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title = "Exploring the pareto fronts of actuation technologies for high performance mechatronic systems"
draft = true
+++
Tags
:
Reference
: (<a href="#citeproc_bib_item_1">Csencsics and Schitter 2020</a>)
Author(s)
: Csencsics, E., &amp; Schitter, G.
Year
: 2020
## Abstract {#abstract}
> This paper proposes a novel method for estimating the limitations of individual actuation technologies for a desired system class based on analytically obtained relations, which can be used to systematically trade off desired range and speed specifications in the design phase.
> The method is presented along the example of **fast steering mirrors** with the tradeoff limit curves estimated for the established **piezoelectric**, **lorentz force** and **hybrid reluctance** actuation technologies.
<a id="figure--fig:csencsics20-fsm-schematic"></a>
{{< figure src="/ox-hugo/csencsics20_fsm_schematic.png" caption="<span class=\"figure-number\">Figure 1: </span>Fast Steering Mirror system. The main components are: mirror, actuators, position sensors and suspension system." >}}
## Fast Steering Mirrors {#fast-steering-mirrors}
### Application area and performance specification {#application-area-and-performance-specification}
<a id="table--tab:fsm-requirements"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:fsm-requirements">Table 1</a>:</span>
FSM performance requirements for two application
</div>
| Application | Pointing | Scanning |
|-------------------|-----------------|----------|
| System Range | large | large |
| System Dimensions | arbitrary | compact |
| Main objective | dist. rejection | tracking |
| Bandwidth | high | high |
| Motion amplitude | small | large |
| Mover inertia | arbitrary | small |
| Precision | high | high |
### Safe operating area {#safe-operating-area}
The concept of the Safe Operating Area (SOA) relates the frequency of a sinusoidal reference to the maximum admissible scan amplitude that still stays within the limits of the system.
From figure [2](#figure--fig:csencsics20-soa) we can already see that piezo are typically used for system with high bandwidth and small range.
<a id="figure--fig:csencsics20-soa"></a>
{{< figure src="/ox-hugo/csencsics20_soa.png" caption="<span class=\"figure-number\">Figure 1: </span>Measured safe operating area of closed-loop FSM systems with sinusoidal reference signals. Piezo actuated in blue, lorentz force actuated in red and hybrid reluctance actuated in green." >}}
## Limitations of actuator technology {#limitations-of-actuator-technology}
### Piezo actuation {#piezo-actuation}
Piezo actuated FMS are in general **high stiffness** system, for which the **bandwidth limitation** for feedback control is typically given by the **first mechanical resonance**.
<a id="figure--fig:csencsics20-typical-piezo-fsm"></a>
{{< figure src="/ox-hugo/csencsics20_typical_piezo_fsm.png" caption="<span class=\"figure-number\">Figure 1: </span>Piezo actuated FSM cross section" >}}
The angular range of the FSM is:
\begin{equation}
\phi = \frac{L/1000}{2 d}
\end{equation}
with \\(L\\) the length of the stack, and d the distance between the stacks and the center of rotation (the factor 1000 is linked to the fact that typical piezo stack have a store equal to 0.1% of their length).
The first resonance frequency is:
\begin{equation}
f\_{PZA} = \frac{1}{2\pi L}\sqrt{\frac{3E}{\rho\_\text{piezo}}}
\end{equation}
with \\(E\\) the elastic modulus and \\(\rho\_\text{piezo}\\) the density of the piezo material.
As the resonance limits the achievable bandwidth, we therefore have that \\(f\_{\text{max,PZA}} \propto 1/\phi\\).
### Lorentz force actuation {#lorentz-force-actuation}
Lorentz force actuated FSM are in general **low stiffness** systems, which typically have a control bandwidth beyond the suspension mode that is usually limited by the **internal modes of the moving part**.
The mover's mass is dominating the dynamics of low stiffness systems beyond the suspension mode.
<a id="figure--fig:csencsics20-typical-lorentz-fsm"></a>
{{< figure src="/ox-hugo/csencsics20_typical_lorentz_fsm.png" caption="<span class=\"figure-number\">Figure 1: </span>Lorentz force actuator designs." >}}
\begin{equation}
f\_\text{max,LFA} = \frac{1}{2\pi} k\_\text{LFA} \sqrt{\frac{1}{\phi J\_\text{init} + \Delta\_J + 2 d \phi^2}}
\end{equation}
### Hybrid reluctance force actuation {#hybrid-reluctance-force-actuation}
<a id="figure--fig:csencsics20-typical-hybrid-reluctance-fsm"></a>
{{< figure src="/ox-hugo/csencsics20_typical_hybrid_reluctance_fsm.png" caption="<span class=\"figure-number\">Figure 1: </span>Hybrid reluctance actuator designs" >}}
## Pareto front estimates for FSM systems {#pareto-front-estimates-for-fsm-systems}
<a id="figure--fig:csencsics20-pareto-estimate"></a>
{{< figure src="/ox-hugo/csencsics20_pareto_estimate.png" caption="<span class=\"figure-number\">Figure 1: </span>Two dimensional performance space for FSM systems showing the tradeoff between range and bandwidth. Commercially available (symbols) as well as academically reported systems (dots) actuated by piezo (blue), Lorentz force (red) and reluctance actuators (green) are depicted." >}}
<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>Csencsics, Ernst, and Georg Schitter. 2020. “Exploring the Pareto Fronts of Actuation Technologies for High Performance Mechatronic Systems.” <i>IEEE/ASME Transactions on Mechatronics</i>. IEEE.</div>
</div>

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title = "The design of high performance mechatronics - third revised edition"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "Awesome book that gives great overview of high performance mechatronic systems"
keywords = ["Metrology", "Mechatronics", "Control"]
draft = false
+++
Tags
: [Reference Books]({{< relref "reference_books" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
: [Reference Books]({{< relref "reference_books.md" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting.md" >}})
Reference
: ([Schmidt, Schitter, and Rankers 2020](#org4e5c703))
: (<a href="#citeproc_bib_item_1">Schmidt, Schitter, and Rankers 2020</a>)
Author(s)
: Schmidt, R. M., Schitter, G., & Rankers, A.
: Schmidt, R. M., Schitter, G., &amp; Rankers, A.
Year
: 2020
@ -66,9 +66,9 @@ Year
#### Electric Field {#electric-field}
<a id="org16b370d"></a>
<a id="figure--fig:schmidt20-electrical-field"></a>
{{< figure src="/ox-hugo/schmidt20_electrical_field.svg" caption="Figure 1: Charges have an electric field" >}}
{{< figure src="/ox-hugo/schmidt20_electrical_field.svg" caption="<span class=\"figure-number\">Figure 1: </span>Charges have an electric field" >}}
##### Potential Difference and Capacitance {#potential-difference-and-capacitance}
@ -172,14 +172,13 @@ The term "rms" refers to Root Mean Square, named from the action of taking the r
The RMS value is a well known term used to characterize the "useful" value of the energy supply with a signal by comparing it with an equivalent DC voltage that would cause the same power in a resistive load.
<div class="exampl">
<div></div>
For a sinusoidal signal \\(V(t) = V\_p \sin(\omega t)\\), the equivalent DC voltage becomes:
\begin{equation}
\begin{aligned}
V\_{\text{rms}} &= \sqrt{\frac{1}{T} \int\_0^T \left< V\_p\sin(\omega t) \right>^2 dt} \\\\\\
&= \dots \\\\\\
V\_{\text{rms}} &= \sqrt{\frac{1}{T} \int\_0^T \left< V\_p\sin(\omega t) \right>^2 dt} \\\\
&= \dots \\\\
&= \frac{V\_p}{\sqrt{2}} \quad [V]
\end{aligned}
\end{equation}
@ -213,7 +212,7 @@ Its inverse, the spatial period length called _wavelength_ \\(\lambda\\) [m] is
The relation between the above defined terms is:
\begin{align}
\lambda &= \frac{1}{\nu} = c\_p T = \frac{c\_p}{f} \quad [m] \\\\\\
\lambda &= \frac{1}{\nu} = c\_p T = \frac{c\_p}{f} \quad [m] \\\\
v\_p &= \frac{f}{\nu} \quad [m/s]
\end{align}
@ -221,11 +220,11 @@ The relation between the above defined terms is:
##### Mechanical Waves {#mechanical-waves}
The propagation speed value of a mechanical wave is mostly determined by the density and elasticity of the medium.
The wave propagation through an elastic material can be qualitatively explained with the help of a simplified lumped element model, consisting of a chain of springs and bodies as shown in Figure [2](#orgaa33fb8).
The wave propagation through an elastic material can be qualitatively explained with the help of a simplified lumped element model, consisting of a chain of springs and bodies as shown in Figure [2](#figure--fig:schmidt20-mechanical-wave).
<a id="orgaa33fb8"></a>
<a id="figure--fig:schmidt20-mechanical-wave"></a>
{{< figure src="/ox-hugo/schmidt20_mechanical_wave.svg" caption="Figure 2: Lumped element model of one wavelength of a mechanical wave." >}}
{{< figure src="/ox-hugo/schmidt20_mechanical_wave.svg" caption="<span class=\"figure-number\">Figure 1: </span>Lumped element model of one wavelength of a mechanical wave." >}}
To explain the principle of energy transfer, the longitudinal wave is taken as example.
When a movement of mass \\(m\_1\\) is introduced in the propagation direction of the chain, this will first cause a compression of the elastic coupling \\(k\_1\\).
@ -234,7 +233,6 @@ This process is repeated over the total chain until the original movement reache
With this mechanism the kinetic energy from mass \\(m\_1\\) is converted into potential energy in \\(k\_1\\), which in turn is transferred into kinetic energy of \\(m\_2\\), and so on.
<div class="important">
<div></div>
This phenomenon of transfer of energy in an elastic body is important in mechatronic systems because driving forces are also transported through the body as a wave and as a consequence will experience a delay between the actuator and the sensor when they are located separately.
@ -247,7 +245,6 @@ The propagation speed \\(c\_p\\) is determined by the density \\(\rho\_m\\) and
\end{equation}
<div class="exampl">
<div></div>
The propagation speed in stainless steel varies between 3500 m/s for transversal waves and 5500 m/s for longitudinal waves.
With for instance half a meter of steel this gives a delay of about 0.1ms, resulting in a phase delay of 36 degrees at 1kHz, which can be significant from a control point of view.
@ -384,7 +381,7 @@ The values are given in Table [1](#table--tab:relation-slope-decade) for a decad
<a id="table--tab:relation-slope-decade"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:relation-slope-decade">Table 1</a></span>:
<span class="table-number"><a href="#table--tab:relation-slope-decade">Table 1</a>:</span>
Relation between the order of the slope of a bode plot and the magnitude ration in dB, amplitude ratio and power ration, per <b>decade</b> (\(f_1 = 10 f_2\))
</div>
@ -398,7 +395,7 @@ The values are given in Table [1](#table--tab:relation-slope-decade) for a decad
<a id="table--tab:relation-slope-octave"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:relation-slope-octave">Table 2</a></span>:
<span class="table-number"><a href="#table--tab:relation-slope-octave">Table 1</a>:</span>
Relation between the order of the slope of a bode plot and the magnitude ration in dB, amplitude ratio and power ration, per <b>octave</b> (\(f_1 = 2 f_2\))
</div>
@ -589,7 +586,8 @@ The values are given in Table [1](#table--tab:relation-slope-decade) for a decad
### Summary on Dynamics {#summary-on-dynamics}
<summary>
<div class="sum">
In this chapter some important lessons have been learned, which are summarised as follows:
- Stiffness, whether it is created mechanically or by means of a control system, is determinative for precision
@ -604,7 +602,8 @@ In this chapter some important lessons have been learned, which are summarised a
- Modal analysis is a powerful and widely applied tool to investigate the dynamics of a mechanical structure.
Finally it can be concluded, that these insights help in designing actively controlled dynamic motion systems with optimally located actuators and sensors, which reduce the sensitivity for modal dynamic problems.
</summary>
</div>
## Motion Control {#motion-control}
@ -612,15 +611,15 @@ Finally it can be concluded, that these insights help in designing actively cont
### A Walk around the Control Loop {#a-walk-around-the-control-loop}
Figure [3](#org6432052) shows a basic control loop of a positioning system.
Figure [1](#figure--fig:schmidt20-walk-control-loop) shows a basic control loop of a positioning system.
First, the A/D and D/A converters are used to translate analog signals into time-discrete digital signals and vice versa.
Secondly, the impact locations of several disturbances are shown, which play a large role in determining what reqwuirements the controller needs to fulfil.
The core of the control system is the _plant_, which is the physical system that needs to be controlled.
<a id="table--tab:walk-control-loop"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:walk-control-loop">Table 3</a></span>:
Symbols used in Figure <a href="#org6432052">10</a>
<span class="table-number"><a href="#table--tab:walk-control-loop">Table 1</a>:</span>
Symbols used in Figure <a href="#orge5a2ac8">3</a>
</div>
| Symbol | Meaning | Unit |
@ -634,16 +633,16 @@ The core of the control system is the _plant_, which is the physical system that
| \\(y\\) | Measured output motion | [m] |
| \\(y\_m\\) | Measurement value | [m] |
<a id="org6432052"></a>
<a id="figure--fig:schmidt20-walk-control-loop"></a>
{{< figure src="/ox-hugo/schmidt20_walk_control_loop.svg" caption="Figure 3: Block diagram of a motion control system, including feedforward and feedback control." >}}
{{< figure src="/ox-hugo/schmidt20_walk_control_loop.svg" caption="<span class=\"figure-number\">Figure 1: </span>Block diagram of a motion control system, including feedforward and feedback control." >}}
The plant combines the mechanical structure, amplifiers and actuators, as they all deal with energy conversion in close interaction (Figure [4](#org21aa9c3)).
The plant combines the mechanical structure, amplifiers and actuators, as they all deal with energy conversion in close interaction (Figure [1](#figure--fig:schmidt20-energy-actuator-system)).
They interact in both directions in such a way that each element not only determines the input of the next element, but also influences the previous element by its dynamic load.
<a id="org21aa9c3"></a>
<a id="figure--fig:schmidt20-energy-actuator-system"></a>
{{< figure src="/ox-hugo/schmidt20_energy_actuator_system.svg" caption="Figure 4: The energy converting part of a mechatronic system consists of a the amplifier, the actuator and the mechanical structure." >}}
{{< figure src="/ox-hugo/schmidt20_energy_actuator_system.svg" caption="<span class=\"figure-number\">Figure 1: </span>The energy converting part of a mechatronic system consists of a the amplifier, the actuator and the mechanical structure." >}}
#### Poles and Zeros in Motion Control {#poles-and-zeros-in-motion-control}
@ -672,7 +671,7 @@ Fortunately the effect is mostly so small that it can be neglected.
#### Overview Feedforward Control {#overview-feedforward-control}
Figure [5](#org4b3a329) shows the typical basic configuration for feedforward control, which is also called _open-loop control_ as it is equal to a situation where the measured output is not connected to the input for feedback.
Figure [1](#figure--fig:schmidt20-feedforward-control-diagram) shows the typical basic configuration for feedforward control, which is also called _open-loop control_ as it is equal to a situation where the measured output is not connected to the input for feedback.
The reference signal \\(r\\) [m] is applied to the controller, which as a reference transfer function \\(C\_{ff}(s)\\) in [N/m].
The output \\(u\\) in [N] of the controller is connected to the input of the motion system, which has a transfer function \\(G(s)\\) in [m/N] giving the output \\(x\\) in [m].
@ -684,12 +683,11 @@ If one would like to achieve perfect control, which means that there is no diffe
G\_{t,ff}(s) = \frac{x}{r} = C\_{ff}(s)G(s) = 1 \quad \Longrightarrow \quad C\_{ff}(s) = G^{-1}(s)
\end{equation}
<a id="org4b3a329"></a>
<a id="figure--fig:schmidt20-feedforward-control-diagram"></a>
{{< figure src="/ox-hugo/schmidt20_feedforward_control_diagram.svg" caption="Figure 5: Block diagram of a feedforward controller motion system with one input and output (SISO)." >}}
{{< figure src="/ox-hugo/schmidt20_feedforward_control_diagram.svg" caption="<span class=\"figure-number\">Figure 1: </span>Block diagram of a feedforward controller motion system with one input and output (SISO)." >}}
<div class="important">
<div></div>
Feedforward control is a very useful and preferred first step in the control of a complex dynamic motion system as it provides the following advantages:
@ -702,7 +700,6 @@ Feedforward control is a very useful and preferred first step in the control of
</div>
<div class="important">
<div></div>
The drawbacks and limitations of feedforward control are:
@ -718,21 +715,20 @@ The drawbacks and limitations of feedforward control are:
In feedback control the actuator status of the motion system is monitored by a sensor and the controller generates a control action based on the difference between the desired motion (reference signal) and the actuator system status (sensor signal).
The block diagram of Figure [6](#org3bccc77) shows a SISO feedback loop for a motion system without the A/D and D/A converters.
The block diagram of Figure [1](#figure--fig:schmidt20-feedback-control-diagram) shows a SISO feedback loop for a motion system without the A/D and D/A converters.
The output \\(x\\) in [m] is the total motion of the plant on all its parts and details, while \\(y\\) is the measured motion with a measured value \\(y\_m\\) measured on a selected location in the plant.
This measured is compared with \\(r\_f\\), which is the reference \\(r\\) after filtering.
The result of this comparison is used as input for the feedback controller.
<div class="note">
<div></div>
The transfer function of any input to any output in a closed-loop feedback controlled dynamic system is equal to the forward path from the input to the output divided by one plus the transfer function of the total feedback path.
</div>
<a id="org3bccc77"></a>
<a id="figure--fig:schmidt20-feedback-control-diagram"></a>
{{< figure src="/ox-hugo/schmidt20_feedback_control_diagram.svg" caption="Figure 6: Block diagram of a SISO feedback controlled motion system." >}}
{{< figure src="/ox-hugo/schmidt20_feedback_control_diagram.svg" caption="<span class=\"figure-number\">Figure 1: </span>Block diagram of a SISO feedback controlled motion system." >}}
In control design, one has the freedom to choose \\(F(s)\\) and particularly \\(C\_{fb}(s)\\) such that the total transfer function fulfills the desired specifications.
Feedback control allows to directly place the system poles at values that are more useful for the operation of the motion system that their natural locations.
@ -743,7 +739,6 @@ It is mainly used to present unwanted signals from entering the system.
This can be signals that drive the system into its "incapability" region where the system can no longer perform as required due to limitations in the hardware.
<div class="important">
<div></div>
Feedback is an addition to feedforward control with the following benefits:
@ -754,7 +749,6 @@ Feedback is an addition to feedforward control with the following benefits:
</div>
<div class="important">
<div></div>
Also, some pitfalls have to be dealt with:
@ -769,9 +763,9 @@ Also, some pitfalls have to be dealt with:
#### Summary {#summary}
<a id="table--tab:feedback-feedforward-summary"></a>
<a id="table--tab:feedback-feedforward-sum"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:feedback-feedforward-summary">Table 4</a></span>:
<span class="table-number"><a href="#table--tab:feedback-feedforward-sum">Table 1</a>:</span>
Summary of Feedback and Feedforward control
</div>
@ -795,11 +789,11 @@ Also, some pitfalls have to be dealt with:
#### Model-Based Feedforward Control {#model-based-feedforward-control}
In the following an example of a model-based feedforward controller is introduced.
The measured frequency-response of the scanning unit taken as as an example is shown in Figure [7](#org77061d7).
The measured frequency-response of the scanning unit taken as as an example is shown in Figure [1](#figure--fig:schmidt20-bode-plot-scanning).
<a id="org77061d7"></a>
<a id="figure--fig:schmidt20-bode-plot-scanning"></a>
{{< figure src="/ox-hugo/schmidt20_bode_plot_scanning.svg" caption="Figure 7: Bode plot of a piezoelectric-actuator based scanning unit for nanometer resolution positioning. It shows the measured response (solid line) and the second order model, which is fitted for the low-frewquency system behaviour (dashed line)." >}}
{{< figure src="/ox-hugo/schmidt20_bode_plot_scanning.svg" caption="<span class=\"figure-number\">Figure 1: </span>Bode plot of a piezoelectric-actuator based scanning unit for nanometer resolution positioning. It shows the measured response (solid line) and the second order model, which is fitted for the low-frewquency system behaviour (dashed line)." >}}
A mathematical model of a seconder-order mass-spring system with a force input is fitted to this measured response:
@ -835,17 +829,17 @@ C\_{ff}(s) = \frac{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2}{(s + \omega\_0)(s^2
Then this controller is connected in series with the scanning unit, the anti-resonance of the controller and the resonance of the piezo-scanner cancel each other out:
\begin{align}
G\_{t,ff}(s) &= G(s)G\_{ff}(s) \\\\\\
&= \frac{C\_f}{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2} \frac{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2}{(s + \omega\_0){s^2 + 2 \omega\_0 s + \omega\_0^2}} \\\\\\
G\_{t,ff}(s) &= G(s)G\_{ff}(s) \\\\
&= \frac{C\_f}{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2} \frac{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2}{(s + \omega\_0){s^2 + 2 \omega\_0 s + \omega\_0^2}} \\\\
&= \frac{C\_f}{(s + \omega\_0){s^2 + 2 \omega\_0 s + \omega\_0^2}}
\end{align}
The bode plot of the resulting dynamics is shown in Figure [8](#org1f513e4).
The bode plot of the resulting dynamics is shown in Figure [1](#figure--fig:schmidt20-bode-plot-feedfoward-example).
The controlled system has low-pass characteristics, rolling of at the scanner's natural frequency.
<a id="org1f513e4"></a>
<a id="figure--fig:schmidt20-bode-plot-feedfoward-example"></a>
{{< figure src="/ox-hugo/schmidt20_bode_plot_feedfoward_example.svg" caption="Figure 8: Bode plot of the feedforward-controlled scanning unit" >}}
{{< figure src="/ox-hugo/schmidt20_bode_plot_feedfoward_example.svg" caption="<span class=\"figure-number\">Figure 1: </span>Bode plot of the feedforward-controlled scanning unit" >}}
#### Input-Shaping {#input-shaping}
@ -861,11 +855,11 @@ The oscillation caused by each individual step are 180 degrees out of phase and
This method is clearly very different form pole-zero cancellation.
In the frequency domain, these sampled adaptations to the input create a frequency spectrum with a multiple of notch filters at the harmonic of the frequency where these adaptations are applied.
Applying input-shaping to the triangular scanning signal results in the introduction of a plateau instead of the sharp peak, where the width of the plateau corresponds to half the period of the scanner's resonance as can be seen in Figure [9](#orgddd25e4).
Applying input-shaping to the triangular scanning signal results in the introduction of a plateau instead of the sharp peak, where the width of the plateau corresponds to half the period of the scanner's resonance as can be seen in Figure [1](#figure--fig:schmidt20-input-shaping-example).
<a id="orgddd25e4"></a>
<a id="figure--fig:schmidt20-input-shaping-example"></a>
{{< figure src="/ox-hugo/schmidt20_input_shaping_example.svg" caption="Figure 9: Input-shaping control of the triangular scanning signal in a scanning probe microscope." >}}
{{< figure src="/ox-hugo/schmidt20_input_shaping_example.svg" caption="<span class=\"figure-number\">Figure 1: </span>Input-shaping control of the triangular scanning signal in a scanning probe microscope." >}}
#### Adaptive Feedforward Control {#adaptive-feedforward-control}
@ -891,13 +885,13 @@ The limitations of the actuators and electronics in a controlled motion system a
Of at least the levels of Jerk and preferable also Snap should be limited.
The standard method to cope with these limitations involves shaping the input of a mechatronic motion system by means of _trajectory profile generation_ or _path-planning_.
Figure [10](#orge30b109) shows a fourth order trajectory profile of a displacement, which means that all derivatives including the fourth derivative are defined in the path planning.
Figure [1](#figure--fig:schmidt20-trajectory-profile) shows a fourth order trajectory profile of a displacement, which means that all derivatives including the fourth derivative are defined in the path planning.
A third order trajectory would show a square profile for the jerk indicating an infinite Snap and the round of the acceleration would be gone.
A second order trajectory would show a square acceleration profile with infinite Jerk and sharp edges on the velocity.
<a id="orge30b109"></a>
<a id="figure--fig:schmidt20-trajectory-profile"></a>
{{< figure src="/ox-hugo/schmidt20_trajectory_profile.svg" caption="Figure 10: Figure caption" >}}
{{< figure src="/ox-hugo/schmidt20_trajectory_profile.svg" caption="<span class=\"figure-number\">Figure 1: </span>Figure caption" >}}
### Feedback Control {#feedback-control}
@ -915,29 +909,29 @@ Feedback control is more complex and critical to design than feedforward control
In general, a feedback controlled motion system is to perform a certain predetermined motion task defined by the reference input \\(r\\), while reducing the effects of other inputs like external vibrations and noise from the electronics.
All these input signals, whether desired of undesired, are treated by the feedback loop as disturbances and it is the sensitivity of the desired output signal to all input signals that determine the performance of the feedback controller.
<a id="org39f635c"></a>
<a id="figure--fig:schmidt20-feedback-full-simplified"></a>
{{< figure src="/ox-hugo/schmidt20_feedback_full_simplified.svg" caption="Figure 11: Full and simplified representation of a feedback loop in order to determine the influence of the reference signal and most important disturbance sources on real motion output of the plant \\(x\\), the feedback controller output \\(u\\) and the measured motion output \\(y\\). \\(y\_m = y\\) when the measurement system is set at unity gain and the sensor disturbance is included in the output disturbance." >}}
{{< figure src="/ox-hugo/schmidt20_feedback_full_simplified.svg" caption="<span class=\"figure-number\">Figure 1: </span>Full and simplified representation of a feedback loop in order to determine the influence of the reference signal and most important disturbance sources on real motion output of the plant \\(x\\), the feedback controller output \\(u\\) and the measured motion output \\(y\\). \\(y\_m = y\\) when the measurement system is set at unity gain and the sensor disturbance is included in the output disturbance." >}}
Several standard sensitivity functions have been defined to quantify the performance of feedback controlled dynamic systems.
There are derived from a simplified version of the generic feedback loop as shown in Figure [11](#org39f635c).
There are derived from a simplified version of the generic feedback loop as shown in Figure [1](#figure--fig:schmidt20-feedback-full-simplified).
The first simplification is made by approximating the measurement system to have a unity-gain transfer function.
For further simplification the sensor disturbance in the measurement system is included in the output disturbance \\(n\\), thereby defining the output of the system \\(y\\) as the measured output.
With this simplified model, the transfer functions of the different inputs of the system to three relevant output variables in the loop are written down in a set of equations.
Six different transfer functions are obtained and summarized in equation \eqref{eq:gang_of_six}.
Six different transfer functions are obtained and summarized in equation <eq:gang_of_six>.
\begin{equation} \label{eq:gang\_of\_six}
\begin{aligned}
\frac{x}{r} &= \frac{y}{r} = \frac{GCF}{1 + GC} \\\\\\
-\frac{x}{n} &= -\frac{u}{d} = \frac{GC}{1 + GC} \\\\\\
\frac{x}{d} &= \frac{y}{d} = \frac{G}{1 + GC} \\\\\\
\frac{u}{r} &= \frac{CF}{1 + GC} \\\\\\
\frac{u}{n} &= \frac{C}{1 + GC} \\\\\\
\frac{x}{r} &= \frac{y}{r} = \frac{GCF}{1 + GC} \\\\
-\frac{x}{n} &= -\frac{u}{d} = \frac{GC}{1 + GC} \\\\
\frac{x}{d} &= \frac{y}{d} = \frac{G}{1 + GC} \\\\
\frac{u}{r} &= \frac{CF}{1 + GC} \\\\
\frac{u}{n} &= \frac{C}{1 + GC} \\\\
\frac{y}{n} &= \frac{1}{1 + GC}
\end{aligned}
\end{equation}
In case no input filter is applied \\(F\\) is equal to one and the set of six equations is reduced to a set of four equations as shown in equation \eqref{eq:gang_of_four}.
In case no input filter is applied \\(F\\) is equal to one and the set of six equations is reduced to a set of four equations as shown in equation <eq:gang_of_four>.
This short set of equations also corresponds to the situation without a reference signal.
The most important transfer function is named the _Sensitivity Function_ (no unit):
@ -968,9 +962,9 @@ For that reason the most relevant motion system performance criteria are the Sen
\begin{equation} \label{eq:gang\_of\_four}
\boxed{\begin{aligned}
\frac{x}{r} &= \frac{y}{r} = -\frac{x}{n} = -\frac{u}{d} = \frac{GC}{1 + GC} \\\\\\
\frac{x}{d} &= \frac{y}{d} = \frac{G}{1 + GC} \\\\\\
\frac{u}{r} &= \frac{u}{n} = \frac{C}{1 + GC} \\\\\\
\frac{x}{r} &= \frac{y}{r} = -\frac{x}{n} = -\frac{u}{d} = \frac{GC}{1 + GC} \\\\
\frac{x}{d} &= \frac{y}{d} = \frac{G}{1 + GC} \\\\
\frac{u}{r} &= \frac{u}{n} = \frac{C}{1 + GC} \\\\
\frac{y}{n} &= \frac{1}{1 + GC}
\end{aligned}}
\end{equation}
@ -1001,17 +995,17 @@ To achieve sufficient robustness against instability in closed-loop feedback con
The condition for robustness of closed-loop stability is that the total phase-lag of the **total feedback-loop**, consisting of the feedback controller in series with the mechatronic system, must be less than 180 degrees in the frequency region of the _unity-gain cross-over frequency_.
The Nyquist plot of the feedback loop, like the example shown in Figure [12](#org6e48553), is most appropriate to analyze the robustness on stability of a feedback system.
The Nyquist plot of the feedback loop, like the example shown in Figure [1](#figure--fig:schmidt20-nyquist-plot-stable), is most appropriate to analyze the robustness on stability of a feedback system.
It is an analysis tool that shows the frequency response of the **feedback-loop** combining magnitude and phase in one plot.
In this figure, two graphs are shown, designed for a different purpose.
The first graph from the left shows margin circles related to the capability of the closed-loop feedback controlled system to follow a reference according to the complementary sensitivity.
The second graph shows a margin circle related to the capability of the closed-loop feedback controlled system to suppress disturbances according to the sensitivity function.
<a id="org6e48553"></a>
<a id="figure--fig:schmidt20-nyquist-plot-stable"></a>
{{< figure src="/ox-hugo/schmidt20_nyquist_plot_stable.svg" caption="Figure 12: Nyquist plot of the feedback-loop response of a stable feedback controlled motion system. Stability is guaranteed as the \\(-1\\) point is kept at the left hand side of the feedback loop repsonse line upon passing with increased frequency, even though the phase-lag is more than 180 degrees at low frequencies." >}}
{{< figure src="/ox-hugo/schmidt20_nyquist_plot_stable.svg" caption="<span class=\"figure-number\">Figure 1: </span>Nyquist plot of the feedback-loop response of a stable feedback controlled motion system. Stability is guaranteed as the \\(-1\\) point is kept at the left hand side of the feedback loop repsonse line upon passing with increased frequency, even though the phase-lag is more than 180 degrees at low frequencies." >}}
Three values are shown in Figure [12](#org6e48553) related to the robustness of the closed-loop feedback system:
Three values are shown in Figure [1](#figure--fig:schmidt20-nyquist-plot-stable) related to the robustness of the closed-loop feedback system:
- **The gain margin** determines by which factor the feedback loop gain additionally can increase before the closed-loop goes unstable.
- **The phase margin** determines how much additional phase-lab at the unity-gain cross-over frequency is acceptable before the closed-loop system becomes unstable.
@ -1024,14 +1018,14 @@ Higher margins corresponds to a higher level of damping.
The Nyquist plot has one significant disadvantage as it does not show directly the frequency along the plot.
For that reason many designers prefer to use the Bode plot.
Fortunately it is also possible to indicate the phase and gain margin in the Bode plot as is shown in Figure [13](#orgc932364).
Fortunately it is also possible to indicate the phase and gain margin in the Bode plot as is shown in Figure [1](#figure--fig:schmidt20-phase-gain-margin-bode).
In many not too complicated cases, these two margins are sufficient to tune a feedback motion controller.
In more complicated control systems, it remains useful to also use the Nyquist plot as it also gives the Modulus margin.
<a id="orgc932364"></a>
<a id="figure--fig:schmidt20-phase-gain-margin-bode"></a>
{{< figure src="/ox-hugo/schmidt20_phase_gain_margin_bode.svg" caption="Figure 13: The gain and phase margin in the Bode plot" >}}
{{< figure src="/ox-hugo/schmidt20_phase_gain_margin_bode.svg" caption="<span class=\"figure-number\">Figure 1: </span>The gain and phase margin in the Bode plot" >}}
### PID Feedback Control {#pid-feedback-control}
@ -1152,11 +1146,11 @@ However, analogue controllers have three important disadvantages:
The digital implementation of filters overcome these problems as well as allows more complex algorithm such as adaptive control, real-time optimization, nonlinear control and learning control methods.
In Figure [14](#org4b95e2a) two elements were introduced, the _analogue-to-digital converter_ (ADC) and the _digital-to-analogue converter_ (DAC), which together transfer the signals between the analogue and the digital domain.
In Figure [1](#figure--fig:schmidt20-digital-implementation) two elements were introduced, the _analogue-to-digital converter_ (ADC) and the _digital-to-analogue converter_ (DAC), which together transfer the signals between the analogue and the digital domain.
<a id="org4b95e2a"></a>
<a id="figure--fig:schmidt20-digital-implementation"></a>
{{< figure src="/ox-hugo/schmidt20_digital_implementation.svg" caption="Figure 14: Overview of a digital implementation of a feedback controller, emphasising the analog-to-digital and digital-to-analog converters with their required analogue filters" >}}
{{< figure src="/ox-hugo/schmidt20_digital_implementation.svg" caption="<span class=\"figure-number\">Figure 1: </span>Overview of a digital implementation of a feedback controller, emphasising the analog-to-digital and digital-to-analog converters with their required analogue filters" >}}
Anti-aliasing filter is needed at the input of the ADC to limit the frequency range at the input to less than half the sampling frequency, according to the Nyquist-Shannon sampling theorem.
@ -1176,7 +1170,7 @@ Fixed point arithmetic has been favored in the past, because of the less complex
A main drawback is, that the developer must pay attention to truncation, overflow, underflow and round-off errors that occur during mathematical operations.
Fixed points numbers are equally spaced over the whole range, separated by the gap which is denoted by the least significant bit.
The two's complement is the most used format for representing positive and negative numbers.
For representing a fixed point fractional number of two's complement notation, the so called \\(Q\_{m,n}\\) format is often used (see Figure [15](#orgf73916a)).
For representing a fixed point fractional number of two's complement notation, the so called \\(Q\_{m,n}\\) format is often used (see Figure [1](#figure--fig:schmidt20-digital-number-representation)).
\\(m\\) denotes the number of integer bits and \\(n\\) denotes the number of fractional bits.
\\(m+n+1=N\\) bits are necessary to store a signed \\(Q\_{m,n}\\) number.
If the binary representation is given, the decimal value can be calculated to:
@ -1185,11 +1179,11 @@ If the binary representation is given, the decimal value can be calculated to:
x = \frac{1}{2^n} \left( -2^{N-1 }b\_{N-1} + \sum\_{i=0}^{N-2} 2^i b\_i \right)
\end{equation}
where \\(b\\) indicate the bit position, starting with \\(b\_0\\) from the right in Figure [15](#orgf73916a).
where \\(b\\) indicate the bit position, starting with \\(b\_0\\) from the right in Figure [1](#figure--fig:schmidt20-digital-number-representation).
<a id="orgf73916a"></a>
<a id="figure--fig:schmidt20-digital-number-representation"></a>
{{< figure src="/ox-hugo/schmidt20_digital_number_representation.svg" caption="Figure 15: Example of a \\(Q\_{m.n}\\) fixed point number representation and a single precision floating point number" >}}
{{< figure src="/ox-hugo/schmidt20_digital_number_representation.svg" caption="<span class=\"figure-number\">Figure 1: </span>Example of a \\(Q\_{m.n}\\) fixed point number representation and a single precision floating point number" >}}
Floating point arithmetic has a higher dynamic range than fixed point arithmetic, given by the largest and smallest number that can be represented, has a higher precision due to the smaller gaps between adjacent numbers, less quantization noise, and it is easier to handle in terms of programming.
A floating point number is represented by a multiplication of a _mantissa_ \\(M\\) with a _base_ \\(b\\) to the power of the _exponent_ \\(q\\):
@ -1209,41 +1203,41 @@ x = -1^i M 2^{E-127}
The term \\(E\\) in the exponent is stored as a positive number ranging from \\(0 \le E < 256\\) with 8 bits.
An offset of \\(-127\\) is added in order to allow very small to very large numbers.
The decimal value is normalized, meaning that only one nonzero digit is noted at the left of the decimal point.
The storage register is divided into three groups, as shown in Figure [15](#orgf73916a).
The storage register is divided into three groups, as shown in Figure [1](#figure--fig:schmidt20-digital-number-representation).
1 bit represents the sign, the exponent term \\(E\\) is represented by 8 bits, and the mantissa is stored in 23 bits.
#### Digital Filter Theory {#digital-filter-theory}
<a id="org3a2480f"></a>
<a id="figure--fig:schmidt20-s-z-planes"></a>
{{< figure src="/ox-hugo/schmidt20_s_z_planes.svg" caption="Figure 16: Corresponding points and area in s and z planes" >}}
{{< figure src="/ox-hugo/schmidt20_s_z_planes.svg" caption="<span class=\"figure-number\">Figure 1: </span>Corresponding points and area in s and z planes" >}}
#### Finite Impulse Response (FIR) Filter {#finite-impulse-response--fir--filter}
<a id="org4983bdd"></a>
<a id="figure--fig:schmidt20-transversal-filter-structure"></a>
{{< figure src="/ox-hugo/schmidt20_transversal_filter_structure.svg" caption="Figure 17: Transversal filter structure of a FIR filter. The term \\(z^{-1}\\) each represent a sampling period which means that \\(b\_0\\) is the gain of the last sample, \\(b\_1\\) is the gain of the precious sample etcetera." >}}
{{< figure src="/ox-hugo/schmidt20_transversal_filter_structure.svg" caption="<span class=\"figure-number\">Figure 1: </span>Transversal filter structure of a FIR filter. The term \\(z^{-1}\\) each represent a sampling period which means that \\(b\_0\\) is the gain of the last sample, \\(b\_1\\) is the gain of the precious sample etcetera." >}}
<a id="org936becf"></a>
<a id="figure--fig:schmidt20-optimized-fir-filter-structure"></a>
{{< figure src="/ox-hugo/schmidt20_optimized_fir_filter_structure.svg" caption="Figure 18: Optimized FIR filter structure with symmetric filter coefficients" >}}
{{< figure src="/ox-hugo/schmidt20_optimized_fir_filter_structure.svg" caption="<span class=\"figure-number\">Figure 1: </span>Optimized FIR filter structure with symmetric filter coefficients" >}}
<a id="org8ea00c7"></a>
<a id="figure--fig:schmidt20-dir-filter-cascaded-sos"></a>
{{< figure src="/ox-hugo/schmidt20_dir_filter_cascaded_sos.svg" caption="Figure 19: Higher-order FIR filter realization with cascade SOS filter structures" >}}
{{< figure src="/ox-hugo/schmidt20_dir_filter_cascaded_sos.svg" caption="<span class=\"figure-number\">Figure 1: </span>Higher-order FIR filter realization with cascade SOS filter structures" >}}
#### Infinite Impulse Response (IIR) Filter {#infinite-impulse-response--iir--filter}
<a id="org696a8aa"></a>
<a id="figure--fig:schmidt20-irr-structure"></a>
{{< figure src="/ox-hugo/schmidt20_irr_structure.svg" caption="Figure 20: (a:) IIR structure in DF-1 realization and (b:) IIR structure in DF-2 realization" >}}
{{< figure src="/ox-hugo/schmidt20_irr_structure.svg" caption="<span class=\"figure-number\">Figure 1: </span>(a:) IIR structure in DF-1 realization and (b:) IIR structure in DF-2 realization" >}}
<a id="orge99cdac"></a>
<a id="figure--fig:schmidt20-irr-sos-structure"></a>
{{< figure src="/ox-hugo/schmidt20_irr_sos_structure.svg" caption="Figure 21: IIR SOS structure in DF-2 realization" >}}
{{< figure src="/ox-hugo/schmidt20_irr_sos_structure.svg" caption="<span class=\"figure-number\">Figure 1: </span>IIR SOS structure in DF-2 realization" >}}
#### Converting Continuous to Discrete-Time Filters {#converting-continuous-to-discrete-time-filters}
@ -1275,7 +1269,8 @@ The storage register is divided into three groups, as shown in Figure [15](#orgf
### Conclusion on Motion Control {#conclusion-on-motion-control}
<summary>
<div class="sum">
Motion control is essential for Precision Mechatronic Systems and consists of two complementary elements:
- **Extremely accurate Feedforward Control** is required when the motion system must execute a user defined motion to within maximum user defined position error limits.
@ -1283,7 +1278,8 @@ Motion control is essential for Precision Mechatronic Systems and consists of tw
- **High Performance Feedback Control** is required when the motion system must be able to follow an unknown motion of a target, stabilize an otherwise unstable system and reduce the impact of disturbing forces and vibrations, such that the position error remains below a maximum user defined level.
Due to the fact that a feedback controller can become unstable, sufficient robustness must be guaranteed.
These is a conflicting relation between stability and performance.
</summary>
</div>
## Electromechanic Actuators {#electromechanic-actuators}
@ -2237,4 +2233,6 @@ Motion control is essential for Precision Mechatronic Systems and consists of tw
## Bibliography {#bibliography}
<a id="org4e5c703"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. _The Design of High Performance Mechatronics - Third Revised Edition_. Ios Press.
<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>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. <i>The Design of High Performance Mechatronics - Third Revised Edition</i>. Ios Press.</div>
</div>

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@ -0,0 +1,443 @@
+++
title = "Robust mass damper design for bandwidth increase of motion stages"
author = ["Dehaeze Thomas"]
draft = true
+++
Tags
:
Reference
: (<a href="#citeproc_bib_item_1">Verbaan 2015</a>)
Author(s)
: Verbaan, C.
Year
: 2015
> This thesis addresses the challenge to increase the modal damping of the bandwidth limiting resonances of motions stages.
> This modal damping increase is realized by adding passive elements, called robust tuned mass dampers, at specific stage locations.
>
> [...]
>
> The damper parameters that have to be determined are mass, stiffness, and damping.
> The optimal parameters are obtained by executing optimization algorithm.
>
> The first motion stage design is optimized based on an open-loop criterion for modal damping increase between 1 and 4kHz.
> Experimental validation shows that a suppression factor of over 24dB is obtained.
## Robust Mass Damper and broad banded damping {#robust-mass-damper-and-broad-banded-damping}
> In high tech motion systems, the finite stiffness of mechanical components results in natural frequencies which limit the bandwidth of the control system.
> This is usually counteracted by increasing the controller complexity by adding notch filters.
> The height of the non-rigid body modes in the frequency response function and the amount of damping significantly affect the achievable bandwidth.
> This chapter described a method to add damping to the flexible behavior of a motion stage, by using robust mass dampers which are mass-spring-damper systems with an **over-critical** damping value.
> This high damping results in robust dynamic behavior with respect to stiffness and damping variations for both the motion stage and the damper mechanisms.
> The main result is a significant increase in modal damping over a broad band of resonance frequencies.
### Tuned mass damper {#tuned-mass-damper}
The effectiveness of the TMD is related to the mass ratio between \\(m\\) and \\(M\\).
To obtain a substantial suppression factor in combination with a relatively small increase in mass, the mass ratio is usually determined to be approximately 5 to 10% of the main structural mass.
The undamped natural frequency of the TMD has to be tuned close to the targeted natural frequency of the main structure.
A drawback of the TMD is the relatively **large sensitivity of the suppression factor for variations in stiffness and damping values**.
This sensitivity also holds for natural frequency variations of the main structure.
<a id="figure--fig:verbaan15-tmd-principle"></a>
{{< figure src="/ox-hugo/verbaan15_tmd_principle.png" caption="<span class=\"figure-number\">Figure 1: </span>TMD principle" >}}
### Damper design and validation {#damper-design-and-validation}
This damper is designed and tested to prove that it is possible to create dampers with over-critical damping values and with natural frequencies that are high enough to be useful.
The spring and damper are assumed to behave linearly.
In addition, the vibration amplitudes of high-tech positioning tables are small, which allows for assuming linear system theory.
These small vibration amplitudes lead to small damper strokes.
Therefore **flexures** can be used to provide for the guidance of the moving mass.
The dimensions of the flexures determine the spring stiffness and therefore the natural frequency of the TMD.
An additional advantage of flexures is the lack of hysteresis, which **enables the damper to work even if the damper strokes are very small**.
The dampers are intended to act purely in z-direction.
The natural frequency in this direction is determined at 1250Hz and the natural frequency in the other directions should be as high as possible.
<a id="figure--fig:verbaan15-tmd-modes"></a>
{{< figure src="/ox-hugo/verbaan15_tmd_modes.png" caption="<span class=\"figure-number\">Figure 2: </span>Natural frequency of the TMD. First natural frequency at 1250Hz and the second at 8100Hz." >}}
The second challenge is to create a damping mechanism with a high damping coefficient in a relatively small volume.
The damper is designed to be **passive**.
This guarantees stability of the damper system itself and preserves from increasing complexity.
As damping concept, a **viscous fuild damper** is chosen due to the following properties:
- the linear time independent behavior
- the ability to create an extremely large damping constant in a small volume
- separation of stiffness and damping
- the supreme damping properties of fuilds with respect to other damping materials
The guild applied is Rocol Kilopoise 0868 and is chosen based on the extremely high viscosity of 220 Pas.
In order to measure the damping the measurement bench shown in Figure [1](#figure--fig:verbaan15-tmd-mech-system) is used.
The measured FRF are shown in Figure [1](#figure--fig:verbaan15-obtained-damping-bench).
The measurement clearly shows that the damper mechanism is over-critically damped.
<a id="figure--fig:verbaan15-tmd-mech-system"></a>
{{< figure src="/ox-hugo/verbaan15_tmd_mech_system.png" caption="<span class=\"figure-number\">Figure 1: </span>Damper test setup to measure the damping characteristics" >}}
<a id="figure--fig:verbaan15-obtained-damping-bench"></a>
{{< figure src="/ox-hugo/verbaan15_obtained_damping_bench.png" caption="<span class=\"figure-number\">Figure 1: </span>Obtained damping results" >}}
## Linear viscoelastic characterisation of an ultra-high viscosity fluid {#linear-viscoelastic-characterisation-of-an-ultra-high-viscosity-fluid}
> This chapter presents the use of a state of the art damper for high precision motion stages as a sliding plate rheometer for measuring linear viscoelastic properties in the frequency range of 10Hz to 10kHz.
> This design is flexure based to minimize parasitic nonlinear forces.
> Design and the damping mechanism are elaborated and a model is presented that describes the dynamic behavior.
The damper shown in Figure [1](#figure--fig:verbaan15-damper-parts) can be used as a sliding plate rheometer to measure the linear viscoelastic properties of ultra-high viscosity fluids in the frequency range 10Hz to 10kHz.
<a id="figure--fig:verbaan15-damper-parts"></a>
{{< figure src="/ox-hugo/verbaan15_damper_parts.png" caption="<span class=\"figure-number\">Figure 1: </span>Damper parts" >}}
The full damper assembly consists of a mass, mounted on two springs and a damper in parallel configuration.
The mass can make small strokes in the x-direction and is fixed in all other directions.
The spring is a double leaf spring guide.
The space between the lead springs is used to accommodate for the damping mechanism.
<a id="figure--fig:verbaan15-tmd-slot-fin-parts"></a>
{{< figure src="/ox-hugo/verbaan15_tmd_slot_fin_parts.png" caption="<span class=\"figure-number\">Figure 1: </span>Exploded view of the damper parts" >}}
A high-viscosity fluid is applied to create a velocity dependent force.
For this purpose, the sliding plate principle is used which induces a **shear flow**: the fluid is placed between two slot plates and a fin is positioned between these two plates (Figure [1](#figure--fig:verbaan15single-double-fin)).
A **flexible encapsulation** is used to hold the fluid between find and slot part.
To study different damping values with the same fluid, two damper designs with different geometries are used (see Figure [1](#figure--fig:verbaan15single-double-fin)).
<a id="figure--fig:verbaan15single-double-fin"></a>
{{< figure src="/ox-hugo/verbaan15single_double_fin.png" caption="<span class=\"figure-number\">Figure 1: </span>Cross-sectional views of the two different damping mechanims. The single fin (left) and double fin (right)." >}}
To excite the damper mass, a voice coil is mounted to the hardware.
The damper position is measured with a laser vibrometer.
A sliding plate damper for high frequencies introduces side effects:
1. geometry related effects
2. frequency dependent effects
A first geometrical effect is due to the **finite length of the plates**.
The ratio length/gap here is more than 100 which makes this effect negligible.
A second geometrical effect is due to the difficulty to get the **plates parallel to each other**, especially with the normal forces acting on the moving fin, induced by the flow.
This design counteracts this problem in two-ways: the damper part is **symmetric**, which means that the fin normal forces cancel each other.
In addition, the double leaf spring mechanism has a **very high lateral stiffness**, which minimizes lateral displacements.
A third geometrical effect is pumping of the fluid, which appears in the case of closed ends and introduces a flow opposite to the fin velocity, and therefore introduces a parasitic damping force.
This problem is avoided by letting the gaps' ends open.
The **fin is shorted than the slot** to maintain the same damping area over the damper stroke.
These effects all arise at low frequencies, at which the flow can be assumed homogeneous.
The ratio between inertial and viscous effects determines up to which frequency the flow can be assumed homogeneous:
\begin{equation}
t\_c = \frac{10 \rho h^2}{\eta}
\end{equation}
in which \\(\rho\\) describes the fluid density in \\(kg/m^3\\), \\(\eta\\) the dynamic viscosity in \\(Pa s\\) and \\(h\\) the gap width in \\(m\\).
Dimensions are provided in Table [1](#table--tab:single-fin-parameters).
This estimate results in a frequency above 100kHz.
It shows that high fluid viscosities and small gap widths enable high frequencies without losing homogeneous flow conditions.
<a id="table--tab:single-fin-parameters"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:single-fin-parameters">Table 1</a>:</span>
Parameters for the single fin design
</div>
| Dimension | Value [mm] |
|----------------|------------|
| Length \\(l\\) | 16 |
| Width \\(w\\) | 8.5 |
| Gap \\(h\\) | 0.12 |
**Conclusion**:
A design of a sliding plate damper that can be used to characterize fluid behavior of high viscosity fluids in the frequency range between 10Hz and 10kHz.
The drawbacks of standard sliding plate devices are taken care off by the mechanical design.
The flexure mechanism very precisely determines the position of the fin with respect to the slot part.
A three mode Maxwell model can accurately describe the behavior.
## Damping optimization of a complex motion stage {#damping-optimization-of-a-complex-motion-stage}
### Stage and damper dynamic models {#stage-and-damper-dynamic-models}
This chapter presents the results of a robust mass damper implementation on a complex motion stage with realistic natural frequencies to increase the modal damping of flexible modes.
A design approach is presented which results in parameter values for the dampers to improve the modal damping over a specified frequency range.
Figure [1](#figure--fig:verbaan15-stage-undamped) shows a collocated FRF of the stage's corner.
The goal is to increase the modal damping of modes 7, 9, 10/11 and 13.
<a id="figure--fig:verbaan15-stage-undamped"></a>
{{< figure src="/ox-hugo/verbaan15_stage_undamped.png" caption="<span class=\"figure-number\">Figure 1: </span>FRF at the stage corner in the z-direction, undamped" >}}
The transfer function \\(T\_i(s)\\) is defined as the contribution of the a single mode \\(i\\) in an input/output transfer function:
\begin{equation}
T\_i(s) = \frac{\phi\_i^{\text{act}} \phi\_i^{\text{sen}}}{s^2 + 2 \xi \omega\_i s + \omega\_i^2} = \frac{1}{m\_i s^2 + c\_i s + k\_i}
\end{equation}
With \\(\phi\_i^{\text{act}}\\) and \\(\phi\_i^{\text{sen}}\\) the modal factors of the actuator and sensor.
From this equation, it appears that the modal mass of a mode in a certain transfer function equals:
\begin{equation}
m\_i = \frac{1}{\phi\_i^{\text{act}} \phi\_i^{\text{sen}}}
\end{equation}
This equation shows that a certain mode's modal mass depends on the locations of the actuator and sensor.
Since a TMD can be seen as a local control loop, the actuator and sensor location are equal.
This results in the following equation for the apparent modal mass for mode \\(i\\) at the TMD location:
\begin{equation}
m\_i = \frac{}{(\phi\_i^{\text{TMD}})^2}
\end{equation}
It is known from literature that the efficiency of a TMD depends on the **mass ratio** of the TMD and the mode that has to be damped.
It follows that the efficiency of a TMD to damp a certain resonance depends on the position of the damper on the stage in a quadratic sense.
The TMD has to be located at the maximum displacement of the mode(s) to be damped.
The damper configuration consists of an inertial mass \\(m\\), a transnational flexible guide designed as a double leaf spring mechanism with total stiffness \\(c\\) and a part that creates the damping force with damping constant \\(d\\) (model shown in Figure [1](#figure--fig:verbaan15-maxwell-fluid-model)).
The velocity dependent damper force is the result of two parameters:
- the fluid's mechanical properties
- the damper geometry
The fluid model is presented in Figure [1](#figure--fig:verbaan15-fluid-lve-model).
This figure shows the viscous and elastic properties of the fluid as a function of the frequency.
The damper principle is chosen to be a parallel plate damper based on the shear principle with the viscous fluid in between the two parallel plates.
In case of a velocity difference between these plates, a velocity gradient is created in the fluid causing a specific force per unit of area, which, multiplied by the effective area submerged in the fluid, leads to a damping force.
The damping can be expressed with a geometrical damping factor (GDF) in meters:
\begin{equation}
\text{GDF} = \frac{A}{h} = \frac{2 n l w}{h}
\end{equation}
with \\(A\\) the total area of the damper fins, \\(n\\) is the number of fins, \\(l\\) is the fin length, \\(w\\) is the fin width and \\(h\\) is the effective gap width in which the fluid is applied.
This GDF, combined with the fluid properties in Pas and Pa, lead to a spring stiffness in N/m and a damping constant in N/(m/s).
In general, larger suppression factors can be obtained with larger TMD masses.
In the example, the modal mass is 3.5kg and the damper mass is 110g (useful inertial mass of 65g).
<a id="figure--fig:verbaan15-maxwell-fluid-model"></a>
{{< figure src="/ox-hugo/verbaan15_maxwell_fluid_model.png" caption="<span class=\"figure-number\">Figure 1: </span>Damper model with multi-mode Maxwell fluid model included" >}}
<a id="figure--fig:verbaan15-fluid-lve-model"></a>
{{< figure src="/ox-hugo/verbaan15_fluid_lve_model.png" caption="<span class=\"figure-number\">Figure 1: </span>Storage and loss modulus of the 3 Maxwell mode LVE fluid model" >}}
### TMD and RMD optimisation {#tmd-and-rmd-optimisation}
An algorithm is used to optimize the damping and is used in two cases:
- a small banded optimisation which includes a single resonance.
This results in a **tuned mass damper** optimal design
- a broad banded optimization which includes a range of resonances.
This results in a **robust mass damper** optimal design
The algorithm is first used to calculate the optimal parameters to suppress a **single** resonance frequency.
The result is shown in Figure [1](#figure--fig:verbaan15-tmd-optimization) and shows **Tuned Mass Damper** behavior.
For this single frequency, stiffness and damping values can be calculated by hand.
<a id="figure--fig:verbaan15-tmd-optimization"></a>
{{< figure src="/ox-hugo/verbaan15_tmd_optimization.png" caption="<span class=\"figure-number\">Figure 1: </span>Result of the optimization procedure. The cost function is specified between 1kHz and 2kHz. This implies that the first mode is suppressed by the damper." >}}
To obtain broad banded damping, the cost function is redefined between 1 and 4kHz.
Figure [1](#figure--fig:verbaan15-broadbanded-damping-results) presents the resulting bode diagram.
<a id="figure--fig:verbaan15-broadbanded-damping-results"></a>
{{< figure src="/ox-hugo/verbaan15_broadbanded_damping_results.png" caption="<span class=\"figure-number\">Figure 1: </span>Result of the optimization procedure with the cost function specified between 1 and 4kHz. The result is a range of resonances that are suppressed by the dampers." >}}
Results of optimizations for increasing damper mass, in the range from 10 to 250g per damper are shown in Figure [1](#figure--fig:verbaan15-results-fct-mass).
<a id="figure--fig:verbaan15-results-fct-mass"></a>
{{< figure src="/ox-hugo/verbaan15_results_fct_mass.png" caption="<span class=\"figure-number\">Figure 1: </span>Optimal damper parameters as a function of the damper mass. The upper graph shows the suppression factor in dB, the second graph shows the natural frequency of the damper in Hz and the lower graph shows the geometrical damping factor in m." >}}
### Damper Design and Validation {#damper-design-and-validation}
A damper mechanism is design which contains the following properties:
- a moving mass \\(m\_d = 65\\,g\\)
- a mounting mass \\(m\_m = 45\\,g\\)
- a natural frequency \\(\omega\_0 = 1270\\,Hz\\)
- other natural frequencies as high as possible
- a geometrical damping factor of 14.3m
- an encapsulation to contain the fluid
Figure [1](#figure--fig:verbaan15-RMD-mechanical-parts) shows an exploded view of the RMD design.
The mechanism part is monolithically designed and consists of:
1. a mounting side
2. leaf spring pair
3. the damper side
The fluid is surrounded by a flexible encapsulation, which prevents it from running out.
<a id="figure--fig:verbaan15-RMD-mechanical-parts"></a>
{{< figure src="/ox-hugo/verbaan15_RMD_mechanical_parts.png" caption="<span class=\"figure-number\">Figure 1: </span>Exploded view of the robust mass damper design with different parts indicated" >}}
<a id="figure--fig:verbaan15-RMD-design-modes"></a>
{{< figure src="/ox-hugo/verbaan15_RMD_design_modes.png" caption="<span class=\"figure-number\">Figure 1: </span>Four lowest natural frequencies and corresponding mode shapes of the RMD while mounted to a stage corner" >}}
<a id="figure--fig:verbaan15-tmd-side-front-views"></a>
{{< figure src="/ox-hugo/verbaan15_tmd_side_front_views.png" caption="<span class=\"figure-number\">Figure 1: </span>A side view and a front view of the fin and slot parts" >}}
| Dimension | Value | Unit |
|-------------|-------|------|
| Length fin | 17 | mm |
| Height fins | 4 | mm |
| Gap width | 50 | um |
| GDF | 14 | m |
<a id="figure--fig:verbaan15-damped-undamped-frf"></a>
{{< figure src="/ox-hugo/verbaan15_damped_undamped_frf.png" caption="<span class=\"figure-number\">Figure 1: </span>Measured undamped and damped FRF" >}}
### Conclusion {#conclusion}
This chapter shows an approach to add damping to a range of resonances of a motion stage by adding robust mass dampers.
Analysis is performed to calculate the damping increase beforehand, and experiments are conducted to validate the behavior of both the damper and the stage with dampers added.
The broadbanded solution shows a resonance suppression of at least 24.3dB between 1kHz and 4kHz.
The overall mass increase is less than 2%.
The robustness, as one of the most important properties of the RMD, is proven: the suppression factor is well predictable despite different errors and estimations:
- stage model errors (the natural frequencies resulting from the FEM are an overestimation of the real frequencies)
- fluid model errors
- a simplified 1DoF model is applied as a damper model
- production tolerances for the dampers
Tuned mass dampers are well known in literature.
The equations are proven to calculate the optimal suppression factor, natural frequency and damping ratio.
In these equations, the damper behavior is assumed to be purely viscous.
We shows that larger suppression factors are possible by using visco-elastic fluids as damping medium.
Although this effect is relatively small for single resonance suppression, it is larger for broadbanded suppression.
The damper benefits from the frequency dependent stiffness of the fluid.
## Conclusion {#conclusion}
In this thesis, the opportunities to increase the performance of high-tech motion systems are investigated by increasing the modal damping of non-rigid body resonances by introducing robust mass dampers (RMD), which provides damping over a broad frequency band.
A combination of techniques is applied to improve the performance of motion stages in a systematical way, including mechanical design, dynamic modeling, material characterization and optimization procedures.
Theoretical improvement factors are calculated and experimental validation is provided to support the theory.
The main conclusions of the previous chapters are summarized and listed by subject.
### Robust Mass Dampers {#robust-mass-dampers}
Robust mass dampers have proven to be able to provide **broad banded damping**.
In addition, **robust behavior** is proven in case of parameter variations of both the motion stage and/or the parameters of the RMDs.
This property explicitly underlines the suitability of RMDs to improve the behavior of motion stages that are operated in closed-loop conditions: parameter sensitive designs will result in a performance decrease and might eventually lead to destabilization of the closed-loop system.
The RMDs in this thesis are **passive and stand-alone devices**.
Advantages of these types of devices are
1. the stabilizing behavior due to the principle of energy dissipation.
2. The stand-alone property implies that no connection between any structural part and the motion stage is created, and no signal or power cables are needed which prevents the introduction of disturbance forces.
3. The damper design by application of LVE behavior enables larger suppression factors than purely viscous fluid behavior.
At least in case of motion stages with a relatively large length-height ratio it appears that an overall mass contribution by the RMDs of 2 % of the stage mass is sufficient to improve the stage performance significantly.
This is proven by experiments.
### Influence on stage dynamics {#influence-on-stage-dynamics}
The relatively high modal damping of the RMDs prevents for visible effects in the rigid body mass line of the frequency response functions.
In other directions, the natural frequencies of the RMDs can be designed above 6 kHz for dampers of 65 g.
This is usually high enough to prevent for detrimental properties in the direction of motion
### RMD locations {#rmd-locations}
The **location of an RMD on the mechanical stage is a significant factor in the performance increase factor**.
The effectiveness of the RMD to improve the modal damping factor scales quadratically with the stage displacement at the damper location.
Therefore, if the limiting natural frequencies are determined, **the locations with large displacements for the corresponding mode shapes have to be found**.
In case of more than one resonance this might be a weighted criterion for the different modes.
This approach is applicable for both open- loop and closed-loop performance criteria.
### The fluid model {#the-fluid-model}
A **linear visco-elastic fluid model** is derived from measurements and applied in the optimization formulations.
The results show that the model quality is good enough to predict the system’s damped behavior quite accurately.
### Open-loop modal damping improvement {#open-loop-modal-damping-improvement}
The principle of **broad banded damping** is well applicable for practical cases: the intended damping range was 1-4 kHz.
In addition, a damping increase is visible up to 6 kHz.
This frequency range abundantly covers the range in which performance limiting flexibilities usually arise in motion stage designs.
An optimization criterion in terms of resonance suppression is applied and works well: this criterion inherently only optimizes the visible resonances at the actuator and sensor location.
The choice which resonances should be suppressed, therefore, is specified in the cost function by the frequency response function.
Robustness of the solution and broad banded effect in practical cases is proven by the experimental validation.
The calculated suppression factor compares well to the measured ones.
The suppression factor amounts approximately 24 dB between 1 and 4 kHz, which indicates a modal damping increase factor of 16.
### Closed-loop performance increase {#closed-loop-performance-increase}
The principle of closed-loop performance increase is formulated in an optimization formulation which accurately estimates the bandwidth improvement factor.
The optimization formulation is non-convex, however, a hybrid optimization procedure is able to solve this specific problem in a limited amount of time.
In addition to the improvements in the intended control loops, other control loops often benefit from the damping increase.
### Advantages in analysis {#advantages-in-analysis}
A more general observation regarding the analyses method is presented.
The approach with separate RMDs is an efficient approach which contains two large advantages: It enables to continue with the current applied mechanical design approach for high natural frequencies and increase the modal damping afterwards.
This enables to still apply the materials with high specific stiffness and low damping.
In the analysis phase the advantages are enormous:
1. Undamped natural frequencies and mode shapes can be calculated and are valid for the low damped stage’s mechanical design.
These algorithms are very efficient and large models can be solved.
2. State space models can be created which contain the complexity of the FEM model and can be validated by calculating the responses by means of superposition of the undamped modes in the FEM software.
3. RMDs can be added at specific locations.
This results in non-proportional damping and complex mode shapes, which are correctly calculated by the state space model.
4. This enables to apply optimization algorithms and compare different RMDs very quickly.
The complete model including dampers can be solved in FEM, however, this approach contains serious drawbacks:
1. The mode shapes change from real normal modes to complex modes due to the damping at specific locations.
This implies that complex solvers have to be applied.
These solvers are much more time consuming than the solvers for real natural modes.
2. The frequency response functions can be calculated using fully harmonic solvers.
This results in the most accurate solution because the model is not truncated as in case of a state space model with a limited number of modes.
However, this algorithm solves the complete model for every frequency point in the frequency response function and, therefore, this approach is extremely time-consuming.
3. Therefore, in this approach the ability to implement different RMD parameters and execute optimization algorithms practically vanishes due to the limitations listed above.
<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>Verbaan, C.A.M. 2015. “Robust mass damper design for bandwidth increase of motion stages.” Mechanical Engineering; Technische Universiteit Eindhoven.</div>
</div>

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title = "Acquisition Systems"
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
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@ -11,6 +10,8 @@ Tags
## Manufacturers {#manufacturers}
<https://dewesoft.com/daq/list-of-data-acquisition-companies>
| Manufacturers | Country |
|----------------------------------------------------------------------------------------------------|----------|
| [Dewesoft](https://dewesoft.com/) | Slovenia |

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{{< youtube b9lxtOJj3yU >}}
Also see (<a href="#citeproc_bib_item_2">Kester 2005</a>).
## Link between required dynamic range and effective number of bits {#link-between-required-dynamic-range-and-effective-number-of-bits}
<a id="figure--fig:dynamic-range-enob"></a>
{{< figure src="/ox-hugo/dynamic_range_enob.png" caption="<span class=\"figure-number\">Figure 2: </span>Relation between Dynamic range and required number of bits (effective)" >}}
## Oversampling {#oversampling}
@ -92,4 +101,5 @@ The quantization is:
<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>Baker, Bonnie. 2011. “How Delta-Sigma Adcs Work, Part.” <i>Analog Applications</i> 7.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Kester, Walt. 2005. “Taking the Mystery out of the Infamous Formula, $snr = 6.02 N + 1.76 Db$, and Why You Should Care.”</div>
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title = "Eddy Current Damping"
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Passive Damping]({{< relref "passive_damping.md" >}})
<https://courses.lumenlearning.com/suny-physics/chapter/23-4-eddy-currents-and-magnetic-damping/>
## Vacuum compatible magnets {#vacuum-compatible-magnets}
<https://www.mceproducts.com/articles/magnets-in-vacuum-applications>
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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title = "Electromagnetism"
draft = false
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Tags
:
## Maxwell equations for magnetics {#maxwell-equations-for-magnetics}
### Gauss law {#gauss-law}
"Magnetic fieldlines are closed loop."
\begin{equation}
\oiint\_S (\bm{B} \cdot \hat{\bm{n}}) dS = 0
\end{equation}
### Faraday's law {#faraday-s-law}
A changing magnetic field causes an electric field over a wire
\begin{equation}
\oint\_L \bm{E} \cdot d\bm{l} = -\frac{d}{dt} \iint\_S(\bm{B} \cdot \bm{n}) dS
\end{equation}
The line-integral of the electrical field over a closed loop L equals the change of the field through the open surface S bounded by the loop L.
This is a voltage source (EMF), where the current is driven in the direction of the electric field.
### Ampère's law {#ampère-s-law}
"Current through a wire gives a magnetic field".
\begin{equation}
\oint\_L \bm{B} \cdot dl = \mu\_0 I
\end{equation}
The line integral of the magnetic field over a closed loop L is proportional to the current through the surface S enclosed by the loop L.
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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title = "Encoders"
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
@ -11,25 +10,18 @@ Tags
There are two main types of encoders: optical encoders, and magnetic encoders.
## Linear Encoders {#linear-encoders}
## Manufacturers {#manufacturers}
### Manufacturers {#manufacturers}
| Manufacturers | Country |
|---------------------------------------------------------------------------------|---------|
| [Heidenhain](https://www.heidenhain.com/en_US/products/linear-encoders/) | Germany |
| [MicroE Systems](https://www.celeramotion.com/microe/products/linear-encoders/) | USA |
| [Renishaw](https://www.renishaw.com/en/browse-encoder-range--6440) | UK |
| [Celera Motion](https://www.celeramotion.com/microe/) | USA |
<https://www.posic.com/EN/>
<https://www.rls.si/eng/products/rotary-magnetic-encoders>
## Angular Encoders {#angular-encoders}
<https://www.maxongroup.com/maxon/view/category/sensor>
| Manufacturers | Country |
|--------------------------------------------------------------------------|-------------|
| [Heidenhain](https://www.heidenhain.com/en_US/products/linear-encoders/) | Germany |
| [Renishaw](https://www.renishaw.com/en/browse-encoder-range--6440) | UK |
| [Celera Motion](https://www.celeramotion.com/microe/) | USA |
| [Magnescale](https://www.magnescale.com/en/) | Japanese |
| [Posic](https://www.posic.com/EN/) | Switzerland |
| [RLS](https://www.rls.si/eng/products/rotary-magnetic-encoders) | Slovenia |
| [AMO](https://www.amo-gmbh.com/en/) | Australia |
| [NumerikJena](https://www.numerikjena.de/en/) | Germany |
## Bibliography {#bibliography}

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title = "EtherCAT"
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
:
## Manufacturers {#manufacturers}
General purpose / PLC:
| Manufacturer |
|-------------------------------------------------------------------------------------|
| [Bechoff](https://www.beckhoff.com/fr-fr/products/i-o/ethercat-terminals/) |
| [Wago](https://www.wago.com/global/i-o-systems/fieldbus-coupler-ethercat/p/750-354) |
Acquisition systems:
| Manufacturer |
|----------------------------------------------------------------------------|
| [National Instrument](https://www.ni.com/fr-fr/support/model.ni-9145.html) |
| [Dewesoft](https://dewesoft.com/products/daq-systems) |
## Cycle Time {#cycle-time}
See (<a href="#citeproc_bib_item_1">Robert et al. 2012</a>).
There is a nice [online calculator](https://developer.acontis.com/ethercat-cycle-time-calculator.html).
## Bibliography {#bibliography}
<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>Robert, Jérémy, Jean-Philippe Georges, Eric Rondeau, and Thierry Divoux. 2012. “Minimum Cycle Time Analysis of Ethernet-Based Real-Time protocols.” <i>International Journal of Computers, Communications and Control</i> 7 (4). Agora University of Oradea: 743–57. <a href="https://hal.archives-ouvertes.fr/hal-00714560">https://hal.archives-ouvertes.fr/hal-00714560</a>.</div>
</div>

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title = "Granite"
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
@ -11,10 +10,11 @@ Tags
## Manufacturers {#manufacturers}
| Manufacturers | Country |
|--------------------------------------------------|---------|
| [Microplan](https://www.microplan-group.com/fr/) | France |
| [Zali](http://zali-precision.it/en/products/) | Italy |
| Manufacturers | Country |
|--------------------------------------------------|-------------|
| [Microplan](https://www.microplan-group.com/fr/) | France |
| [Zali](http://zali-precision.it/en/products/) | Italy |
| [Mytri](https://www.mytri.nl/en) | Netherlands |
## Bibliography {#bibliography}

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title = "Linear Brushless Motor"
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
:
: [Motors]({{< relref "motors.md" >}})
## Ironcore VS Ironless {#ironcore-vs-ironless}
- Ironcore: more torque/force density
- Ironless: less cogging
## Manufacturesr {#manufacturesr}
@ -27,6 +29,7 @@ Tags
| [Hiwin](https://www.hiwin.de/fr/Produits/c/3952) | Germany |
| [Baumeuller](https://www.baumueller.com/en/products/motors/linear-motors) | Germany |
| [Rexroth](https://www.boschrexroth.com/en/xc/products/product-groups/electric-drives-and-controls/motors-and-gearboxes/synchronous-linear-motors) | Germany |
| [Kollmorgen](https://www.kollmorgen.com/fr-fr/products/motors/direct-drive/direct-drive-linear/moteurs-lin%C3%A9aires-accouplement-direct/) | Germany |
| [PBA Systems](https://www.pbasystems.com.sg/product-category/precision-robotics/direct-drive-motors/) | Singapore |
| [Akribis](https://www.akribis-sys.de/en/produkte/1/linear-motors/) | Singapore |
| [Chieftek](http://www.chieftek.com/product-lm.asp) | Taiwan |

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author = ["Dehaeze Thomas"]
draft = false
+++
@ -13,7 +12,7 @@ Tags
Let's consider Figure [1](#figure--fig:mass-spring-damper-system) where:
- \\(m\\) is the mass in [kg]
- \\(ḱ\\) is the spring stiffness in [N/m]
- \\(k\\) is the spring stiffness in [N/m]
- \\(c\\) is the damping coefficient in [N/(m/s)]
- \\(F\\) is the actuator force in [N]
- \\(F\_d\\) is external force applied to the mass in [N]
@ -42,14 +41,34 @@ with:
- \\(\xi\\) the damping ratio
## Transmissibility {#transmissibility}
## Transfer function {#transfer-function}
### Voice Coil Actuator with flexible guiding {#voice-coil-actuator-with-flexible-guiding}
```matlab
%% Mechanical properties
m = 1; % Mobile mass [kg]
k = 1e6; % stiffness [N/m]
xi = 0.01; % Modal Damping
c = 2*xi*sqrt(k*m);
```
```matlab
%% Transfer function from F [N] to x [m]
G = 1/(m*s^2 + c*s + k);
```
### Transmissibility {#transmissibility}
\begin{equation}
\frac{x}{w}(s) = \frac{1}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1}
\end{equation}
## Compliance {#compliance}
### Compliance {#compliance}
\begin{equation}
\frac{x}{F\_d}(s) = \frac{1/k}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1}

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draft = false
+++
Tags
: [Motors]({{< relref "motors.md" >}})
## Sensors {#sensors}
- Hall effect sensors
- [Encoders]({{< relref "encoders.md" >}})
## Electrical Commutation {#electrical-commutation}
For a 3 phase motor (linear or angular), the force constant is a function of the position.
The motor can be designed in such a way that the relation is close to a sinusoidal function of a trapezoidal function.
### "Hard" commutation {#hard-commutation}
<a id="figure--fig:motor-hard-commutation"></a>
{{< figure src="/ox-hugo/motor_hard_commutation.png" caption="<span class=\"figure-number\">Figure 1: </span>By changing the direction of the current at the zero force positions of each coil (dashed), an almost constant force-constant of the total actuator is obtained." >}}
### Sinusoidal Commutation {#sinusoidal-commutation}
<a id="figure--fig:motor-sin-commutation"></a>
{{< figure src="/ox-hugo/motor_sin_commutation.png" caption="<span class=\"figure-number\">Figure 2: </span>Three phase commutation with a sinusoidal control of the currents in each coil segment (\\(I\_R, I\_S, I\_T\\)) in phase with their spatial sinusoidal force-constant \\(B l = k\\) values (\\(k\_R, k\_S, k\_T\\)) results in a force per segment with a spatial frequency that is double the original spatial frequency of the coils. The resulting total force of the three coil segments is the sum of the values of the force in each segment and is independent of the position." >}}
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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author = ["Dehaeze Thomas"]
draft = false
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@ -14,24 +13,19 @@ Reviews:
## Linear Motors {#linear-motors}
### Short Stroke {#short-stroke}
[Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}})
### Long Stroke {#long-stroke}
[Voice Coil Actuators]({{< relref "voice_coil_actuators.md" >}})
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}})
- [Voice Coil Actuators]({{< relref "voice_coil_actuators.md" >}})
- [Linear Brushless Motor]({{< relref "linear_brushless_motor.md" >}})
## Angular Motors {#angular-motors}
[Stepper Motor]({{< relref "stepper_motor.md" >}})
- [Stepper Motor]({{< relref "stepper_motor.md" >}})
- [Torque Motor]({{< relref "torque_motor.md" >}})
## Bibliography {#bibliography}
<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>Murugesan, S. 1981. “An Overview of Electric Motors for Space Applications.” <i>Ieee Transactions on Industrial Electronics and Control Instrumentation</i> IECI-28 (4): 260–65. doi:<a href="https://doi.org/10.1109/TIECI.1981.351050">10.1109/TIECI.1981.351050</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Murugesan, S. 1981. “An Overview of Electric Motors for Space Applications.” <i>IEEE Transactions on Industrial Electronics and Control Instrumentation</i> IECI-28 (4): 260–65. doi:<a href="https://doi.org/10.1109/TIECI.1981.351050">10.1109/TIECI.1981.351050</a>.</div>
</div>

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@ -165,21 +165,21 @@ This is due to the fact that voltage amplifier has a limitation on the deliverab
<a id="figure--fig:piezoelectric-capacitance-voltage-max"></a>
{{< figure src="/ox-hugo/piezoelectric_capacitance_voltage_max.png" caption="<span class=\"figure-number\">Figure 2: </span>Maximum sin-wave amplitude as a function of frequency for several piezoelectric capacitance" >}}
{{< figure src="/ox-hugo/piezoelectric_capacitance_voltage_max.png" caption="<span class=\"figure-number\">Figure 1: </span>Maximum sin-wave amplitude as a function of frequency for several piezoelectric capacitance" >}}
## Piezoelectric actuator experiencing a mass load {#piezoelectric-actuator-experiencing-a-mass-load}
When the piezoelectric actuator is supporting a payload, it will experience a static deflection due to its finite stiffness \\(\Delta l\_n = \frac{mg}{k\_p}\\), but its stroke will remain unchanged (Figure [3](#figure--fig:piezoelectric-mass-load)).
When the piezoelectric actuator is supporting a payload, it will experience a static deflection due to its finite stiffness \\(\Delta l\_n = \frac{mg}{k\_p}\\), but its stroke will remain unchanged (Figure [1](#figure--fig:piezoelectric-mass-load)).
<a id="figure--fig:piezoelectric-mass-load"></a>
{{< figure src="/ox-hugo/piezoelectric_mass_load.png" caption="<span class=\"figure-number\">Figure 3: </span>Motion of a piezoelectric stack actuator under external constant force" >}}
{{< figure src="/ox-hugo/piezoelectric_mass_load.png" caption="<span class=\"figure-number\">Figure 1: </span>Motion of a piezoelectric stack actuator under external constant force" >}}
## Piezoelectric actuator in contact with a spring load {#piezoelectric-actuator-in-contact-with-a-spring-load}
Then the piezoelectric actuator is in contact with a spring load \\(k\_e\\), its maximum stroke \\(\Delta L\\) is less than its free stroke \\(\Delta L\_f\\) (Figure [4](#figure--fig:piezoelectric-spring-load)):
Then the piezoelectric actuator is in contact with a spring load \\(k\_e\\), its maximum stroke \\(\Delta L\\) is less than its free stroke \\(\Delta L\_f\\) (Figure [1](#figure--fig:piezoelectric-spring-load)):
\begin{equation}
\Delta L = \Delta L\_f \frac{k\_p}{k\_p + k\_e}
@ -187,16 +187,16 @@ Then the piezoelectric actuator is in contact with a spring load \\(k\_e\\), its
<a id="figure--fig:piezoelectric-spring-load"></a>
{{< figure src="/ox-hugo/piezoelectric_spring_load.png" caption="<span class=\"figure-number\">Figure 4: </span>Motion of a piezoelectric stack actuator in contact with a stiff environment" >}}
{{< figure src="/ox-hugo/piezoelectric_spring_load.png" caption="<span class=\"figure-number\">Figure 1: </span>Motion of a piezoelectric stack actuator in contact with a stiff environment" >}}
For piezo actuators, force and displacement are inversely related (Figure [5](#figure--fig:piezoelectric-force-displ-relation)).
For piezo actuators, force and displacement are inversely related (Figure [1](#figure--fig:piezoelectric-force-displ-relation)).
Maximum, or blocked, force (\\(F\_b\\)) occurs when there is no displacement.
Likewise, at maximum displacement, or free stroke, (\\(\Delta L\_f\\)) no force is generated.
When an external load is applied, the stiffness of the load (\\(k\_e\\)) determines the displacement (\\(\Delta L\_A\\)) and force (\\(\Delta F\_A\\)) that can be produced.
<a id="figure--fig:piezoelectric-force-displ-relation"></a>
{{< figure src="/ox-hugo/piezoelectric_force_displ_relation.png" caption="<span class=\"figure-number\">Figure 5: </span>Relation between the maximum force and displacement" >}}
{{< figure src="/ox-hugo/piezoelectric_force_displ_relation.png" caption="<span class=\"figure-number\">Figure 1: </span>Relation between the maximum force and displacement" >}}
## Piezoelectric stiffness - Electrical Boundaries {#piezoelectric-stiffness-electrical-boundaries}
@ -211,13 +211,14 @@ Therefore, if the piezoelectric actuator is driven by a charge amplifier (i.e. h
Piezoelectric actuators can be driven either using a voltage to charge converter or a [Voltage Amplifier]({{< relref "voltage_amplifier.md" >}}).
Limitations of the electronics is discussed in [Design, modeling and control of nanopositioning systems]({{< relref "fleming14_desig_model_contr_nanop_system.md" >}}).
Also see (<a href="#citeproc_bib_item_4">Liu et al. 2007</a>).
## Bibliography {#bibliography}
<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_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>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>Liu, W. Q., Z. H. Feng, R. B. Liu, and J. Zhang. 2007. “The Influence of Preamplifiers on the Piezoelectric Sensor’s Dynamic Property.” <i>Review of Scientific Instruments</i> 78 (12): 125107. doi:<a href="https://doi.org/10.1063/1.2825404">10.1063/1.2825404</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_5"></a>Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”</div>

5
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@ -1,6 +1,5 @@
+++
title = "Signal to Noise Ratio"
author = ["Dehaeze Thomas"]
draft = false
+++
@ -15,7 +14,7 @@ From (<a href="#citeproc_bib_item_2">Jabben 2007</a>) (Section 3.3.2):
> Electronic equipment does most often not come with detailed electric schemes, in which case the PSD should be determined from measurements.
> In the design phase however, one has to rely on information provided by specification sheets from the manufacturer.
> The noise performance of components like sensors, amplifiers, converters, etc., is often specified in terms of a **Signal to Noise Ratio** (SNR).
> The SNR gives the ratio of the RMS value of a sine that covers the full range of the channel through which the signal is propagating over the RMS value of the electrical noise.
> The SNR gives the **ratio of the RMS value of a sine that covers the full range** of the channel through which the signal is propagating **over the RMS value of the electrical noise**.
>
> Usually, the SNR is specified up to a certain cut-off frequency.
> If no information on the colouring of the noise is available, then the corresponding **PSD can be assumed to be white up to the cut-off frequency** \\(f\_c\\):
@ -95,6 +94,6 @@ The peak-to-peak noise will be approximately \\(6 \sigma = 1.7 nm\\)
## Bibliography {#bibliography}
<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>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_1"></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_2"></a>Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.</div>
</div>

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@ -0,0 +1,26 @@
+++
title = "Torque Motor"
draft = false
+++
Tags
: [Motors]({{< relref "motors.md" >}})
## Manufacturers {#manufacturers}
| Manufacturers | Country |
|--------------------------------------------------------------------------------------------------------------------|-------------|
| [Tecnotion](https://www.tecnotion.com/product-category/torque-motors/) | Netherlands |
| [MagneticInnovations](https://www.magneticinnovations.com/direct-drive-electric-motors/torque-motor-direct-drive/) | Netherlands |
| [Etel](https://www.etel.ch/torque-motors/overview/) | Switzerland |
| [TDS](https://www.tds-pp.com/en/products/torque-motors/) | Switzerland |
| [Aerotech](https://www.aerotech.com/product/motors/s-series-brushless-frameless-torque-motor/) | USA |
| [ThinGap](https://www.thingap.com/) | USA |
| [CeleraMotion](https://www.celeramotion.com/applimotion/products/direct-drive-frameless-rotary-motors/) | |
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
</div>

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@ -1,6 +1,5 @@
+++
title = "Transconductance Amplifiers"
author = ["Dehaeze Thomas"]
draft = false
category = "equipment"
+++
@ -18,46 +17,6 @@ Such a converter is called a voltage-to-current converter, also named a voltage-
Such amplifier is used to control motors (e.g. voice coil, BLDC, stepper motors, ...).
## Specifications {#specifications}
### Noise {#noise}
```matlab
BL = 20; % [N/A]
m = 1; % [kg]
```
```matlab
freq = logspace(0,4,1000); % [Hz]
%% Current noise of the amplifier
I_asd = 1e-6*ones(size(freq)); % [A/sqrt(Hz)]
```
```matlab
x_asd = I_asd*(BL/m)./(2*pi*freq).^2;
```
```matlab
figure;
plot(freq, x_asd)
xlabel("Frequency [Hz]");
ylabel("ASD [$m/\sqrt{Hz}$]");
set(gca, 'Xscale', 'log');
set(gca, 'Yscale', 'log');
```
```matlab
figure;
plot(freq, sqrt(flip(-cumtrapz(flip(freq), flip(x_asd.^2)))))
xlabel("Frequency [Hz]");
ylabel("Cumulative Amplitude Spectrum [m rms]");
set(gca, 'Xscale', 'log');
set(gca, 'Yscale', 'log');
```
## Manufacturers {#manufacturers}
<a id="table--tab:table-name"></a>
@ -71,9 +30,8 @@ set(gca, 'Yscale', 'log');
| [Apogee](https://prodrive-technologies.com/motion/products/servo-drives/apogee-kepler-series/) | Prodrive | PWM | 1 to 3 | +/-10V 16bits | Encoder | 7kHz | 1e-6 |
| [S3-400/8](https://prodrive-technologies.com/motion/products/servo-drives/cygnus-series/) | Prodrive | PWM | 1 | +/-10V | Encoder | 1kHz | 1e-4 |
| [LWM7S](https://www.maccon.co.uk/linear-servo-amplifier.html) | Macon | Linear | 1 | | Encoder/Hall | | |
| [Soloist ML](https://www.aerotech.com/product/motion-control-platforms/soloist-ml-controller-and-linear-digital-drive/) | Aerotech | Linear | 1 | +/-10V 16bits | Encoder/Hall | | |
| [Automation1 XL4s](https://www.aerotech.com/product/motion-control-platforms/automation1-xl4s-high-performance-voice-coil-drive/) | Aerotech | Linear | 1 (voice coil) | +/-10V 16bits | ? | | |
| [Automation1 XL2e](https://www.aerotech.com/product/motion-control-platforms/automation1-xl4s-high-performance-voice-coil-drive/) | Aerotech | Linear | 1 | +/-10V 16bits | Encoder/Hall | 2.5kHz | |
| [Automation1 XL4s](https://www.aerotech.com/product/motion-control-platforms/automation1-xl4s-high-performance-voice-coil-drive/) | Aerotech | Linear | 1 (voice coil) | +/-10V 16bits | ? | | |
| [EM-356B](https://electromen.com/en/products/item/motor-controllers/brushless-dc-motor/EM-356B) | Electromen | PWM | 1 | 0-10V | Hall | | |
| [azbh10a4](https://www.a-m-c.com/product/azbh10a4/) | AMC | PWM | 1 | +/-10V | Hall | | |
| [X-MCC](https://www.zaber.com/products/controllers-joysticks/X-MCC) | Zaber | ?? | 1 to 4 | | | | |
@ -88,14 +46,276 @@ set(gca, 'Yscale', 'log');
| Model | Manufacturer | Linear / PWM | Axes | Interfaces | Current Bandwidth | Max Current | ASD at 1kHz [A/sqrt(Hz)] |
|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-----------------|--------------|------------------------|------------|-------------------|-------------|--------------------------|
| [LA300](https://varedan.com/product/analog-linear-servo-amplifiers/la-300-analog-linear-servo-amplifier/) | Varedan | Linear | 3 | +/-10V | 10kHz | 4A | |
| [LA24](https://www.cedrat-technologies.com/en/technologies/actuators/magnetic-actuators-motors.html) | Cedrat | Linear | 3 | +/-10V | 35kHz | 1.5A | |
| [CMAu10](https://www.cedrat-technologies.com/en/products/magnetic-controllers/oem-amplifiers.html) | Cedrat | Linear | 1 | +/-10V | 5kHz | 0.5A | |
| [TA115](https://www.trustautomation.com/products/linear-drives/ta115-linear-drive/) and [TA105](https://www.trustautomation.com/products/linear-drives/ta105-linear-drive/) | TrustAutomation | Linear | 1 | +/-10V | 5kHz | | 1e-6 |
| [SMA6520](https://www.glentek.com/shop/?swoof=1&product_cat=linear-brushless-series&really_curr_tax=21-product_cat) | Glentek | Linear | 1 Brushless (3 phases) | +/-10V | 10kHz | | |
| [SMA5005](https://www.glentek.com/shop/?swoof=1&product_cat=linear-brush-series&really_curr_tax=21-product_cat) | Glentek | Linear | 1 | +/-10V | 10kHz | | |
## Required properties {#required-properties}
Main required properties are (taken from (<a href="#citeproc_bib_item_1">Schmidt, Schitter, and Rankers 2020</a>)):
- **Power delivery capability**
- **Dynamic properties**
- **Linearity**
- **Voltage or current drive**
- **Efficiency**
- **Four quadrant operation**
## Four Quadrant Operation {#four-quadrant-operation}
The self-inductance of an electromagnetic actuator also causes another problem when the actuator is driven with a period signal, because for a sinusoidal signal the current is out of phase with the voltage.
In the extreme case of a purely reactive load, the maximum current needs to be delivered at zero voltage, while at a quarter of the period a positive current is delivered with a negative voltage and another quarter it is just the other way around.
In mechatronic positioning systems with a high moving mass, the real problem is caused by the kinetic energy that is involved.
At acceleration, the motion voltage of the actuator increases in phase with the current and electric power is inserted in the system and converted into kinetic energy.
The deceleration phase is however completely the opposite.
While the motion voltage still has the same sign as during constant motion, the current needs to be reversed in order to reverse the energy flow.
This means that the full amount of kinetic energy has to be absorbed by the amplifier.
## How to size a linear drive? {#how-to-size-a-linear-drive}
### Why it is important to properly choose a linear drive? {#why-it-is-important-to-properly-choose-a-linear-drive}
From a TrustAutomation [white paper](https://www.trustautomation.com/resources/engineering-blog/how-to-size-a-linear-drive-for-precision-positioning-applications/):
> The price you'll pay for the improved precision (i.e. thanks to the linear drive as compared to a PWM one) will mostly come in the form of heat.
> Linear drive typically maintain small amounts of power inside the drive circuits, increasing heat.
> **Excess voltage not needed by the motor is also dissipated as heat**.
### Determine required currents and voltages {#determine-required-currents-and-voltages}
In order to properly choose a linear amplifier, it is important to determine the voltage and torque that has to be generated.
The required current is based on the force (resp. torque) constant \\(K\_f\\) and peak force (resp. torque).
The required voltage is based on the back EMF constant \\(K\_u\\), peak velocity \\(v\_\text{peak}\\), peak current \\(I\_\text{peak}\\) and winding resistance \\(R\\).
<div class="exampl">
Consider a linear brushless motor with a force constant \\(K\_f\\) equal to 30 N/A, a BEMF constant \\(K\_u\\) equal to \\(18\\,\frac{Vrms}{m/s}\\) (i.e. \\(25\\,\frac{V}{m/s}\\)) and a electrical resistance \\(R\\) of \\(20\\,\Omega\\).
The peak velocity \\(v\_\max\\) is 1 mm/s and the wanted applied peak force \\(F\_\text{peak}\\) is 50 N.
The peak current required is:
\\[ I\_\text{peak} = F\_\text{peak}/K\_f \\]
And we obtain a peak current of 1.7 A.
The peak voltage is:
\\[ V\_\text{peak} = K\_u \cdot v\_\text{peak} + R \cdot I\_\text{peak} + V\_\text{margin} \\]
With \\(V\_\text{margin}\\) of 10 V, we obtain \\(V\_\text{peak} = 45\\,V\\).
From this simple calculation, it is possible to obtain the required capability of the amplifier.
</div>
### Determine safe operating area {#determine-safe-operating-area}
There are two danger scenarios: a stalled motor and a dynamic stopping motion
#### Stalled motor {#stalled-motor}
Consider the voltage supply to the drive \\(V\_\text{supply}\\) and the peak current \\(V\_\text{peak}\\).
Now suppose the motor is pushing against a hard stop, the power \\(W\_\text{drive}\\) that the drive must dissipate is equal to:
\\[ W\_\text{drive} = I\_\text{peak} \cdot V\_\text{drive} \\]
with:
\\[ V\_\text{drive} = V\_\text{supply} - I\_\text{peak} R \\]
<div class="exampl">
For our current application, \\(V\_\text{supply} = 45\\,V\\), \\(R = 20\\,\Omega\\) and \\(I\_\text{peak} = 1.7\\,A\\) which gives:
\\[ W\_\text{drive} = 19\\,W \\]
Then, it should be checked that the amplifier can dissipate this amount of power.
</div>
#### Dynamic Stopping. {#dynamic-stopping-dot}
With a linear drive, the kinetic energy is absorbed by the drive itself, but must be dissipated as heat.
This energy must be added to the energy required by the drive to stop all motion.
The kinetic energy \\(E\_K\\) is (expressed in Joules):
\\[ E\_K = \frac{1}{2} m v\_\text{peak}^2 \\]
with \\(m\\) the payload mass.
During the linear deceleration phase, the power \\(W\_d\\) that has to be dissipate by the drive is:
\\[ W\_d = \frac{E\_K}{t\_\text{dec}} \\]
with \\(t\_\text{dec}\\) the deceleration time.
<div class="exampl">
Consider a mass of 5 kg with a peak velocity of 1 mm/s and a deceleration time of 0.1s, the power to be dissipated in the drive is:
\\[ W\_d = 25\\,\mu W \\]
which is quite negligible.
If a velocity of 1 m/s is considered instead, we obtain \\(W\_d = 25\\,W\\).
</div>
### Matlab Script to size a linear drive {#matlab-script-to-size-a-linear-drive}
```matlab
%% Motor properties
Kt = 28; % Force constant [N/A] or Torque constant [Nm/A]
Ku = 28; % BEMF in [V/(m/s)] or in [V/(rad/s)]
R = 8.5; % Winding resistance [Ohm]
%% Motion property
Fp = 100; % Peak force [N] or Peak torque [Nm]
vp = 10e-3; % Peak Velocity [m/s] or peak rotation [rad/s]
m = 5; % Mass of the payload [kg]
td = 0.1; % Deceleration time [s]
```
```matlab
%% Driver wanted properties
V_margin = 10; % Power supply margin [V]
Imax = Fp / Kt; % Peak current to be supplied by the driver [A]
Vmax = vp * Ku + R * Imax + V_margin; % Peak voltage to be generated by the driver [V]
```
```text
Imax = 3.6 [A], Vmax = 41 [V]
```
```matlab
%% Stalled Motor
W_stalled = Imax * (Vmax - Imax * R); % [W]
```
```text
W_stalled = 37 [W]
```
```matlab
%% Dynamic Stopping
W_stop = 0.5*m*vp^2 / t_dec; % [W]
```
```text
W_stop = 0.0025 [W]
```
## Estimation of the required current noise {#estimation-of-the-required-current-noise}
### Voice Coil Actuator with flexible guiding {#voice-coil-actuator-with-flexible-guiding}
```matlab
%% Frequency vector used for the analysis
freqs = logspace(0, 4, 1000); % [Hz]
%% Motor properties
Kt = 28; % Force constant [N/A]
%% Amplifier Noise
In = 1e-6.*ones(size(freqs)); % Current noise density [A/sqrt(Hz)]
%% DAC Noise
Vn = (20/2^20)^2/12*1e4*ones(size(freqs)); % DAC output noise in [V/sqrt(Hz)]
Vn = 3e-8
Gi = 0.2; % Amplifier Gain [A/V]
%% Mechanical properties
m = 200e-3; % Mobile mass [kg]
k = 1e3; % Guiding stiffness [N/m]
xi = 0.05; % Modal Damping
```
```matlab
%% Transfer function from F [N] to x [m]
Gx = 1/(m*s^2);
```
```matlab
%% Transfer function from I [A] to x [m]
x_asd_i = In.*abs(squeeze(freqresp(Gx*Kt, freqs, 'Hz')))';
x_asd_v = Vn.*abs(squeeze(freqresp(Gx*Gi*Kt, freqs, 'Hz')))';
%% Cumulative amplitude spectrum
figure;
tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
nexttile();
hold on;
plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(x_asd_i.^2)))) , '-');
plot(freqs, sqrt(flip(-cumtrapz(flip(freqs), flip(x_asd_v.^2)))) , '-');
plot(freqs, ex , '-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); xlabel('Frequency [Hz]');
xlim([0, 1e3]);
```
### Approximate analytical formula {#approximate-analytical-formula}
Parameters:
- `Kt`: motor force constant in N/A
- `In`: current noise density of the amplifier in \\(A/\sqrt{Hz}\\)
- `m`: mass in kg
- `fb`: the feedback bandwidth in Hz
We have that the residual motion when the feedback controller is closed is approximately equal to:
\begin{equation}
\epsilon\_x = \sqrt{\int\_\infty^{f\_b} \left(\frac{K\_t I\_n}{m \omega^2}\right)^2 d\omega}
\end{equation}
\begin{equation}
\epsilon\_x = \frac{K\_t I\_n}{m (2\pi)^2} \sqrt{\frac{1}{3 f\_b^3}}
\end{equation}
Therefore, this formula can be used to:
-
<!--listend-->
```matlab
%% Estimate the position stability from the current noise and system parameters
m = 1; % [kg]
In = 1e-6; % [A/sqrt(Hz)]
Kt = 10; % [N/A]
fb = 10; % [Hz]
ex = In*Kt/m/(2*pi)^2*sqrt(1./(3*fb^3));
```
```text
epsilon x = 4.6 [nm RMS]
```
-
<!--listend-->
```matlab
%% Estimate the required current noise from the wanted position stability and the parameters of the system
m = 1; % [kg]
Kt = 10; % [N/A]
fb = 50; % [Hz]
ex = 1e-9; % [m RMS]
In = ex*m*(2*pi)^2/Kt * sqrt(3*fb^3);
```
```text
In = 2.4e-06 [A/sqrt(Hz)]
```
## Bibliography {#bibliography}
<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>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. <i>The Design of High Performance Mechatronics - Third Revised Edition</i>. Ios Press.</div>
</div>

25
content/zettels/tuned_mass_damper.md
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@ -19,6 +19,17 @@ The TMD then has large internal damping such that the energy is dissipated (i.e.
{{< youtube qDzGCgLu59A >}}
## How to properly apply a TMD? {#how-to-properly-apply-a-tmd}
Few questions:
- What damping mechanism to use?
Eddy current damping?
Viscous damping?
- How to optimize parameters of the TMD (i.e. mass, stiffness and damping)?
- Where to fix the TMD to the structure?
## Tuned Mass Damper Optimization {#tuned-mass-damper-optimization}
The optimal parameters of the tuned mass damper can be roughly estimated as follows:
@ -100,18 +111,18 @@ The following mass ratios are tested:
mus = [0.01, 0.02, 0.05, 0.1];
```
The obtained transfer functions are shown in Figure [3](#figure--fig:tuned-mass-damper-mass-effect).
The obtained transfer functions are shown in Figure [1](#figure--fig:tuned-mass-damper-mass-effect).
<a id="figure--fig:tuned-mass-damper-mass-effect"></a>
{{< figure src="/ox-hugo/tuned_mass_damper_mass_effect.png" caption="<span class=\"figure-number\">Figure 3: </span>Effect of the TMD mass on its efficiency" >}}
{{< figure src="/ox-hugo/tuned_mass_damper_mass_effect.png" caption="<span class=\"figure-number\">Figure 1: </span>Effect of the TMD mass on its efficiency" >}}
The maximum amplification (i.e. \\(\mathcal{H}\_\infty\\) norm) of the transmissibility as a function of the mass ratio is shown in Figure [4](#figure--fig:tuned-mass-damper-effect-mass-ratio).
The maximum amplification (i.e. \\(\mathcal{H}\_\infty\\) norm) of the transmissibility as a function of the mass ratio is shown in Figure [1](#figure--fig:tuned-mass-damper-effect-mass-ratio).
This relation can help to determine the minimum mass of the TMD that will give acceptable results.
<a id="figure--fig:tuned-mass-damper-effect-mass-ratio"></a>
{{< figure src="/ox-hugo/tuned_mass_damper_effect_mass_ratio.png" caption="<span class=\"figure-number\">Figure 4: </span>Maximum amplification due to resonance as a function of the mass ratio" >}}
{{< figure src="/ox-hugo/tuned_mass_damper_effect_mass_ratio.png" caption="<span class=\"figure-number\">Figure 1: </span>Maximum amplification due to resonance as a function of the mass ratio" >}}
## Manufacturers {#manufacturers}
@ -126,7 +137,11 @@ This relation can help to determine the minimum mass of the TMD that will give a
Possible damping sources:
- Magnetic (eddy current)
- Viscous
- Viscous fluid
| Fuild | Reference |
|----------------------|---------------------------------------------------|
| Rocol Kilopoise 0868 | (<a href="#citeproc_bib_item_2">Verbaan 2015</a>) |
<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>Elias, Said, and Vasant Matsagar. 2017. “Research Developments in Vibration Control of Structures Using Passive Tuned Mass Dampers.” <i>Annual Reviews in Control</i> 44 (nil): 129–56. doi:<a href="https://doi.org/10.1016/j.arcontrol.2017.09.015">10.1016/j.arcontrol.2017.09.015</a>.</div>

30
content/zettels/voice_coil_actuators.md
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@ -27,20 +27,18 @@ As the force is proportional to the current, a [Transconductance Amplifiers]({{<
## Manufacturers {#manufacturers}
| Manufacturers | Country |
|-------------------------------------------------------------------------------------------------------------------------------------|-------------|
| [Thorlabs](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=14116) | |
| [Geeplus](https://www.geeplus.com/) | UK |
| [Maccon](https://www.maccon.de/en.html) | Germany |
| [TDS PP](https://www.tds-pp.com/en/product/linear-voice-coil-actuators-avm/) | Switzerland |
| [PBA Systems](https://www.pbasystems.com.sg/product/circular-voice-coil-motor-cvc/) | Singapore |
| [Magnetic Innovations](https://www.magneticinnovations.com/) | Netherlands |
| [H2tech](https://www.h2wtech.com/) | USA |
| [Beikimco](http://www.beikimco.com/) | USA |
| [Monticont](http://www.moticont.com/) | USA |
| [Thorlabs](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=14116) | USA |
| [Akribis](https://akribis-sys.com/products/voice-coil-motors/avm-series) | USA |
| [Celera](https://www.celeramotion.com/applimotion/products/direct-drive-frameless-linear-motors/voice-coil/juke-series-round-body/) | |
| Manufacturers | Country |
|-------------------------------------------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------------------------------------------------------|
| [Akribis](https://akribis-sys.com/products/voice-coil-motors/avm-series) | Singapore (european distributors: [Maccon](https://www.maccon.de/en.html), [TDS PP](https://www.tds-pp.com/en/product/linear-voice-coil-actuators-avm/)) |
| [Thorlabs](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=14116) | |
| [Geeplus](https://www.geeplus.com/) | UK |
| [PBA Systems](https://www.pbasystems.com.sg/product/circular-voice-coil-motor-cvc/) | Singapore |
| [Magnetic Innovations](https://www.magneticinnovations.com/) | Netherlands |
| [H2tech](https://www.h2wtech.com/) | USA |
| [Beikimco](http://www.beikimco.com/) | USA |
| [Monticont](http://www.moticont.com/) | USA |
| [Thorlabs](https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=14116) | USA |
| [Celera](https://www.celeramotion.com/applimotion/products/direct-drive-frameless-linear-motors/voice-coil/juke-series-round-body/) | |
## Voice Coil Stages {#voice-coil-stages}
@ -135,11 +133,11 @@ Dg = m * g ./ k; % [m]
<a id="figure--fig:voice-coil-resonance-fct-stroke"></a>
{{< figure src="/ox-hugo/voice_coil_resonance_fct_stroke.png" caption="<span class=\"figure-number\">Figure 3: </span>Resonance frequency and deflection due to gravity as a function of the wanted stroke (Max voice coil force is 50N and payload mass is 5kg)" >}}
{{< figure src="/ox-hugo/voice_coil_resonance_fct_stroke.png" caption="<span class=\"figure-number\">Figure 1: </span>Resonance frequency and deflection due to gravity as a function of the wanted stroke (Max voice coil force is 50N and payload mass is 5kg)" >}}
<a id="figure--fig:voice-coil-stiffness-fct-stroke"></a>
{{< figure src="/ox-hugo/voice_coil_stiffness_fct_stroke.png" caption="<span class=\"figure-number\">Figure 4: </span>Resonance frequency and deflection due to gravity as a function of the wanted stroke (Max voice coil force is 50N and payload mass is 5kg)" >}}
{{< figure src="/ox-hugo/voice_coil_stiffness_fct_stroke.png" caption="<span class=\"figure-number\">Figure 1: </span>Resonance frequency and deflection due to gravity as a function of the wanted stroke (Max voice coil force is 50N and payload mass is 5kg)" >}}
## Bibliography {#bibliography}

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