternating poles #+end_subfigure #+end_figure -*** Nano-Hexapod Model Dynamics +*** Model Dynamics <> **** Introduction :ignore: @@ -12535,7 +12535,7 @@ Both the "direct" terms (Section\nbsp{}ref:ssec:test_nhexa_comp_model) and "coup Second, it should also represents how the system dynamics changes when a payload is fixed to the top platform. This is checked in Section\nbsp{}ref:ssec:test_nhexa_comp_model_masses. -**** Nano-Hexapod model dynamics +**** Nano-Hexapod Model Dynamics <> The multi-body model of the nano-hexapod was first configured with 4-DoF flexible joints, 2-DoF acrshort:apa, and rigid top and bottom plates. @@ -12567,7 +12567,7 @@ At higher frequencies, no resonances can be observed in the model, as the top pl #+end_subfigure #+end_figure -**** Dynamical coupling +**** Dynamical Coupling <> Another desired feature of the model is that it effectively represents coupling in the system, as this is often the limiting factor for the control of acrshort:mimo systems. @@ -12591,7 +12591,7 @@ Therefore, if the modes of the struts are to be modeled, the /super-element/ of #+attr_latex: :scale 0.8 [[file:figs/test_nhexa_comp_simscape_de_all_flex.png]] -**** Effect of payload mass +**** Effect of Payload Mass <> Another important characteristic of the model is that it should represents the dynamics of the system well for all considered payloads. @@ -12787,7 +12787,7 @@ The five equations\nbsp{}eqref:eq:test_id31_metrology_kinematics can be written \end{bmatrix} \end{equation} -**** Rough alignment of the reference spheres +**** Rough Alignment of the Reference Spheres <> The two reference spheres must be well aligned with the rotation axis of the spindle. @@ -12801,7 +12801,7 @@ With this setup, the alignment accuracy of both spheres with the spindle axis wa The accuracy was probably limited by the poor coaxiality between the cylinders and the spheres. However, this first alignment should be sufficient to position the two sphere within the acceptance range of the interferometers. -**** Tip-Tilt adjustment of the interferometers +**** Tip-Tilt Adjustment of the Interferometers <> The short-stroke metrology system was placed on top of the main granite using granite blocs (Figure\nbsp{}ref:fig:test_id31_short_stroke_metrology_overview). @@ -12824,7 +12824,7 @@ This is achieved by maximizing the intensity of the reflected light of each inte After the alignment procedure, the top interferometer should coincide with the spindle axis, and the lateral interferometers should all be in the horizontal plane and intersect the centers of the spheres. -**** Fine Alignment of reference spheres using interferometers +**** Fine Alignment of Reference Spheres using Interferometers <> Thanks to the first alignment of the two reference spheres with the spindle axis (Section\nbsp{}ref:ssec:test_id31_metrology_sphere_rought_alignment) and to the fine adjustment of the interferometer orientations (Section\nbsp{}ref:ssec:test_id31_metrology_alignment), the spindle can perform complete rotations while still having interference for all five interferometers. @@ -12856,7 +12856,7 @@ The remaining errors after alignment are in the order of $\pm5\,\mu\text{rad}$ i #+end_subfigure #+end_figure -**** Estimated measurement volume +**** Estimated Measurement Volume <> Because the interferometers point to spheres and not flat surfaces, the lateral acceptance is limited. @@ -12877,7 +12877,7 @@ The obtained lateral acceptance for pure displacements in any direction is estim | $d_4$ (x) | $>2\,\text{mm}$ | $0.99\,\text{mm}$ | $0.94\,\text{mm}$ | | $d_5$ (z) | $1.33\,\text{mm}$ | $1.06\,\text{mm}$ | $>2\,\text{mm}$ | -**** Estimated measurement errors +**** Estimated Measurement Errors <> When using the NASS, the accuracy of the sample positioning is determined by the accuracy of the external metrology. @@ -13255,7 +13255,7 @@ The experimental results validated the model predictions, showing a reduction in Although higher gains could achieve better damping performance for lighter payloads, the chosen fixed-gain configuration represents a robust compromise that maintains stability and performance under all operating conditions. The good correlation between the modeled and measured damped plants confirms the effectiveness of using the multi-body model for both controller design and performance prediction. -*** High Authority Control in the frame of the struts +*** High Authority Control in the Frame of the Struts <> **** Introduction :ignore: @@ -13359,7 +13359,7 @@ However, small stability margins were observed for the highest mass, indicating #+end_subfigure #+end_figure -**** Performance estimation with simulation of Tomography scans +**** Performance - Tomography Scans <> To estimate the performances that can be expected with this acrshort:haclac architecture and the designed controller, simulations of tomography experiments were performed[fn:test_id31_4]. @@ -13385,7 +13385,7 @@ The obtained closed-loop positioning accuracy was found to comply with the requi #+end_subfigure #+end_figure -**** Robustness estimation with simulation of Tomography scans +**** Robustness - Tomography Scans <> To verify the robustness against payload mass variations, four simulations of tomography experiments were performed with payloads as shown Figure\nbsp{}ref:fig:test_id31_picture_masses (i.e. $0\,\text{kg}$, $13\,\text{kg}$, $26\,\text{kg}$ and $39\,\text{kg}$). @@ -13418,7 +13418,7 @@ With no payload at $180\,\text{deg/s}$, the NASS successfully maintained the sam At $6\,\text{deg/s}$, although the positioning errors increased with the payload mass (particularly in the lateral direction), the system remained stable. These results demonstrate both the effectiveness and limitations of implementing control in the frame of the struts. -*** Validation with Scientific experiments +*** Validation with Scientific Experiments <> **** Introduction :ignore: @@ -13454,7 +13454,7 @@ Results obtained for all experiments are summarized and compared to the specific **** Tomography Scans <> -***** Slow Tomography scans +***** Slow Tomography Scans First, tomography scans were performed with a rotational velocity of $6\,\text{deg/s}$ for all considered payload masses (shown in Figure\nbsp{}ref:fig:test_id31_picture_masses). Each experimental sequence consisted of two complete spindle rotations: an initial open-loop rotation followed by a closed-loop rotation. @@ -13492,7 +13492,7 @@ These experimental findings are consistent with the predictions from the tomogra #+attr_latex: :scale 0.8 [[file:figs/test_id31_tomo_Wz36_results.png]] -***** Fast Tomography scans +***** Fast Tomography Scans A tomography experiment was then performed with the highest rotational velocity of the Spindle: $180\,\text{deg/s}$[fn:test_id31_7]. The trajectory of the acrshort:poi during the fast tomography scan is shown in Figure\nbsp{}ref:fig:test_id31_tomo_m0_30rpm_robust_hac_iff_exp. @@ -13592,7 +13592,7 @@ The results confirmed that the NASS successfully maintained the acrshort:poi wit In some cases, samples are composed of several atomic "layers" that are first aligned in the horizontal plane through $R_x$ and $R_y$ positioning, followed by vertical scanning with precise $D_z$ motion. These vertical scans can be executed either continuously or in a step-by-step manner. -***** Step by Step $D_z$ motion +***** Step by Step $D_z$ Motion The vertical step motion was performed exclusively with the nano-hexapod. Testing was conducted across step sizes ranging from $10\,\text{nm}$ to $1\,\mu\text{m}$. @@ -13627,7 +13627,7 @@ The settling duration typically decreases for smaller step sizes. #+end_subfigure #+end_figure -***** Continuous $D_z$ motion: Dirty Layer Scans +***** Continuous $D_z$ Motion: Dirty Layer Scans For these and subsequent experiments, the NASS performs "ramp scans" (constant velocity scans). To eliminate tracking errors, the feedback controller incorporates two integrators, compensating for the plant's lack of integral action at low frequencies. @@ -13695,7 +13695,7 @@ The stepper motor controller[fn:test_id31_5] generates a setpoint that is transm Within the Speedgoat, the system computes the positioning error by comparing the measured $D_y$ sample position against the received setpoint, and the Nano-Hexapod compensates for positioning errors introduced during $T_y$ stage scanning. The scanning range is constrained $\pm 100\,\mu\text{m}$ due to the limited acceptance of the metrology system. -***** Slow scan +***** Slow Scan Initial testing were made with a scanning velocity of $10\,\mu\text{m/s}$, which is typical for these experiments. Figure\nbsp{}ref:fig:test_id31_dy_10ums compares the positioning errors between open-loop (without NASS) and closed-loop operation. @@ -13809,7 +13809,7 @@ Alternatively, a feedforward controller could improve the lateral positioning ac #+end_subfigure #+end_figure -**** Feedback control using Complementary Filters +**** Feedback Control using Complementary Filters <> # TODO - Add link to section @@ -14227,7 +14227,7 @@ Therefore, adopting a design approach using dynamic error budgets, cascading fro [fn:test_nhexa_7]PCB 356B18. Sensitivity is $1\,\text{V/g}$, measurement range is $\pm 5\,\text{g}$ and bandwidth is $0.5$ to $5\,\text{kHz}$. [fn:test_nhexa_6]"SZ8005 20 x 044" from Steinel. The spring rate is specified at $17.8\,\text{N/mm}$ -[fn:test_nhexa_5]The 450 mm x 450 mm x 60 mm Nexus B4545A from Thorlabs. +[fn:test_nhexa_5]The $450\,\text{mm} \times 450\,\text{mm} \times 60\,\text{mm}$ Nexus B4545A from Thorlabs. [fn:test_nhexa_4]As the accuracy of the FARO arm is $\pm 13\,\mu\text{m}$, the true straightness is probably better than the values indicated. The limitation of the instrument is here reached. [fn:test_nhexa_3]The height dimension is better than $40\,\mu\text{m}$. The diameter fitting of 182g6 and 24g6 with the two plates is verified. [fn:test_nhexa_2]Location of all the interface surfaces with the flexible joints were checked. The fittings (182H7 and 24H8) with the interface element were also checked.