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content/zettels/actuators.md
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:
Links to specific actuators:
## Short Stroke Actuators {#short-stroke-actuators}
For short stroke and very high dynamic applications, mainly two types of actuators can be used:
- [Voice Coil Actuators]({{< relref "voice_coil_actuators.md" >}})
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}})
## Long Stroke Actuators {#long-stroke-actuators}
- Linear motors
- Piezoelectric Walking Drive ([PI](https://www.physikinstrumente.com/en/expertise/technology/piezoelectric-drives/piezowalk-piezo-motors/))
Rotational drives can be combined with ball-screw mechanisms for long (infinite) axial motion:
- Brush-less DC Motor. See (<a href="#citeproc_bib_item_2">Yedamale 2003</a>) and this [working principle](https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html).
- [Stepper Motor]({{< relref "stepper_motor.md" >}})
## How to choose the correct actuator for my application? {#how-to-choose-the-correct-actuator-for-my-application}
For vibration isolation:
@@ -20,16 +34,6 @@ For vibration isolation:
- In (<a href="#citeproc_bib_item_1">Ito and Schitter 2016</a>), the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat.md" >}}))
## Brush-less DC Motor {#brush-less-dc-motor}
- (<a href="#citeproc_bib_item_2">Yedamale 2003</a>)
<https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html>
## [Stepper Motor]({{< relref "stepper_motor.md" >}}) {#stepper-motor--stepper-motor-dot-md}
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">

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title = "Continuous Transfer Functions"
author = ["Dehaeze Thomas"]
draft = false
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:
## Typical Transfer Functions {#typical-transfer-functions}

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title = "Digital Filters"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
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@@ -9,4 +9,8 @@ Tags
A nice open access book on digital filter is accessible here: <https://ccrma.stanford.edu/~jos/filters/filters.html>
<./biblio/references.bib>
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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title = "Discrete Transfer Functions"
author = ["Dehaeze Thomas"]
draft = false
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Tags
: [Digital Filters]({{< relref "digital_filters.md" >}})
## Typical Transfer functions {#typical-transfer-functions}
### Delay {#delay}
### First Order Low Pass {#first-order-low-pass}
### First Order High Pass {#first-order-high-pass}
### Integrator {#integrator}
### Derivator {#derivator}
### Second Order Low Pass {#second-order-low-pass}
### PID {#pid}
### Notch {#notch}
### Moving Average {#moving-average}

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Tags
:
TODOS:
- [X] Electronics circuits containing input voltage, output voltage, R L and C components
- [ ] Bode plot of the filter from input voltage to output voltage
- [ ] Equation of the transfer functions with nice parameters (\\(\omega\_c\\), \\(\xi\\))
## First Order Low Pass Filter {#first-order-low-pass-filter}
<a id="figure--fig:elec-passive-first-order-low-pass-filter"></a>
<a id="figure--fig:analog-filt-first-order-lpf"></a>
{{< figure src="/ox-hugo/elec_passive_first_order_low_pass_filter.png" caption="<span class=\"figure-number\">Figure 1: </span>First Order Low Pass Filter using an RC circuit" >}}
{{< figure src="/ox-hugo/analog_filt_first_order_lpf.png" caption="<span class=\"figure-number\">Figure 1: </span>First Order Low Pass Filter using an RC circuit" >}}
\begin{equation}
\boxed{\frac{V\_o}{V\_i}(s) = \frac{1}{1 + \frac{s}{\omega\_0}}, \quad \omega\_0 = \frac{1}{RC}}
\end{equation}
\begin{equation}
Z\_f(s) = \frac{V\_i}{i\_i}(s) = \frac{1 + RC \cdot s}{C \cdot s}
\end{equation}
```matlab
%% First Order Low Pass Filter
R = 1e3; % [Ohm]
C = 1e-6; % [F]
```
```matlab
G = 1/(1 + R*C*s); % Filter Transfer Function
Z = (1 + R*C*s)/(C*s); % Filter Impedance
```
<a id="figure--fig:analog-filt-first-order-lpf-characteristics"></a>
{{< figure src="/ox-hugo/analog_filt_first_order_lpf_characteristics.png" caption="<span class=\"figure-number\">Figure 2: </span>First Order Low Pass Filter - Filter transfer function and filter impedance" >}}
## First Order High Pass Filter {#first-order-high-pass-filter}
<a id="figure--fig:elec-passive-first-order-high-pass-filter"></a>
<a id="figure--fig:analog-filt-first-order-hpf"></a>
{{< figure src="/ox-hugo/elec_passive_first_order_high_pass_filter.png" caption="<span class=\"figure-number\">Figure 2: </span>First Order High Pass Filter using an RC circuit" >}}
{{< figure src="/ox-hugo/analog_filt_first_order_hpf.png" caption="<span class=\"figure-number\">Figure 3: </span>First Order High Pass Filter using an RC circuit" >}}
\begin{equation}
\boxed{\frac{V\_o}{V\_i}(s) = \frac{\frac{s}{\omega\_0}}{1 + \frac{s}{\omega\_0}}, \quad \omega\_0 = \frac{1}{RC}}
\end{equation}
\begin{equation}
Z\_f(s) = \frac{V\_i}{i\_i}(s) = \frac{1 + RC \cdot s}{C \cdot s}
\end{equation}
```matlab
%% First Order High Pass Filter
R = 1e3; % [Ohm]
C = 1e-6; % [F]
```
```matlab
G = R*C*s/(1 + R*C*s); % Filter Transfer Function
Z = (1 + R*C*s)/(C*s); % Filter Impedance
```
<a id="figure--fig:analog-filt-first-order-hpf-characteristics"></a>
{{< figure src="/ox-hugo/analog_filt_first_order_hpf_characteristics.png" caption="<span class=\"figure-number\">Figure 4: </span>First Order High Pass Filter - Filter transfer function and filter impedance" >}}
## Second Order Low Pass Filter {#second-order-low-pass-filter}
<a id="figure--fig:elec-passive-second-order-low-pass-filter"></a>
<a id="figure--fig:analog-filt-second-order-lpf"></a>
{{< figure src="/ox-hugo/elec_passive_second_order_low_pass_filter.png" caption="<span class=\"figure-number\">Figure 3: </span>Second Order Low Pass Filter using an RLC circuit" >}}
{{< figure src="/ox-hugo/analog_filt_second_order_lpf.png" caption="<span class=\"figure-number\">Figure 5: </span>Second Order Low Pass Filter using an RLC circuit" >}}
\begin{equation}
\boxed{\frac{V\_o}{V\_i}(s) = \frac{1}{1 + 2 \xi \frac{s}{\omega\_0} + \frac{s^2}{\omega\_0^2}}, \quad \omega\_0 = \frac{1}{\sqrt{LC}},\quad \xi = \frac{1}{2R\sqrt{LC}}}
\end{equation}
\begin{equation}
Z\_f(s) = \frac{V\_i}{i\_i}(s) = L \cdot s + \frac{R}{1 + RC \cdot s}
\end{equation}
```matlab
%% Second Order Low Pass Filter
R = 1e2; % [Ohm]
C = 1e-5; % [F]
L = 1e-2; % [H]
```
```matlab
G = 1/(1 + L/R*s + C*L*s^2); % Filter Transfer Function
Z = L*s + (R)/(1 + R*C*s); % Filter Impedance
```
<a id="figure--fig:analog-filt-second-order-lpf-characteristics"></a>
{{< figure src="/ox-hugo/analog_filt_second_order_lpf_characteristics.png" caption="<span class=\"figure-number\">Figure 6: </span>Second Order Low Pass Filter - Filter transfer function and filter impedance" >}}
## Second Order High Pass Filter {#second-order-high-pass-filter}
<a id="figure--fig:elec-passive-second-order-high-pass-filter"></a>
<a id="figure--fig:analog-filt-second-order-hpf"></a>
{{< figure src="/ox-hugo/elec_passive_second_order_high_pass_filter.png" caption="<span class=\"figure-number\">Figure 4: </span>Second Order High Pass Filter using an RLC circuit" >}}
{{< figure src="/ox-hugo/analog_filt_second_order_hpf.png" caption="<span class=\"figure-number\">Figure 7: </span>Second Order High Pass Filter using an RLC circuit" >}}
## Bibliography {#bibliography}

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title = "Laplace Transform"
author = ["Dehaeze Thomas"]
draft = false
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Tags
: [Continuous Transfer Functions]({{< relref "continuous_transfer_functions.md" >}})
## Definition {#definition}
Laplace transform:
\begin{equation}
\boxed{F(s) = \mathcal{L}{f(t)} = \int\_0^\infty e^{-st} f(t) dt}
\end{equation}
## Laplace transform of signals {#laplace-transform-of-signals}
<https://en.wikipedia.org/wiki/List_of_Laplace_transforms>
## Laplace transform of functions {#laplace-transform-of-functions}
<https://en.wikibooks.org/wiki/Signals_and_Systems/Table_of_Laplace_Transforms>
## Solving Linear Differential equations {#solving-linear-differential-equations}
### Mechanical System {#mechanical-system}
### Electrical System {#electrical-system}

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@@ -62,11 +62,13 @@ The Amplified Piezo Actuators principle is presented in (<a href="#citeproc_bib_
> The displacement amplification effect is related in a first approximation to the ratio of the shell long axis length to the short axis height.
> The flatter is the actuator, the higher is the amplification.
A model of an amplified piezoelectric actuator is described in (<a href="#citeproc_bib_item_3">Lucinskis and Mangeot 2016</a>).
A model of an amplified piezoelectric actuator is described in (<a href="#citeproc_bib_item_4">Lucinskis and Mangeot 2016</a>).
Typical topology of mechanically amplified piezoelectric actuators are displayed in Figure [1](#figure--fig:ling16-topology-piezo-mechanism-types) (from (<a href="#citeproc_bib_item_3">Ling et al. 2016</a>)).
<a id="figure--fig:ling16-topology-piezo-mechanism-types"></a>
{{< figure src="/ox-hugo/ling16_topology_piezo_mechanism_types.png" caption="<span class=\"figure-number\">Figure 1: </span>Topology of several types of compliant mechanisms <ling16_enhan_mathem_model_displ_amplif>" >}}
{{< figure src="/ox-hugo/ling16_topology_piezo_mechanism_types.png" caption="<span class=\"figure-number\">Figure 1: </span>Topology of several types of compliant mechanisms" >}}
| Manufacturers | Country |
|----------------------------------------------------------------------------------------------------|-----------|
@@ -208,5 +210,6 @@ Limitations of the electronics is discussed in [Design, modeling and control of
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static &#38; Dynamic Applications.” <i>Ferroelectrics</i> 351 (1): 3–14. doi:<a href="https://doi.org/10.1080/00150190701351865">10.1080/00150190701351865</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>Ieee/Asme Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_3"></a>Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”</div>
<div class="csl-entry"><a id="citeproc_bib_item_3"></a>Ling, Mingxiang, Junyi Cao, Minghua Zeng, Jing Lin, and Daniel J Inman. 2016. “Enhanced Mathematical Modeling of the Displacement Amplification Ratio for Piezoelectric Compliant Mechanisms.” <i>Smart Materials and Structures</i> 25 (7): 075022. doi:<a href="https://doi.org/10.1088/0964-1726/25/7/075022">10.1088/0964-1726/25/7/075022</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_4"></a>Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”</div>
</div>

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:
## 2 phase VS 5 phase stepper motor {#2-phase-vs-5-phase-stepper-motor}
## Errors between steps (micro-stepping) {#errors-between-steps--micro-stepping}
<https://www.orientalmotor.com/stepper-motors/technology/2-phase-vs-5-phase-stepper-motors.html>
For a two phase stepper motor, there are (typically) **200 steps per revolution** (i.e. 1.8% per step).
Between each step, even when using some micro-stepping, there are some position errors that are due to non-perfect magnetic and electromagnetic fields.
## Errors {#errors}
The period of this error is corresponding to 200 period/revolution.
For a two phase stepper motor, there are (typically) 200 steps per revolution.
Errors with a period of 200 period/revolution can be expected.
Then scanning, this can lead to high frequency vibrations.
This is what is typically limiting the accuracy of the stepper motor (usually specified in between 3% and 5% of the step increment).
This is approximately corresponding to **1mrad**.
<div class="exampl">
Consider a stepper motor with 200 steps by turn attached to a ball-screw with a pitch of 1mm per turn.
A rotation of 1 turn per second will induce vibrations at 200Hz with an amplitude of \\(1\\,\mu m\\).
</div>
Note that this error is not a pure sine, it also has some harmonics with corresponding periods of 1/100 revolution and 1/50 revolution.
This error should repeat every turn and can be calibrated provided it is repeatable over time.
One way to reduce these errors is to use a ball-screw mechanism with a smaller pitch.
The price to pay is smaller velocity.
## Manufacturers {#manufacturers}
@@ -25,3 +42,14 @@ Errors with a period of 200 period/revolution can be expected.
|--------------------------------------------------------------------------|----------------|
| [AML](https://arunmicro.com/) | United Kingdom |
| [Sanyo](https://www.sanyodenki.com/catalogs/servo/stepping_systems.html) | |
## 2 phase VS 5 phase stepper motor {#2-phase-vs-5-phase-stepper-motor}
<https://www.orientalmotor.com/stepper-motors/technology/2-phase-vs-5-phase-stepper-motors.html>
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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title = "Z-Transform"
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Discrete Transfer Functions]({{< relref "discrete_transfer_functions.md" >}})
## From Continuous to Discrete transfer function {#from-continuous-to-discrete-transfer-function}
A continuous transfer function (i.e. in the Laplace domain) can easily be converted to the discrete time domain (i.e. in the z-domain) using the `c2d` Matlab command ([doc](https://fr.mathworks.com/help/control/ref/lti.c2d.html;jsessionid=206bd0fc8950c5f8cb6b568d7393#mw_53fc4689-2099-41d0-93b3-de1e51a174c1)).
Several methods can be used, each with some advantages and drawbacks, see [this document](https://fr.mathworks.com/help/control/ug/continuous-discrete-conversion-methods.html).