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@ -1794,16 +1794,17 @@ While this modeling approach provides accurate predictions of component behavior
This is exemplified by the nano-hexapod configuration, where the implementation of six Amplified Piezoelectric Actuators, each modeled with 48 degrees of freedom, yields 288 degrees of freedom only for the actuators.
However, the methodology remains valuable for system analysis, as the extraction of frequency domain characteristics can be efficiently performed even with such high-order models.
* Actuator
* Actuator Selection
<<sec:detail_fem_actuator>>
** Introduction :ignore:
Goals:
- Based on dynamical models and previous studies, extract specifications for the actuators to be included in the nano-hexapod.
Then choose the most appropriate actuator based on specifications (Section ref:ssec:detail_fem_actuator_specifications)
- Model this actuator accurately using a "reduced order flexible body" to check the dynamics and validate the choice of actuator
and validate this choice with simulations
- Development of a 2DoF model for lower order models (i.e. for simulations)
The selection and modeling of actuators constitutes a critical step in the development of the nano-hexapod.
This chapter presents the approach to actuator selection and modeling.
First, specifications for the nano-hexapod actuators are derived from previous analyses, leading to the selection of the actuator type and ultimately to a specific model (Section ref:ssec:detail_fem_actuator_specifications).
Then, the chosen actuator is modeled using the reduced-order flexible body approach developed in the previous section, enabling validation of this selection through detailed dynamical analysis (Section ref:ssec:detail_fem_actuator_apa300ml).
Finally, a simplified two-degree-of-freedom model is developed to facilitate time-domain simulations while maintaining accurate representation of the actuator's essential characteristics (Section ref:ssec:detail_fem_actuator_apa300ml_2dof).
# TODO Add link to other sections
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -1843,47 +1844,19 @@ gs = -5.08e6; % [V/m]
** Choice of the Actuator based on Specifications
<<ssec:detail_fem_actuator_specifications>>
From previous analysis:
- Actuator stiffness has major impact on the system dynamics and performances due to several factors:
- Spindle rotation: modification of plant dynamics and coupling increase due to Gyroscopic effects
This require to have stiffness above ~
- Limited micro-station compliance / complex dynamics:
The actuator stiffness should be small enough such that the suspension modes of the nano-hexapod are below the problematic modes of the micro-stations.
- There is therefore an intermediate stiffness that is foreseen to give the best compromise, and it is around $1\,N/\mu m$
- HAC-LAC strategy:
Actuator must include a force sensor
Because of the rotation, some stiffness should be present in parallel to the force sensor
- Limited space:
As the maximum height of the nano-hexapod is 95mm, and each strut has a flexible joint at each end, it is estimated that the maximum height of the actuator should be less than 50mm
- Stroke:
The stroke of the each actuator should be large enough such that the nano-hexapod mobility exceed the micro-station positioning errors.
Some margins should be included for mounting errors, and further flexibility of the system (for instance to perform scans with the nano-hexapod, or to align the point of interest with the rotation axis)
The actuator selection process was driven by several critical requirements derived from previous dynamic analyses.
A primary consideration is the actuator stiffness, which significantly impacts system dynamics through multiple mechanisms.
The spindle rotation induces gyroscopic effects that modify plant dynamics and increase coupling, necessitating sufficient stiffness.
Conversely, the actuator stiffness must be carefully limited to ensure the nano-hexapod's suspension modes remain below the problematic modes of the micro-stations to limit the coupling between the two structures.
These competing requirements suggest an optimal stiffness of approximately $1\,N/\mu m$.
Actuator specifications:
- Height (<50mm)
- Stroke (~100um)
- Stiffness (0.1-1 N/um)
- Blocked force?
- Force sensor
Options:
- Two main options: piezoelectric actuators and Lorentz actuator (also known as Voice coil actuators).
Variable reluctance actuators were not considered, even though they have better efficiency than voice coil actuators, they are non linear and induce additional control complexity.
- Voice coil + relatively soft flexible guiding (1N/um):
- required force ~100N for 100um correction
This constant force/current would induce large thermal loads, that may negatively impact system's stability
Advantages of voice coil (longer strokes than piezo + allow for very low stiffness in the direction of actuation, extremely linear for high performance feedforward) are not used here.
- Piezoelectric stack actuators:
- PZT: stroke ~0.1% of its length.
- 50mm length => 50um stroke which is barely enough
- Extremely stiff, in the order of $100\,N/\mu m$, which is not wanted here.
- Amplified Piezoelectric Actuator:
- shell is used to pre-stress the piezoelectric stacks and amplify the motion (roughly by the ratio of the width over the height)
- This also reduce the stiffness in the direction of motion
- This make this design quick compact in the direction of motion (i.e. in height)
- When several stacks are used, one of them can be used as a force sensor, which is therefore very well collocated with the actuators
- Therefore, this actuator is well suited for decentralized IFF, already applied for a Stewart platform with APA [[cite:&hanieh03_activ_stewar]]
Additional specifications arise from the control strategy and physical constraints.
The implementation of a HAC-LAC (High Authority Control-Low Authority Control) architecture necessitates integrated force sensing capability.
The system's geometric constraints limit the actuator height to 50mm, given the nano-hexapod's maximum height of 95mm 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, which is estimated at $\approx 100\,\mu m$.
Three actuator technologies were evaluated (examples are shown in Figure ref:fig:detail_fem_actuator_pictures): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators.
Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce unnecessary control complexity.
#+name: fig:detail_fem_actuator_pictures
#+caption: Example of actuators considered for the nano-hexapod. Voice coil from Sensata Technologies (\subref{fig:detail_fem_voice_coil_picture}). Piezoelectric stack actuator from Physik Instrumente (\subref{fig:detail_fem_piezo_picture}). Amplified Piezoelectric Actuator from DSM (\subref{fig:detail_fem_fpa_picture}).
@ -1909,15 +1882,23 @@ Options:
#+end_subfigure
#+end_figure
Based on previous analysis, it was decided to use amplified piezoelectric actuators for the nano-hexapod.
Table ref:tab:detail_fem_piezo_act_models: compares few models that fulfill specifications.
It was decided to go for the APA300ML (shown in Figure ref:fig:detail_fem_apa300ml_picture).
One reason is that we already had experience with APA from Cedrat technologies, and the Finite Element Model was validated experimentally, so we are confident to model the APA300ML with FEA and include it in the NASS model for validation.
Voice coil actuators (shown in Figure ref:fig:detail_fem_voice_coil_picture), when combined with flexure guides of wanted stiffness $\approx 1\,N/\mu m$, would require forces above $100\,N$ to achieve the specified $100\,\mu m$ displacement.
While these actuators offer excellent linearity and long strokes, the constant force requirement would result in significant steady-state current, leading to thermal loads that could compromise system stability.
Their advantages were not considered adapted for this application, diminishing their benefits relative to piezoelectric solutions.
Conventional piezoelectric stack actuators (shown in Figure ref:fig:detail_fem_piezo_picture) present two significant limitations for the current application.
Their stroke is inherently limited to approximately $0.1\,\%$ of their length, meaning that even with the maximum allowable height of $50\,mm$, the achievable stroke would only be $50\,\mu m$, insufficient for the application.
Additionally, their extremely high stiffness, typically around $100\,N/\mu m$, exceeds the desired specifications by two orders of magnitude.
- Talk about piezoelectric actuator? bandwidth? noise?
- Resolution: really depends on the electrical noise (induced by DAC and voltage amplifier).
They will be chosen appropriately
Amplified Piezoelectric Actuators (APAs) emerged as the optimal solution by addressing these limitations through an specific mechanical design.
The incorporation of a shell structure serves multiple purposes: it provides mechanical amplification of the piezoelectric displacement, reduces the effective axial stiffness to more suitable levels for the application, and creates a compact vertical profile.
Furthermore, the multi-stack configuration enables one stack to be dedicated to force sensing, ensuring excellent collocation with the actuator stacks, a critical feature for implementing robust decentralized control strategies.
Moreover, using APA for active damping has been successfully demonstrated in similar applications [[cite:&hanieh03_activ_stewar]].
Several specific APA models were evaluated against the established specifications (Table ref:tab:detail_fem_piezo_act_models).
The APA300ML emerged as the optimal choice.
This selection was further reinforced by previous experience with APAs from the same manufacturer[fn:2], and particularly by the successful validation of the modeling methodology with a similar actuator (Section ref:ssec:detail_fem_super_element_example).
The demonstrated accuracy of the modeling approach for the APA95ML provides confidence in the reliable prediction of the APA300ML's dynamic characteristics, thereby supporting both the selection decision and subsequent dynamical analyses.
#+name: tab:detail_fem_piezo_act_models
#+caption: List of some amplified piezoelectric actuators that could be used for the nano-hexapod
@ -1934,10 +1915,11 @@ One reason is that we already had experience with APA from Cedrat technologies,
** APA300ML - Reduced Order Flexible Body
<<ssec:detail_fem_actuator_apa300ml>>
To validate the choice of the APA300ML (Shown in Figure ref:fig:detail_fem_apa300ml_picture):
- the APA300ML is modeled using a Finite Element Software
- a /super element/ is exported and imported in Simscape where its dynamic is studied
- similarly to what was done with the APA95ML, frames defined for the /super element/ are shown in figure ref:fig:detail_fem_apa300ml_frames
The validation of the APA300ML started by incorporating a "reduced order flexible body" into the multi-body model as explained in Section ref:sec:detail_fem_super_element.
The FEA model was developed with particular attention to the placement of reference frames, as illustrated in Figure ref:fig:detail_fem_apa300ml_frames.
Seven distinct frames were defined, with blue frames designating the force sensor stack interfaces for strain measurement, red frames denoting the actuator stack interfaces for force application and green frames for connecting to other elements.
120 additional modes were added during the modal reduction for a total order of 162.
While this high order provides excellent accuracy for validation purposes, it proves computationally intensive for simulations.
#+name: fig:detail_fem_apa300ml
#+caption: Amplified Piezoelectric Actuator APA300ML. Picture shown in (\subref{fig:detail_fem_apa300ml_picture}). Frames (or "remote points") used for the modal reduction are shown in (\subref{fig:detail_fem_apa300ml_frames}).
@ -1957,65 +1939,42 @@ To validate the choice of the APA300ML (Shown in Figure ref:fig:detail_fem_apa30
#+end_subfigure
#+end_figure
- For this reduced order model, 7 frames are defined and 120 additional modes are modelled for a total matrix size of 162.
- This is very large and will not be practical for simulations, but the best model accuracy was wanted for validation
- The blue frames are used to model the force sensor stack: the relative motion between the two frame is measured
- The red frames are used to model the two actuator stacks: /internal force/ are added
- One mass is fixed at one end of the piezo-electric stack actuator (remove point F), the other end is fixed to the world frame (remote point G).
- The link between mechanical properties and electrical properties was discussed in Section ref:ssec:detail_fem_super_element_validation.
As the stacks are the same between the APA300ML and the APA95ML, the values estimated for $g_a$ and $g_s$ are used for the APA300ML.
The sensor and actuator "constants" ($g_s$ and $g_a$) derived in Section ref:ssec:detail_fem_super_element_example for the APA95ML were used for the APA300ML model, as both actuators employ identical piezoelectric stacks.
** Simpler 2DoF Model of the APA300ML
<<sec:apa_model>>
**** Introduction :ignore:
<<ssec:detail_fem_actuator_apa300ml_2dof>>
- /super-element/ order is quite large, and therefore not practical for simulations
- the goal here is to develop a low order model, that still represents wanted characteristics of the APA300ML:
- axial stiffness
- actuator and force sensor characteristics
- what is not modelled:
- higher order modes
- the flexibility of the APA in the other directions
- Therefore this model can be useful for simulations as it contains a very limited number of states, but when more complex dynamics of the APA is to be modelled, a flexible model will be used.
To facilitate efficient time-domain simulations while maintaining essential dynamic characteristics, a simplified two-degree-of-freedom model was developed, adapted from [[cite:&souleille18_concep_activ_mount_space_applic]].
**** 2DoF Model
The model is adapted from cite:souleille18_concep_activ_mount_space_applic.
It can be decomposed into three components:
- the shell whose axial properties are represented by $k_1$ and $c_1$
- the actuator stacks whose contribution to the axial stiffness is represented by $k_a$ and $c_a$.
The force source $f$ represents the axial force induced by the force sensor stacks.
The sensitivity $g_a$ (in $N/m$) is used to convert the applied voltage $V_a$ to the axial force $f$
- the sensor stack whose contribution to the axial stiffness is represented by $k_e$ and $c_e$.
A sensor measures the stack strain $d_e$ which is then converted to a voltage $V_s$ using a sensitivity $g_s$ (in $V/m$)
Such a simple model has some limitations:
- it only represents the axial characteristics of the APA as it is modeled as infinitely rigid in the other directions
- some physical insights are lost, such as the amplification factor and the real stress and strain in the piezoelectric stacks
- the creep and hysteresis of the piezoelectric stacks are not modeled as the model is linear
The main advantage is that this model is very simple, only adds 4 states
This model, illustrated in Figure ref:fig:detail_fem_apa_2dof_model, comprises three components.
The mechanical shell is characterized by its axial stiffness $k_1$ and damping $c_1$.
The actuator is modelled with stiffness $k_a$ and damping $c_a$, incorporating a force source $f$.
This force is related to the applied voltage $V_a$ through the actuator constant $g_a$.
The sensor stack is modeled with stiffness $k_e$ and damping $c_e$, with its deformation $d_L$ being converted to the output voltage $V_s$ through the sensor sensitivity $g_s$.
#+name: fig:detail_fem_apa_2dof_model
#+caption: Schematic of the 2DoF model of the Amplified Piezoelectric Actuator
[[file:figs/detail_fem_apa_2dof_model.png]]
**** Parameter Tuning
9 parameters ($m$, $k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$ and $g_a$) have to be tuned such that the dynamics of the model (Figure ref:fig:detail_fem_apa_2dof_model) well represents the identified dynamics using the FEM.
While providing computational efficiency, this simplified model has inherent limitations.
It considers only axial behavior, treating the actuator as infinitely rigid in other directions.
Several physical characteristics are not explicitly represented, including the mechanical amplification factor and the actual stress the piezoelectric stacks.
Nevertheless, the model's primary advantage lies in its simplicity, adding only four states to the system model.
- Mass is 5kg (similar to the test bench)
- Tune the parameters:
- From the first zero of the transfer function from Va to Vs, k1 and c1 are tuned
- From the first pole of the transfer function from Va to y, ka, ca, ke, ce are tuned
- because the actuator and sensor stacks are physically the same, we suppose
Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics.
Therefore, we have $k_e = 2 k_a$ and $c_e = 2 c_a$ as the actuator stack is composed of two stacks in series.
- In the last step, $g_s$ and $g_a$ for the 2DoF motion can be tuned to match the gain of the transfer functions extracted from the FEM
- Found parameters are summarized in Table ref:tab:detail_fem_apa300ml_2dof_parameters
- Comparison of the transfer functions extracted from the high order flexible model with the 4th order (2DoF) model is done in Figure ref:fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof.
Good match is obtained.
Of course, higher order modes are not represented by the 2DoF model, nor the limited stiffness in the other directions.
The model requires tuning of 8 parameters ($k_1$, $c_1$, $k_e$, $c_e$, $k_a$, $c_a$, $g_s$, and $g_a$) to match the dynamics extracted from the finite element analysis.
The shell parameters $k_1$ and $c_1$ were determined first through analysis of the zero in the $V_a$ to $V_s$ transfer function.
The physical interpretation of this zero can be understood through Root Locus analysis: as controller gain increases, the poles of a closed-loop system converge to the open-loop zeros.
In this context, the zero corresponds to the poles of the system with a theoretical infinite-gain controller that ensures zero force in the sensor stack.
This condition effectively represents the dynamics of an APA without the force sensor stack.
This physical interpretation enables straightforward parameter tuning: $k_1$ determines the frequency of the zero, while $c_1$ defines its damping characteristic.
The stack parameters ($k_a$, $c_a$, $k_e$, $c_e$) were then derived from the first pole of the $V_a$ to $y$ response.
Given that identical piezoelectric stacks are used for both sensing and actuation, the relationships $k_e = 2k_a$ and $c_e = 2c_a$ were enforced, reflecting the series configuration of the dual actuator stacks.
Finally, the sensitivities $g_s$ and $g_a$ were adjusted to match the DC gains of the respective transfer functions.
The resulting parameters, documented in Table ref:tab:detail_fem_apa300ml_2dof_parameters, yield dynamic behavior that closely matches the high-order finite element model, as demonstrated in Figure ref:fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof.
While higher-order modes and non-axial flexibility are not captured, the model accurately represents the fundamental dynamics within the operational frequency range.
#+name: tab:detail_fem_apa300ml_2dof_parameters
#+caption: Summary of the obtained parameters for the 2 DoF APA300ML model
@ -2189,16 +2148,14 @@ exportFig('figs/detail_fem_apa300ml_comp_fem_2dof_force_sensor.pdf', 'width', 'h
#+end_figure
** Electrical characteristics of the APA
<<ssec:detail_fem_actuator_apa300ml_electrical>>
- Mechanical equations and electrical equations are coupled
- This means for instance, that the stiffness of the piezoelectric stack (i.e. the APA) depends on the electrical boundaries of the stacks:
- Short circuited stacks are less stiff than open-circuited ones
- This effect is quite small: example with the APA95ML (Figure ref:fig:detail_fem_apa95ml_effect_electrical_boundaries)
transfer function from Va to di are estimated with the force sensor stack being short circuited or open-circuited.
- In the model used, the electrical phenomena are not modelled.
But as this effect is small, it should be fine
- The electrical characteristics of the APA are very important both from the voltage amplifier side and the ADC measuring the force sensor voltage.
This will be discussed in chapter "instrumentation"
The behavior of piezoelectric actuators is characterized by coupled constitutive equations that establish relationships between electrical properties (charges, voltages) and mechanical properties (stress, strain) [[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapter 5.5]].
To evaluate the impact of electrical boundary conditions on the system dynamics, experimental measurements were conducted using the APA95ML, comparing the transfer function from $V_a$ to $y$ under two distinct configurations.
With the force sensor stack in open-circuit condition (analogous to voltage measurement with high input impedance) and in short-circuit condition (similar to charge measurement with low output impedance).
As demonstrated in Figure ref:fig:detail_fem_apa95ml_effect_electrical_boundaries, short-circuiting the force sensor stack results in a minor decrease in resonance frequency.
This relatively modest effect validates the simplifying assumption made in the model of the APA.
#+begin_src matlab
%% Effect of electrical boundaries on the
@ -2230,7 +2187,7 @@ hold on;
plot(f, abs(G_oc), '-', 'DisplayName', sprintf('Open-Circuit - $f_0 = %.1f Hz$', f(i_oc)))
plot(f, abs(G_sc), '-', 'DisplayName', sprintf('Short-Circuit - $f_0 = %.1f Hz$', f(i_sc)))
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-6, 1e-4]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
@ -2260,18 +2217,25 @@ exportFig('figs/detail_fem_apa95ml_effect_electrical_boundaries.pdf', 'width', '
#+RESULTS:
[[file:figs/detail_fem_apa95ml_effect_electrical_boundaries.png]]
** Validation with the Nano-Hexapod
NASS model + FEM model (or just 2DoF) of APA300ML => validation (based on what?)
However, the electrical characteristics of the APA remain crucial for instrumentation design.
Proper consideration must be given to voltage amplifier specifications and force sensor signal conditioning requirements.
These aspects, being fundamental to system implementation, will be addressed in the instrumentation chapter.
- Compare 2DoF model and FEM (Figure ref:fig:detail_fem_actuator_fem_vs_perfect_plants)
- HAC plant
- IFF Plant
- Very similar => can use 2nd order actuator models
- Talk about model order
- 2DoF actuators: 24 states
- FEM actuators:
here matrices have a size of 36
36*6+12 => ~300
** Validation with the Nano-Hexapod
<<ssec:detail_fem_actuator_apa300ml_validation>>
The integration of the APA300ML model within the nano-hexapod simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with APA modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full FEM implementation.
The dynamic characteristics predicted using the flexible body model align well with the design requirements established during the conceptual phase.
The dynamics from $\bm{u}$ to $\bm{V}_s$ exhibits the desired alternating pole-zero pattern (Figure ref:fig:detail_fem_actuator_fem_vs_perfect_hac_plant), a critical characteristic for implementing robust decentralized Integral Force Feedback.
Additionally, the model predicts no problematic high-frequency modes in the dynamics from $\bm{u}$ to $\bm{\epsilon}_{\mathcal{L}}$ (Figure ref:fig:detail_fem_actuator_fem_vs_perfect_iff_plant), maintaining consistency with earlier conceptual simulations.
These findings suggest that the control performance targets established during the conceptual phase remain achievable with the selected actuator.
Comparative analysis between the high-order FEM implementation and the simplified 2DoF model (Figure ref:fig:detail_fem_actuator_fem_vs_perfect_plants) demonstrates remarkable agreement in the frequency range of interest.
This validates the use of the simplified model for time-domain simulations, where computational efficiency is paramount.
The reduction in model order is substantial: while the FEM implementation results in approximately 300 states (36 states per actuator plus 12 additional states), the 2DoF model requires only 24 states for the complete nano-hexapod.
These results validate both the selection of the APA300ML and the effectiveness of the simplified modeling approach for the nano-hexapod.
#+begin_src matlab
%% Compare Dynamics between "Reduced Order" flexible joints and "2-dof and 3-dof" joints
@ -2421,9 +2385,7 @@ exportFig('figs/detail_fem_actuator_fem_vs_perfect_iff_plant.pdf', 'width', 'hal
#+end_subfigure
#+end_figure
* Flexible Joint
* Flexible Joint Design
<<sec:detail_fem_joint>>
** Notes :noexport:
@ -3905,5 +3867,5 @@ end
#+end_src
* Footnotes
[fn:2]Cedrat technologies
[fn:1]The manufacturer of the APA95ML was not willing to share the piezoelectric material properties of the stack.

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@ -1,4 +1,4 @@
% Created 2025-02-26 Wed 15:42
% Created 2025-02-26 Wed 23:11
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -43,7 +43,7 @@ To do so, Reduced Order Flexible Bodies are used (Section \ref{sec:detail_fem_su
\end{itemize}
\chapter{Reduced order flexible bodies}
\label{sec:org5704c94}
\label{sec:orgeb21f99}
\label{sec:detail_fem_super_element}
Components exhibiting complex dynamical behavior are frequently found to be unsuitable for direct implementation within multi-body models.
These components are traditionally analyzed using Finite Element Analysis (FEA) software.
@ -55,7 +55,7 @@ First, the fundamental principles and methodological approaches of this modeling
It is then illustrated through its practical application to the modelling of an Amplified Piezoelectric Actuator (APA) (Section \ref{ssec:detail_fem_super_element_example}).
Finally, the validity of this modeling approach is demonstrated through experimental validation, wherein the obtained dynamics from the hybrid modelling approach is compared with measurements (Section \ref{ssec:detail_fem_super_element_validation}).
\section{Procedure}
\label{sec:orga74dca6}
\label{sec:org494af25}
\label{ssec:detail_fem_super_element_theory}
In this modeling approach, some components within the multi-body framework are represented as \emph{reduced-order flexible bodies}, wherein their modal behavior is characterized through reduced mass and stiffness matrices derived from finite element analysis (FEA) models.
@ -79,7 +79,7 @@ m = 6 \times n + p
\end{equation}
\section{Example with an Amplified Piezoelectric Actuator}
\label{sec:org3e7c2ec}
\label{sec:orgad4f3ec}
\label{ssec:detail_fem_super_element_example}
The presented modeling framework was first applied to an Amplified Piezoelectric Actuator (APA) for several reasons.
Primarily, this actuator represents an excellent candidate for implementation within the nano-hexapod, as will be elaborated in Section \ref{sec:detail_fem_actuator}.
@ -110,7 +110,7 @@ Stiffness & \(21\,N/\mu m\)\\
\captionof{table}{\label{tab:detail_fem_apa95ml_specs}APA95ML specifications}
\end{minipage}
\paragraph{Finite Element Model}
\label{sec:org491eeae}
\label{sec:orgb4da286}
The development of the finite element model for the APA95ML necessitated the specification of appropriate material properties, as summarized in Table \ref{tab:detail_fem_material_properties}.
The finite element mesh, shown in Figure \ref{fig:detail_fem_apa95ml_mesh}, was then generated.
@ -151,7 +151,7 @@ The modal reduction procedure was then executed, yielding the reduced mass and s
\end{figure}
\paragraph{Super Element in the Multi-Body Model}
\label{sec:org29dd028}
\label{sec:org2f8920b}
Previously computed reduced order mass and stiffness matrices were imported in a multi-body model block called ``Reduced Order Flexible Solid''.
This block has several interface frames corresponding to the ones defined in the FEA software.
@ -163,7 +163,7 @@ This is illustrated in Figure \ref{fig:detail_fem_apa_model_schematic}.
However, to have access to the physical voltage input of the actuators stacks \(V_a\) and to the generated voltage by the force sensor \(V_s\), conversion between the electrical and mechanical domains need to be determined.
\paragraph{Sensor and Actuator ``constants''}
\label{sec:org1329f1a}
\label{sec:org13dbb26}
To link the electrical domain to the mechanical domain, an ``actuator constant'' \(g_a\) and a ``sensor constant'' \(g_s\) were introduced as shown in Figure \ref{fig:detail_fem_apa_model_schematic}.
@ -222,7 +222,7 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine
\end{table}
\paragraph{Identification of the APA Characteristics}
\label{sec:org5512e6c}
\label{sec:org83f2d6f}
Initial validation of the finite element model and its integration as a reduced-order flexible model within the multi-body model was accomplished through comparative analysis of key actuator characteristics against manufacturer specifications.
@ -248,7 +248,7 @@ Through the established amplification factor of 1.5, this translates to a predic
The high degree of concordance observed across multiple performance metrics provides a first validation of the ability to include FEM into multi-body model.
\section{Experimental Validation}
\label{sec:org8627abc}
\label{sec:org92e845e}
\label{ssec:detail_fem_super_element_validation}
Further validation of the reduced-order flexible body methodology was undertaken through experimental investigation.
The goal is to measure the dynamics of the APA95ML and compared it with predictions derived from the multi-body model incorporating the actuator as a flexible element.
@ -274,7 +274,7 @@ Measurement of the sensor stack voltage \(V_s\) was performed using an analog-to
\caption{\label{fig:detail_fem_apa95ml_bench}Test bench used to validate ``reduced order solid bodies'' using an APA95ML. Picture of the bench is shown in (\subref{fig:detail_fem_apa95ml_bench_picture}). Schematic is shown in (\subref{fig:detail_fem_apa95ml_bench_schematic}).}
\end{figure}
\paragraph{Comparison of the dynamics}
\label{sec:orgb3fa207}
\label{sec:orgfabbfd3}
Frequency domain system identification techniques were used to characterize the dynamic behavior of the APA95ML.
The identification procedure necessitated careful choice of the excitation signal \cite[, chap. 5]{pintelon12_system_ident}.
@ -309,7 +309,7 @@ Regarding the amplitude characteristics, the constants \(g_a\) and \(g_s\) could
\end{figure}
\paragraph{Integral Force Feedback with APA}
\label{sec:org182828d}
\label{sec:orgd052e81}
To further validate this modeling methodology, its ability to predict closed-loop behavior was verified experimentally.
Integral Force Feedback (IFF) was implemented using the force sensor stack, and the measured dynamics of the damped system were compared with model predictions across multiple feedback gains.
@ -343,7 +343,7 @@ The close agreement between experimental measurements and theoretical prediction
\end{figure}
\section*{Conclusion}
\label{sec:org7af3b1c}
\label{sec:org0aca0da}
The modeling procedure presented in this section will demonstrate significant utility for the optimization of complex mechanical components within multi-body systems, particularly in the design of actuators (Section \ref{sec:detail_fem_actuator}) and flexible joints (Section \ref{sec:detail_fem_joint}).
Through experimental validation using an Amplified Piezoelectric Actuator, the methodology has been shown to accurately predict both open-loop and closed-loop dynamic behavior, thereby establishing its reliability for component design and system analysis.
@ -352,76 +352,31 @@ While this modeling approach provides accurate predictions of component behavior
This is exemplified by the nano-hexapod configuration, where the implementation of six Amplified Piezoelectric Actuators, each modeled with 48 degrees of freedom, yields 288 degrees of freedom only for the actuators.
However, the methodology remains valuable for system analysis, as the extraction of frequency domain characteristics can be efficiently performed even with such high-order models.
\chapter{Actuator}
\label{sec:orgcb23435}
\chapter{Actuator Selection}
\label{sec:orgb6d5574}
\label{sec:detail_fem_actuator}
Goals:
\begin{itemize}
\item Based on dynamical models and previous studies, extract specifications for the actuators to be included in the nano-hexapod.
Then choose the most appropriate actuator based on specifications (Section \ref{ssec:detail_fem_actuator_specifications})
\item Model this actuator accurately using a ``reduced order flexible body'' to check the dynamics and validate the choice of actuator
and validate this choice with simulations
\item Development of a 2DoF model for lower order models (i.e. for simulations)
\end{itemize}
The selection and modeling of actuators constitutes a critical step in the development of the nano-hexapod.
This chapter presents the approach to actuator selection and modeling.
First, specifications for the nano-hexapod actuators are derived from previous analyses, leading to the selection of the actuator type and ultimately to a specific model (Section \ref{ssec:detail_fem_actuator_specifications}).
Then, the chosen actuator is modeled using the reduced-order flexible body approach developed in the previous section, enabling validation of this selection through detailed dynamical analysis (Section \ref{ssec:detail_fem_actuator_apa300ml}).
Finally, a simplified two-degree-of-freedom model is developed to facilitate time-domain simulations while maintaining accurate representation of the actuator's essential characteristics (Section \ref{ssec:detail_fem_actuator_apa300ml_2dof}).
\section{Choice of the Actuator based on Specifications}
\label{sec:org6a6861c}
\label{sec:org021ab2f}
\label{ssec:detail_fem_actuator_specifications}
From previous analysis:
\begin{itemize}
\item Actuator stiffness has major impact on the system dynamics and performances due to several factors:
\begin{itemize}
\item Spindle rotation: modification of plant dynamics and coupling increase due to Gyroscopic effects
This require to have stiffness above \textasciitilde{}
\item Limited micro-station compliance / complex dynamics:
The actuator stiffness should be small enough such that the suspension modes of the nano-hexapod are below the problematic modes of the micro-stations.
\item There is therefore an intermediate stiffness that is foreseen to give the best compromise, and it is around \(1\,N/\mu m\)
\end{itemize}
\item HAC-LAC strategy:
Actuator must include a force sensor
Because of the rotation, some stiffness should be present in parallel to the force sensor
\item Limited space:
As the maximum height of the nano-hexapod is 95mm, and each strut has a flexible joint at each end, it is estimated that the maximum height of the actuator should be less than 50mm
\item Stroke:
The stroke of the each actuator should be large enough such that the nano-hexapod mobility exceed the micro-station positioning errors.
Some margins should be included for mounting errors, and further flexibility of the system (for instance to perform scans with the nano-hexapod, or to align the point of interest with the rotation axis)
\end{itemize}
The actuator selection process was driven by several critical requirements derived from previous dynamic analyses.
A primary consideration is the actuator stiffness, which significantly impacts system dynamics through multiple mechanisms.
The spindle rotation induces gyroscopic effects that modify plant dynamics and increase coupling, necessitating sufficient stiffness.
Conversely, the actuator stiffness must be carefully limited to ensure the nano-hexapod's suspension modes remain below the problematic modes of the micro-stations to limit the coupling between the two structures.
These competing requirements suggest an optimal stiffness of approximately \(1\,N/\mu m\).
Actuator specifications:
\begin{itemize}
\item Height (<50mm)
\item Stroke (\textasciitilde{}100um)
\item Stiffness (0.1-1 N/um)
\item Blocked force?
\item Force sensor
\end{itemize}
Options:
\begin{itemize}
\item Two main options: piezoelectric actuators and Lorentz actuator (also known as Voice coil actuators).
Variable reluctance actuators were not considered, even though they have better efficiency than voice coil actuators, they are non linear and induce additional control complexity.
\item Voice coil + relatively soft flexible guiding (1N/um):
\begin{itemize}
\item required force \textasciitilde{}100N for 100um correction
This constant force/current would induce large thermal loads, that may negatively impact system's stability
Advantages of voice coil (longer strokes than piezo + allow for very low stiffness in the direction of actuation, extremely linear for high performance feedforward) are not used here.
\end{itemize}
\item Piezoelectric stack actuators:
\begin{itemize}
\item PZT: stroke \textasciitilde{}0.1\% of its length.
\item 50mm length => 50um stroke which is barely enough
\item Extremely stiff, in the order of \(100\,N/\mu m\), which is not wanted here.
\end{itemize}
\item Amplified Piezoelectric Actuator:
\begin{itemize}
\item shell is used to pre-stress the piezoelectric stacks and amplify the motion (roughly by the ratio of the width over the height)
\item This also reduce the stiffness in the direction of motion
\item This make this design quick compact in the direction of motion (i.e. in height)
\item When several stacks are used, one of them can be used as a force sensor, which is therefore very well collocated with the actuators
\item Therefore, this actuator is well suited for decentralized IFF, already applied for a Stewart platform with APA \cite{hanieh03_activ_stewar}
\end{itemize}
\end{itemize}
Additional specifications arise from the control strategy and physical constraints.
The implementation of a HAC-LAC (High Authority Control-Low Authority Control) architecture necessitates integrated force sensing capability.
The system's geometric constraints limit the actuator height to 50mm, given the nano-hexapod's maximum height of 95mm 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, which is estimated at \(\approx 100\,\mu m\).
Three actuator technologies were evaluated (examples are shown in Figure \ref{fig:detail_fem_actuator_pictures}): voice coil actuators, piezoelectric stack actuators, and amplified piezoelectric actuators.
Variable reluctance actuators were not considered despite their superior efficiency compared to voice coil actuators, as their inherent nonlinearity would introduce unnecessary control complexity.
\begin{figure}[htbp]
\begin{subfigure}{0.25\textwidth}
@ -445,17 +400,23 @@ Advantages of voice coil (longer strokes than piezo + allow for very low stiffne
\caption{\label{fig:detail_fem_actuator_pictures}Example of actuators considered for the nano-hexapod. Voice coil from Sensata Technologies (\subref{fig:detail_fem_voice_coil_picture}). Piezoelectric stack actuator from Physik Instrumente (\subref{fig:detail_fem_piezo_picture}). Amplified Piezoelectric Actuator from DSM (\subref{fig:detail_fem_fpa_picture}).}
\end{figure}
Based on previous analysis, it was decided to use amplified piezoelectric actuators for the nano-hexapod.
Table \ref{tab:detail_fem_piezo_act_models}: compares few models that fulfill specifications.
It was decided to go for the APA300ML (shown in Figure \ref{fig:detail_fem_apa300ml_picture}).
One reason is that we already had experience with APA from Cedrat technologies, and the Finite Element Model was validated experimentally, so we are confident to model the APA300ML with FEA and include it in the NASS model for validation.
Voice coil actuators (shown in Figure \ref{fig:detail_fem_voice_coil_picture}), when combined with flexure guides of wanted stiffness \(\approx 1\,N/\mu m\), would require forces above \(100\,N\) to achieve the specified \(100\,\mu m\) displacement.
While these actuators offer excellent linearity and long strokes, the constant force requirement would result in significant steady-state current, leading to thermal loads that could compromise system stability.
Their advantages were not considered adapted for this application, diminishing their benefits relative to piezoelectric solutions.
Conventional piezoelectric stack actuators (shown in Figure \ref{fig:detail_fem_piezo_picture}) present two significant limitations for the current application.
Their stroke is inherently limited to approximately \(0.1\,\%\) of their length, meaning that even with the maximum allowable height of \(50\,mm\), the achievable stroke would only be \(50\,\mu m\), insufficient for the application.
Additionally, their extremely high stiffness, typically around \(100\,N/\mu m\), exceeds the desired specifications by two orders of magnitude.
\begin{itemize}
\item Talk about piezoelectric actuator? bandwidth? noise?
\item Resolution: really depends on the electrical noise (induced by DAC and voltage amplifier).
They will be chosen appropriately
\end{itemize}
Amplified Piezoelectric Actuators (APAs) emerged as the optimal solution by addressing these limitations through an specific mechanical design.
The incorporation of a shell structure serves multiple purposes: it provides mechanical amplification of the piezoelectric displacement, reduces the effective axial stiffness to more suitable levels for the application, and creates a compact vertical profile.
Furthermore, the multi-stack configuration enables one stack to be dedicated to force sensing, ensuring excellent collocation with the actuator stacks, a critical feature for implementing robust decentralized control strategies.
Moreover, using APA for active damping has been successfully demonstrated in similar applications \cite{hanieh03_activ_stewar}.
Several specific APA models were evaluated against the established specifications (Table \ref{tab:detail_fem_piezo_act_models}).
The APA300ML emerged as the optimal choice.
This selection was further reinforced by previous experience with APAs from the same manufacturer\footnote{Cedrat technologies}, and particularly by the successful validation of the modeling methodology with a similar actuator (Section \ref{ssec:detail_fem_super_element_example}).
The demonstrated accuracy of the modeling approach for the APA95ML provides confidence in the reliable prediction of the APA300ML's dynamic characteristics, thereby supporting both the selection decision and subsequent dynamical analyses.
\begin{table}[htbp]
\caption{\label{tab:detail_fem_piezo_act_models}List of some amplified piezoelectric actuators that could be used for the nano-hexapod}
@ -475,15 +436,14 @@ Height \(< 50\, [mm]\) & 22 & 30 & 24 & 27 & 16\\
\end{table}
\section{APA300ML - Reduced Order Flexible Body}
\label{sec:org56a3ff1}
\label{sec:orgbc872e3}
\label{ssec:detail_fem_actuator_apa300ml}
To validate the choice of the APA300ML (Shown in Figure \ref{fig:detail_fem_apa300ml_picture}):
\begin{itemize}
\item the APA300ML is modeled using a Finite Element Software
\item a \emph{super element} is exported and imported in Simscape where its dynamic is studied
\item similarly to what was done with the APA95ML, frames defined for the \emph{super element} are shown in figure \ref{fig:detail_fem_apa300ml_frames}
\end{itemize}
The validation of the APA300ML started by incorporating a ``reduced order flexible body'' into the multi-body model as explained in Section \ref{sec:detail_fem_super_element}.
The FEA model was developed with particular attention to the placement of reference frames, as illustrated in Figure \ref{fig:detail_fem_apa300ml_frames}.
Seven distinct frames were defined, with blue frames designating the force sensor stack interfaces for strain measurement, red frames denoting the actuator stack interfaces for force application and green frames for connecting to other elements.
120 additional modes were added during the modal reduction for a total order of 162.
While this high order provides excellent accuracy for validation purposes, it proves computationally intensive for simulations.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
@ -501,56 +461,19 @@ To validate the choice of the APA300ML (Shown in Figure \ref{fig:detail_fem_apa3
\caption{\label{fig:detail_fem_apa300ml}Amplified Piezoelectric Actuator APA300ML. Picture shown in (\subref{fig:detail_fem_apa300ml_picture}). Frames (or ``remote points'') used for the modal reduction are shown in (\subref{fig:detail_fem_apa300ml_frames}).}
\end{figure}
\begin{itemize}
\item For this reduced order model, 7 frames are defined and 120 additional modes are modelled for a total matrix size of 162.
\item This is very large and will not be practical for simulations, but the best model accuracy was wanted for validation
\item The blue frames are used to model the force sensor stack: the relative motion between the two frame is measured
\item The red frames are used to model the two actuator stacks: \emph{internal force} are added
\item One mass is fixed at one end of the piezo-electric stack actuator (remove point F), the other end is fixed to the world frame (remote point G).
\item The link between mechanical properties and electrical properties was discussed in Section \ref{ssec:detail_fem_super_element_validation}.
As the stacks are the same between the APA300ML and the APA95ML, the values estimated for \(g_a\) and \(g_s\) are used for the APA300ML.
\end{itemize}
The sensor and actuator ``constants'' (\(g_s\) and \(g_a\)) derived in Section \ref{ssec:detail_fem_super_element_example} for the APA95ML were used for the APA300ML model, as both actuators employ identical piezoelectric stacks.
\section{Simpler 2DoF Model of the APA300ML}
\label{sec:orgfabc6b8}
\label{sec:apa_model}
\begin{itemize}
\item \emph{super-element} order is quite large, and therefore not practical for simulations
\item the goal here is to develop a low order model, that still represents wanted characteristics of the APA300ML:
\begin{itemize}
\item axial stiffness
\item actuator and force sensor characteristics
\end{itemize}
\item what is not modelled:
\begin{itemize}
\item higher order modes
\item the flexibility of the APA in the other directions
\end{itemize}
\item Therefore this model can be useful for simulations as it contains a very limited number of states, but when more complex dynamics of the APA is to be modelled, a flexible model will be used.
\end{itemize}
\paragraph{2DoF Model}
\label{sec:org5962dd3}
\label{sec:org798e5bf}
\label{ssec:detail_fem_actuator_apa300ml_2dof}
The model is adapted from \cite{souleille18_concep_activ_mount_space_applic}.
To facilitate efficient time-domain simulations while maintaining essential dynamic characteristics, a simplified two-degree-of-freedom model was developed, adapted from \cite{souleille18_concep_activ_mount_space_applic}.
It can be decomposed into three components:
\begin{itemize}
\item the shell whose axial properties are represented by \(k_1\) and \(c_1\)
\item the actuator stacks whose contribution to the axial stiffness is represented by \(k_a\) and \(c_a\).
The force source \(f\) represents the axial force induced by the force sensor stacks.
The sensitivity \(g_a\) (in \(N/m\)) is used to convert the applied voltage \(V_a\) to the axial force \(f\)
\item the sensor stack whose contribution to the axial stiffness is represented by \(k_e\) and \(c_e\).
A sensor measures the stack strain \(d_e\) which is then converted to a voltage \(V_s\) using a sensitivity \(g_s\) (in \(V/m\))
\end{itemize}
Such a simple model has some limitations:
\begin{itemize}
\item it only represents the axial characteristics of the APA as it is modeled as infinitely rigid in the other directions
\item some physical insights are lost, such as the amplification factor and the real stress and strain in the piezoelectric stacks
\item the creep and hysteresis of the piezoelectric stacks are not modeled as the model is linear
\end{itemize}
The main advantage is that this model is very simple, only adds 4 states
This model, illustrated in Figure \ref{fig:detail_fem_apa_2dof_model}, comprises three components.
The mechanical shell is characterized by its axial stiffness \(k_1\) and damping \(c_1\).
The actuator is modelled with stiffness \(k_a\) and damping \(c_a\), incorporating a force source \(f\).
This force is related to the applied voltage \(V_a\) through the actuator constant \(g_a\).
The sensor stack is modeled with stiffness \(k_e\) and damping \(c_e\), with its deformation \(d_L\) being converted to the output voltage \(V_s\) through the sensor sensitivity \(g_s\).
\begin{figure}[htbp]
\centering
@ -558,26 +481,25 @@ The main advantage is that this model is very simple, only adds 4 states
\caption{\label{fig:detail_fem_apa_2dof_model}Schematic of the 2DoF model of the Amplified Piezoelectric Actuator}
\end{figure}
\paragraph{Parameter Tuning}
\label{sec:org7bd1971}
9 parameters (\(m\), \(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\) and \(g_a\)) have to be tuned such that the dynamics of the model (Figure \ref{fig:detail_fem_apa_2dof_model}) well represents the identified dynamics using the FEM.
While providing computational efficiency, this simplified model has inherent limitations.
It considers only axial behavior, treating the actuator as infinitely rigid in other directions.
Several physical characteristics are not explicitly represented, including the mechanical amplification factor and the actual stress the piezoelectric stacks.
Nevertheless, the model's primary advantage lies in its simplicity, adding only four states to the system model.
\begin{itemize}
\item Mass is 5kg (similar to the test bench)
\item Tune the parameters:
\begin{itemize}
\item From the first zero of the transfer function from Va to Vs, k1 and c1 are tuned
\item From the first pole of the transfer function from Va to y, ka, ca, ke, ce are tuned
\item because the actuator and sensor stacks are physically the same, we suppose
Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics.
Therefore, we have \(k_e = 2 k_a\) and \(c_e = 2 c_a\) as the actuator stack is composed of two stacks in series.
\item In the last step, \(g_s\) and \(g_a\) for the 2DoF motion can be tuned to match the gain of the transfer functions extracted from the FEM
\item Found parameters are summarized in Table \ref{tab:detail_fem_apa300ml_2dof_parameters}
\end{itemize}
\item Comparison of the transfer functions extracted from the high order flexible model with the 4th order (2DoF) model is done in Figure \ref{fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof}.
Good match is obtained.
Of course, higher order modes are not represented by the 2DoF model, nor the limited stiffness in the other directions.
\end{itemize}
The model requires tuning of 8 parameters (\(k_1\), \(c_1\), \(k_e\), \(c_e\), \(k_a\), \(c_a\), \(g_s\), and \(g_a\)) to match the dynamics extracted from the finite element analysis.
The shell parameters \(k_1\) and \(c_1\) were determined first through analysis of the zero in the \(V_a\) to \(V_s\) transfer function.
The physical interpretation of this zero can be understood through Root Locus analysis: as controller gain increases, the poles of a closed-loop system converge to the open-loop zeros.
In this context, the zero corresponds to the poles of the system with a theoretical infinite-gain controller that ensures zero force in the sensor stack.
This condition effectively represents the dynamics of an APA without the force sensor stack.
This physical interpretation enables straightforward parameter tuning: \(k_1\) determines the frequency of the zero, while \(c_1\) defines its damping characteristic.
The stack parameters (\(k_a\), \(c_a\), \(k_e\), \(c_e\)) were then derived from the first pole of the \(V_a\) to \(y\) response.
Given that identical piezoelectric stacks are used for both sensing and actuation, the relationships \(k_e = 2k_a\) and \(c_e = 2c_a\) were enforced, reflecting the series configuration of the dual actuator stacks.
Finally, the sensitivities \(g_s\) and \(g_a\) were adjusted to match the DC gains of the respective transfer functions.
The resulting parameters, documented in Table \ref{tab:detail_fem_apa300ml_2dof_parameters}, yield dynamic behavior that closely matches the high-order finite element model, as demonstrated in Figure \ref{fig:detail_fem_apa300ml_comp_fem_2dof_fem_2dof}.
While higher-order modes and non-axial flexibility are not captured, the model accurately represents the fundamental dynamics within the operational frequency range.
\begin{table}[htbp]
\caption{\label{tab:detail_fem_apa300ml_2dof_parameters}Summary of the obtained parameters for the 2 DoF APA300ML model}
@ -615,21 +537,15 @@ Of course, higher order modes are not represented by the 2DoF model, nor the lim
\end{figure}
\section{Electrical characteristics of the APA}
\label{sec:org5cdd335}
\label{sec:org67d72f6}
\label{ssec:detail_fem_actuator_apa300ml_electrical}
\begin{itemize}
\item Mechanical equations and electrical equations are coupled
\item This means for instance, that the stiffness of the piezoelectric stack (i.e. the APA) depends on the electrical boundaries of the stacks:
\begin{itemize}
\item Short circuited stacks are less stiff than open-circuited ones
\item This effect is quite small: example with the APA95ML (Figure \ref{fig:detail_fem_apa95ml_effect_electrical_boundaries})
transfer function from Va to di are estimated with the force sensor stack being short circuited or open-circuited.
\end{itemize}
\item In the model used, the electrical phenomena are not modelled.
But as this effect is small, it should be fine
\item The electrical characteristics of the APA are very important both from the voltage amplifier side and the ADC measuring the force sensor voltage.
This will be discussed in chapter ``instrumentation''
\end{itemize}
The behavior of piezoelectric actuators is characterized by coupled constitutive equations that establish relationships between electrical properties (charges, voltages) and mechanical properties (stress, strain) \cite[, chapter 5.5]{schmidt20_desig_high_perfor_mechat_third_revis_edition}.
To evaluate the impact of electrical boundary conditions on the system dynamics, experimental measurements were conducted using the APA95ML, comparing the transfer function from \(V_a\) to \(y\) under two distinct configurations.
With the force sensor stack in open-circuit condition (analogous to voltage measurement with high input impedance) and in short-circuit condition (similar to charge measurement with low output impedance).
As demonstrated in Figure \ref{fig:detail_fem_apa95ml_effect_electrical_boundaries}, short-circuiting the force sensor stack results in a minor decrease in resonance frequency.
This relatively modest effect validates the simplifying assumption made in the model of the APA.
\begin{figure}[htbp]
\centering
@ -637,25 +553,26 @@ This will be discussed in chapter ``instrumentation''
\caption{\label{fig:detail_fem_apa95ml_effect_electrical_boundaries}Effect of the electrical bondaries of the force sensor stack on the APA95ML resonance frequency}
\end{figure}
\section{Validation with the Nano-Hexapod}
\label{sec:orgf89bb46}
NASS model + FEM model (or just 2DoF) of APA300ML => validation (based on what?)
However, the electrical characteristics of the APA remain crucial for instrumentation design.
Proper consideration must be given to voltage amplifier specifications and force sensor signal conditioning requirements.
These aspects, being fundamental to system implementation, will be addressed in the instrumentation chapter.
\begin{itemize}
\item Compare 2DoF model and FEM (Figure \ref{fig:detail_fem_actuator_fem_vs_perfect_plants})
\begin{itemize}
\item HAC plant
\item IFF Plant
\item Very similar => can use 2nd order actuator models
\end{itemize}
\item Talk about model order
\begin{itemize}
\item 2DoF actuators: 24 states
\item FEM actuators:
here matrices have a size of 36
36*6+12 => \textasciitilde{}300
\end{itemize}
\end{itemize}
\section{Validation with the Nano-Hexapod}
\label{sec:orgd7e1728}
\label{ssec:detail_fem_actuator_apa300ml_validation}
The integration of the APA300ML model within the nano-hexapod simulation framework served two validation objectives: to validate the APA300ML choice through analysis of system dynamics with APA modelled as flexible bodies, and to validate the simplified 2DoF model through comparative analysis with the full FEM implementation.
The dynamic characteristics predicted using the flexible body model align well with the design requirements established during the conceptual phase.
The dynamics from \(\bm{u}\) to \(\bm{V}_s\) exhibits the desired alternating pole-zero pattern (Figure \ref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}), a critical characteristic for implementing robust decentralized Integral Force Feedback.
Additionally, the model predicts no problematic high-frequency modes in the dynamics from \(\bm{u}\) to \(\bm{\epsilon}_{\mathcal{L}}\) (Figure \ref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}), maintaining consistency with earlier conceptual simulations.
These findings suggest that the control performance targets established during the conceptual phase remain achievable with the selected actuator.
Comparative analysis between the high-order FEM implementation and the simplified 2DoF model (Figure \ref{fig:detail_fem_actuator_fem_vs_perfect_plants}) demonstrates remarkable agreement in the frequency range of interest.
This validates the use of the simplified model for time-domain simulations, where computational efficiency is paramount.
The reduction in model order is substantial: while the FEM implementation results in approximately 300 states (36 states per actuator plus 12 additional states), the 2DoF model requires only 24 states for the complete nano-hexapod.
These results validate both the selection of the APA300ML and the effectiveness of the simplified modeling approach for the nano-hexapod.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
@ -673,10 +590,8 @@ here matrices have a size of 36
\caption{\label{fig:detail_fem_actuator_fem_vs_perfect_plants}Comparison of the dynamics obtained between a nano-hexpod having the actuators modeled with FEM and a nano-hexapod having actuators modelled a 2DoF system. Both from actuator force \(\bm{f}\) to strut motion measured by external metrology \(\bm{\epsilon}_{\mathcal{L}}\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_iff_plant}) and to the force sensors \(\bm{f}_m\) (\subref{fig:detail_fem_actuator_fem_vs_perfect_hac_plant}).}
\end{figure}
\chapter{Flexible Joint}
\label{sec:org8601117}
\chapter{Flexible Joint Design}
\label{sec:orgce87a36}
\label{sec:detail_fem_joint}
The flexible joints have few advantages compared to conventional joints such as the \textbf{absence of wear, friction and backlash} which allows extremely high-precision (predictable) motion.
The parasitic bending and torsional stiffness of these joints usually induce some \textbf{limitation on the control performance}. \cite{mcinroy02_model_desig_flexur_joint_stewar}
@ -705,7 +620,7 @@ Say that for simplicity (reduced number of parts, etc.), we consider the same jo
\item Implementation of flexible elements in the Simscape model: close to simplified model
\end{itemize}
\section{Flexible joints for Stewart platforms}
\label{sec:orgd5923ee}
\label{sec:orgebc6043}
Review of different types of flexible joints for Stewart plaftorms (see Figure \ref{fig:detail_fem_joints_examples}).
@ -748,7 +663,7 @@ Typical values?
\end{figure}
\section{Bending and Torsional Stiffness}
\label{sec:orgf11a334}
\label{sec:orgcea815a}
\label{sec:joints_rot_stiffness}
Because of bending stiffness of the flexible joints, the forces applied by the struts are no longer aligned with the struts (additional forces applied by the ``spring force'' of the flexible joints).
@ -829,7 +744,7 @@ Conclusion:
\end{itemize}
\section{Axial Stiffness}
\label{sec:orgd08fa7c}
\label{sec:org0768c98}
\label{sec:joints_trans_stiffness}
\begin{itemize}
@ -897,7 +812,7 @@ Conclusion:
\end{itemize}
\section{Obtained design / Specifications}
\label{sec:org93383a9}
\label{sec:org17da38c}
\begin{itemize}
\item Summary of specifications (Table \ref{tab:detail_fem_joints_specs})
@ -957,7 +872,7 @@ Bending Stroke & \(> 1\,\text{mrad}\) & 24.5\\
\end{figure}
\section{Validation with the Nano-Hexapod}
\label{sec:org9751c8e}
\label{sec:orgd711eb0}
To validate the designed flexible joint:
\begin{itemize}
@ -1031,7 +946,7 @@ Talk about model order:
\end{figure}
\chapter*{Conclusion}
\label{sec:org57f9ca5}
\label{sec:orgd540d5a}
\label{sec:detail_fem_conclusion}
\printbibliography[heading=bibintoc,title={Bibliography}]