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Thomas Dehaeze 2024-04-30 17:25:33 +02:00
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@ -145,25 +145,25 @@ CLOSED: [2024-04-04 Thu 10:42]
| drga | DRGA | Dynamical Relative Gain Array | | drga | DRGA | Dynamical Relative Gain Array |
| hpf | HPF | high-pass filter | | hpf | HPF | high-pass filter |
| lpf | LPF | low-pass filter | | lpf | LPF | low-pass filter |
| dof | DoF | Degrees of Freedom | | dof | DoF | degrees-of-freedom |
* Introduction :ignore: * Introduction :ignore:
In this chapter, the goal is to make sure that the received APA300ML (shown in Figure ref:fig:test_apa_received) are complying with the requirements and that dynamical models of the actuator are well representing its dynamics. In this chapter, the goal is to ensure that the received APA300ML (shown in Figure ref:fig:test_apa_received) are complying with the requirements and that the dynamical models of the actuator accurately represent its dynamics.
In section ref:sec:test_apa_basic_meas, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks, the achievable stroke. In section ref:sec:test_apa_basic_meas, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks and the achievable stroke.
Flexible modes of the APA300ML which were estimated using a finite element model are compared with measurements. The flexible modes of the APA300ML, which were estimated using a finite element model, are compared with measurements.
Using a dedicated test bench, dynamical measurements are performed (Section ref:sec:test_apa_dynamics). Using a dedicated test bench, dynamical measurements are performed (Section ref:sec:test_apa_dynamics).
The dynamics from the generated DAC voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated. The dynamics from the generated DAC voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated.
Integral Force Feedback is experimentally applied and the damped plants are estimated for several feedback gains. Integral Force Feedback is experimentally applied, and the damped plants are estimated for several feedback gains.
Two different models of the APA300ML are then presented. Two different models of the APA300ML are presented.
First, in Section ref:sec:test_apa_model_2dof, a two degrees of freedom model is presented, tuned and compared with the measured dynamics. First, in Section ref:sec:test_apa_model_2dof, a two degrees-of-freedom model is presented, tuned, and compared with the measured dynamics.
This model is proven to accurately represents the APA300ML's axial dynamics while having low complexity. This model is proven to accurately represent the APA300ML's axial dynamics while having low complexity.
Then, in Section ref:sec:test_apa_model_flexible, a /super element/ of the APA300ML is extracted using a finite element model and imported in Simscape. Then, in Section ref:sec:test_apa_model_flexible, a /super element/ of the APA300ML is extracted using a finite element model and imported into Simscape.
This more complex model is also shown to well capture the axial dynamics of the APA300ML. This more complex model also captures well capture the axial dynamics of the APA300ML.
#+name: fig:test_apa_received #+name: fig:test_apa_received
#+attr_latex: :width 0.7\linewidth #+attr_latex: :width 0.7\linewidth
@ -189,7 +189,7 @@ This more complex model is also shown to well capture the axial dynamics of the
** Introduction :ignore: ** Introduction :ignore:
Before measuring the dynamical characteristics of the APA300ML, first simple measurements are performed. Before measuring the dynamical characteristics of the APA300ML, simple measurements are performed.
First, the tolerances (especially flatness) of the mechanical interfaces are checked in Section ref:ssec:test_apa_geometrical_measurements. First, the tolerances (especially flatness) of the mechanical interfaces are checked in Section ref:ssec:test_apa_geometrical_measurements.
Then, the capacitance of the piezoelectric stacks is measured in Section ref:ssec:test_apa_electrical_measurements. Then, the capacitance of the piezoelectric stacks is measured in Section ref:ssec:test_apa_electrical_measurements.
The achievable stroke of the APA300ML is measured using a displacement probe in Section ref:ssec:test_apa_stroke_measurements. The achievable stroke of the APA300ML is measured using a displacement probe in Section ref:ssec:test_apa_stroke_measurements.
@ -220,9 +220,9 @@ Finally, in Section ref:ssec:test_apa_spurious_resonances, the flexible modes of
<<ssec:test_apa_geometrical_measurements>> <<ssec:test_apa_geometrical_measurements>>
To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with excellent flatness. To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with excellent flatness.
As shown in Figure ref:fig:test_apa_flatness_setup, the APA is fixed to a clamp while a measuring probe[fn:3] is used to measure the height of 4 points on each of the APA300ML interfaces. As shown in Figure ref:fig:test_apa_flatness_setup, the APA is fixed to a clamp while a measuring probe[fn:3] is used to measure the height of four points on each of the APA300ML interfaces.
From the X-Y-Z coordinates of the measured 8 points, the flatness is estimated by best fitting[fn:4] a plane through all the points. From the X-Y-Z coordinates of the measured eight points, the flatness is estimated by best fitting[fn:4] a plane through all the points.
The measured flatness, summarized in Table ref:tab:test_apa_flatness_meas, are within the specifications. The measured flatness values, summarized in Table ref:tab:test_apa_flatness_meas, are within the specifications.
#+begin_src matlab #+begin_src matlab
%% Measured height for all the APA at the 8 locations %% Measured height for all the APA at the 8 locations
@ -296,14 +296,14 @@ data2orgtable(1e6*apa_d', {'APA 1', 'APA 2', 'APA 3', 'APA 4', 'APA 5', 'APA 6',
From the documentation of the APA300ML, the total capacitance of the three stacks should be between $18\,\mu F$ and $26\,\mu F$ with a nominal capacitance of $20\,\mu F$. From the documentation of the APA300ML, the total capacitance of the three stacks should be between $18\,\mu F$ and $26\,\mu F$ with a nominal capacitance of $20\,\mu F$.
The piezoelectric stacks capacitance of the APA300ML have been measured with the LCR meter[fn:1] shown in Figure ref:fig:test_apa_lcr_meter. The capacitance of the APA300ML piezoelectric stacks was measured with the LCR meter[fn:1] shown in Figure ref:fig:test_apa_lcr_meter.
The two stacks used as an actuator and the stack used as a force sensor are measured separately. The two stacks used as the actuator and the stack used as the force sensor were measured separately.
The measured capacitance are summarized in Table ref:tab:test_apa_capacitance and the average capacitance of one stack is $\approx 5 \mu F$. The measured capacitance values are summarized in Table ref:tab:test_apa_capacitance and the average capacitance of one stack is $\approx 5 \mu F$.
However, the measured capacitance of the stacks of "APA 3" is only half of the expected capacitance. However, the measured capacitance of the stacks of "APA 3" is only half of the expected capacitance.
This may indicate a manufacturing defect. This may indicate a manufacturing defect.
The measured capacitance is found to be lower than the specified one. The measured capacitance is found to be lower than the specified value.
This may be due to the fact that the manufacturer measures the capacitance with large signals ($-20\,V$ to $150\,V$) while it was here measured with small signals [[cite:&wehrsdorfer95_large_signal_measur_piezoel_stack]]. This may be because the manufacturer measures the capacitance with large signals ($-20\,V$ to $150\,V$), whereas it was here measured with small signals [[cite:&wehrsdorfer95_large_signal_measur_piezoel_stack]].
#+attr_latex: :options [b]{0.49\linewidth} #+attr_latex: :options [b]{0.49\linewidth}
#+begin_minipage #+begin_minipage
@ -333,10 +333,10 @@ This may be due to the fact that the manufacturer measures the capacitance with
** Stroke and Hysteresis Measurement ** Stroke and Hysteresis Measurement
<<ssec:test_apa_stroke_measurements>> <<ssec:test_apa_stroke_measurements>>
In order to verify that the stroke of the APA300ML is as specified in the datasheet, one side of the APA is fixed to the granite, and a displacement probe[fn:2] is located on the other side as shown in Figure ref:fig:test_apa_stroke_bench. To compare the stroke of the APA300ML with the datasheet specifications, one side of the APA is fixed to the granite, and a displacement probe[fn:2] is located on the other side as shown in Figure ref:fig:test_apa_stroke_bench.
Then, the voltage across the two actuator stacks is varied from $-20\,V$ to $150\,V$ using a DAC[fn:12] and a voltage amplifier[fn:13]. The voltage across the two actuator stacks is varied from $-20\,V$ to $150\,V$ using a DAC[fn:12] and a voltage amplifier[fn:13].
Note that the voltage is here slowly varied as the displacement probe has a very low measurement bandwidth (see Figure ref:fig:test_apa_stroke_voltage). Note that the voltage is slowly varied as the displacement probe has a very low measurement bandwidth (see Figure ref:fig:test_apa_stroke_voltage).
#+name: fig:test_apa_stroke_bench #+name: fig:test_apa_stroke_bench
#+caption: Bench to measure the APA stroke #+caption: Bench to measure the APA stroke
@ -347,12 +347,12 @@ The measured APA displacement is shown as a function of the applied voltage in F
Typical hysteresis curves for piezoelectric stack actuators can be observed. Typical hysteresis curves for piezoelectric stack actuators can be observed.
The measured stroke is approximately $250\,\mu m$ when using only two of the three stacks. The measured stroke is approximately $250\,\mu m$ when using only two of the three stacks.
This is even above what is specified as the nominal stroke in the data-sheet ($304\,\mu m$, therefore $\approx 200\,\mu m$ if only two stacks are used). This is even above what is specified as the nominal stroke in the data-sheet ($304\,\mu m$, therefore $\approx 200\,\mu m$ if only two stacks are used).
For the NASS, this stroke is sufficient as the positioning errors to be corrected using the nano-hexapod are expected to be in the order of $10\,\mu m$. For the NASS, this stroke is sufficient because the positioning errors to be corrected using the nano-hexapod are expected to be in the order of $10\,\mu m$.
It is clear from Figure ref:fig:test_apa_stroke_hysteresis that "APA 3" has an issue compared to the other units. It is clear from Figure ref:fig:test_apa_stroke_hysteresis that "APA 3" has an issue compared with the other units.
This confirms the abnormal electrical measurements made in Section ref:ssec:test_apa_electrical_measurements. This confirms the abnormal electrical measurements made in Section ref:ssec:test_apa_electrical_measurements.
This unit was send sent back to Cedrat and a new one was shipped back. This unit was sent sent back to Cedrat, and a new one was shipped back.
From now on, only the six APA that behave as expected will be used. From now on, only the six remaining amplified piezoelectric actuators that behave as expected will be used.
#+begin_src matlab #+begin_src matlab
%% Load the measured strokes %% Load the measured strokes
@ -389,7 +389,7 @@ exportFig('figs/test_apa_stroke_hysteresis.pdf', 'width', 'half', 'height', 'nor
#+end_src #+end_src
#+name: fig:test_apa_stroke #+name: fig:test_apa_stroke
#+caption: Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (\subref{fig:test_apa_stroke_voltage}). Measured displacement as a function of the applied voltage (\subref{fig:test_apa_stroke_hysteresis}) #+caption: Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (\subref{fig:test_apa_stroke_voltage}). Measured displacement as a function of applied voltage (\subref{fig:test_apa_stroke_hysteresis})
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_stroke_voltage}Applied voltage for stroke estimation} #+attr_latex: :caption \subcaption{\label{fig:test_apa_stroke_voltage}Applied voltage for stroke estimation}
@ -410,11 +410,11 @@ exportFig('figs/test_apa_stroke_hysteresis.pdf', 'width', 'half', 'height', 'nor
<<ssec:test_apa_spurious_resonances>> <<ssec:test_apa_spurious_resonances>>
In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model. In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model.
To experimentally estimate these modes, the APA is fixed on one end (see Figure ref:fig:test_apa_meas_setup_modes). To experimentally estimate these modes, the APA is fixed at one end (see Figure ref:fig:test_apa_meas_setup_modes).
A Laser Doppler Vibrometer[fn:6] is used to measure the difference of motion between two "red" points and an instrumented hammer[fn:7] is used to excite the flexible modes. A Laser Doppler Vibrometer[fn:6] is used to measure the difference of motion between two "red" points and an instrumented hammer[fn:7] is used to excite the flexible modes.
Using this setup, the transfer function from the injected force to the measured rotation can be computed in different conditions and the frequency and mode shapes of the flexible modes can be estimated. Using this setup, the transfer function from the injected force to the measured rotation can be computed under different conditions, and the frequency and mode shapes of the flexible modes can be estimated.
The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software and the results are shown in Figure ref:fig:test_apa_mode_shapes. The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software, and the results are shown in Figure ref:fig:test_apa_mode_shapes.
#+name: fig:test_apa_mode_shapes #+name: fig:test_apa_mode_shapes
#+caption: First three modes of the APA300ML in a fix-free condition estimated from a Finite Element Model #+caption: First three modes of the APA300ML in a fix-free condition estimated from a Finite Element Model
@ -441,7 +441,7 @@ The flexible modes for the same condition (i.e. one mechanical interface of the
#+end_figure #+end_figure
#+name: fig:test_apa_meas_setup_modes #+name: fig:test_apa_meas_setup_modes
#+caption: Experimental setup to measure flexible modes of the APA300ML. For the bending in the $X$ direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is located at the back of the top measurement point. For the bending in the $Y$ direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points). #+caption: Experimental setup to measure the flexible modes of the APA300ML. For the bending in the $X$ direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is at the back of the top measurement point. For the bending in the $Y$ direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_X_bending}$X$ bending} #+attr_latex: :caption \subcaption{\label{fig:test_apa_meas_setup_X_bending}$X$ bending}
@ -482,8 +482,8 @@ bending_Y = load('apa300ml_bending_Y_top.mat');
The measured frequency response functions computed from the experimental setups of figures ref:fig:test_apa_meas_setup_X_bending and ref:fig:test_apa_meas_setup_Y_bending are shown in Figure ref:fig:test_apa_meas_freq_compare. The measured frequency response functions computed from the experimental setups of figures ref:fig:test_apa_meas_setup_X_bending and ref:fig:test_apa_meas_setup_Y_bending are shown in Figure ref:fig:test_apa_meas_freq_compare.
The $y$ bending mode is observed at $280\,\text{Hz}$ and the $x$ bending mode is at $412\,\text{Hz}$. The $y$ bending mode is observed at $280\,\text{Hz}$ and the $x$ bending mode is at $412\,\text{Hz}$.
These modes are measured at higher frequencies than the estimated frequencies from the Finite Element Model (see frequencies in Figure ref:fig:test_apa_mode_shapes). These modes are measured at higher frequencies than the frequencies estimated from the Finite Element Model (see frequencies in Figure ref:fig:test_apa_mode_shapes).
This is opposite to what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model). This is the opposite of what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model).
This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used). This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used).
Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades. Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades.
@ -506,7 +506,7 @@ exportFig('figs/test_apa_meas_freq_compare.pdf', 'width', 'wide', 'height', 'nor
#+end_src #+end_src
#+name: fig:test_apa_meas_freq_compare #+name: fig:test_apa_meas_freq_compare
#+caption: Frequency response functions for the 2 tests with the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at $280\,\text{Hz}$ and the X-bending mode at $412\,\text{Hz}$ #+caption: Frequency response functions for the two tests using the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at $280\,\text{Hz}$ and the X-bending mode at $412\,\text{Hz}$
#+RESULTS: #+RESULTS:
[[file:figs/test_apa_meas_freq_compare.png]] [[file:figs/test_apa_meas_freq_compare.png]]
@ -516,10 +516,10 @@ exportFig('figs/test_apa_meas_freq_compare.pdf', 'width', 'wide', 'height', 'nor
:END: :END:
<<sec:test_apa_dynamics>> <<sec:test_apa_dynamics>>
** Introduction :ignore: ** Introduction :ignore:
After the measurements on the APA were performed in Section ref:sec:test_apa_basic_meas, a new test bench is used to better characterize the dynamics of the APA300ML. After the measurements on the APA were performed in Section ref:sec:test_apa_basic_meas, a new test bench was used to better characterize the dynamics of the APA300ML.
This test bench, depicted in Figure ref:fig:test_bench_apa, comprises the APA300ML fixed at one end to a stationary granite block, and at the other end to a 5kg granite block that is vertically guided by an air bearing. This test bench, depicted in Figure ref:fig:test_bench_apa, comprises the APA300ML fixed at one end to a stationary granite block and at the other end to a 5kg granite block that is vertically guided by an air bearing.
That way, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors. Thus, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors.
An encoder[fn:8] is utilized to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the APA. An encoder[fn:8] is used to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the APA.
#+name: fig:test_bench_apa #+name: fig:test_bench_apa
#+caption: Schematic of the test bench used to estimate the dynamics of the APA300ML #+caption: Schematic of the test bench used to estimate the dynamics of the APA300ML
@ -539,11 +539,11 @@ An encoder[fn:8] is utilized to measure the relative movement between the two gr
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
The bench is schematically shown in Figure ref:fig:test_apa_schematic with all the associated signals. The bench is schematically shown in Figure ref:fig:test_apa_schematic with the associated signals.
It will be first used to estimate the hysteresis from the piezoelectric stack (Section ref:ssec:test_apa_hysteresis) as well as the axial stiffness of the APA300ML (Section ref:ssec:test_apa_stiffness). It will be first used to estimate the hysteresis from the piezoelectric stack (Section ref:ssec:test_apa_hysteresis) as well as the axial stiffness of the APA300ML (Section ref:ssec:test_apa_stiffness).
Then, the frequency response functions from the DAC voltage $u$ to the displacement $d_e$ and to the voltage $V_s$ are measured in Section ref:ssec:test_apa_meas_dynamics. The frequency response functions from the DAC voltage $u$ to the displacement $d_e$ and to the voltage $V_s$ are measured in Section ref:ssec:test_apa_meas_dynamics.
The presence of a non minimum phase zero found on the transfer function from $u$ to $V_s$ is investigated in Section ref:ssec:test_apa_non_minimum_phase. The presence of a non-minimum phase zero found on the transfer function from $u$ to $V_s$ is investigated in Section ref:ssec:test_apa_non_minimum_phase.
In order to limit the low frequency gain of the transfer function from $u$ to $V_s$, a resistor is added across the force sensor stack (Section ref:ssec:test_apa_resistance_sensor_stack). To limit the low-frequency gain of the transfer function from $u$ to $V_s$, a resistor is added across the force sensor stack (Section ref:ssec:test_apa_resistance_sensor_stack).
Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section ref:ssec:test_apa_iff_locus. Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section ref:ssec:test_apa_iff_locus.
#+name: fig:test_apa_schematic #+name: fig:test_apa_schematic
@ -575,10 +575,10 @@ Finally, the Integral Force Feedback is implemented, and the amount of damping a
** Hysteresis ** Hysteresis
<<ssec:test_apa_hysteresis>> <<ssec:test_apa_hysteresis>>
As the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload. Because the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload.
Do to so, a quasi static[fn:9] sinusoidal excitation $V_a$ with an offset of $65\,V$ (halfway between $-20\,V$ and $150\,V$) and with an amplitude varying from $4\,V$ up to $80\,V$ is generated using the DAC. A quasi static[fn:9] sinusoidal excitation $V_a$ with an offset of $65\,V$ (halfway between $-20\,V$ and $150\,V$) and with an amplitude varying from $4\,V$ up to $80\,V$ is generated using the DAC.
For each excitation amplitude, the vertical displacement $d_e$ of the mass is measured and displayed as a function of the applied voltage in Figure ref:fig:test_apa_meas_hysteresis. For each excitation amplitude, the vertical displacement $d_e$ of the mass is measured and displayed as a function of the applied voltage in Figure ref:fig:test_apa_meas_hysteresis.
This is the typical behavior expected from a PZT stack actuator where the hysteresis increases as a function of the applied voltage amplitude [[cite:&fleming14_desig_model_contr_nanop_system chap. 1.4]]. This is the typical behavior expected from a PZT stack actuator, where the hysteresis increases as a function of the applied voltage amplitude [[cite:&fleming14_desig_model_contr_nanop_system chap. 1.4]].
#+begin_src matlab #+begin_src matlab
%% Load measured data - hysteresis %% Load measured data - hysteresis
@ -619,7 +619,7 @@ exportFig('figs/test_apa_meas_hysteresis.pdf', 'width', 'wide', 'height', 'norma
** Axial stiffness ** Axial stiffness
<<ssec:test_apa_stiffness>> <<ssec:test_apa_stiffness>>
In order to estimate the stiffness of the APA, a weight with known mass $m_a = 6.4\,\text{kg}$ is added on top of the suspended granite and the deflection $\Delta d_e$ is measured using the encoder. To estimate the stiffness of the APA, a weight with known mass $m_a = 6.4\,\text{kg}$ is added on top of the suspended granite and the deflection $\Delta d_e$ is measured using the encoder.
The APA stiffness can then be estimated from equation eqref:eq:test_apa_stiffness, with $g \approx 9.8\,m/s^2$ the acceleration of gravity. The APA stiffness can then be estimated from equation eqref:eq:test_apa_stiffness, with $g \approx 9.8\,m/s^2$ the acceleration of gravity.
\begin{equation} \label{eq:test_apa_stiffness} \begin{equation} \label{eq:test_apa_stiffness}
@ -640,10 +640,10 @@ added_mass = 6.4; % Added mass [kg]
#+end_src #+end_src
The measured displacement $d_e$ as a function of time is shown in Figure ref:fig:test_apa_meas_stiffness_time. The measured displacement $d_e$ as a function of time is shown in Figure ref:fig:test_apa_meas_stiffness_time.
It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep) and the that displacement does not come back to the initial position after the mass is removed (probably due to piezoelectric hysteresis). It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep), and that the displacement does not return to the initial position after the mass is removed (probably due to piezoelectric hysteresis).
These two effects induce some uncertainties in the measured stiffness. These two effects induce some uncertainties in the measured stiffness.
The stiffnesses are computed for all the APA from the two displacements $d_1$ and $d_2$ (see Figure ref:fig:test_apa_meas_stiffness_time) leading to two stiffness estimations $k_1$ and $k_2$. The stiffnesses are computed for all APAs from the two displacements $d_1$ and $d_2$ (see Figure ref:fig:test_apa_meas_stiffness_time) leading to two stiffness estimations $k_1$ and $k_2$.
These estimated stiffnesses are summarized in Table ref:tab:test_apa_measured_stiffnesses and are found to be close to the specified nominal stiffness of the APA300ML $k = 1.8\,N/\mu m$. These estimated stiffnesses are summarized in Table ref:tab:test_apa_measured_stiffnesses and are found to be close to the specified nominal stiffness of the APA300ML $k = 1.8\,N/\mu m$.
#+begin_src matlab :exports none #+begin_src matlab :exports none
@ -710,16 +710,16 @@ The stiffness can also be computed using equation eqref:eq:test_apa_res_freq by
\omega_z = \sqrt{\frac{k}{m_{\text{sus}}}} \omega_z = \sqrt{\frac{k}{m_{\text{sus}}}}
\end{equation} \end{equation}
The obtained stiffness is $k \approx 2\,N/\mu m$ which is close to the values found in the documentation and by the "static deflection" method. The obtained stiffness is $k \approx 2\,N/\mu m$ which is close to the values found in the documentation and using the "static deflection" method.
It is important to note that changes to the electrical impedance connected to the piezoelectric stacks impacts the mechanical compliance (or stiffness) of the piezoelectric stack [[cite:&reza06_piezoel_trans_vibrat_contr_dampin chap. 2]]. It is important to note that changes to the electrical impedance connected to the piezoelectric stacks affect the mechanical compliance (or stiffness) of the piezoelectric stack [[cite:&reza06_piezoel_trans_vibrat_contr_dampin chap. 2]].
To estimate this effect for the APA300ML, its stiffness is estimated using the "static deflection" method in two cases: To estimate this effect for the APA300ML, its stiffness is estimated using the "static deflection" method in two cases:
- $k_{\text{os}}$: piezoelectric stacks left unconnected (or connect to the high impedance ADC) - $k_{\text{os}}$: piezoelectric stacks left unconnected (or connect to the high impedance ADC)
- $k_{\text{sc}}$: piezoelectric stacks short circuited (or connected to the voltage amplifier with small output impedance) - $k_{\text{sc}}$: piezoelectric stacks short-circuited (or connected to the voltage amplifier with small output impedance)
It is found that the open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,N/\mu m$ while the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,N/\mu m$. The open-circuit stiffness is estimated at $k_{\text{oc}} \approx 2.3\,N/\mu m$ while the closed-circuit stiffness $k_{\text{sc}} \approx 1.7\,N/\mu m$.
#+begin_src matlab #+begin_src matlab
%% Load Data %% Load Data
@ -788,24 +788,24 @@ save('mat/meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
In this section, the dynamics from the excitation voltage $u$ to the encoder measured displacement $d_e$ and to the force sensor voltage $V_s$ is identified. In this section, the dynamics from the excitation voltage $u$ to the encoder measured displacement $d_e$ and to the force sensor voltage $V_s$ is identified.
First, the dynamics from $u$ to $d_e$ for the six APA300ML are compared in Figure ref:fig:test_apa_frf_encoder. First, the dynamics from $u$ to $d_e$ for the six APA300ML are compared in Figure ref:fig:test_apa_frf_encoder.
The obtained frequency response functions are similar to that of a (second order) mass-spring-damper system with: The obtained frequency response functions are similar to those of a (second order) mass-spring-damper system with:
- A "stiffness line" indicating a static gain equal to $\approx -17\,\mu m/V$. - A "stiffness line" indicating a static gain equal to $\approx -17\,\mu m/V$.
The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the APA The negative sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the APA
- A lightly damped resonance at $95\,\text{Hz}$ - A lightly damped resonance at $95\,\text{Hz}$
- A "mass line" up to $\approx 800\,\text{Hz}$, above which additional resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support. - A "mass line" up to $\approx 800\,\text{Hz}$, above which additional resonances appear. These additional resonances might be due to the limited stiffness of the encoder support or from the limited compliance of the APA support.
Flexible modes studied in section ref:ssec:test_apa_spurious_resonances seem not to impact the measured axial motion of the actuator. The flexible modes studied in section ref:ssec:test_apa_spurious_resonances seem not to impact the measured axial motion of the actuator.
The dynamics from $u$ to the measured voltage across the sensor stack $V_s$ for the six APA300ML are compared in Figure ref:fig:test_apa_frf_force. The dynamics from $u$ to the measured voltage across the sensor stack $V_s$ for the six APA300ML are compared in Figure ref:fig:test_apa_frf_force.
A lightly damped resonance (pole) is observed at $95\,\text{Hz}$ and a lightly damped anti-resonance (zero) at $41\,\text{Hz}$. A lightly damped resonance (pole) is observed at $95\,\text{Hz}$ and a lightly damped anti-resonance (zero) at $41\,\text{Hz}$.
No additional resonances is present up to at least $2\,\text{kHz}$ indicating that Integral Force Feedback can be applied without stability issues from high frequency flexible modes. No additional resonances are present up to at least $2\,\text{kHz}$ indicating that Integral Force Feedback can be applied without stability issues from high-frequency flexible modes.
The zero at $41\,\text{Hz}$ seems to be non-minimum phase (the phase /decreases/ by 180 degrees whereas it should have /increased/ by 180 degrees for a minimum phase zero). The zero at $41\,\text{Hz}$ seems to be non-minimum phase (the phase /decreases/ by 180 degrees whereas it should have /increased/ by 180 degrees for a minimum phase zero).
This is investigated in Section ref:ssec:test_apa_non_minimum_phase. This is investigated in Section ref:ssec:test_apa_non_minimum_phase.
As illustrated by the Root Locus, the poles of the /closed-loop/ system converges to the zeros of the /open-loop/ plant as the feedback gain increases. As illustrated by the Root Locus plot, the poles of the /closed-loop/ system converges to the zeros of the /open-loop/ plant as the feedback gain increases.
The significance of this behavior varies on the type of sensor used as explained in [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chap. 7.6]]. The significance of this behavior varies with the type of sensor used, as explained in [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chap. 7.6]].
Considering the transfer function from $u$ to $V_s$, if a controller with a very high gain is applied such that the sensor stack voltage $V_s$ is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain. Considering the transfer function from $u$ to $V_s$, if a controller with a very high gain is applied such that the sensor stack voltage $V_s$ is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain.
Consequently, the closed-loop system would virtually corresponds to one where the piezoelectric stacks are absent, leaving only the mechanical shell. Consequently, the closed-loop system virtually corresponds to one in which the piezoelectric stacks are absent, leaving only the mechanical shell.
From this analysis, it can be inferred that the axial stiffness of the shell is $k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m$ (which is close to what is found using a finite element model). From this analysis, it can be inferred that the axial stiffness of the shell is $k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m$ (which is close to what is found using a finite element model).
All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure ref:fig:test_apa_frf_encoder and at the force sensor in Figure ref:fig:test_apa_frf_force) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell. All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure ref:fig:test_apa_frf_encoder and at the force sensor in Figure ref:fig:test_apa_frf_force) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell.
@ -905,16 +905,15 @@ exportFig('figs/test_apa_frf_force.pdf', 'width', 'half', 'height', 'tall');
** Non Minimum Phase Zero? ** Non Minimum Phase Zero?
<<ssec:test_apa_non_minimum_phase>> <<ssec:test_apa_non_minimum_phase>>
It was surprising to observe a non-minimum phase behavior for the zero on the transfer function from $u$ to $V_s$ (Figure ref:fig:test_apa_frf_force). It was surprising to observe a non-minimum phase zero on the transfer function from $u$ to $V_s$ (Figure ref:fig:test_apa_frf_force).
It was initially thought that this non-minimum phase behavior is an artifact coming from the measurement. It was initially thought that this non-minimum phase behavior was an artifact arising from the measurement.
A longer measurement was performed with different excitation signals (noise, slow sine sweep, etc.) to see it the phase behavior of the zero changes. A longer measurement was performed using different excitation signals (noise, slow sine sweep, etc.) to determine if the phase behavior of the zero changes (Figure ref:fig:test_apa_non_minimum_phase).
Results of one long measurement is shown in Figure ref:fig:test_apa_non_minimum_phase.
The coherence (Figure ref:fig:test_apa_non_minimum_phase_coherence) is good even in the vicinity of the lightly damped zero, and the phase (Figure ref:fig:test_apa_non_minimum_phase_zoom) clearly indicates non-minimum phase behavior. The coherence (Figure ref:fig:test_apa_non_minimum_phase_coherence) is good even in the vicinity of the lightly damped zero, and the phase (Figure ref:fig:test_apa_non_minimum_phase_zoom) clearly indicates non-minimum phase behavior.
Such non-minimum phase zero when using load cells has also been observed on other mechanical systems [[cite:&spanos95_soft_activ_vibrat_isolat;&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]]. Such non-minimum phase zero when using load cells has also been observed on other mechanical systems [[cite:&spanos95_soft_activ_vibrat_isolat;&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]].
It could be induced to small non-linearity in the system, but the reason of this non-minimum phase for the APA300ML is not yet clear. It could be induced to small non-linearity in the system, but the reason for this non-minimum phase for the APA300ML is not yet clear.
However, this is not so important here as the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure ref:fig:test_apa_iff_root_locus) should not be unstable except for very large controller gains that will never be applied in practice. However, this is not so important here because the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure ref:fig:test_apa_iff_root_locus) should not be unstable, except for very large controller gains that will never be applied in practice.
#+begin_src matlab #+begin_src matlab
%% Long measurement %% Long measurement
@ -982,7 +981,7 @@ exportFig('figs/test_apa_non_minimum_phase_zoom.pdf', 'width', 'half', 'height',
#+end_src #+end_src
#+name: fig:test_apa_non_minimum_phase #+name: fig:test_apa_non_minimum_phase
#+caption: Measurement of the anti-resonance found on the transfer function from $u$ to $V_s$. The coherence (\subref{fig:test_apa_non_minimum_phase_coherence}) is quite good around the anti-resonance frequency. The phase (\subref{fig:test_apa_non_minimum_phase_zoom}) shoes a non-minimum phase behavior. #+caption: Measurement of the anti-resonance found in the transfer function from $u$ to $V_s$. The coherence (\subref{fig:test_apa_non_minimum_phase_coherence}) is quite good around the anti-resonance frequency. The phase (\subref{fig:test_apa_non_minimum_phase_zoom}) shoes a non-minimum phase behavior.
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_non_minimum_phase_coherence} Coherence} #+attr_latex: :caption \subcaption{\label{fig:test_apa_non_minimum_phase_coherence} Coherence}
@ -1003,12 +1002,12 @@ exportFig('figs/test_apa_non_minimum_phase_zoom.pdf', 'width', 'half', 'height',
** Effect of the resistor on the IFF Plant ** Effect of the resistor on the IFF Plant
<<ssec:test_apa_resistance_sensor_stack>> <<ssec:test_apa_resistance_sensor_stack>>
A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack which has the effect to form a high pass filter with the capacitance of the piezoelectric stack (capacitance estimated at $\approx 5\,\mu F$). A resistor $R \approx 80.6\,k\Omega$ is added in parallel with the sensor stack, which forms a high-pass filter with the capacitance of the piezoelectric stack (capacitance estimated at $\approx 5\,\mu F$).
As explain before, this is done to limit the voltage offset due to the input bias current of the ADC as well as to limit the low frequency gain. As explained before, this is done to limit the voltage offset due to the input bias current of the ADC as well as to limit the low frequency gain.
The (low frequency) transfer function from $u$ to $V_s$ with and without this resistor have been measured and are compared in Figure ref:fig:test_apa_effect_resistance. The (low frequency) transfer function from $u$ to $V_s$ with and without this resistor were measured and compared in Figure ref:fig:test_apa_effect_resistance.
It is confirmed that the added resistor as the effect of adding an high pass filter with a cut-off frequency of $\approx 0.39\,\text{Hz}$. It is confirmed that the added resistor has the effect of adding a high-pass filter with a cut-off frequency of $\approx 0.39\,\text{Hz}$.
#+begin_src matlab #+begin_src matlab
%% Load the data %% Load the data
@ -1094,14 +1093,14 @@ Noverlap = floor(Nfft/2);
[G_iff, f] = tfestimate(data.id_plant, data.Vs, win, Noverlap, Nfft, 1/Ts); [G_iff, f] = tfestimate(data.id_plant, data.Vs, win, Noverlap, Nfft, 1/Ts);
#+end_src #+end_src
In order to implement the Integral Force Feedback strategy, the measured frequency response function from $u$ to $V_s$ (Figure ref:fig:test_apa_frf_force) is fitted using the transfer function shown in equation eqref:eq:test_apa_iff_manual_fit. To implement the Integral Force Feedback strategy, the measured frequency response function from $u$ to $V_s$ (Figure ref:fig:test_apa_frf_force) is fitted using the transfer function shown in equation eqref:eq:test_apa_iff_manual_fit.
The parameters are manually tuned, and the obtained values are $\omega_{\textsc{hpf}} = 0.4\, \text{Hz}$, $\omega_{z} = 42.7\, \text{Hz}$, $\xi_{z} = 0.4\,\%$, $\omega_{p} = 95.2\, \text{Hz}$, $\xi_{p} = 2\,\%$ and $g_0 = 0.64$. The parameters were manually tuned, and the obtained values are $\omega_{\textsc{hpf}} = 0.4\, \text{Hz}$, $\omega_{z} = 42.7\, \text{Hz}$, $\xi_{z} = 0.4\,\%$, $\omega_{p} = 95.2\, \text{Hz}$, $\xi_{p} = 2\,\%$ and $g_0 = 0.64$.
\begin{equation} \label{eq:test_apa_iff_manual_fit} \begin{equation} \label{eq:test_apa_iff_manual_fit}
G_{\textsc{iff},m}(s) = g_0 \cdot \frac{1 + 2 \xi_z \frac{s}{\omega_z} + \frac{s^2}{\omega_z^2}}{1 + 2 \xi_p \frac{s}{\omega_p} + \frac{s^2}{\omega_p^2}} \cdot \frac{s}{\omega_{\textsc{hpf}} + s} G_{\textsc{iff},m}(s) = g_0 \cdot \frac{1 + 2 \xi_z \frac{s}{\omega_z} + \frac{s^2}{\omega_z^2}}{1 + 2 \xi_p \frac{s}{\omega_p} + \frac{s^2}{\omega_p^2}} \cdot \frac{s}{\omega_{\textsc{hpf}} + s}
\end{equation} \end{equation}
The comparison between the identified plant and the manually tuned transfer function is done in Figure ref:fig:test_apa_iff_plant_comp_manual_fit. A comparison between the identified plant and the manually tuned transfer function is shown in Figure ref:fig:test_apa_iff_plant_comp_manual_fit.
#+begin_src matlab #+begin_src matlab
%% Basic manually tuned model %% Basic manually tuned model
@ -1148,12 +1147,12 @@ exportFig('figs/test_apa_iff_plant_comp_manual_fit.pdf', 'width', 'wide', 'heigh
#+end_src #+end_src
#+name: fig:test_apa_iff_plant_comp_manual_fit #+name: fig:test_apa_iff_plant_comp_manual_fit
#+caption: Identified IFF plant and manually tuned model of the plant (a time delay of $200\,\mu s$ is added to the model of the plant to better match the identified phase). Note that a minimum-phase zero is here identified even though the coherence is not good arround the frequency of the zero. #+caption: Identified IFF plant and manually tuned model of the plant (a time delay of $200\,\mu s$ is added to the model of the plant to better match the identified phase). Note that a minimum-phase zero is identified here even though the coherence is not good around the frequency of the zero.
#+RESULTS: #+RESULTS:
[[file:figs/test_apa_iff_plant_comp_manual_fit.png]] [[file:figs/test_apa_iff_plant_comp_manual_fit.png]]
The implemented Integral Force Feedback Controller transfer function is shown in equation eqref:eq:test_apa_Kiff_formula. The implemented Integral Force Feedback Controller transfer function is shown in equation eqref:eq:test_apa_Kiff_formula.
It contains an high pass filter (cut-off frequency of $2\,\text{Hz}$) to limit the low frequency gain, a low pass filter to add integral action above $20\,\text{Hz}$, a second low pass filter to add robustness to high frequency resonances and a tunable gain $g$. It contains a high-pass filter (cut-off frequency of $2\,\text{Hz}$) to limit the low-frequency gain, a low-pass filter to add integral action above $20\,\text{Hz}$, a second low-pass filter to add robustness to high-frequency resonances, and a tunable gain $g$.
\begin{equation} \label{eq:test_apa_Kiff_formula} \begin{equation} \label{eq:test_apa_Kiff_formula}
K_{\textsc{iff}}(s) = -10 \cdot g \cdot \frac{s}{s + 2\pi \cdot 2} \cdot \frac{1}{s + 2\pi \cdot 20} \cdot \frac{1}{s + 2\pi\cdot 2000} K_{\textsc{iff}}(s) = -10 \cdot g \cdot \frac{s}{s + 2\pi \cdot 2} \cdot \frac{1}{s + 2\pi \cdot 20} \cdot \frac{1}{s + 2\pi\cdot 2000}
@ -1198,12 +1197,14 @@ for i = 1:length(i_kept)
end end
#+end_src #+end_src
The identified dynamics are then fitted by second order transfer functions[fn:10]. The identified dynamics were then fitted by second order transfer functions[fn:10].
The comparison between the identified damped dynamics and the fitted second order transfer functions is done in Figure ref:fig:test_apa_identified_damped_plants for different gains $g$. A comparison between the identified damped dynamics and the fitted second-order transfer functions is shown in Figure ref:fig:test_apa_identified_damped_plants for different gains $g$.
It is clear that large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies. It is clear that a large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies.
The evolution of the pole in the complex plane as a function of the controller gain $g$ (i.e. the "root locus") is computed both using the IFF plant model eqref:eq:test_apa_iff_manual_fit and the implemented controller eqref:eq:test_apa_Kiff_formula and from the fitted transfer functions of the damped plants experimentally identified for several controller gains. The evolution of the pole in the complex plane as a function of the controller gain $g$ (i.e. the "root locus") is computed in two cases.
The two obtained root loci are compared in Figure ref:fig:test_apa_iff_root_locus and are in good agreement considering that the damped plants were only fitted using a second order transfer function. First using the IFF plant model eqref:eq:test_apa_iff_manual_fit and the implemented controller eqref:eq:test_apa_Kiff_formula.
Second using the fitted transfer functions of the damped plants experimentally identified for several controller gains.
The two obtained root loci are compared in Figure ref:fig:test_apa_iff_root_locus and are in good agreement considering that the damped plants were fitted using only a second-order transfer function.
#+begin_src matlab #+begin_src matlab
%% Fit the data with 2nd order transfer function using vectfit3 %% Fit the data with 2nd order transfer function using vectfit3
@ -1301,7 +1302,7 @@ exportFig('figs/test_apa_iff_root_locus.pdf', 'width', 'half', 'height', 'tall')
#+caption: Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus}) corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula} #+caption: Experimental results of applying Integral Force Feedback to the APA300ML. Obtained damped plant (\subref{fig:test_apa_identified_damped_plants}) and Root Locus (\subref{fig:test_apa_iff_root_locus}) corresponding to the implemented IFF controller \eqref{eq:test_apa_Kiff_formula}
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_apa_identified_damped_plants}Measured frequency response functions of damped plants for several IFF gains (solid lines). Identified 2nd order plants to match the experimental data (dashed lines)} #+attr_latex: :caption \subcaption{\label{fig:test_apa_identified_damped_plants}Measured frequency response functions of damped plants for several IFF gains (solid lines). Identified 2nd order plants that match the experimental data (dashed lines)}
#+attr_latex: :options {0.59\textwidth} #+attr_latex: :options {0.59\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :height 8cm #+attr_latex: :height 8cm
@ -1316,17 +1317,17 @@ exportFig('figs/test_apa_iff_root_locus.pdf', 'width', 'half', 'height', 'tall')
#+end_figure #+end_figure
* APA300ML - 2 Degrees of Freedom Model * APA300ML - 2 degrees-of-freedom Model
:PROPERTIES: :PROPERTIES:
:header-args:matlab+: :tangle matlab/test_apa_3_model_2dof.m :header-args:matlab+: :tangle matlab/test_apa_3_model_2dof.m
:END: :END:
<<sec:test_apa_model_2dof>> <<sec:test_apa_model_2dof>>
**** Introduction :ignore: **** Introduction :ignore:
In this section, a Simscape model (Figure ref:fig:test_apa_bench_model) of the measurement bench is used to tune the 2 degrees of freedom model of the APA using the measured frequency response functions. In this section, a Simscape model (Figure ref:fig:test_apa_bench_model) of the measurement bench is used to tune the two degrees-of-freedom model of the APA using the measured frequency response functions.
This 2 degrees of freedom model is developed to accurately represent the APA300ML dynamics while having low complexity and low number of associated states. This two degrees-of-freedom model is developed to accurately represent the APA300ML dynamics while having low complexity and a low number of associated states.
After the model presented, the procedure to tune the model is described and the obtained model dynamics is compared with the measurements. After the model is presented, the procedure for tuning the model is described, and the obtained model dynamics is compared with the measurements.
#+name: fig:test_apa_bench_model #+name: fig:test_apa_bench_model
#+caption: Screenshot of the Simscape model #+caption: Screenshot of the Simscape model
@ -1373,24 +1374,24 @@ io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
freqs = 5*logspace(0, 3, 1000); freqs = 5*logspace(0, 3, 1000);
#+end_src #+end_src
**** Two Degrees of Freedom APA Model **** Two degrees-of-freedom APA Model
The model of the amplified piezoelectric actuator is shown in Figure ref:fig:test_apa_2dof_model. The model of the amplified piezoelectric actuator is shown in Figure ref:fig:test_apa_2dof_model.
It can be decomposed into three components: It can be decomposed into three components:
- the shell whose axial properties are represented by $k_1$ and $c_1$ - the shell whose axial properties are represented by $k_1$ and $c_1$
- the actuator stacks whose contribution in the axial stiffness is represented by $k_a$ and $c_a$. - the actuator stacks whose contribution to the axial stiffness is represented by $k_a$ and $c_a$.
A force source $\tau$ represents the axial force induced by the force sensor stacks. The force source $\tau$ 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 $\tau$ The sensitivity $g_a$ (in $N/m$) is used to convert the applied voltage $V_a$ to the axial force $\tau$
- the sensor stack whose contribution in the axial stiffness is represented by $k_e$ and $c_e$. - 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 $V/m$) 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 simple model has some limitations: Such a simple model has some limitations:
- it only represents the axial characteristics of the APA as it is modelled as infinitely rigid in the other directions - 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, the real stress and strain in the piezoelectric stacks - some physical insights are lost, such as the amplification factor and the real stress and strain in the piezoelectric stacks
- it is fully linear and therefore the creep and hysteresis of the piezoelectric stacks are not modelled - the creep and hysteresis of the piezoelectric stacks are not modeled as the model is linear
#+name: fig:test_apa_2dof_model #+name: fig:test_apa_2dof_model
#+caption: Schematic of the two degrees of freedom model of the APA300ML, adapted from cite:souleille18_concep_activ_mount_space_applic #+caption: Schematic of the two degrees-of-freedom model of the APA300ML, adapted from cite:souleille18_concep_activ_mount_space_applic
[[file:figs/test_apa_2dof_model.png]] [[file:figs/test_apa_2dof_model.png]]
**** Tuning of the APA model :ignore: **** Tuning of the APA model :ignore:
@ -1398,7 +1399,7 @@ Such simple model has some limitations:
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:test_apa_2dof_model_Simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics. 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:test_apa_2dof_model_Simscape) well represents the identified dynamics in Section ref:sec:test_apa_dynamics.
#+name: fig:test_apa_2dof_model_Simscape #+name: fig:test_apa_2dof_model_Simscape
#+caption: Schematic of the two degrees of freedom model of the APA300ML with input $V_a$ and outputs $d_e$ and $V_s$ #+caption: Schematic of the two degrees-of-freedom model of the APA300ML with input $V_a$ and outputs $d_e$ and $V_s$
[[file:figs/test_apa_2dof_model_Simscape.png]] [[file:figs/test_apa_2dof_model_Simscape.png]]
#+begin_src matlab #+begin_src matlab
@ -1456,14 +1457,14 @@ gs = -abs(iff_frf(i_f,1))./abs(evalfr(G_norm('Vs', 'u'), 1i*2*pi*fs))/ga;
#+end_src #+end_src
First, the mass $m$ supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale. First, the mass $m$ supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale.
Both methods leads to an estimated mass of $m = 5.7\,\text{kg}$. Both methods lead to an estimated mass of $m = 5.7\,\text{kg}$.
Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,N/\mu m$ in Section ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure ref:fig:test_apa_frf_force. Then, the axial stiffness of the shell was estimated at $k_1 = 0.38\,N/\mu m$ in Section ref:ssec:test_apa_meas_dynamics from the frequency of the anti-resonance seen on Figure ref:fig:test_apa_frf_force.
Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resonance and is found to be close to $20\,Ns/m$. Similarly, $c_1$ can be estimated from the damping ratio of the same anti-resonance and is found to be close to $20\,Ns/m$.
Then, it is reasonable to make the assumption that the sensor stacks and the two actuator stacks have identical mechanical characteristics[fn:5]. Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics[fn:5].
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. 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 that case, the total stiffness of the APA model is described by eqref:eq:test_apa_2dof_stiffness. In this case, the total stiffness of the APA model is described by eqref:eq:test_apa_2dof_stiffness.
\begin{equation}\label{eq:test_apa_2dof_stiffness} \begin{equation}\label{eq:test_apa_2dof_stiffness}
k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a
@ -1478,7 +1479,7 @@ Knowing from eqref:eq:test_apa_tot_stiffness that the total stiffness is $k_{\te
Then, $c_a$ (and therefore $c_e = 2 c_a$) can be tuned to match the damping ratio of the identified resonance. Then, $c_a$ (and therefore $c_e = 2 c_a$) can be tuned to match the damping ratio of the identified resonance.
$c_a = 100\,Ns/m$ and $c_e = 200\,Ns/m$ are obtained. $c_a = 100\,Ns/m$ and $c_e = 200\,Ns/m$ are obtained.
Finally $g_s$ and $g_a$ can be tuned to match the gain of the identified transfer functions. In the last step, $g_s$ and $g_a$ can be tuned to match the gain of the identified transfer functions.
The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model_Simscape are summarized in Table ref:tab:test_apa_2dof_parameters. The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model_Simscape are summarized in Table ref:tab:test_apa_2dof_parameters.
@ -1500,8 +1501,8 @@ The obtained parameters of the model shown in Figure ref:fig:test_apa_2dof_model
**** Obtained Dynamics :ignore: **** Obtained Dynamics :ignore:
The dynamics of the two degrees of freedom model of the APA300ML is now extracted using optimized parameters (listed in Table ref:tab:test_apa_2dof_parameters) from the Simscape model. The dynamics of the two degrees-of-freedom model of the APA300ML are extracted using optimized parameters (listed in Table ref:tab:test_apa_2dof_parameters) from the Simscape model.
It is compared with the experimental data in Figure ref:fig:test_apa_2dof_comp_frf. This is compared with the experimental data in Figure ref:fig:test_apa_2dof_comp_frf.
A good match can be observed between the model and the experimental data, both for the encoder (Figure ref:fig:test_apa_2dof_comp_frf_enc) and for the force sensor (Figure ref:fig:test_apa_2dof_comp_frf_force). A good match can be observed between the model and the experimental data, both for the encoder (Figure ref:fig:test_apa_2dof_comp_frf_enc) and for the force sensor (Figure ref:fig:test_apa_2dof_comp_frf_force).
This indicates that this model represents well the axial dynamics of the APA300ML. This indicates that this model represents well the axial dynamics of the APA300ML.
@ -1630,15 +1631,14 @@ exportFig('figs/test_apa_2dof_comp_frf_force.pdf', 'width', 'half', 'height', 't
**** Introduction :ignore: **** Introduction :ignore:
In this section, a /super element/ of the APA300ML is computed using a finite element software[fn:11]. In this section, a /super element/ of the APA300ML is computed using a finite element software[fn:11].
It is then imported in Simscape (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in ref:sec:test_apa_model_2dof. It is then imported into Simscape (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in ref:sec:test_apa_model_2dof.
This procedure is illustrated in Figure ref:fig:test_apa_super_element_Simscape. This procedure is illustrated in Figure ref:fig:test_apa_super_element_Simscape.
Several /remote points/ are defined in the finite element model (here illustrated by colorful planes and numbers from =1= to =5=) and are then make accessible in the Simscape model as shown at the right by the "frames" =F1= to =F5=. Several /remote points/ are defined in the finite element model (here illustrated by colorful planes and numbers from =1= to =5=) and are then made accessible in the Simscape model as shown at the right by the "frames" =F1= to =F5=.
For the APA300ML /super element/, 5 /remote points/ are defined. For the APA300ML /super element/, 5 /remote points/ are defined.
Two /remote points/ (=1= and =2=) are fixed to the top and bottom mechanical interfaces of the APA300ML and will be used for connecting the APA300ML with other mechanical elements. Two /remote points/ (=1= and =2=) are fixed to the top and bottom mechanical interfaces of the APA300ML and will be used to connect the APA300ML with other mechanical elements.
Two /remote points/ (=3= and =4=) are located across two piezoelectric stacks and will be used to apply internal forces representing the actuator stacks. Two /remote points/ (=3= and =4=) are located across two piezoelectric stacks and are used to apply internal forces representing the actuator stacks.
Finally, two /remote points/ (=4= and =4=) are located across the third piezoelectric stack. Finally, two /remote points/ (=4= and =4=) are located across the third piezoelectric stack, and will be used to measured the strain of the sensor stack.
It will be used to measure the strain experience by this stack, and model the sensor stack.
#+name: fig:test_apa_super_element_Simscape #+name: fig:test_apa_super_element_Simscape
#+attr_latex: :width 1.0\linewidth #+attr_latex: :width 1.0\linewidth
@ -1688,7 +1688,7 @@ freqs = 5*logspace(0, 3, 1000);
**** Identification of the Actuator and Sensor constants **** Identification of the Actuator and Sensor constants
Once the APA300ML /super element/ is included in the Simscape model, the transfer function from $F_a$ to $d_L$ and $d_e$ can be extracted. Once the APA300ML /super element/ is included in the Simscape 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 be tuned such that the gain of the transfer functions are matching the identified ones. The gains $g_a$ and $g_s$ are then tuned such that the gains of the transfer functions match the identified ones.
By doing so, $g_s = 4.9\,V/\mu m$ and $g_a = 23.2\,N/V$ are obtained. By doing so, $g_s = 4.9\,V/\mu m$ and $g_a = 23.2\,N/V$ are obtained.
#+begin_src matlab #+begin_src matlab
@ -1716,7 +1716,7 @@ ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(G_norm('de', 'u'));
gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(G_norm('Vs', 'u'), 1e3, 'Hz')))); gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(G_norm('Vs', 'u'), 1e3, 'Hz'))));
#+end_src #+end_src
To make sure the sensitivities $g_a$ and $g_s$ are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material. To ensure that the sensitivities $g_a$ and $g_s$ are physically valid, it is possible to estimate them from the physical properties of the piezoelectric stack material.
From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by eqref:eq:test_apa_piezo_strain_to_voltage and from [[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by eqref:eq:test_apa_piezo_voltage_to_force. From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation between relative displacement $d_L$ of the sensor stack and generated voltage $V_s$ is given by eqref:eq:test_apa_piezo_strain_to_voltage and from [[cite:&fleming10_integ_strain_force_feedb_high]] the relation between the force $F_a$ and the applied voltage $V_a$ is given by eqref:eq:test_apa_piezo_voltage_to_force.
@ -1728,10 +1728,10 @@ From [[cite:&fleming14_desig_model_contr_nanop_system p. 123]], the relation bet
\end{subequations} \end{subequations}
Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML. Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML.
However, based on available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties. However, based on the available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate "THP5H" from Thorlabs seemed to match quite well the observed properties.
The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table ref:tab:test_apa_piezo_properties. The properties of this "THP5H" material used to compute $g_a$ and $g_s$ are listed in Table ref:tab:test_apa_piezo_properties.
From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained which are close to the identified constants using the experimentally identified transfer functions. From these parameters, $g_s = 5.1\,V/\mu m$ and $g_a = 26\,N/V$ were obtained, which are close to the constants identified using the experimentally identified transfer functions.
#+name: tab:test_apa_piezo_properties #+name: tab:test_apa_piezo_properties
#+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities #+caption: Piezoelectric properties used for the estimation of the sensor and actuators sensitivities
@ -1771,10 +1771,10 @@ ga_th = d33*n*ka; % Actuator Constant [N/V]
**** 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 frequency response functions in Figure ref:fig:test_apa_super_element_comp_frf. The obtained dynamics using the /super element/ with the tuned "sensor sensitivity" and "actuator sensitivity" are compared with the experimentally identified frequency response functions in Figure ref:fig:test_apa_super_element_comp_frf.
A good match between the model and the experimental results is observed. A good match between the model and the experimental results was observed.
It is however surprising that the model is "softer" than the measured system as finite element models are usually overestimating the stiffness (see Section ref:ssec:test_apa_spurious_resonances for possible explanations). It is however surprising that the model is "softer" than the measured system, as finite element models usually overestimate the stiffness (see Section ref:ssec:test_apa_spurious_resonances for possible explanations).
Using this simple test bench, it can be concluded that the /super element/ model of the APA300ML well captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever). Using this simple test bench, it can be concluded that the /super element/ model of the APA300ML captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever).
#+begin_src matlab #+begin_src matlab
%% Idenfify the dynamics of the Simscape model with correct actuator and sensor "constants" %% Idenfify the dynamics of the Simscape model with correct actuator and sensor "constants"
@ -1889,24 +1889,24 @@ exportFig('figs/test_apa_super_element_comp_frf_force.pdf', 'width', 'half', 'he
* Conclusion * Conclusion
<<sec:test_apa_conclusion>> <<sec:test_apa_conclusion>>
In this study, the amplified piezoelectric actuators "APA300ML" have been characterized to make sure they are fulfilling all the requirements determined during the detailed design phase. In this study, the amplified piezoelectric actuators "APA300ML" have been characterized to ensure that they fulfill all the requirements determined during the detailed design phase.
Geometrical features such as the flatness of its interfaces, electrical capacitance and achievable strokes were measured in Section ref:sec:test_apa_basic_meas. Geometrical features such as the flatness of its interfaces, electrical capacitance, and achievable strokes were measured in Section ref:sec:test_apa_basic_meas.
These simple measurements allowed for early detection of a manufacturing defect in one of the APA300ML. These simple measurements allowed for the early detection of a manufacturing defect in one of the APA300ML.
Then in Section ref:sec:test_apa_dynamics, using a dedicated test bench, the dynamics of all the APA300ML were measured and were found to all match very well (Figure ref:fig:test_apa_frf_dynamics). Then in Section ref:sec:test_apa_dynamics, using a dedicated test bench, the dynamics of all the APA300ML were measured and were found to all match very well (Figure ref:fig:test_apa_frf_dynamics).
This consistency indicates good manufacturing tolerances, facilitating the modeling and control of the nano-hexapod. This consistency indicates good manufacturing tolerances, facilitating the modeling and control of the nano-hexapod.
Although a non-minimum zero was identified in the transfer function from $u$ to $V_s$ (Figure ref:fig:test_apa_non_minimum_phase), it was found not to be problematic as large amount of damping could be added using the integral force feedback strategy (Figure ref:fig:test_apa_iff). Although a non-minimum zero was identified in the transfer function from $u$ to $V_s$ (Figure ref:fig:test_apa_non_minimum_phase), it was found not to be problematic because a large amount of damping could be added using the integral force feedback strategy (Figure ref:fig:test_apa_iff).
Then, two different models were used to represent the APA300ML dynamics. Then, two different models were used to represent the APA300ML dynamics.
In Section ref:sec:test_apa_model_2dof, a simple two degrees of freedom mass-spring-damper model was presented and tuned based on the measured dynamics. In Section ref:sec:test_apa_model_2dof, a simple two degrees-of-freedom mass-spring-damper model was presented and tuned based on the measured dynamics.
After following a tuning procedure, the model dynamics was shown to match very well with the experiment. After following a tuning procedure, the model dynamics was shown to match very well with the experiment.
However, it is important to note that this model only represents the axial dynamics of the actuators, assuming infinite stiffness in other directions. However, this model only represents the axial dynamics of the actuators, assuming infinite stiffness in other directions.
In Section ref:sec:test_apa_model_flexible, a /super element/ extracted from a finite element model was used to model the APA300ML. In Section ref:sec:test_apa_model_flexible, a /super element/ extracted from a finite element model was used to model the APA300ML.
This time, the /super element/ represents the dynamics of the APA300ML in all directions. Here, the /super element/ represents the dynamics of the APA300ML in all directions.
However, only the axial dynamics could be compared with the experimental results yielding a good match. However, only the axial dynamics could be compared with the experimental results, yielding a good match.
The benefit of employing this model over the two degrees of freedom model is not immediately apparent due to its increased complexity and the larger number of model states involved. The benefit of employing this model over the two degrees-of-freedom model is not immediately apparent due to its increased complexity and the larger number of model states involved.
Nonetheless, the /super element/ model's value will become clear in subsequent sections, when its capacity to accurately model the APA300ML's flexibility across various directions will be important. Nonetheless, the /super element/ model's value will become clear in subsequent sections, when its capacity to accurately model the APA300ML's flexibility across various directions will be important.
* Bibliography :ignore: * Bibliography :ignore:
@ -2133,8 +2133,8 @@ actuator.cs = args.cs; % Damping of one stack [N/m]
[fn:8]Renishaw Vionic, resolution of $2.5\,nm$ [fn:8]Renishaw Vionic, resolution of $2.5\,nm$
[fn:7]Kistler 9722A [fn:7]Kistler 9722A
[fn:6]Polytec controller 3001 with sensor heads OFV512 [fn:6]Polytec controller 3001 with sensor heads OFV512
[fn:5]Note that this is not fully correct as it was shown in Section ref:ssec:test_apa_stiffness that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited. [fn:5]Note that this is not completely correct as it was shown in Section ref:ssec:test_apa_stiffness that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited.
[fn:4]The Matlab =fminsearch= command is used to fit the plane [fn:4]The Matlab =fminsearch= command is used to fit the plane
[fn:3]Heidenhain MT25, specified accuracy of $\pm 0.5\,\mu m$ [fn:3]Heidenhain MT25, specified accuracy of $\pm 0.5\,\mu m$
[fn:2]Millimar 1318 probe, specified linearity better than $1\,\mu m$ [fn:2]Millimar 1318 probe, specified linearity better than $1\,\mu m$
[fn:1]LCR-819 from Gwinstek, specified accuracy of $0.05\%$, measured frequency is set at $1\,\text{kHz}$ [fn:1]LCR-819 from Gwinstek, with a specified accuracy of $0.05\%$. The measured frequency is set at $1\,\text{kHz}$

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@ -1,4 +1,4 @@
% Created 2024-04-30 Tue 16:37 % Created 2024-04-30 Tue 17:24
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -16,7 +16,7 @@
\newacronym{drga}{DRGA}{Dynamical Relative Gain Array} \newacronym{drga}{DRGA}{Dynamical Relative Gain Array}
\newacronym{hpf}{HPF}{high-pass filter} \newacronym{hpf}{HPF}{high-pass filter}
\newacronym{lpf}{LPF}{low-pass filter} \newacronym{lpf}{LPF}{low-pass filter}
\newacronym{dof}{DoF}{Degrees of Freedom} \newacronym{dof}{DoF}{degrees-of-freedom}
\newglossaryentry{psdx}{name=\ensuremath{\Phi_{x}},description={{Power spectral density of signal $x$}}} \newglossaryentry{psdx}{name=\ensuremath{\Phi_{x}},description={{Power spectral density of signal $x$}}}
\newglossaryentry{asdx}{name=\ensuremath{\Gamma_{x}},description={{Amplitude spectral density of signal $x$}}} \newglossaryentry{asdx}{name=\ensuremath{\Gamma_{x}},description={{Amplitude spectral density of signal $x$}}}
\newglossaryentry{cpsx}{name=\ensuremath{\Phi_{x}},description={{Cumulative Power Spectrum of signal $x$}}} \newglossaryentry{cpsx}{name=\ensuremath{\Phi_{x}},description={{Cumulative Power Spectrum of signal $x$}}}
@ -42,21 +42,21 @@
\clearpage \clearpage
In this chapter, the goal is to make sure that the received APA300ML (shown in Figure \ref{fig:test_apa_received}) are complying with the requirements and that dynamical models of the actuator are well representing its dynamics. In this chapter, the goal is to ensure that the received APA300ML (shown in Figure \ref{fig:test_apa_received}) are complying with the requirements and that the dynamical models of the actuator accurately represent its dynamics.
In section \ref{sec:test_apa_basic_meas}, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks, the achievable stroke. In section \ref{sec:test_apa_basic_meas}, the mechanical tolerances of the APA300ML interfaces are checked together with the electrical properties of the piezoelectric stacks and the achievable stroke.
Flexible modes of the APA300ML which were estimated using a finite element model are compared with measurements. The flexible modes of the APA300ML, which were estimated using a finite element model, are compared with measurements.
Using a dedicated test bench, dynamical measurements are performed (Section \ref{sec:test_apa_dynamics}). Using a dedicated test bench, dynamical measurements are performed (Section \ref{sec:test_apa_dynamics}).
The dynamics from the generated DAC voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated. The dynamics from the generated DAC voltage (going through the voltage amplifier and then to two actuator stacks) to the induced axial displacement and to the measured voltage across the force sensor stack are estimated.
Integral Force Feedback is experimentally applied and the damped plants are estimated for several feedback gains. Integral Force Feedback is experimentally applied, and the damped plants are estimated for several feedback gains.
Two different models of the APA300ML are then presented. Two different models of the APA300ML are presented.
First, in Section \ref{sec:test_apa_model_2dof}, a two degrees of freedom model is presented, tuned and compared with the measured dynamics. First, in Section \ref{sec:test_apa_model_2dof}, a two degrees-of-freedom model is presented, tuned, and compared with the measured dynamics.
This model is proven to accurately represents the APA300ML's axial dynamics while having low complexity. This model is proven to accurately represent the APA300ML's axial dynamics while having low complexity.
Then, in Section \ref{sec:test_apa_model_flexible}, a \emph{super element} of the APA300ML is extracted using a finite element model and imported in Simscape. Then, in Section \ref{sec:test_apa_model_flexible}, a \emph{super element} of the APA300ML is extracted using a finite element model and imported into Simscape.
This more complex model is also shown to well capture the axial dynamics of the APA300ML. This more complex model also captures well capture the axial dynamics of the APA300ML.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -67,7 +67,7 @@ This more complex model is also shown to well capture the axial dynamics of the
\chapter{First Basic Measurements} \chapter{First Basic Measurements}
\label{sec:test_apa_basic_meas} \label{sec:test_apa_basic_meas}
Before measuring the dynamical characteristics of the APA300ML, first simple measurements are performed. Before measuring the dynamical characteristics of the APA300ML, simple measurements are performed.
First, the tolerances (especially flatness) of the mechanical interfaces are checked in Section \ref{ssec:test_apa_geometrical_measurements}. First, the tolerances (especially flatness) of the mechanical interfaces are checked in Section \ref{ssec:test_apa_geometrical_measurements}.
Then, the capacitance of the piezoelectric stacks is measured in Section \ref{ssec:test_apa_electrical_measurements}. Then, the capacitance of the piezoelectric stacks is measured in Section \ref{ssec:test_apa_electrical_measurements}.
The achievable stroke of the APA300ML is measured using a displacement probe in Section \ref{ssec:test_apa_stroke_measurements}. The achievable stroke of the APA300ML is measured using a displacement probe in Section \ref{ssec:test_apa_stroke_measurements}.
@ -77,9 +77,9 @@ Finally, in Section \ref{ssec:test_apa_spurious_resonances}, the flexible modes
\label{ssec:test_apa_geometrical_measurements} \label{ssec:test_apa_geometrical_measurements}
To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with excellent flatness. To measure the flatness of the two mechanical interfaces of the APA300ML, a small measurement bench is installed on top of a metrology granite with excellent flatness.
As shown in Figure \ref{fig:test_apa_flatness_setup}, the APA is fixed to a clamp while a measuring probe\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\mu m\)} is used to measure the height of 4 points on each of the APA300ML interfaces. As shown in Figure \ref{fig:test_apa_flatness_setup}, the APA is fixed to a clamp while a measuring probe\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\mu m\)} is used to measure the height of four points on each of the APA300ML interfaces.
From the X-Y-Z coordinates of the measured 8 points, the flatness is estimated by best fitting\footnote{The Matlab \texttt{fminsearch} command is used to fit the plane} a plane through all the points. From the X-Y-Z coordinates of the measured eight points, the flatness is estimated by best fitting\footnote{The Matlab \texttt{fminsearch} command is used to fit the plane} a plane through all the points.
The measured flatness, summarized in Table \ref{tab:test_apa_flatness_meas}, are within the specifications. The measured flatness values, summarized in Table \ref{tab:test_apa_flatness_meas}, are within the specifications.
\begin{minipage}[b]{0.49\linewidth} \begin{minipage}[b]{0.49\linewidth}
\begin{center} \begin{center}
@ -113,14 +113,14 @@ APA 7 & 18.7\\
From the documentation of the APA300ML, the total capacitance of the three stacks should be between \(18\,\mu F\) and \(26\,\mu F\) with a nominal capacitance of \(20\,\mu F\). From the documentation of the APA300ML, the total capacitance of the three stacks should be between \(18\,\mu F\) and \(26\,\mu F\) with a nominal capacitance of \(20\,\mu F\).
The piezoelectric stacks capacitance of the APA300ML have been measured with the LCR meter\footnote{LCR-819 from Gwinstek, specified accuracy of \(0.05\%\), measured frequency is set at \(1\,\text{kHz}\)} shown in Figure \ref{fig:test_apa_lcr_meter}. The capacitance of the APA300ML piezoelectric stacks was measured with the LCR meter\footnote{LCR-819 from Gwinstek, with a specified accuracy of \(0.05\%\). The measured frequency is set at \(1\,\text{kHz}\)} shown in Figure \ref{fig:test_apa_lcr_meter}.
The two stacks used as an actuator and the stack used as a force sensor are measured separately. The two stacks used as the actuator and the stack used as the force sensor were measured separately.
The measured capacitance are summarized in Table \ref{tab:test_apa_capacitance} and the average capacitance of one stack is \(\approx 5 \mu F\). The measured capacitance values are summarized in Table \ref{tab:test_apa_capacitance} and the average capacitance of one stack is \(\approx 5 \mu F\).
However, the measured capacitance of the stacks of ``APA 3'' is only half of the expected capacitance. However, the measured capacitance of the stacks of ``APA 3'' is only half of the expected capacitance.
This may indicate a manufacturing defect. This may indicate a manufacturing defect.
The measured capacitance is found to be lower than the specified one. The measured capacitance is found to be lower than the specified value.
This may be due to the fact that the manufacturer measures the capacitance with large signals (\(-20\,V\) to \(150\,V\)) while it was here measured with small signals \cite{wehrsdorfer95_large_signal_measur_piezoel_stack}. This may be because the manufacturer measures the capacitance with large signals (\(-20\,V\) to \(150\,V\)), whereas it was here measured with small signals \cite{wehrsdorfer95_large_signal_measur_piezoel_stack}.
\begin{minipage}[b]{0.49\linewidth} \begin{minipage}[b]{0.49\linewidth}
\begin{center} \begin{center}
@ -152,10 +152,10 @@ APA 7 & 4.85 & 9.85\\
\section{Stroke and Hysteresis Measurement} \section{Stroke and Hysteresis Measurement}
\label{ssec:test_apa_stroke_measurements} \label{ssec:test_apa_stroke_measurements}
In order to verify that the stroke of the APA300ML is as specified in the datasheet, one side of the APA is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\mu m\)} is located on the other side as shown in Figure \ref{fig:test_apa_stroke_bench}. To compare the stroke of the APA300ML with the datasheet specifications, one side of the APA is fixed to the granite, and a displacement probe\footnote{Millimar 1318 probe, specified linearity better than \(1\,\mu m\)} is located on the other side as shown in Figure \ref{fig:test_apa_stroke_bench}.
Then, the voltage across the two actuator stacks is varied from \(-20\,V\) to \(150\,V\) using a DAC\footnote{The DAC used is the one included in the IO133 card sold by Speedgoat. It has an output range of \(\pm 10\,V\) and 16-bits resolution} and a voltage amplifier\footnote{PD200 from PiezoDrive. The gain is \(20\,V/V\)}. The voltage across the two actuator stacks is varied from \(-20\,V\) to \(150\,V\) using a DAC\footnote{The DAC used is the one included in the IO133 card sold by Speedgoat. It has an output range of \(\pm 10\,V\) and 16-bits resolution} and a voltage amplifier\footnote{PD200 from PiezoDrive. The gain is \(20\,V/V\)}.
Note that the voltage is here slowly varied as the displacement probe has a very low measurement bandwidth (see Figure \ref{fig:test_apa_stroke_voltage}). Note that the voltage is slowly varied as the displacement probe has a very low measurement bandwidth (see Figure \ref{fig:test_apa_stroke_voltage}).
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -167,12 +167,12 @@ The measured APA displacement is shown as a function of the applied voltage in F
Typical hysteresis curves for piezoelectric stack actuators can be observed. Typical hysteresis curves for piezoelectric stack actuators can be observed.
The measured stroke is approximately \(250\,\mu m\) when using only two of the three stacks. The measured stroke is approximately \(250\,\mu m\) when using only two of the three stacks.
This is even above what is specified as the nominal stroke in the data-sheet (\(304\,\mu m\), therefore \(\approx 200\,\mu m\) if only two stacks are used). This is even above what is specified as the nominal stroke in the data-sheet (\(304\,\mu m\), therefore \(\approx 200\,\mu m\) if only two stacks are used).
For the NASS, this stroke is sufficient as the positioning errors to be corrected using the nano-hexapod are expected to be in the order of \(10\,\mu m\). For the NASS, this stroke is sufficient because the positioning errors to be corrected using the nano-hexapod are expected to be in the order of \(10\,\mu m\).
It is clear from Figure \ref{fig:test_apa_stroke_hysteresis} that ``APA 3'' has an issue compared to the other units. It is clear from Figure \ref{fig:test_apa_stroke_hysteresis} that ``APA 3'' has an issue compared with the other units.
This confirms the abnormal electrical measurements made in Section \ref{ssec:test_apa_electrical_measurements}. This confirms the abnormal electrical measurements made in Section \ref{ssec:test_apa_electrical_measurements}.
This unit was send sent back to Cedrat and a new one was shipped back. This unit was sent sent back to Cedrat, and a new one was shipped back.
From now on, only the six APA that behave as expected will be used. From now on, only the six remaining amplified piezoelectric actuators that behave as expected will be used.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth} \begin{subfigure}{0.49\textwidth}
@ -187,18 +187,18 @@ From now on, only the six APA that behave as expected will be used.
\end{center} \end{center}
\subcaption{\label{fig:test_apa_stroke_hysteresis}Hysteresis curves of the APA} \subcaption{\label{fig:test_apa_stroke_hysteresis}Hysteresis curves of the APA}
\end{subfigure} \end{subfigure}
\caption{\label{fig:test_apa_stroke}Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (\subref{fig:test_apa_stroke_voltage}). Measured displacement as a function of the applied voltage (\subref{fig:test_apa_stroke_hysteresis})} \caption{\label{fig:test_apa_stroke}Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML (\subref{fig:test_apa_stroke_voltage}). Measured displacement as a function of applied voltage (\subref{fig:test_apa_stroke_hysteresis})}
\end{figure} \end{figure}
\section{Flexible Mode Measurement} \section{Flexible Mode Measurement}
\label{ssec:test_apa_spurious_resonances} \label{ssec:test_apa_spurious_resonances}
In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model. In this section, the flexible modes of the APA300ML are investigated both experimentally and using a Finite Element Model.
To experimentally estimate these modes, the APA is fixed on one end (see Figure \ref{fig:test_apa_meas_setup_modes}). To experimentally estimate these modes, the APA is fixed at one end (see Figure \ref{fig:test_apa_meas_setup_modes}).
A Laser Doppler Vibrometer\footnote{Polytec controller 3001 with sensor heads OFV512} is used to measure the difference of motion between two ``red'' points and an instrumented hammer\footnote{Kistler 9722A} is used to excite the flexible modes. A Laser Doppler Vibrometer\footnote{Polytec controller 3001 with sensor heads OFV512} is used to measure the difference of motion between two ``red'' points and an instrumented hammer\footnote{Kistler 9722A} is used to excite the flexible modes.
Using this setup, the transfer function from the injected force to the measured rotation can be computed in different conditions and the frequency and mode shapes of the flexible modes can be estimated. Using this setup, the transfer function from the injected force to the measured rotation can be computed under different conditions, and the frequency and mode shapes of the flexible modes can be estimated.
The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software and the results are shown in Figure \ref{fig:test_apa_mode_shapes}. The flexible modes for the same condition (i.e. one mechanical interface of the APA300ML fixed) are estimated using a finite element software, and the results are shown in Figure \ref{fig:test_apa_mode_shapes}.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.36\textwidth} \begin{subfigure}{0.36\textwidth}
@ -235,28 +235,28 @@ The flexible modes for the same condition (i.e. one mechanical interface of the
\end{center} \end{center}
\subcaption{\label{fig:test_apa_meas_setup_Y_bending}$Y$ Bending} \subcaption{\label{fig:test_apa_meas_setup_Y_bending}$Y$ Bending}
\end{subfigure} \end{subfigure}
\caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measure flexible modes of the APA300ML. For the bending in the \(X\) direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is located at the back of the top measurement point. For the bending in the \(Y\) direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).} \caption{\label{fig:test_apa_meas_setup_modes}Experimental setup to measure the flexible modes of the APA300ML. For the bending in the \(X\) direction (\subref{fig:test_apa_meas_setup_X_bending}), the impact point is at the back of the top measurement point. For the bending in the \(Y\) direction (\subref{fig:test_apa_meas_setup_Y_bending}), the impact point is located on the back surface of the top interface (on the back of the 2 measurements points).}
\end{figure} \end{figure}
The measured frequency response functions computed from the experimental setups of figures \ref{fig:test_apa_meas_setup_X_bending} and \ref{fig:test_apa_meas_setup_Y_bending} are shown in Figure \ref{fig:test_apa_meas_freq_compare}. The measured frequency response functions computed from the experimental setups of figures \ref{fig:test_apa_meas_setup_X_bending} and \ref{fig:test_apa_meas_setup_Y_bending} are shown in Figure \ref{fig:test_apa_meas_freq_compare}.
The \(y\) bending mode is observed at \(280\,\text{Hz}\) and the \(x\) bending mode is at \(412\,\text{Hz}\). The \(y\) bending mode is observed at \(280\,\text{Hz}\) and the \(x\) bending mode is at \(412\,\text{Hz}\).
These modes are measured at higher frequencies than the estimated frequencies from the Finite Element Model (see frequencies in Figure \ref{fig:test_apa_mode_shapes}). These modes are measured at higher frequencies than the frequencies estimated from the Finite Element Model (see frequencies in Figure \ref{fig:test_apa_mode_shapes}).
This is opposite to what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model). This is the opposite of what is usually observed (i.e. having lower resonance frequencies in practice than the estimation from a finite element model).
This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used). This could be explained by underestimation of the Young's modulus of the steel used for the shell (190 GPa was used for the model, but steel with Young's modulus of 210 GPa could have been used).
Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades. Another explanation is the shape difference between the manufactured APA300ML and the 3D model, for instance thicker blades.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/test_apa_meas_freq_compare.png} \includegraphics[scale=1]{figs/test_apa_meas_freq_compare.png}
\caption{\label{fig:test_apa_meas_freq_compare}Frequency response functions for the 2 tests with the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at \(280\,\text{Hz}\) and the X-bending mode at \(412\,\text{Hz}\)} \caption{\label{fig:test_apa_meas_freq_compare}Frequency response functions for the two tests using the instrumented hammer and the laser vibrometer. The Y-bending mode is measured at \(280\,\text{Hz}\) and the X-bending mode at \(412\,\text{Hz}\)}
\end{figure} \end{figure}
\chapter{Dynamical measurements} \chapter{Dynamical measurements}
\label{sec:test_apa_dynamics} \label{sec:test_apa_dynamics}
After the measurements on the APA were performed in Section \ref{sec:test_apa_basic_meas}, a new test bench is used to better characterize the dynamics of the APA300ML. After the measurements on the APA were performed in Section \ref{sec:test_apa_basic_meas}, a new test bench was used to better characterize the dynamics of the APA300ML.
This test bench, depicted in Figure \ref{fig:test_bench_apa}, comprises the APA300ML fixed at one end to a stationary granite block, and at the other end to a 5kg granite block that is vertically guided by an air bearing. This test bench, depicted in Figure \ref{fig:test_bench_apa}, comprises the APA300ML fixed at one end to a stationary granite block and at the other end to a 5kg granite block that is vertically guided by an air bearing.
That way, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors. Thus, there is no friction when actuating the APA300ML, and it will be easier to characterize its behavior independently of other factors.
An encoder\footnote{Renishaw Vionic, resolution of \(2.5\,nm\)} is utilized to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the APA. An encoder\footnote{Renishaw Vionic, resolution of \(2.5\,nm\)} is used to measure the relative movement between the two granite blocks, thereby measuring the axial displacement of the APA.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.3\textwidth} \begin{subfigure}{0.3\textwidth}
@ -274,11 +274,11 @@ An encoder\footnote{Renishaw Vionic, resolution of \(2.5\,nm\)} is utilized to m
\caption{\label{fig:test_bench_apa}Schematic of the test bench used to estimate the dynamics of the APA300ML} \caption{\label{fig:test_bench_apa}Schematic of the test bench used to estimate the dynamics of the APA300ML}
\end{figure} \end{figure}
The bench is schematically shown in Figure \ref{fig:test_apa_schematic} with all the associated signals. The bench is schematically shown in Figure \ref{fig:test_apa_schematic} with the associated signals.
It will be first used to estimate the hysteresis from the piezoelectric stack (Section \ref{ssec:test_apa_hysteresis}) as well as the axial stiffness of the APA300ML (Section \ref{ssec:test_apa_stiffness}). It will be first used to estimate the hysteresis from the piezoelectric stack (Section \ref{ssec:test_apa_hysteresis}) as well as the axial stiffness of the APA300ML (Section \ref{ssec:test_apa_stiffness}).
Then, the frequency response functions from the DAC voltage \(u\) to the displacement \(d_e\) and to the voltage \(V_s\) are measured in Section \ref{ssec:test_apa_meas_dynamics}. The frequency response functions from the DAC voltage \(u\) to the displacement \(d_e\) and to the voltage \(V_s\) are measured in Section \ref{ssec:test_apa_meas_dynamics}.
The presence of a non minimum phase zero found on the transfer function from \(u\) to \(V_s\) is investigated in Section \ref{ssec:test_apa_non_minimum_phase}. The presence of a non-minimum phase zero found on the transfer function from \(u\) to \(V_s\) is investigated in Section \ref{ssec:test_apa_non_minimum_phase}.
In order to limit the low frequency gain of the transfer function from \(u\) to \(V_s\), a resistor is added across the force sensor stack (Section \ref{ssec:test_apa_resistance_sensor_stack}). To limit the low-frequency gain of the transfer function from \(u\) to \(V_s\), a resistor is added across the force sensor stack (Section \ref{ssec:test_apa_resistance_sensor_stack}).
Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section \ref{ssec:test_apa_iff_locus}. Finally, the Integral Force Feedback is implemented, and the amount of damping added is experimentally estimated in Section \ref{ssec:test_apa_iff_locus}.
\begin{figure}[htbp] \begin{figure}[htbp]
@ -289,10 +289,10 @@ Finally, the Integral Force Feedback is implemented, and the amount of damping a
\section{Hysteresis} \section{Hysteresis}
\label{ssec:test_apa_hysteresis} \label{ssec:test_apa_hysteresis}
As the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload. Because the payload is vertically guided without friction, the hysteresis of the APA can be estimated from the motion of the payload.
Do to so, a quasi static\footnote{Frequency of the sinusoidal wave is \(1\,\text{Hz}\)} sinusoidal excitation \(V_a\) with an offset of \(65\,V\) (halfway between \(-20\,V\) and \(150\,V\)) and with an amplitude varying from \(4\,V\) up to \(80\,V\) is generated using the DAC. A quasi static\footnote{Frequency of the sinusoidal wave is \(1\,\text{Hz}\)} sinusoidal excitation \(V_a\) with an offset of \(65\,V\) (halfway between \(-20\,V\) and \(150\,V\)) and with an amplitude varying from \(4\,V\) up to \(80\,V\) is generated using the DAC.
For each excitation amplitude, the vertical displacement \(d_e\) of the mass is measured and displayed as a function of the applied voltage in Figure \ref{fig:test_apa_meas_hysteresis}. For each excitation amplitude, the vertical displacement \(d_e\) of the mass is measured and displayed as a function of the applied voltage in Figure \ref{fig:test_apa_meas_hysteresis}.
This is the typical behavior expected from a PZT stack actuator where the hysteresis increases as a function of the applied voltage amplitude \cite[chap. 1.4]{fleming14_desig_model_contr_nanop_system}. This is the typical behavior expected from a PZT stack actuator, where the hysteresis increases as a function of the applied voltage amplitude \cite[chap. 1.4]{fleming14_desig_model_contr_nanop_system}.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -303,7 +303,7 @@ This is the typical behavior expected from a PZT stack actuator where the hyster
\section{Axial stiffness} \section{Axial stiffness}
\label{ssec:test_apa_stiffness} \label{ssec:test_apa_stiffness}
In order to estimate the stiffness of the APA, a weight with known mass \(m_a = 6.4\,\text{kg}\) is added on top of the suspended granite and the deflection \(\Delta d_e\) is measured using the encoder. To estimate the stiffness of the APA, a weight with known mass \(m_a = 6.4\,\text{kg}\) is added on top of the suspended granite and the deflection \(\Delta d_e\) is measured using the encoder.
The APA stiffness can then be estimated from equation \eqref{eq:test_apa_stiffness}, with \(g \approx 9.8\,m/s^2\) the acceleration of gravity. The APA stiffness can then be estimated from equation \eqref{eq:test_apa_stiffness}, with \(g \approx 9.8\,m/s^2\) the acceleration of gravity.
\begin{equation} \label{eq:test_apa_stiffness} \begin{equation} \label{eq:test_apa_stiffness}
@ -311,10 +311,10 @@ The APA stiffness can then be estimated from equation \eqref{eq:test_apa_stiffne
\end{equation} \end{equation}
The measured displacement \(d_e\) as a function of time is shown in Figure \ref{fig:test_apa_meas_stiffness_time}. The measured displacement \(d_e\) as a function of time is shown in Figure \ref{fig:test_apa_meas_stiffness_time}.
It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep) and the that displacement does not come back to the initial position after the mass is removed (probably due to piezoelectric hysteresis). It can be seen that there are some drifts in the measured displacement (probably due to piezoelectric creep), and that the displacement does not return to the initial position after the mass is removed (probably due to piezoelectric hysteresis).
These two effects induce some uncertainties in the measured stiffness. These two effects induce some uncertainties in the measured stiffness.
The stiffnesses are computed for all the APA from the two displacements \(d_1\) and \(d_2\) (see Figure \ref{fig:test_apa_meas_stiffness_time}) leading to two stiffness estimations \(k_1\) and \(k_2\). The stiffnesses are computed for all APAs from the two displacements \(d_1\) and \(d_2\) (see Figure \ref{fig:test_apa_meas_stiffness_time}) leading to two stiffness estimations \(k_1\) and \(k_2\).
These estimated stiffnesses are summarized in Table \ref{tab:test_apa_measured_stiffnesses} and are found to be close to the specified nominal stiffness of the APA300ML \(k = 1.8\,N/\mu m\). These estimated stiffnesses are summarized in Table \ref{tab:test_apa_measured_stiffnesses} and are found to be close to the specified nominal stiffness of the APA300ML \(k = 1.8\,N/\mu m\).
\begin{minipage}[b]{0.57\linewidth} \begin{minipage}[b]{0.57\linewidth}
@ -349,18 +349,18 @@ The stiffness can also be computed using equation \eqref{eq:test_apa_res_freq} b
\omega_z = \sqrt{\frac{k}{m_{\text{sus}}}} \omega_z = \sqrt{\frac{k}{m_{\text{sus}}}}
\end{equation} \end{equation}
The obtained stiffness is \(k \approx 2\,N/\mu m\) which is close to the values found in the documentation and by the ``static deflection'' method. The obtained stiffness is \(k \approx 2\,N/\mu m\) which is close to the values found in the documentation and using the ``static deflection'' method.
It is important to note that changes to the electrical impedance connected to the piezoelectric stacks impacts the mechanical compliance (or stiffness) of the piezoelectric stack \cite[chap. 2]{reza06_piezoel_trans_vibrat_contr_dampin}. It is important to note that changes to the electrical impedance connected to the piezoelectric stacks affect the mechanical compliance (or stiffness) of the piezoelectric stack \cite[chap. 2]{reza06_piezoel_trans_vibrat_contr_dampin}.
To estimate this effect for the APA300ML, its stiffness is estimated using the ``static deflection'' method in two cases: To estimate this effect for the APA300ML, its stiffness is estimated using the ``static deflection'' method in two cases:
\begin{itemize} \begin{itemize}
\item \(k_{\text{os}}\): piezoelectric stacks left unconnected (or connect to the high impedance ADC) \item \(k_{\text{os}}\): piezoelectric stacks left unconnected (or connect to the high impedance ADC)
\item \(k_{\text{sc}}\): piezoelectric stacks short circuited (or connected to the voltage amplifier with small output impedance) \item \(k_{\text{sc}}\): piezoelectric stacks short-circuited (or connected to the voltage amplifier with small output impedance)
\end{itemize} \end{itemize}
It is found that the open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,N/\mu m\) while the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,N/\mu m\). The open-circuit stiffness is estimated at \(k_{\text{oc}} \approx 2.3\,N/\mu m\) while the closed-circuit stiffness \(k_{\text{sc}} \approx 1.7\,N/\mu m\).
\section{Dynamics} \section{Dynamics}
\label{ssec:test_apa_meas_dynamics} \label{ssec:test_apa_meas_dynamics}
@ -368,26 +368,26 @@ It is found that the open-circuit stiffness is estimated at \(k_{\text{oc}} \app
In this section, the dynamics from the excitation voltage \(u\) to the encoder measured displacement \(d_e\) and to the force sensor voltage \(V_s\) is identified. In this section, the dynamics from the excitation voltage \(u\) to the encoder measured displacement \(d_e\) and to the force sensor voltage \(V_s\) is identified.
First, the dynamics from \(u\) to \(d_e\) for the six APA300ML are compared in Figure \ref{fig:test_apa_frf_encoder}. First, the dynamics from \(u\) to \(d_e\) for the six APA300ML are compared in Figure \ref{fig:test_apa_frf_encoder}.
The obtained frequency response functions are similar to that of a (second order) mass-spring-damper system with: The obtained frequency response functions are similar to those of a (second order) mass-spring-damper system with:
\begin{itemize} \begin{itemize}
\item A ``stiffness line'' indicating a static gain equal to \(\approx -17\,\mu m/V\). \item A ``stiffness line'' indicating a static gain equal to \(\approx -17\,\mu m/V\).
The minus sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the APA The negative sign comes from the fact that an increase in voltage stretches the piezoelectric stack which reduces the height of the APA
\item A lightly damped resonance at \(95\,\text{Hz}\) \item A lightly damped resonance at \(95\,\text{Hz}\)
\item A ``mass line'' up to \(\approx 800\,\text{Hz}\), above which additional resonances appear. These additional resonances might be coming from the limited stiffness of the encoder support or from the limited compliance of the APA support. \item A ``mass line'' up to \(\approx 800\,\text{Hz}\), above which additional resonances appear. These additional resonances might be due to the limited stiffness of the encoder support or from the limited compliance of the APA support.
Flexible modes studied in section \ref{ssec:test_apa_spurious_resonances} seem not to impact the measured axial motion of the actuator. The flexible modes studied in section \ref{ssec:test_apa_spurious_resonances} seem not to impact the measured axial motion of the actuator.
\end{itemize} \end{itemize}
The dynamics from \(u\) to the measured voltage across the sensor stack \(V_s\) for the six APA300ML are compared in Figure \ref{fig:test_apa_frf_force}. The dynamics from \(u\) to the measured voltage across the sensor stack \(V_s\) for the six APA300ML are compared in Figure \ref{fig:test_apa_frf_force}.
A lightly damped resonance (pole) is observed at \(95\,\text{Hz}\) and a lightly damped anti-resonance (zero) at \(41\,\text{Hz}\). A lightly damped resonance (pole) is observed at \(95\,\text{Hz}\) and a lightly damped anti-resonance (zero) at \(41\,\text{Hz}\).
No additional resonances is present up to at least \(2\,\text{kHz}\) indicating that Integral Force Feedback can be applied without stability issues from high frequency flexible modes. No additional resonances are present up to at least \(2\,\text{kHz}\) indicating that Integral Force Feedback can be applied without stability issues from high-frequency flexible modes.
The zero at \(41\,\text{Hz}\) seems to be non-minimum phase (the phase \emph{decreases} by 180 degrees whereas it should have \emph{increased} by 180 degrees for a minimum phase zero). The zero at \(41\,\text{Hz}\) seems to be non-minimum phase (the phase \emph{decreases} by 180 degrees whereas it should have \emph{increased} by 180 degrees for a minimum phase zero).
This is investigated in Section \ref{ssec:test_apa_non_minimum_phase}. This is investigated in Section \ref{ssec:test_apa_non_minimum_phase}.
As illustrated by the Root Locus, the poles of the \emph{closed-loop} system converges to the zeros of the \emph{open-loop} plant as the feedback gain increases. As illustrated by the Root Locus plot, the poles of the \emph{closed-loop} system converges to the zeros of the \emph{open-loop} plant as the feedback gain increases.
The significance of this behavior varies on the type of sensor used as explained in \cite[chap. 7.6]{preumont18_vibrat_contr_activ_struc_fourt_edition}. The significance of this behavior varies with the type of sensor used, as explained in \cite[chap. 7.6]{preumont18_vibrat_contr_activ_struc_fourt_edition}.
Considering the transfer function from \(u\) to \(V_s\), if a controller with a very high gain is applied such that the sensor stack voltage \(V_s\) is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain. Considering the transfer function from \(u\) to \(V_s\), if a controller with a very high gain is applied such that the sensor stack voltage \(V_s\) is kept at zero, the sensor (and by extension, the actuator stacks since they are in series) experiences negligible stress and strain.
Consequently, the closed-loop system would virtually corresponds to one where the piezoelectric stacks are absent, leaving only the mechanical shell. Consequently, the closed-loop system virtually corresponds to one in which the piezoelectric stacks are absent, leaving only the mechanical shell.
From this analysis, it can be inferred that the axial stiffness of the shell is \(k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m\) (which is close to what is found using a finite element model). From this analysis, it can be inferred that the axial stiffness of the shell is \(k_{\text{shell}} = m \omega_0^2 = 5.7 \cdot (2\pi \cdot 41)^2 = 0.38\,N/\mu m\) (which is close to what is found using a finite element model).
All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure \ref{fig:test_apa_frf_encoder} and at the force sensor in Figure \ref{fig:test_apa_frf_force}) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell. All the identified dynamics of the six APA300ML (both when looking at the encoder in Figure \ref{fig:test_apa_frf_encoder} and at the force sensor in Figure \ref{fig:test_apa_frf_force}) are almost identical, indicating good manufacturing repeatability for the piezoelectric stacks and the mechanical shell.
@ -411,16 +411,15 @@ All the identified dynamics of the six APA300ML (both when looking at the encode
\section{Non Minimum Phase Zero?} \section{Non Minimum Phase Zero?}
\label{ssec:test_apa_non_minimum_phase} \label{ssec:test_apa_non_minimum_phase}
It was surprising to observe a non-minimum phase behavior for the zero on the transfer function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_frf_force}). It was surprising to observe a non-minimum phase zero on the transfer function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_frf_force}).
It was initially thought that this non-minimum phase behavior is an artifact coming from the measurement. It was initially thought that this non-minimum phase behavior was an artifact arising from the measurement.
A longer measurement was performed with different excitation signals (noise, slow sine sweep, etc.) to see it the phase behavior of the zero changes. A longer measurement was performed using different excitation signals (noise, slow sine sweep, etc.) to determine if the phase behavior of the zero changes (Figure \ref{fig:test_apa_non_minimum_phase}).
Results of one long measurement is shown in Figure \ref{fig:test_apa_non_minimum_phase}.
The coherence (Figure \ref{fig:test_apa_non_minimum_phase_coherence}) is good even in the vicinity of the lightly damped zero, and the phase (Figure \ref{fig:test_apa_non_minimum_phase_zoom}) clearly indicates non-minimum phase behavior. The coherence (Figure \ref{fig:test_apa_non_minimum_phase_coherence}) is good even in the vicinity of the lightly damped zero, and the phase (Figure \ref{fig:test_apa_non_minimum_phase_zoom}) clearly indicates non-minimum phase behavior.
Such non-minimum phase zero when using load cells has also been observed on other mechanical systems \cite{spanos95_soft_activ_vibrat_isolat,thayer02_six_axis_vibrat_isolat_system,hauge04_sensor_contr_space_based_six}. Such non-minimum phase zero when using load cells has also been observed on other mechanical systems \cite{spanos95_soft_activ_vibrat_isolat,thayer02_six_axis_vibrat_isolat_system,hauge04_sensor_contr_space_based_six}.
It could be induced to small non-linearity in the system, but the reason of this non-minimum phase for the APA300ML is not yet clear. It could be induced to small non-linearity in the system, but the reason for this non-minimum phase for the APA300ML is not yet clear.
However, this is not so important here as the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure \ref{fig:test_apa_iff_root_locus}) should not be unstable except for very large controller gains that will never be applied in practice. However, this is not so important here because the zero is lightly damped (i.e. very close to the imaginary axis), and the closed loop poles (see the Root Locus plot in Figure \ref{fig:test_apa_iff_root_locus}) should not be unstable, except for very large controller gains that will never be applied in practice.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth} \begin{subfigure}{0.49\textwidth}
@ -435,19 +434,19 @@ However, this is not so important here as the zero is lightly damped (i.e. very
\end{center} \end{center}
\subcaption{\label{fig:test_apa_non_minimum_phase_zoom} Zoom on the non-minimum phase zero} \subcaption{\label{fig:test_apa_non_minimum_phase_zoom} Zoom on the non-minimum phase zero}
\end{subfigure} \end{subfigure}
\caption{\label{fig:test_apa_non_minimum_phase}Measurement of the anti-resonance found on the transfer function from \(u\) to \(V_s\). The coherence (\subref{fig:test_apa_non_minimum_phase_coherence}) is quite good around the anti-resonance frequency. The phase (\subref{fig:test_apa_non_minimum_phase_zoom}) shoes a non-minimum phase behavior.} \caption{\label{fig:test_apa_non_minimum_phase}Measurement of the anti-resonance found in the transfer function from \(u\) to \(V_s\). The coherence (\subref{fig:test_apa_non_minimum_phase_coherence}) is quite good around the anti-resonance frequency. The phase (\subref{fig:test_apa_non_minimum_phase_zoom}) shoes a non-minimum phase behavior.}
\end{figure} \end{figure}
\section{Effect of the resistor on the IFF Plant} \section{Effect of the resistor on the IFF Plant}
\label{ssec:test_apa_resistance_sensor_stack} \label{ssec:test_apa_resistance_sensor_stack}
A resistor \(R \approx 80.6\,k\Omega\) is added in parallel with the sensor stack which has the effect to form a high pass filter with the capacitance of the piezoelectric stack (capacitance estimated at \(\approx 5\,\mu F\)). A resistor \(R \approx 80.6\,k\Omega\) is added in parallel with the sensor stack, which forms a high-pass filter with the capacitance of the piezoelectric stack (capacitance estimated at \(\approx 5\,\mu F\)).
As explain before, this is done to limit the voltage offset due to the input bias current of the ADC as well as to limit the low frequency gain. As explained before, this is done to limit the voltage offset due to the input bias current of the ADC as well as to limit the low frequency gain.
The (low frequency) transfer function from \(u\) to \(V_s\) with and without this resistor have been measured and are compared in Figure \ref{fig:test_apa_effect_resistance}. The (low frequency) transfer function from \(u\) to \(V_s\) with and without this resistor were measured and compared in Figure \ref{fig:test_apa_effect_resistance}.
It is confirmed that the added resistor as the effect of adding an high pass filter with a cut-off frequency of \(\approx 0.39\,\text{Hz}\). It is confirmed that the added resistor has the effect of adding a high-pass filter with a cut-off frequency of \(\approx 0.39\,\text{Hz}\).
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -458,23 +457,23 @@ It is confirmed that the added resistor as the effect of adding an high pass fil
\section{Integral Force Feedback} \section{Integral Force Feedback}
\label{ssec:test_apa_iff_locus} \label{ssec:test_apa_iff_locus}
In order to implement the Integral Force Feedback strategy, the measured frequency response function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_frf_force}) is fitted using the transfer function shown in equation \eqref{eq:test_apa_iff_manual_fit}. To implement the Integral Force Feedback strategy, the measured frequency response function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_frf_force}) is fitted using the transfer function shown in equation \eqref{eq:test_apa_iff_manual_fit}.
The parameters are manually tuned, and the obtained values are \(\omega_{\textsc{hpf}} = 0.4\, \text{Hz}\), \(\omega_{z} = 42.7\, \text{Hz}\), \(\xi_{z} = 0.4\,\%\), \(\omega_{p} = 95.2\, \text{Hz}\), \(\xi_{p} = 2\,\%\) and \(g_0 = 0.64\). The parameters were manually tuned, and the obtained values are \(\omega_{\textsc{hpf}} = 0.4\, \text{Hz}\), \(\omega_{z} = 42.7\, \text{Hz}\), \(\xi_{z} = 0.4\,\%\), \(\omega_{p} = 95.2\, \text{Hz}\), \(\xi_{p} = 2\,\%\) and \(g_0 = 0.64\).
\begin{equation} \label{eq:test_apa_iff_manual_fit} \begin{equation} \label{eq:test_apa_iff_manual_fit}
G_{\textsc{iff},m}(s) = g_0 \cdot \frac{1 + 2 \xi_z \frac{s}{\omega_z} + \frac{s^2}{\omega_z^2}}{1 + 2 \xi_p \frac{s}{\omega_p} + \frac{s^2}{\omega_p^2}} \cdot \frac{s}{\omega_{\textsc{hpf}} + s} G_{\textsc{iff},m}(s) = g_0 \cdot \frac{1 + 2 \xi_z \frac{s}{\omega_z} + \frac{s^2}{\omega_z^2}}{1 + 2 \xi_p \frac{s}{\omega_p} + \frac{s^2}{\omega_p^2}} \cdot \frac{s}{\omega_{\textsc{hpf}} + s}
\end{equation} \end{equation}
The comparison between the identified plant and the manually tuned transfer function is done in Figure \ref{fig:test_apa_iff_plant_comp_manual_fit}. A comparison between the identified plant and the manually tuned transfer function is shown in Figure \ref{fig:test_apa_iff_plant_comp_manual_fit}.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/test_apa_iff_plant_comp_manual_fit.png} \includegraphics[scale=1]{figs/test_apa_iff_plant_comp_manual_fit.png}
\caption{\label{fig:test_apa_iff_plant_comp_manual_fit}Identified IFF plant and manually tuned model of the plant (a time delay of \(200\,\mu s\) is added to the model of the plant to better match the identified phase). Note that a minimum-phase zero is here identified even though the coherence is not good arround the frequency of the zero.} \caption{\label{fig:test_apa_iff_plant_comp_manual_fit}Identified IFF plant and manually tuned model of the plant (a time delay of \(200\,\mu s\) is added to the model of the plant to better match the identified phase). Note that a minimum-phase zero is identified here even though the coherence is not good around the frequency of the zero.}
\end{figure} \end{figure}
The implemented Integral Force Feedback Controller transfer function is shown in equation \eqref{eq:test_apa_Kiff_formula}. The implemented Integral Force Feedback Controller transfer function is shown in equation \eqref{eq:test_apa_Kiff_formula}.
It contains an high pass filter (cut-off frequency of \(2\,\text{Hz}\)) to limit the low frequency gain, a low pass filter to add integral action above \(20\,\text{Hz}\), a second low pass filter to add robustness to high frequency resonances and a tunable gain \(g\). It contains a high-pass filter (cut-off frequency of \(2\,\text{Hz}\)) to limit the low-frequency gain, a low-pass filter to add integral action above \(20\,\text{Hz}\), a second low-pass filter to add robustness to high-frequency resonances, and a tunable gain \(g\).
\begin{equation} \label{eq:test_apa_Kiff_formula} \begin{equation} \label{eq:test_apa_Kiff_formula}
K_{\textsc{iff}}(s) = -10 \cdot g \cdot \frac{s}{s + 2\pi \cdot 2} \cdot \frac{1}{s + 2\pi \cdot 20} \cdot \frac{1}{s + 2\pi\cdot 2000} K_{\textsc{iff}}(s) = -10 \cdot g \cdot \frac{s}{s + 2\pi \cdot 2} \cdot \frac{1}{s + 2\pi \cdot 20} \cdot \frac{1}{s + 2\pi\cdot 2000}
@ -489,19 +488,21 @@ The transfer function from the ``damped'' plant input \(u\prime\) to the encoder
\caption{\label{fig:test_apa_iff_schematic}Implementation of Integral Force Feedback in the Speedgoat. The damped plant has a new input \(u\prime\)} \caption{\label{fig:test_apa_iff_schematic}Implementation of Integral Force Feedback in the Speedgoat. The damped plant has a new input \(u\prime\)}
\end{figure} \end{figure}
The identified dynamics are then fitted by second order transfer functions\footnote{The transfer function fitting was computed using the \texttt{vectfit3} routine, see \cite{gustavsen99_ration_approx_frequen_domain_respon}}. The identified dynamics were then fitted by second order transfer functions\footnote{The transfer function fitting was computed using the \texttt{vectfit3} routine, see \cite{gustavsen99_ration_approx_frequen_domain_respon}}.
The comparison between the identified damped dynamics and the fitted second order transfer functions is done in Figure \ref{fig:test_apa_identified_damped_plants} for different gains \(g\). A comparison between the identified damped dynamics and the fitted second-order transfer functions is shown in Figure \ref{fig:test_apa_identified_damped_plants} for different gains \(g\).
It is clear that large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies. It is clear that a large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies.
The evolution of the pole in the complex plane as a function of the controller gain \(g\) (i.e. the ``root locus'') is computed both using the IFF plant model \eqref{eq:test_apa_iff_manual_fit} and the implemented controller \eqref{eq:test_apa_Kiff_formula} and from the fitted transfer functions of the damped plants experimentally identified for several controller gains. The evolution of the pole in the complex plane as a function of the controller gain \(g\) (i.e. the ``root locus'') is computed in two cases.
The two obtained root loci are compared in Figure \ref{fig:test_apa_iff_root_locus} and are in good agreement considering that the damped plants were only fitted using a second order transfer function. First using the IFF plant model \eqref{eq:test_apa_iff_manual_fit} and the implemented controller \eqref{eq:test_apa_Kiff_formula}.
Second using the fitted transfer functions of the damped plants experimentally identified for several controller gains.
The two obtained root loci are compared in Figure \ref{fig:test_apa_iff_root_locus} and are in good agreement considering that the damped plants were fitted using only a second-order transfer function.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.59\textwidth} \begin{subfigure}{0.59\textwidth}
\begin{center} \begin{center}
\includegraphics[scale=1,height=8cm]{figs/test_apa_identified_damped_plants.png} \includegraphics[scale=1,height=8cm]{figs/test_apa_identified_damped_plants.png}
\end{center} \end{center}
\subcaption{\label{fig:test_apa_identified_damped_plants}Measured frequency response functions of damped plants for several IFF gains (solid lines). Identified 2nd order plants to match the experimental data (dashed lines)} \subcaption{\label{fig:test_apa_identified_damped_plants}Measured frequency response functions of damped plants for several IFF gains (solid lines). Identified 2nd order plants that match the experimental data (dashed lines)}
\end{subfigure} \end{subfigure}
\begin{subfigure}{0.39\textwidth} \begin{subfigure}{0.39\textwidth}
\begin{center} \begin{center}
@ -513,12 +514,12 @@ The two obtained root loci are compared in Figure \ref{fig:test_apa_iff_root_loc
\end{figure} \end{figure}
\chapter{APA300ML - 2 Degrees of Freedom Model} \chapter{APA300ML - 2 degrees-of-freedom Model}
\label{sec:test_apa_model_2dof} \label{sec:test_apa_model_2dof}
In this section, a Simscape model (Figure \ref{fig:test_apa_bench_model}) of the measurement bench is used to tune the 2 degrees of freedom model of the APA using the measured frequency response functions. In this section, a Simscape model (Figure \ref{fig:test_apa_bench_model}) of the measurement bench is used to tune the two degrees-of-freedom model of the APA using the measured frequency response functions.
This 2 degrees of freedom model is developed to accurately represent the APA300ML dynamics while having low complexity and low number of associated states. This two degrees-of-freedom model is developed to accurately represent the APA300ML dynamics while having low complexity and a low number of associated states.
After the model presented, the procedure to tune the model is described and the obtained model dynamics is compared with the measurements. After the model is presented, the procedure for tuning the model is described, and the obtained model dynamics is compared with the measurements.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -526,30 +527,30 @@ After the model presented, the procedure to tune the model is described and the
\caption{\label{fig:test_apa_bench_model}Screenshot of the Simscape model} \caption{\label{fig:test_apa_bench_model}Screenshot of the Simscape model}
\end{figure} \end{figure}
\paragraph{Two Degrees of Freedom APA Model} \paragraph{Two degrees-of-freedom APA Model}
The model of the amplified piezoelectric actuator is shown in Figure \ref{fig:test_apa_2dof_model}. The model of the amplified piezoelectric actuator is shown in Figure \ref{fig:test_apa_2dof_model}.
It can be decomposed into three components: It can be decomposed into three components:
\begin{itemize} \begin{itemize}
\item the shell whose axial properties are represented by \(k_1\) and \(c_1\) \item the shell whose axial properties are represented by \(k_1\) and \(c_1\)
\item the actuator stacks whose contribution in the axial stiffness is represented by \(k_a\) and \(c_a\). \item the actuator stacks whose contribution to the axial stiffness is represented by \(k_a\) and \(c_a\).
A force source \(\tau\) represents the axial force induced by the force sensor stacks. The force source \(\tau\) 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 \(\tau\) The sensitivity \(g_a\) (in \(N/m\)) is used to convert the applied voltage \(V_a\) to the axial force \(\tau\)
\item the sensor stack whose contribution in the axial stiffness is represented by \(k_e\) and \(c_e\). \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 \(V/m\)) 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} \end{itemize}
Such simple model has some limitations: Such a simple model has some limitations:
\begin{itemize} \begin{itemize}
\item it only represents the axial characteristics of the APA as it is modelled as infinitely rigid in the other directions \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, the real stress and strain in the piezoelectric stacks \item some physical insights are lost, such as the amplification factor and the real stress and strain in the piezoelectric stacks
\item it is fully linear and therefore the creep and hysteresis of the piezoelectric stacks are not modelled \item the creep and hysteresis of the piezoelectric stacks are not modeled as the model is linear
\end{itemize} \end{itemize}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/test_apa_2dof_model.png} \includegraphics[scale=1]{figs/test_apa_2dof_model.png}
\caption{\label{fig:test_apa_2dof_model}Schematic of the two degrees of freedom model of the APA300ML, adapted from \cite{souleille18_concep_activ_mount_space_applic}} \caption{\label{fig:test_apa_2dof_model}Schematic of the two degrees-of-freedom model of the APA300ML, adapted from \cite{souleille18_concep_activ_mount_space_applic}}
\end{figure} \end{figure}
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:test_apa_2dof_model_Simscape}) well represents the identified dynamics in Section \ref{sec:test_apa_dynamics}. 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:test_apa_2dof_model_Simscape}) well represents the identified dynamics in Section \ref{sec:test_apa_dynamics}.
@ -557,18 +558,18 @@ Such simple model has some limitations:
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/test_apa_2dof_model_Simscape.png} \includegraphics[scale=1]{figs/test_apa_2dof_model_Simscape.png}
\caption{\label{fig:test_apa_2dof_model_Simscape}Schematic of the two degrees of freedom model of the APA300ML with input \(V_a\) and outputs \(d_e\) and \(V_s\)} \caption{\label{fig:test_apa_2dof_model_Simscape}Schematic of the two degrees-of-freedom model of the APA300ML with input \(V_a\) and outputs \(d_e\) and \(V_s\)}
\end{figure} \end{figure}
First, the mass \(m\) supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale. First, the mass \(m\) supported by the APA300ML can be estimated from the geometry and density of the different parts or by directly measuring it using a precise weighing scale.
Both methods leads to an estimated mass of \(m = 5.7\,\text{kg}\). Both methods lead to an estimated mass of \(m = 5.7\,\text{kg}\).
Then, the axial stiffness of the shell was estimated at \(k_1 = 0.38\,N/\mu m\) in Section \ref{ssec:test_apa_meas_dynamics} from the frequency of the anti-resonance seen on Figure \ref{fig:test_apa_frf_force}. Then, the axial stiffness of the shell was estimated at \(k_1 = 0.38\,N/\mu m\) in Section \ref{ssec:test_apa_meas_dynamics} from the frequency of the anti-resonance seen on Figure \ref{fig:test_apa_frf_force}.
Similarly, \(c_1\) can be estimated from the damping ratio of the same anti-resonance and is found to be close to \(20\,Ns/m\). Similarly, \(c_1\) can be estimated from the damping ratio of the same anti-resonance and is found to be close to \(20\,Ns/m\).
Then, it is reasonable to make the assumption that the sensor stacks and the two actuator stacks have identical mechanical characteristics\footnote{Note that this is not fully correct as it was shown in Section \ref{ssec:test_apa_stiffness} that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited.}. Then, it is reasonable to assume that the sensor stacks and the two actuator stacks have identical mechanical characteristics\footnote{Note that this is not completely correct as it was shown in Section \ref{ssec:test_apa_stiffness} that the electrical boundaries of the piezoelectric stack impacts its stiffness and that the sensor stack is almost open-circuited while the actuator stacks are almost short-circuited.}.
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. 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 that case, the total stiffness of the APA model is described by \eqref{eq:test_apa_2dof_stiffness}. In this case, the total stiffness of the APA model is described by \eqref{eq:test_apa_2dof_stiffness}.
\begin{equation}\label{eq:test_apa_2dof_stiffness} \begin{equation}\label{eq:test_apa_2dof_stiffness}
k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a k_{\text{tot}} = k_1 + \frac{k_e k_a}{k_e + k_a} = k_1 + \frac{2}{3} k_a
@ -583,7 +584,7 @@ Knowing from \eqref{eq:test_apa_tot_stiffness} that the total stiffness is \(k_{
Then, \(c_a\) (and therefore \(c_e = 2 c_a\)) can be tuned to match the damping ratio of the identified resonance. Then, \(c_a\) (and therefore \(c_e = 2 c_a\)) can be tuned to match the damping ratio of the identified resonance.
\(c_a = 100\,Ns/m\) and \(c_e = 200\,Ns/m\) are obtained. \(c_a = 100\,Ns/m\) and \(c_e = 200\,Ns/m\) are obtained.
Finally \(g_s\) and \(g_a\) can be tuned to match the gain of the identified transfer functions. In the last step, \(g_s\) and \(g_a\) can be tuned to match the gain of the identified transfer functions.
The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_model_Simscape} are summarized in Table \ref{tab:test_apa_2dof_parameters}. The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_model_Simscape} are summarized in Table \ref{tab:test_apa_2dof_parameters}.
@ -608,8 +609,8 @@ The obtained parameters of the model shown in Figure \ref{fig:test_apa_2dof_mode
\end{table} \end{table}
The dynamics of the two degrees of freedom model of the APA300ML is now extracted using optimized parameters (listed in Table \ref{tab:test_apa_2dof_parameters}) from the Simscape model. The dynamics of the two degrees-of-freedom model of the APA300ML are extracted using optimized parameters (listed in Table \ref{tab:test_apa_2dof_parameters}) from the Simscape model.
It is compared with the experimental data in Figure \ref{fig:test_apa_2dof_comp_frf}. This is compared with the experimental data in Figure \ref{fig:test_apa_2dof_comp_frf}.
A good match can be observed between the model and the experimental data, both for the encoder (Figure \ref{fig:test_apa_2dof_comp_frf_enc}) and for the force sensor (Figure \ref{fig:test_apa_2dof_comp_frf_force}). A good match can be observed between the model and the experimental data, both for the encoder (Figure \ref{fig:test_apa_2dof_comp_frf_enc}) and for the force sensor (Figure \ref{fig:test_apa_2dof_comp_frf_force}).
This indicates that this model represents well the axial dynamics of the APA300ML. This indicates that this model represents well the axial dynamics of the APA300ML.
@ -632,15 +633,14 @@ This indicates that this model represents well the axial dynamics of the APA300M
\chapter{APA300ML - Super Element} \chapter{APA300ML - Super Element}
\label{sec:test_apa_model_flexible} \label{sec:test_apa_model_flexible}
In this section, a \emph{super element} of the APA300ML is computed using a finite element software\footnote{Ansys\textsuperscript{\textregistered} was used}. In this section, a \emph{super element} of the APA300ML is computed using a finite element software\footnote{Ansys\textsuperscript{\textregistered} was used}.
It is then imported in Simscape (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in \ref{sec:test_apa_model_2dof}. It is then imported into Simscape (in the form of a stiffness matrix and a mass matrix) and included in the same model that was used in \ref{sec:test_apa_model_2dof}.
This procedure is illustrated in Figure \ref{fig:test_apa_super_element_Simscape}. This procedure is illustrated in Figure \ref{fig:test_apa_super_element_Simscape}.
Several \emph{remote points} are defined in the finite element model (here illustrated by colorful planes and numbers from \texttt{1} to \texttt{5}) and are then make accessible in the Simscape model as shown at the right by the ``frames'' \texttt{F1} to \texttt{F5}. Several \emph{remote points} are defined in the finite element model (here illustrated by colorful planes and numbers from \texttt{1} to \texttt{5}) and are then made accessible in the Simscape model as shown at the right by the ``frames'' \texttt{F1} to \texttt{F5}.
For the APA300ML \emph{super element}, 5 \emph{remote points} are defined. For the APA300ML \emph{super element}, 5 \emph{remote points} are defined.
Two \emph{remote points} (\texttt{1} and \texttt{2}) are fixed to the top and bottom mechanical interfaces of the APA300ML and will be used for connecting the APA300ML with other mechanical elements. Two \emph{remote points} (\texttt{1} and \texttt{2}) are fixed to the top and bottom mechanical interfaces of the APA300ML and will be used to connect the APA300ML with other mechanical elements.
Two \emph{remote points} (\texttt{3} and \texttt{4}) are located across two piezoelectric stacks and will be used to apply internal forces representing the actuator stacks. Two \emph{remote points} (\texttt{3} and \texttt{4}) are located across two piezoelectric stacks and are used to apply internal forces representing the actuator stacks.
Finally, two \emph{remote points} (\texttt{4} and \texttt{4}) are located across the third piezoelectric stack. Finally, two \emph{remote points} (\texttt{4} and \texttt{4}) are located across the third piezoelectric stack, and will be used to measured the strain of the sensor stack.
It will be used to measure the strain experience by this stack, and model the sensor stack.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -651,10 +651,10 @@ It will be used to measure the strain experience by this stack, and model the se
\paragraph{Identification of the Actuator and Sensor constants} \paragraph{Identification of the Actuator and Sensor constants}
Once the APA300ML \emph{super element} is included in the Simscape model, the transfer function from \(F_a\) to \(d_L\) and \(d_e\) can be extracted. Once the APA300ML \emph{super element} is included in the Simscape 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 be tuned such that the gain of the transfer functions are matching the identified ones. The gains \(g_a\) and \(g_s\) are then tuned such that the gains of the transfer functions match the identified ones.
By doing so, \(g_s = 4.9\,V/\mu m\) and \(g_a = 23.2\,N/V\) are obtained. By doing so, \(g_s = 4.9\,V/\mu m\) and \(g_a = 23.2\,N/V\) are obtained.
To make sure the sensitivities \(g_a\) and \(g_s\) are physically valid, it is possible to estimate them from physical properties of the piezoelectric stack material. To ensure that the sensitivities \(g_a\) and \(g_s\) are physically valid, it is possible to estimate them from the physical properties of the piezoelectric stack material.
From \cite[p. 123]{fleming14_desig_model_contr_nanop_system}, the relation between relative displacement \(d_L\) of the sensor stack and generated voltage \(V_s\) is given by \eqref{eq:test_apa_piezo_strain_to_voltage} and from \cite{fleming10_integ_strain_force_feedb_high} the relation between the force \(F_a\) and the applied voltage \(V_a\) is given by \eqref{eq:test_apa_piezo_voltage_to_force}. From \cite[p. 123]{fleming14_desig_model_contr_nanop_system}, the relation between relative displacement \(d_L\) of the sensor stack and generated voltage \(V_s\) is given by \eqref{eq:test_apa_piezo_strain_to_voltage} and from \cite{fleming10_integ_strain_force_feedb_high} the relation between the force \(F_a\) and the applied voltage \(V_a\) is given by \eqref{eq:test_apa_piezo_voltage_to_force}.
@ -666,10 +666,10 @@ From \cite[p. 123]{fleming14_desig_model_contr_nanop_system}, the relation betwe
\end{subequations} \end{subequations}
Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML. Unfortunately, the manufacturer of the stack was not willing to share the piezoelectric material properties of the stack used in the APA300ML.
However, based on available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties. However, based on the available properties of the APA300ML stacks in the data-sheet, the soft Lead Zirconate Titanate ``THP5H'' from Thorlabs seemed to match quite well the observed properties.
The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) are listed in Table \ref{tab:test_apa_piezo_properties}. The properties of this ``THP5H'' material used to compute \(g_a\) and \(g_s\) are listed in Table \ref{tab:test_apa_piezo_properties}.
From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtained which are close to the identified constants using the experimentally identified transfer functions. From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtained, which are close to the constants identified using the experimentally identified transfer functions.
\begin{table}[htbp] \begin{table}[htbp]
\centering \centering
@ -693,10 +693,10 @@ From these parameters, \(g_s = 5.1\,V/\mu m\) and \(g_a = 26\,N/V\) were obtaine
\paragraph{Comparison of the obtained dynamics} \paragraph{Comparison of the obtained dynamics}
The obtained dynamics using the \emph{super element} with the tuned ``sensor sensitivity'' and ``actuator sensitivity'' are compared with the experimentally identified frequency response functions in Figure \ref{fig:test_apa_super_element_comp_frf}. The obtained dynamics using the \emph{super element} with the tuned ``sensor sensitivity'' and ``actuator sensitivity'' are compared with the experimentally identified frequency response functions in Figure \ref{fig:test_apa_super_element_comp_frf}.
A good match between the model and the experimental results is observed. A good match between the model and the experimental results was observed.
It is however surprising that the model is ``softer'' than the measured system as finite element models are usually overestimating the stiffness (see Section \ref{ssec:test_apa_spurious_resonances} for possible explanations). It is however surprising that the model is ``softer'' than the measured system, as finite element models usually overestimate the stiffness (see Section \ref{ssec:test_apa_spurious_resonances} for possible explanations).
Using this simple test bench, it can be concluded that the \emph{super element} model of the APA300ML well captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever). Using this simple test bench, it can be concluded that the \emph{super element} model of the APA300ML captures the axial dynamics of the actuator (the actuator stacks, the force sensor stack as well as the shell used as a mechanical lever).
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth} \begin{subfigure}{0.49\textwidth}
@ -717,24 +717,24 @@ Using this simple test bench, it can be concluded that the \emph{super element}
\chapter{Conclusion} \chapter{Conclusion}
\label{sec:test_apa_conclusion} \label{sec:test_apa_conclusion}
In this study, the amplified piezoelectric actuators ``APA300ML'' have been characterized to make sure they are fulfilling all the requirements determined during the detailed design phase. In this study, the amplified piezoelectric actuators ``APA300ML'' have been characterized to ensure that they fulfill all the requirements determined during the detailed design phase.
Geometrical features such as the flatness of its interfaces, electrical capacitance and achievable strokes were measured in Section \ref{sec:test_apa_basic_meas}. Geometrical features such as the flatness of its interfaces, electrical capacitance, and achievable strokes were measured in Section \ref{sec:test_apa_basic_meas}.
These simple measurements allowed for early detection of a manufacturing defect in one of the APA300ML. These simple measurements allowed for the early detection of a manufacturing defect in one of the APA300ML.
Then in Section \ref{sec:test_apa_dynamics}, using a dedicated test bench, the dynamics of all the APA300ML were measured and were found to all match very well (Figure \ref{fig:test_apa_frf_dynamics}). Then in Section \ref{sec:test_apa_dynamics}, using a dedicated test bench, the dynamics of all the APA300ML were measured and were found to all match very well (Figure \ref{fig:test_apa_frf_dynamics}).
This consistency indicates good manufacturing tolerances, facilitating the modeling and control of the nano-hexapod. This consistency indicates good manufacturing tolerances, facilitating the modeling and control of the nano-hexapod.
Although a non-minimum zero was identified in the transfer function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_non_minimum_phase}), it was found not to be problematic as large amount of damping could be added using the integral force feedback strategy (Figure \ref{fig:test_apa_iff}). Although a non-minimum zero was identified in the transfer function from \(u\) to \(V_s\) (Figure \ref{fig:test_apa_non_minimum_phase}), it was found not to be problematic because a large amount of damping could be added using the integral force feedback strategy (Figure \ref{fig:test_apa_iff}).
Then, two different models were used to represent the APA300ML dynamics. Then, two different models were used to represent the APA300ML dynamics.
In Section \ref{sec:test_apa_model_2dof}, a simple two degrees of freedom mass-spring-damper model was presented and tuned based on the measured dynamics. In Section \ref{sec:test_apa_model_2dof}, a simple two degrees-of-freedom mass-spring-damper model was presented and tuned based on the measured dynamics.
After following a tuning procedure, the model dynamics was shown to match very well with the experiment. After following a tuning procedure, the model dynamics was shown to match very well with the experiment.
However, it is important to note that this model only represents the axial dynamics of the actuators, assuming infinite stiffness in other directions. However, this model only represents the axial dynamics of the actuators, assuming infinite stiffness in other directions.
In Section \ref{sec:test_apa_model_flexible}, a \emph{super element} extracted from a finite element model was used to model the APA300ML. In Section \ref{sec:test_apa_model_flexible}, a \emph{super element} extracted from a finite element model was used to model the APA300ML.
This time, the \emph{super element} represents the dynamics of the APA300ML in all directions. Here, the \emph{super element} represents the dynamics of the APA300ML in all directions.
However, only the axial dynamics could be compared with the experimental results yielding a good match. However, only the axial dynamics could be compared with the experimental results, yielding a good match.
The benefit of employing this model over the two degrees of freedom model is not immediately apparent due to its increased complexity and the larger number of model states involved. The benefit of employing this model over the two degrees-of-freedom model is not immediately apparent due to its increased complexity and the larger number of model states involved.
Nonetheless, the \emph{super element} model's value will become clear in subsequent sections, when its capacity to accurately model the APA300ML's flexibility across various directions will be important. Nonetheless, the \emph{super element} model's value will become clear in subsequent sections, when its capacity to accurately model the APA300ML's flexibility across various directions will be important.
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