Publications: add paper pages (dehaeze18, brumund21, dehaeze20, dehaeze21 x2), drop Fastjack, look for PDFs in journal/
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title = "Multibody Simulations with Reduced Order Flexible Bodies obtained by FEA"
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author = ["Dehaeze Thomas"]
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draft = false
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venue = "MEDSI 2020"
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year = 2021
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pubtype = "conference"
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doi = "10.18429/JACoW-MEDSI2020-WEPB08"
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code = "https://git.tdehaeze.xyz/tdehaeze/brumund21_multib_simul_reduc_order_flexib_bodies_fea"
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+++
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> **Abstract**:
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>
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> Tighter specifications in synchrotron instrumentation development force the design engineers more and more often to choose a mechatronics design approach.
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> This includes actively controlled systems that need to be properly designed.
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> The new Nano Active Stabilization System (NASS) for the ESRF beamline ID31 was designed with such an approach.
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>
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> We chose a multi-body design modelling approach for the development of the NASS end-station.
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> Significance of such models depend strongly on its input and consideration of the right stiffness of the system's components and subsystems.
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> For that matter, we considered sub-components in the multi-body model as _reduced order flexible bodies_ representing the component's modal behaviour with reduced mass and stiffness matrices obtained from finite element analysis (FEA) models.
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> These matrices were created from FEA models via modal reduction techniques, more specifically the component mode synthesis (CMS).
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> This makes this design approach a combined multibody-FEA technique.
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>
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> We validated the technique with a test bench that confirmed the good modelling capabilities using reduced order flexible body models obtained from FEA for an amplified piezoelectric actuator (APA).
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## Conference Paper ([pdf](paper/brumund21_multib_simul_reduc_order_flexib_bodies_fea.pdf)) {#conference-paper--pdf-paper-brumund21-multib-simul-reduc-order-flexib-bodies-fea-dot-pdf}
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## Cite this work {#cite-this-work}
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To cite the conference paper use the following bibTeX code.
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```bibtex
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@inproceedings{brumund21_multib_simul_reduc_order_flexib_bodies_fea,
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author = {Philipp Brumund and Thomas Dehaeze},
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title = {Multibody Simulations with Reduced Order Flexible Bodies
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obtained by {FEA}},
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booktitle = {MEDSI'20},
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year = 2021,
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language = {english},
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publisher = {JACoW Publishing},
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series = {Mechanical Engineering Design of Synchrotron Radiation
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Equipment and Instrumentation},
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venue = {Chicago, USA},
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}
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```
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You can also use the formatted citation below.
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> Brumund, P., & Dehaeze, T., Multibody simulations with reduced order flexible bodies obtained by FEA, In MEDSI'20 (2021), JACoW Publishing.
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title = "Sample Stabilization for Tomography Experiments in Presence of Large Plant Uncertainty"
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author = ["Dehaeze Thomas"]
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draft = false
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venue = "MEDSI 2018"
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year = 2018
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pubtype = "conference"
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doi = "10.18429/JACoW-MEDSI2018-WEOAMA02"
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code = "https://github.com/tdehaeze/dehaeze18_sampl_stabil_for_tomog_exper"
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+++
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> **Abstract**:
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>
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> A new low emittance lattice storage ring is under construction at the ESRF.
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> In this new instrument, an upgraded end station for ID31 beamline must allow to position the samples along complex trajectories with a nanometer precision.
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> In order to reach these requirements, samples have to be mounted on high precision stages, combining a capability of large stroke, spin motion, and active rejection of disturbances.
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> First, the end station will be presented with the associated requirements. However, the precision is limited by thermal expansion and various imperfections that are not actively compensated.
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> Our approach is to add a Nano Active Stabilization System (NASS) which is composed of a 6DoF Stewart platform and a 6 DoF metrology system.
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> A 3D model of the end station updated with experimental data is developed.
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> As the mass of the samples may vary by up to two orders of magnitudes, robust control strategies are required to address such plant uncertainty.
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> The proposed control strategy are presented and applied on the developed model by conducting time domain simulations of tomography experiment in presence of instrumentation noise and system uncertainty.
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## Paper ([link](paper/dehaeze18_sampl_stabil_for_tomog_exper.pdf)) {#paper--link-paper-dehaeze18-sampl-stabil-for-tomog-exper-dot-pdf}
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The paper has been created [Org Mode](https://orgmode.org/) (generating [LaTeX](https://www.latex-project.org/) code) under [Emacs](https://www.gnu.org/software/emacs/).
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## Tikz Figures ([link]({{< relref "tikz/_index.md" >}})) {#tikz-figures--link-tikz-index-dot-md}
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All the figures for the paper have been generated using [TikZ](https://sourceforge.net/projects/pgf/).
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## Poster ([link](poster/dehaeze18_sampl_stabil_for_tomog_exper_poster.pdf)) {#poster--link-poster-dehaeze18-sampl-stabil-for-tomog-exper-poster-dot-pdf}
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The poster has been created using the [tikzposter](https://www.ctan.org/pkg/tikzposter) package for [beamer](https://sourceforge.net/projects/latex-beamer/).
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## Talk ([link](talk/dehaeze18_sampl_stabil_for_tomog_exper_talk.pdf)) {#talk--link-talk-dehaeze18-sampl-stabil-for-tomog-exper-talk-dot-pdf}
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This work has been presented at [MEDSI 2018](https://indico.cern.ch/event/680538/).
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## How to cite this paper {#how-to-cite-this-paper}
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To cite this paper use the following bibtex code.
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```bibtex
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@inproceedings{dehaeze18_sampl_stabil_for_tomog_exper,
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author = {Thomas Dehaeze and M. Magnin Mattenet and Christophe Collette},
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title = {Sample Stabilization For Tomography Experiments In Presence Of
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Large Plant Uncertainty},
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booktitle = {MEDSI'18},
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year = 2018,
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number = 10,
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pages = {153--157},
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doi = {10.18429/JACoW-MEDSI2018-WEOAMA02},
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url = {https://doi.org/10.18429/JACoW-MEDSI2018-WEOAMA02},
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address = {Geneva, Switzerland},
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isbn = {978-3-95450-207-3},
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language = {english},
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month = {Dec},
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publisher = {JACoW Publishing},
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series = {Mechanical Engineering Design of Synchrotron Radiation
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Equipment and Instrumentation},
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venue = {Paris, France},
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}
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```
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You can also use the formatted citation below.
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> Dehaeze, T., Mattenet, M. M., & Collette, C., Sample Stabilization For Tomography Experiments In Presence Of Large Plant Uncertainty, In MEDSI'18 (pp. 153–157) (2018). Geneva, Switzerland
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title = "Sample Stabilization for Tomography Experiments in Presence of Large Plant Uncertainty - Tikz Figures"
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author = ["Dehaeze Thomas"]
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draft = false
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+++
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Configuration file is accessible [here]({{< relref "config.md" >}}).
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## Fig 1: Schematic representation of the ID31 end station {#fig-1-schematic-representation-of-the-id31-end-station}
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<a id="figure--fig:schematic-sys-without-nass"></a>
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{{< figure src="figs/schematic_sys_without_nass.png" caption="<span class='figure-number'>Figure 1: </span>Schematic representation of the ID31 end station ([png](figs/schematic_sys_without_nass.png), [pdf](figs/schematic_sys_without_nass.pdf), [tex](./figs/schematic_sys_without_nass.tex))." >}}
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## Fig 2: CAD View of the ID31 end station {#fig-2-cad-view-of-the-id31-end-station}
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```latex
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\graphicspath{{~/Cloud/tikz/org/img/}}
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\begin{tikzpicture}
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\tikzstyle{legend}=[draw, text width=4.2cm, align=center]
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\node[inner sep=0pt, anchor=south west] (assemblage) at (0,0)
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{\includegraphics[width=0.42\textwidth]{/home/thomas/Cloud/thesis/papers/dehaeze18_sampl_stabil_for_tomog_exper/tikz/img/assemblage_img.png}};
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\coordinate[] (aheight) at (assemblage.north west);
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\coordinate[] (awidth) at (assemblage.south east);
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\coordinate[] (xrightlabel) at (-0.2, 0);
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\coordinate[] (xleftlabel) at ($(awidth)+(0.2, 0)$);
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% Translation Stage
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\coordinate[] (ty) at ($0.5*(aheight)+0.1*(awidth)$);
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\draw[<-] (ty) -- (ty-|xrightlabel) node[left, legend]{Translation Stage\\$\SI{-5}{m\metre} < T_y < \SI{5}{m\metre}$};
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% Sample Interface
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\coordinate[] (sampleint) at ($0.77*(aheight)+0.5*(awidth)$);
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\coordinate[] (sampleintmid) at ($(sampleint)+(-1, -0.5)$);
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\draw[<-] (sampleint) -- (sampleintmid) -- (sampleintmid-|xrightlabel) node[left, legend]{Sample Interface};
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% Sample
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\coordinate[] (sample) at ($0.9*(aheight)+0.5*(awidth)$);
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\draw[<-] (sample) -- (sample-|xrightlabel) node[left, legend]{Sample Environment\\$\SI{1}{\kg} < M < \SI{50}{\kg}$};
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% Tilt Stage
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\coordinate[] (tilt) at ($0.55*(aheight)+0.78*(awidth)$);
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\coordinate[] (tiltmid) at ($(tilt)+(1, 0.5)$);
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\draw[<-] (tilt) -- (tiltmid) -- (tiltmid-|xleftlabel) node[right, legend]{Tilt Stage\\$\ang{-3} < \theta_y < \ang{3}$};
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% Spindle
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\coordinate[] (spindle) at ($0.53*(aheight)+0.33*(awidth)$);
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\coordinate[] (spindlemid) at ($(spindle)+(-1, -1.5)$);
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\draw[<-] (spindle) -- (spindlemid) -- (spindlemid-|xrightlabel) node[left, legend]{Spindle\\$\SI{1}{rpm} < \dot{\theta_z} < \SI{60}{rpm}$};
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% Center of gravity compensation
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\coordinate[] (axisc) at ($0.65*(aheight)+0.65*(awidth)$);
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\coordinate[] (axiscmid) at ($(axisc)+(1, 1.5)$);
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\draw[<-] (axisc) -- (axiscmid) -- (axiscmid-|xleftlabel) node[right, legend]{Center of gravity\\compensation system};
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% Micro Hexapod
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\coordinate[] (hexapod) at ($0.52*(aheight)+0.6*(awidth)$);
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\coordinate[] (hexapodmid) at ($(hexapod)+(1, -1.0)$);
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\draw[<-] (hexapod) -- (hexapodmid) -- (hexapodmid-|xleftlabel) node[right, legend]{Long Stroke Hexapod\\$\SI{-10}{m\metre} < T_{x y z} < \SI{10}{m\metre}$\\$\ang{-3} < \theta_{x y z} < \ang{3}$};
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% Frame
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\coordinate[] (frame) at ($0.14*(aheight)+0.65*(awidth)$);
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\draw[<-] (frame) -- (frame-|xleftlabel) node[right, legend]{Frame fixed\\on the granite};
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% X-Ray
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\draw[color=red, ->-=0.7] ($0.92*(aheight)+0.8*(awidth)$) -- node[above, color=black]{X-ray} ++(190:1.8);
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% Size of the setup
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\draw[dashed, <->, color=black!70, line width=0.5pt] ($0.03*(aheight)+0.35*(awidth)$) -- node[below, color=black, pos=0.6]{$\approx\SI{1}{m}$} ($0.14*(aheight)+0.98*(awidth)$);
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\draw[dashed, <->, color=black!70, line width=0.5pt] ($0.032*(aheight)+0.32*(awidth)$) -- node[left, color=black, pos=0.4]{$\approx\SI{1}{m}$} ($0.305*(aheight)+0.0*(awidth)$);
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% Axis
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\begin{scope}[shift={(0.0, 0.7)}]
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\draw[->] (0, 0) -- ++(195:0.8) node[above] {$x$};
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\draw[->] (0, 0) -- ++(90:0.9) node[right] {$z$};
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\draw[->] (0, 0) -- ++(-40:0.7) node[above] {$y$};
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\end{scope}
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\end{tikzpicture}
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```
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<a id="figure--fig:assemblage"></a>
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{{< figure src="figs/assemblage.png" caption="<span class='figure-number'>Figure 2: </span>CAD View of the ID31 end station ([png](figs/assemblage.png), [pdf](figs/assemblage.pdf), [tex](./figs/assemblage.tex))." >}}
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## Fig 3: Picture of the ID31 end station {#fig-3-picture-of-the-id31-end-station}
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```latex
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\begin{tikzpicture}
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\node[inner sep=0pt, anchor=south west] (photo) at (0,0)
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{\includegraphics[width=0.39\textwidth]{/home/thomas/Cloud/thesis/papers/dehaeze18_sampl_stabil_for_tomog_exper/tikz/img/exp_setup_photo.png}};
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\coordinate[] (aheight) at (photo.north west);
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\coordinate[] (awidth) at (photo.south east);
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\coordinate[] (granite) at ($0.1*(aheight)+0.1*(awidth)$);
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\coordinate[] (trans) at ($0.5*(aheight)+0.4*(awidth)$);
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\coordinate[] (tilt) at ($0.65*(aheight)+0.75*(awidth)$);
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\coordinate[] (hexapod) at ($0.7*(aheight)+0.5*(awidth)$);
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\coordinate[] (sample) at ($0.9*(aheight)+0.55*(awidth)$);
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% Granite
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\node[labelc] at (granite) {1};
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% Translation stage
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\node[labelc] at (trans) {2};
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% Tilt Stage
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\node[labelc] at (tilt) {3};
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% Micro-Hexapod
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\node[labelc] at (hexapod) {4};
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% Sample
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\node[labelc] at (sample) {5};
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% Axis
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\begin{scope}[shift={($0.07*(aheight)+0.87*(awidth)$)}]
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\draw[->] (0, 0) -- ++(55:0.7) node[above] {$y$};
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\draw[->] (0, 0) -- ++(90:0.9) node[left] {$z$};
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\draw[->] (0, 0) -- ++(-20:0.7) node[above] {$x$};
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\end{scope}
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\end{tikzpicture}
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```
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<a id="figure--fig:exp-setup"></a>
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{{< figure src="figs/exp_setup.png" caption="<span class='figure-number'>Figure 3: </span>Picture of the ID31 end station ([png](figs/exp_setup.png), [pdf](figs/exp_setup.pdf), [tex](./figs/exp_setup.tex))." >}}
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## Fig 4: Schematic representation of the NASS added below the sample and the control architecture used {#fig-4-schematic-representation-of-the-nass-added-below-the-sample-and-the-control-architecture-used}
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<a id="figure--fig:system-control"></a>
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{{< figure src="figs/system_control.png" caption="<span class='figure-number'>Figure 4: </span>Schematic representation of the NASS added below the sample and the control architecture used ([png](figs/system_control.png), [pdf](figs/system_control.pdf), [tex](./figs/system_control.tex))." >}}
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## Fig 5: Transfer function from a force applied by the NASS to the displacement of the sample {#fig-5-transfer-function-from-a-force-applied-by-the-nass-to-the-displacement-of-the-sample}
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<a id="figure--fig:G-x-mass"></a>
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{{< figure src="figs/G_x_mass.png" caption="<span class='figure-number'>Figure 5: </span>Transfer function from a force applied by the NASS to the displacement of the sample ([png](figs/G_x_mass.png), [pdf](figs/G_x_mass.pdf))." >}}
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## Fig 6: General control configuration applied to the end station {#fig-6-general-control-configuration-applied-to-the-end-station}
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```latex
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\begin{tikzpicture}
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% Blocs
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\node[block={2.5cm}{2cm}] (P) {P};
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\node[block={2.5cm}{2cm}, below=1 of P, scale=0.6] (K) {\[%
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\begin{pmatrix}
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K_{T_x} & 0 & \cdots & 0 \\
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0 & \ddots & \ddots & \vdots \\
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\vdots & \ddots & \ddots & 0 \\
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0 & \cdots & 0 & K_{\theta_z} \\
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\end{pmatrix}
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\]};
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% Block names
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\node[above] at (P.north) {End Station};
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\node[above] at (K.north) {Controller};
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% Input and outputs coordinates
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\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west)$);
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\coordinate[] (inputu) at ($(P.south west)!0.25!(P.north west)$);
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\coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east)$);
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\coordinate[] (outputv) at ($(P.south east)!0.25!(P.north east)$);
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% Connections and labels
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\draw[<-] (inputw) node[above left]{$w$} -- ++(-0.8, 0);
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\draw[<-] (inputu) node[above left]{$F$} -- ++(-0.8, 0) |- (K.west);
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\draw[->] (outputz) node[above right]{$z$} -- ++(0.8, 0);
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\draw[->] (outputv) node[above right]{$d$} -- ++(0.8, 0) |- (K.east);
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\end{tikzpicture}
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```
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<a id="figure--fig:general-conf-K"></a>
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{{< figure src="figs/general_conf_K.png" caption="<span class='figure-number'>Figure 6: </span>General control configuration applied to the end station ([png](figs/general_conf_K.png), [pdf](figs/general_conf_K.pdf), [tex](./figs/general_conf_K.tex))." >}}
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## Fig 7: Bode plot of the loop gain for the control in the x direction {#fig-7-bode-plot-of-the-loop-gain-for-the-control-in-the-x-direction}
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<a id="figure--fig:loopgain"></a>
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{{< figure src="figs/loopgain.png" caption="<span class='figure-number'>Figure 7: </span>Bode plot of the loop gain for the control in the x direction ([png](figs/loopgain.png), [pdf](figs/loopgain.pdf), [tex](./figs/loopgain.tex))." >}}
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## Fig 8: Positioning error of the sample in the x and y direction during the simulation of a tomography experiment {#fig-8-positioning-error-of-the-sample-in-the-x-and-y-direction-during-the-simulation-of-a-tomography-experiment}
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<a id="figure--fig:exp-w-wo-nass-xy"></a>
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{{< figure src="figs/exp_w_wo_nass_xy.png" caption="<span class='figure-number'>Figure 8: </span>Positioning error of the sample in the x and y direction during the simulation of a tomography experiment ([png](figs/exp_w_wo_nass_xy.png), [pdf](figs/exp_w_wo_nass_xy.pdf))." >}}
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## Fig 1: Schematic of the Tomography Experiment (Poster) {#fig-1-schematic-of-the-tomography-experiment--poster}
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<a id="figure--fig:exp-full-setup"></a>
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{{< figure src="figs/exp_full_setup.png" caption="<span class='figure-number'>Figure 9: </span>Schematic of the Tomography Experiment ([png](figs/exp_full_setup.png), [pdf](figs/exp_full_setup.pdf), [tex](./figs/exp_full_setup.tex))." >}}
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@@ -0,0 +1,773 @@
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+++
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||||
title = "LaTeX Configuration for Tikz Figures"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
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+++
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## Packages {#packages}
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|
||||
```latex
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[T1]{fontenc}
|
||||
|
||||
\usepackage[french, english]{babel} % Last language is main language
|
||||
|
||||
\usepackage{lmodern} % Latin Modern Font
|
||||
\usepackage{gensymb} % Generic symbols for both text and math mode
|
||||
|
||||
\usepackage{standalone} % Used to generate standalone Tikz
|
||||
|
||||
\usepackage{amsmath} % Main math Package
|
||||
\usepackage{mathtools} % Extension package to amsmath
|
||||
\usepackage{amsthm} % Typesetting theorems (AMS style)
|
||||
\usepackage{amsfonts} % More fonts from the AMS
|
||||
\usepackage{textcomp} % provide many text symbols
|
||||
\usepackage{steinmetz} % For phase symbol
|
||||
|
||||
\usepackage{xstring} % Utils to manipulate strings
|
||||
\usepackage{etoolbox} % Add basic if/then
|
||||
\usepackage{esvect} % Beautyfull vectors
|
||||
\usepackage{graphicx} % Enhanced support for graphics
|
||||
\usepackage{grffile} % Used by matlab2tikz
|
||||
|
||||
\usepackage{microtype} % typographic tuning
|
||||
\usepackage{setspace} % for line spacing, e.g. \onehalfspacing
|
||||
\usepackage{tabularx} % table features
|
||||
\usepackage{enumitem} % for simple list modifications
|
||||
\usepackage{booktabs} % better table support
|
||||
|
||||
\usepackage{stackengine} %
|
||||
|
||||
\usepackage[load-configurations=abbreviations]{siunitx} % SI units
|
||||
\sisetup{
|
||||
locale = US,
|
||||
detect-all,
|
||||
range-phrase=--,
|
||||
range-units=single
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Tikz related packages {#tikz-related-packages}
|
||||
|
||||
```latex
|
||||
\usepackage{tikz} % Tikz
|
||||
\usepackage{tikzscale} % Used to scale Tikz graphics
|
||||
\usepackage{adjustbox} % Used to proper positioning of tikz pictures
|
||||
\usepackage{circuitikz} % Draw electronic circuits
|
||||
\usepackage{pgfpages} % Needed to use notes
|
||||
\usepackage{pgfplots} % Used to plot functions
|
||||
```
|
||||
|
||||
|
||||
## Tikz Libraries {#tikz-libraries}
|
||||
|
||||
```latex
|
||||
\usetikzlibrary{arrows} % Arrow tip library
|
||||
\usetikzlibrary{arrows.meta} % Add some arrows
|
||||
\usetikzlibrary{calc} % The library allows advanced Coordinate Calculations
|
||||
\usetikzlibrary{intersections} % calculate intersections of paths
|
||||
\usetikzlibrary{matrix} %
|
||||
\usetikzlibrary{patterns} %
|
||||
\usetikzlibrary{shapes} % Defines circle and rectangle
|
||||
\usetikzlibrary{shapes.geometric} % Use for the shape diamond and isosceles triangle
|
||||
\usetikzlibrary{snakes} % snake=coil and snake=zigzag using segment amplitude=10pt
|
||||
\usetikzlibrary{positioning} % Additional options for placing nodes
|
||||
\usetikzlibrary{3d} % Plot 3D shapes
|
||||
\usetikzlibrary{spy} % Creating a magnified area
|
||||
\usetikzlibrary{decorations.text} % Used to make text follows a curve
|
||||
\usetikzlibrary{decorations.pathmorphing} % deformation of a path
|
||||
\usetikzlibrary{decorations.markings} % Used for spring and damper
|
||||
\usetikzlibrary{babel} % A tiny library that make the interaction with the babel package easier
|
||||
\usetikzlibrary{plotmarks} % This library defines a number of plot marks
|
||||
\usetikzlibrary{fit} % Used to make rectangle as nodes by specifying two points
|
||||
\usetikzlibrary{backgrounds} % Used to put things under others
|
||||
```
|
||||
|
||||
|
||||
## PGF Plot libraries and config {#pgf-plot-libraries-and-config}
|
||||
|
||||
```latex
|
||||
\usepgfplotslibrary{patchplots}
|
||||
\usepgfplotslibrary{groupplots}
|
||||
|
||||
\pgfplotsset{compat=newest}
|
||||
\pgfplotsset{plot coordinates/math parser=false}
|
||||
```
|
||||
|
||||
|
||||
## Setup size of figures {#setup-size-of-figures}
|
||||
|
||||
```latex
|
||||
\newlength{\fheight}
|
||||
\newlength{\fwidth}
|
||||
|
||||
\setlength{\fwidth}{85mm}
|
||||
\setlength{\fheight}{112mm}
|
||||
```
|
||||
|
||||
|
||||
## Setup Arrows style {#setup-arrows-style}
|
||||
|
||||
```latex
|
||||
\tikzset{>=Stealth}
|
||||
% Setup default Linewidth
|
||||
\tikzset{every path/.style={line width=1pt}}
|
||||
```
|
||||
|
||||
|
||||
## Colors {#colors}
|
||||
|
||||
```latex
|
||||
\usepackage{xcolor}% Color extension
|
||||
|
||||
\definecolor{mycolor1}{RGB}{79,115,193}
|
||||
\definecolor{mycolor2}{RGB}{213,91,53}
|
||||
\definecolor{mycolor3}{RGB}{152,126,49}
|
||||
```
|
||||
|
||||
|
||||
## Control {#control}
|
||||
|
||||
|
||||
### Blocks {#blocks}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
block/.style n args={2}{%
|
||||
draw,
|
||||
fill=white,
|
||||
minimum width = #1,
|
||||
minimum height = #2,
|
||||
},
|
||||
block/.default={1.2cm}{1.0cm}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Branches {#branches}
|
||||
|
||||
```latex
|
||||
\tikzstyle{branch}=[fill,shape=circle,minimum size=4pt,inner sep=0pt]
|
||||
\tikzstyle{->top}=[-{Stealth[color=black, scale=0.8]}, draw=white, double=black, double distance=1pt, line width=1pt]
|
||||
\tikzstyle{<-top}=[{stealth[color=black, scale=0.8]}-, draw=white, double=black, double distance=1pt, line width=1pt]
|
||||
```
|
||||
|
||||
|
||||
### Hand Writen Style {#hand-writen-style}
|
||||
|
||||
Usefull for schematic plots
|
||||
|
||||
```latex
|
||||
\tikzstyle{handwriten}=[decorate,decoration={random steps,amplitude=0.1pt,segment length=0.8pt}]
|
||||
```
|
||||
|
||||
|
||||
### DAC {#dac}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
DAC/.style={%
|
||||
draw,
|
||||
signal,
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### ADC {#adc}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
ADC/.style={%
|
||||
draw,
|
||||
signal,
|
||||
signal to = west,
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Gain {#gain}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
gain right/.style={%
|
||||
draw,
|
||||
regular polygon,
|
||||
regular polygon sides = 3,
|
||||
inner sep = 2pt,
|
||||
shape border rotate=-90
|
||||
},
|
||||
gain left/.style={%
|
||||
draw,
|
||||
regular polygon,
|
||||
regular polygon sides = 3,
|
||||
inner sep = 2pt,
|
||||
shape border rotate=90
|
||||
},
|
||||
gain top/.style={%
|
||||
draw,
|
||||
regular polygon,
|
||||
regular polygon sides = 3,
|
||||
inner sep = 2pt,
|
||||
shape border rotate=0
|
||||
},
|
||||
gain bottom/.style={%
|
||||
draw,
|
||||
regular polygon,
|
||||
regular polygon sides = 3,
|
||||
inner sep = 2pt,
|
||||
shape border rotate=180
|
||||
},
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Add / Substract / Divide / Multiply block {#add-substract-divide-multiply-block}
|
||||
|
||||
```latex
|
||||
\tikzset{% Add block with Circled operations
|
||||
addc/.style n args={5}{%
|
||||
draw,
|
||||
fill=white,
|
||||
circle,
|
||||
outer sep = 0pt,
|
||||
inner sep = 0pt,
|
||||
minimum size = 2em,
|
||||
execute at begin node={\LARGE $#1$},
|
||||
append after command={\pgfextra{\let\mainnode=\tikzlastnode}
|
||||
\ifx#2\empty\else
|
||||
node[draw, circle, outer sep=6pt, inner sep=0pt, above left] at (\mainnode.west) {$#2$}%
|
||||
\fi
|
||||
\ifx#3\empty\else
|
||||
node[draw, circle, outer sep=6pt, inner sep=0pt, above right] at (\mainnode.north) {$#3$}%
|
||||
\fi
|
||||
\ifx#4\empty\else
|
||||
node[draw, circle, outer sep=6pt, inner sep=0pt, below right] at (\mainnode.east) {$#4$}%
|
||||
\fi
|
||||
\ifx#5\empty\else
|
||||
node[draw, circle, outer sep=6pt, inner sep=0pt, below left] at (\mainnode.south) {$#5$}%
|
||||
\fi
|
||||
}
|
||||
},
|
||||
addc/.default={+}{}{}{}{},
|
||||
}
|
||||
```
|
||||
|
||||
```latex
|
||||
\tikzset{% Add Block
|
||||
addb/.style n args={5}{%
|
||||
draw,
|
||||
fill=white,
|
||||
circle,
|
||||
outer sep = 0pt,
|
||||
inner sep = 0pt,
|
||||
minimum size = 2em,
|
||||
execute at begin node={\LARGE $#1$},
|
||||
append after command={\pgfextra{\let\mainnode=\tikzlastnode}
|
||||
\ifx#2\empty\else
|
||||
node[outer sep=2pt, inner sep=0pt, above left] at (\mainnode.west) {$#2$}%
|
||||
\fi
|
||||
\ifx#3\empty\else
|
||||
node[outer sep=2pt, inner sep=0pt, above right] at (\mainnode.north) {$#3$}%
|
||||
\fi
|
||||
\ifx#4\empty\else
|
||||
node[outer sep=2pt, inner sep=0pt, below right] at (\mainnode.east) {$#4$}%
|
||||
\fi
|
||||
\ifx#5\empty\else
|
||||
node[outer sep=2pt, inner sep=0pt, below left] at (\mainnode.south) {$#5$}%
|
||||
\fi
|
||||
}
|
||||
},
|
||||
addb/.default={+}{}{}{}{},
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Plots {#plots}
|
||||
|
||||
|
||||
### Default line caps {#default-line-caps}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{
|
||||
every axis plot/.append style={line join=round},
|
||||
every axis plot/.append style={line cap=round},
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Grid {#grid}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{grid style={black}}
|
||||
\pgfplotsset{major grid style={black!30!white}}
|
||||
\pgfplotsset{minor grid style={black!10!white}}
|
||||
\pgfplotsset{xmajorgrids}
|
||||
\pgfplotsset{ymajorgrids}
|
||||
```
|
||||
|
||||
|
||||
### Lines {#lines}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{separate axis lines=false} % draw axis as rectangle and not as 4 lines
|
||||
\pgfplotsset{every outer x axis line/.append style={black}}
|
||||
\pgfplotsset{every outer y axis line/.append style={black}}
|
||||
\pgfplotsset{axis background/.style={fill=white}}
|
||||
\pgfplotsset{axis x line*=bottom} % solid line on the bottom with thin on the top
|
||||
\pgfplotsset{axis y line*=left} % solid line on the left with thin on the right
|
||||
```
|
||||
|
||||
|
||||
### Ticks {#ticks}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{every y tick label/.append style={font=\color{black}}}
|
||||
\pgfplotsset{every y tick/.append style={black}}
|
||||
\pgfplotsset{every x tick label/.append style={font=\color{black}}}
|
||||
\pgfplotsset{every x tick/.append style={black}}
|
||||
```
|
||||
|
||||
|
||||
### Size {#size}
|
||||
|
||||
If `scale only axis=false` (the default), pgfplots will try to produce the desired width including labels, titles and ticks.
|
||||
|
||||
```latex
|
||||
\pgfplotsset{scale only axis=true}
|
||||
```
|
||||
|
||||
|
||||
### Label {#label}
|
||||
|
||||
Used to align all of ylabel of one figure.
|
||||
|
||||
```latex
|
||||
\pgfplotsset{ylabel absolute}
|
||||
```
|
||||
|
||||
|
||||
### Legend {#legend}
|
||||
|
||||
```latex
|
||||
% https://tex.stackexchange.com/questions/54794/using-a-pgfplots-style-legend-in-a-plain-old-tikzpicture#54834
|
||||
|
||||
% argument #1: any options
|
||||
\newenvironment{customlegend}[1][]{%
|
||||
\begingroup
|
||||
% inits/clears the lists (which might be populated from previous
|
||||
% axes):
|
||||
\csname pgfplots@init@cleared@structures\endcsname
|
||||
\pgfplotsset{#1}%
|
||||
}{%
|
||||
% draws the legend:
|
||||
\csname pgfplots@createlegend\endcsname
|
||||
\endgroup
|
||||
}%
|
||||
|
||||
% makes \addlegendimage available (typically only available within an
|
||||
% axis environment):
|
||||
\def\addlegendimage{\csname pgfplots@addlegendimage\endcsname}
|
||||
|
||||
% definition to insert numbers
|
||||
% \pgfkeys{/pgfplots/number in legend/.style={%
|
||||
% /pgfplots/legend image code/.code={%
|
||||
% \node at (0.125,-0.0225){#1}; % <= changed x value
|
||||
% },%
|
||||
% },
|
||||
% }
|
||||
\pgfplotsset{
|
||||
every legend to name picture/.style={west}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Upper and Lower bounds {#upper-and-lower-bounds}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{upperbound}=[line cap=round, postaction={decorate,draw,decoration={border, segment length=0.2cm, amplitude=0.3cm, angle=60}}]
|
||||
\pgfplotsset{lowerbound}=[line cap=round, postaction={decorate,draw,decoration={border, segment length=0.2cm, amplitude=0.3cm, angle=-60}}]
|
||||
```
|
||||
|
||||
And we add the corresdonding
|
||||
|
||||
```latex
|
||||
\pgfplotsset{
|
||||
/pgfplots/upperbound/.style 1 args={
|
||||
legend image code/.code={
|
||||
\draw[##1, upperbound]
|
||||
plot coordinates {
|
||||
(0cm,0cm)
|
||||
(0.6cm,0cm)
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Pole {#pole}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
pole/.style{%
|
||||
color=red,
|
||||
cross out,
|
||||
draw,
|
||||
inner sep=0pt,
|
||||
outer sep=0pt,
|
||||
minimum size=#1pt
|
||||
},
|
||||
pole/.default={4}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Zero {#zero}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
zero/.style{%
|
||||
color=red,
|
||||
circle,
|
||||
draw,
|
||||
inner sep=0pt,
|
||||
outer sep=0pt,
|
||||
minimum size=#1pt
|
||||
},
|
||||
zero/.default={4}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Mechanical {#mechanical}
|
||||
|
||||
|
||||
### Spring {#spring}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
spring/.style={%
|
||||
thick,
|
||||
decoration={
|
||||
zigzag,
|
||||
pre length = #1cm,
|
||||
post length = #1cm,
|
||||
segment length = 6
|
||||
},
|
||||
decorate
|
||||
},
|
||||
spring/.default={0.2}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Coil {#coil}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
coil/.style n args={2}{%
|
||||
thick,
|
||||
decoration={
|
||||
coil,
|
||||
pre length = #1cm,
|
||||
post length = #2cm,
|
||||
segment length = 4
|
||||
},
|
||||
decorate
|
||||
},
|
||||
coil/.default={0.3}{0.3}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Damper {#damper}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
damper/.style n args={2}{%
|
||||
thick,
|
||||
decoration={markings, mark connection node=dmp, mark=at position 0.5 with {
|
||||
\node (dmp) [thick,
|
||||
inner sep = 0pt,
|
||||
transform shape,
|
||||
rotate =-90,
|
||||
minimum width = #1pt,
|
||||
minimum height = #2pt,
|
||||
draw=none] {};
|
||||
\draw [thick] ($(dmp.north east)+(0.6*#2pt,0)$) -- (dmp.south east) -- (dmp.south west) -- ($(dmp.north west)+(0.6*#2pt,0)$);
|
||||
\draw [thick] ($(dmp.north)+(0,-0.3*#1pt)$) -- ($(dmp.north)+(0,0.3*#1pt)$);
|
||||
}
|
||||
},
|
||||
decorate
|
||||
},
|
||||
damper/.default={12}{3}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Actuator {#actuator}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
actuator/.style n args={2}{%
|
||||
thick,
|
||||
draw=none,
|
||||
decoration={
|
||||
markings,
|
||||
mark connection node=my node,
|
||||
mark=at position .5 with {
|
||||
\node [draw, inner sep=0pt, minimum width=#1cm, minimum height=#2cm,
|
||||
transform shape, fill=white] (my node) {};
|
||||
},
|
||||
mark=at position .0 with {
|
||||
\draw[<-] (0, 0) -- (my node);
|
||||
},
|
||||
mark=at position 1.0 with {
|
||||
\draw[<-] (0, 0) -- (my node);
|
||||
}
|
||||
},
|
||||
decorate
|
||||
},
|
||||
actuator/.default={0.5}{0.2}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Ground {#ground}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
ground/.style n args={2}{%
|
||||
fill,
|
||||
pattern = north east lines,
|
||||
draw = none,
|
||||
anchor = north,
|
||||
minimum width = #1cm,
|
||||
minimum height = #2cm,
|
||||
append after command={
|
||||
(\tikzlastnode.north west) edge (\tikzlastnode.north east)
|
||||
}
|
||||
},
|
||||
ground/.default={2.5}{0.3}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Force Sensor {#force-sensor}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
forcesensor/.style n args={2}{%
|
||||
rectangle,
|
||||
outer sep=0pt,
|
||||
inner sep=0pt,
|
||||
draw=black,
|
||||
fill=white!60!black,
|
||||
anchor=south,
|
||||
minimum width =#1cm,
|
||||
minimum height=#2cm,
|
||||
append after command={
|
||||
[every edge/.append style={
|
||||
thick,
|
||||
black,
|
||||
}]
|
||||
(\tikzlastnode.north west) edge (\tikzlastnode.south east)
|
||||
(\tikzlastnode.north east) edge (\tikzlastnode.south west)
|
||||
}
|
||||
},
|
||||
forcesensor/.default={2.0}{0.5}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Inertial Sensor {#inertial-sensor}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
inertialsensor/.style={%
|
||||
rectangle,
|
||||
outer sep=0pt,
|
||||
inner sep=0pt,
|
||||
draw=black,
|
||||
fill=white!60!black,
|
||||
anchor=south east,
|
||||
minimum size=#1cm,
|
||||
append after command={
|
||||
[every edge/.append style={
|
||||
thick,
|
||||
black,
|
||||
}]
|
||||
(\tikzlastnode.north west) edge (\tikzlastnode.south east)
|
||||
(\tikzlastnode.north east) edge (\tikzlastnode.south west)
|
||||
}
|
||||
},
|
||||
inertialsensor/.default={0.3}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Cross {#cross}
|
||||
|
||||
```latex
|
||||
\tikzstyle{cross}=[path picture={
|
||||
\draw[black]
|
||||
(path picture bounding box.south east) -- (path picture bounding box.north west) (path picture bounding box.south west) -- (path picture bounding box.north east);
|
||||
}]
|
||||
|
||||
```
|
||||
|
||||
|
||||
### Piezoelectric actuator {#piezoelectric-actuator}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
piezo/.style n args={3}{%
|
||||
draw,
|
||||
rectangle,
|
||||
minimum width = #1cm,
|
||||
minimum height = #2cm,
|
||||
fill=blue!10!white,
|
||||
anchor=center,
|
||||
append after command={
|
||||
[every edge/.append style={
|
||||
thick,
|
||||
black,
|
||||
}]
|
||||
\foreach \i in {1,...,#3}{
|
||||
(${\i/(1+#3)}*(\tikzlastnode.north west)+{(1+#3-\i)/(1+#3)}*(\tikzlastnode.south west)+0.1*(#1,0)$) edge (${\i/(1+#3)}*(\tikzlastnode.north east)+{(1+#3-\i)/(1+#3)}*(\tikzlastnode.south east)-0.1*(#1,0)$)
|
||||
}
|
||||
}
|
||||
},
|
||||
piezo/.default={2}{4}{10}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Voice coil {#voice-coil}
|
||||
|
||||
```latex
|
||||
\def\voicecoil#1#2#3{
|
||||
% ======================
|
||||
% Parameters
|
||||
% ======================
|
||||
\def\voicecoilw{#1} % Total Width
|
||||
\def\voicecoilh{#2} % Total Height
|
||||
|
||||
\def\magnetw{\voicecoilw} % Width of the magnet
|
||||
\def\magneth{\voicecoilh/1.4} % Height of the magnet
|
||||
|
||||
\def\magnetwb{0.15*\magnetw} % Width of the borders of the magnet
|
||||
\def\magnetmw{0.15*\magnetw} % Width of the middle part of the magnet
|
||||
\def\magnetwg{0.5*\magnetw} % Width of the gap of the magnet
|
||||
|
||||
\def\magnethl{\magnetwb} % Height of the low part of the magnet
|
||||
\def\magnetmh{0.15*\magneth} % Height of the middle part of the magnet
|
||||
\def\magnethg{0.2*\magneth} % Height of the gap of the magnet
|
||||
% ======================
|
||||
|
||||
\begin{scope}[shift={(0.5*\voicecoilw, 0.5*\voicecoilh)}, rotate=#3, shift={(0, -0.5*\voicecoilh)}]
|
||||
% ======================
|
||||
% Magnet
|
||||
% ======================
|
||||
\draw[fill=white] (0, 0) -| ++(0.5*\magnetw, \magneth) -| ++(-0.5*\magnetw+0.5*\magnetwg, -\magnethg) -| (0.5*\magnetw-\magnetwb, \magnethl) -| (-0.5*\magnetw+\magnetwb, \magneth-\magnethg) -| (-0.5*\magnetwg, \magneth) -| (-0.5*\magnetw, 0) -- (cycle);
|
||||
\begin{scope}[shift={(0, \magnethl)}]
|
||||
\draw[fill=red] (-0.5*\magnetmw, 0) rectangle (0.5*\magnetmw, \magnetmh);
|
||||
\draw[fill=blue] (-0.5*\magnetmw, \magnetmh) rectangle (0.5*\magnetmw, 2*\magnetmh);
|
||||
% Top conductive Magnet
|
||||
\draw[fill=white] (-0.5*\magnetmw, 2*\magnetmh) -| (0.5*\magnetmw, -\magnethl+\magneth-\magnethg) -| ++(0.1, \magnethg) -| ++(-0.2-\magnetmw, -\magnethg) -| (-0.5*\magnetmw, \magnetmh);
|
||||
\end{scope}
|
||||
% ======================
|
||||
|
||||
% ======================
|
||||
% Coil
|
||||
% ======================
|
||||
\pgfmathsetmacro{\coilwidth}{0.5*0.5*\magnetmw+0.5*0.1+0.25*\magnetwg}%
|
||||
\draw[] ( \coilwidth, 0.5*\magneth) -- ++(0, 0.7*\magneth);
|
||||
\draw[] (-\coilwidth, 0.5*\magneth) -- ++(0, 0.7*\magneth);
|
||||
% Point on the coil
|
||||
\foreach \x in {0,1,...,9}
|
||||
{
|
||||
\node[circle,inner sep=0.6pt,fill] at ( \coilwidth, \x*0.7*\magneth/10+0.5*\magneth);
|
||||
\node[circle,inner sep=0.6pt,fill] at (-\coilwidth, \x*0.7*\magneth/10+0.5*\magneth);
|
||||
}
|
||||
\draw[fill=white] (-0.5*\magnetw, 1.2*\magneth) rectangle ++(\magnetw, \magnethg);
|
||||
% ======================
|
||||
|
||||
% ======================
|
||||
% Coordinates
|
||||
% ======================
|
||||
% Force
|
||||
\coordinate[] (vc_force) at (0, \magneth-0.5*\magnethg);
|
||||
% Coil
|
||||
\coordinate[] (vc_coil) at (0, \voicecoilh);
|
||||
% Magnet
|
||||
\coordinate[] (vc_magnet) at (0, 0);
|
||||
% Coil Wires
|
||||
\coordinate[] (vc_wire_one) at ( \coilwidth, 1.2*\magneth);
|
||||
\coordinate[] (vc_wire_two) at (-\coilwidth, 1.2*\magneth);
|
||||
% ======================
|
||||
\end{scope}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Axis Rotator {#axis-rotator}
|
||||
|
||||
```latex
|
||||
\newcommand{\AxisRotator}[1][rotate=0]{%
|
||||
\tikz [x=0.1cm,y=0.30cm,-stealth,#1] \draw (0,0) arc (-150:150:1 and 1);%
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Optics {#optics}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
->-/.style={
|
||||
decoration={
|
||||
markings,
|
||||
mark = at position #1 with {\arrow{>}
|
||||
}
|
||||
},
|
||||
postaction={decorate}
|
||||
}
|
||||
}
|
||||
\tikzset{%
|
||||
-<-/.style={
|
||||
decoration={
|
||||
markings,
|
||||
mark = at position #1 with {\arrow{<}
|
||||
}
|
||||
},
|
||||
postaction={decorate}
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Misc {#misc}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
labelc/.style= {%
|
||||
draw,
|
||||
fill=white,
|
||||
shape=circle,
|
||||
inner sep=2pt,
|
||||
outer sep=6pt,
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## More Defaults specific to this paper {#more-defaults-specific-to-this-paper}
|
||||
|
||||
```latex
|
||||
\tikzset{block/.default={0.8cm}{0.8cm}}
|
||||
\tikzset{addb/.append style={scale=0.7}}
|
||||
\tikzset{node distance=0.6}
|
||||
```
|
||||
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@@ -0,0 +1,57 @@
|
||||
+++
|
||||
title = "Active Damping of Rotating Platforms using Integral Force Feedback"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
venue = "ISMA 2020"
|
||||
year = 2020
|
||||
pubtype = "conference"
|
||||
code = "https://git.tdehaeze.xyz/tdehaeze/dehaeze20_activ_dampin_rotat_platf_integ_force_feedb"
|
||||
video = "https://www.youtube.com/watch?v=F9j2-ge2FPE"
|
||||
+++
|
||||
|
||||
> **Abstract**:
|
||||
>
|
||||
> This paper investigates the use of Integral Force Feedback (IFF) for the active damping of rotating mechanical systems.
|
||||
> Guaranteed stability, typical benefit of IFF, is lost as soon as the system is rotating due to gyroscopic effects.
|
||||
> To overcome this issue, two modifications of the classical IFF control scheme are proposed.
|
||||
> The first consists of slightly modifying the control law while the second consists of adding springs in parallel with the force sensors.
|
||||
> Conditions for stability and optimal parameters are derived.
|
||||
> The results reveal that, despite their different implementations, both modified IFF control scheme have almost identical damping authority on suspension modes.
|
||||
|
||||
|
||||
## Conference Paper ([pdf](paper/dehaeze20_activ_dampin_rotat_platf_integ_force_feedb.pdf)) {#conference-paper--pdf-paper-dehaeze20-activ-dampin-rotat-platf-integ-force-feedb-dot-pdf}
|
||||
|
||||
To cite this conference paper use the following bibtex code.
|
||||
|
||||
```bibtex
|
||||
@inproceedings{dehaeze20_activ_dampin_rotat_platf_integ_force_feedb,
|
||||
author = {Dehaeze, T. and Collette, C.},
|
||||
title = {Active Damping of Rotating Platforms using Integral Force
|
||||
Feedback},
|
||||
booktitle = {Proceedings of the International Conference on Modal
|
||||
Analysis Noise and Vibration Engineering (ISMA)},
|
||||
year = 2020,
|
||||
}
|
||||
```
|
||||
|
||||
You can also use the formatted citation below.
|
||||
|
||||
> Dehaeze, T., & Collette, C., Active damping of rotating platforms using integral force feedback, In , Proceedings of the International Conference on Modal Analysis Noise and Vibration Engineering (ISMA) (pp. ) (2020)
|
||||
|
||||
|
||||
## Matlab Scripts ([link]({{< relref "matlab/index.md" >}})) {#matlab-scripts--link-matlab-index-dot-md}
|
||||
|
||||
The Matlab scripts that permits to obtain all the results presented in the paper are accessible [here]({{< relref "matlab/index.md" >}}).
|
||||
|
||||
|
||||
## Figures ([link]({{< relref "tikz/_index.md" >}})) {#figures--link-tikz-index-dot-md}
|
||||
|
||||
All the figures in the paper are generated using either [TikZ](https://sourceforge.net/projects/pgf/) or [Inkscape](https://inkscape.org/). The code snippets that was used to generate the figures are accessible [here]({{< relref "tikz/_index.md" >}}).
|
||||
|
||||
|
||||
## Talk ([link](talk/dehaeze20_activ_dampin_rotat_platf_integ_force_feedb_talk.pdf)) {#talk--link-talk-dehaeze20-activ-dampin-rotat-platf-integ-force-feedb-talk-dot-pdf}
|
||||
|
||||
<iframe width="720"
|
||||
height="540"
|
||||
src="https://www.youtube.com/embed/F9j2-ge2FPE"
|
||||
frameborder="0" allowfullscreen> </iframe>
|
||||
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|
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+959
@@ -0,0 +1,959 @@
|
||||
+++
|
||||
title = "Active Damping of Rotating Platforms using Integral Force Feedback - Matlab Computation"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
<hr>
|
||||
<p>This report is also available as a <a href="./index.pdf">pdf</a>.</p>
|
||||
<hr>
|
||||
|
||||
This document gathers the Matlab code used to for the conference paper (<a href="#citeproc_bib_item_1">Dehaeze and Collette 2020</a>) and the journal paper (<a href="#citeproc_bib_item_3">Dehaeze and Collette 2021</a>).
|
||||
|
||||
It is structured in several sections:
|
||||
|
||||
- Section : presents a simple model of a rotating suspended platform that will be used throughout this study.
|
||||
- Section : explains how the unconditional stability of IFF is lost due to Gyroscopic effects induced by the rotation.
|
||||
- Section : suggests a simple modification of the control law such that damping can be added to the suspension modes in a robust way.
|
||||
- Section : proposes to add springs in parallel with the force sensors to regain the unconditional stability of IFF.
|
||||
- Section : compares both proposed modifications to the classical IFF in terms of damping authority and closed-loop system behavior.
|
||||
- Section : contains the notations used for both the Matlab code and the paper
|
||||
|
||||
The matlab code is accessible on [Zonodo](https://zenodo.org/record/3894343) and [Github](https://github.com/tdehaeze/dehaeze20_contr_stewa_platf) (<a href="#citeproc_bib_item_2">Dehaeze 2020</a>). It can also be download as a `.zip` file [here](https://git.tdehaeze.xyz/tdehaeze/dehaeze20_activ_dampin_rotat_platf_integ_force_feedb/archive/master.zip).
|
||||
|
||||
To run the Matlab code, go in the `matlab` directory and run the following Matlab files corresponding to each section.
|
||||
|
||||
<div class="table-caption">
|
||||
<span class="table-number">Table 1:</span>
|
||||
Paper's sections and corresponding Matlab files
|
||||
</div>
|
||||
|
||||
| Sections | Matlab File |
|
||||
|----------|----------------------------|
|
||||
| Section | `s1_system_description.m` |
|
||||
| Section | `s2_iff_pure_int.m` |
|
||||
| Section | `s3_iff_hpf.m` |
|
||||
| Section | `s4_iff_kp.m` |
|
||||
| Section | `s5_act_damp_comparison.m` |
|
||||
|
||||
|
||||
## System Description and Analysis {#system-description-and-analysis}
|
||||
|
||||
<span class="org-target" id="org-target--sec-system-description"></span>
|
||||
|
||||
|
||||
### System description {#system-description}
|
||||
|
||||
The system consists of one 2 degree of freedom translation stage on top of a spindle (figure [Figure 1](#figure--fig:system)).
|
||||
|
||||
<a id="figure--fig:system"></a>
|
||||
|
||||
{{< figure src="system.png" caption="<span class='figure-number'>Figure 1: </span>Schematic of the studied system" >}}
|
||||
|
||||
The control inputs are the forces applied by the actuators of the translation stage (\\(F\_u\\) and \\(F\_v\\)).
|
||||
As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along \\(\vec{i}\_u\\) and \\(\vec{i}\_v\\).
|
||||
|
||||
|
||||
### Equations {#equations}
|
||||
|
||||
Based on the Figure [Figure 1](#figure--fig:system), the equations of motions are:
|
||||
|
||||
<div class="important">
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} d\_u \\\ d\_v \end{bmatrix} =
|
||||
\bm{G}\_d
|
||||
\begin{bmatrix} F\_u \\\ F\_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
Where \\(\bm{G}\_d\\) is a \\(2 \times 2\\) transfer function matrix.
|
||||
|
||||
\begin{equation}
|
||||
\bm{G}\_d = \frac{1}{k} \frac{1}{G\_{dp}}
|
||||
\begin{bmatrix}
|
||||
G\_{dz} & G\_{dc} \\\\
|
||||
-G\_{dc} & G\_{dz}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
With:
|
||||
|
||||
\begin{align}
|
||||
G\_{dp} &= \left( \frac{s^2}{{\omega\_0}^2} + 2 \xi \frac{s}{\omega\_0} + 1 - \frac{{\Omega}^2}{{\omega\_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega\_0} \frac{s}{\omega\_0} \right)^2 \\\\
|
||||
G\_{dz} &= \frac{s^2}{{\omega\_0}^2} + 2 \xi \frac{s}{\omega\_0} + 1 - \frac{{\Omega}^2}{{\omega\_0}^2} \\\\
|
||||
G\_{dc} &= 2 \frac{\Omega}{\omega\_0} \frac{s}{\omega\_0}
|
||||
\end{align}
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
### Numerical Values {#numerical-values}
|
||||
|
||||
Let's define initial values for the model.
|
||||
|
||||
```matlab
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
```
|
||||
|
||||
```matlab
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
```
|
||||
|
||||
|
||||
### Campbell Diagram {#campbell-diagram}
|
||||
|
||||
The Campbell Diagram displays the evolution of the real and imaginary parts of the system as a function of the rotating speed.
|
||||
|
||||
It is shown in Figures [Figure 2](#figure--fig:campbell-diagram-real) and [Figure 3](#figure--fig:campbell-diagram-imag), and one can see that the system becomes unstable for \\(\Omega > \omega\_0\\) (the real part of one of the poles becomes positive).
|
||||
|
||||
<a id="figure--fig:campbell-diagram-real"></a>
|
||||
|
||||
{{< figure src="figs/campbell_diagram_real.png" caption="<span class='figure-number'>Figure 2: </span>Campbell Diagram - Real Part" >}}
|
||||
|
||||
<a id="figure--fig:campbell-diagram-imag"></a>
|
||||
|
||||
{{< figure src="figs/campbell_diagram_imag.png" caption="<span class='figure-number'>Figure 3: </span>Campbell Diagram - Imaginary Part" >}}
|
||||
|
||||
|
||||
### Simscape Model {#simscape-model}
|
||||
|
||||
In order to validate all the equations of motion, a Simscape model of the same system has been developed.
|
||||
The dynamics of the system can be identified from the Simscape model and compare with the analytical model.
|
||||
|
||||
The rotating speed for the Simscape Model is defined.
|
||||
|
||||
```matlab
|
||||
W = 0.1; % Rotation Speed [rad/s]
|
||||
```
|
||||
|
||||
```matlab
|
||||
open('rotating_frame.slx');
|
||||
```
|
||||
|
||||
The transfer function from \\([F\_u, F\_v]\\) to \\([d\_u, d\_v]\\) is identified from the Simscape model.
|
||||
|
||||
```matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/G'], 2, 'openoutput'); io_i = io_i + 1;
|
||||
```
|
||||
|
||||
```matlab
|
||||
G = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G.InputName = {'Fu', 'Fv'};
|
||||
G.OutputName = {'du', 'dv'};
|
||||
```
|
||||
|
||||
The same transfer function from \\([F\_u, F\_v]\\) to \\([d\_u, d\_v]\\) is written down from the analytical model.
|
||||
|
||||
```matlab
|
||||
Gth = (1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
|
||||
-2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)];
|
||||
```
|
||||
|
||||
Both transfer functions are compared in Figure [Figure 4](#figure--fig:plant-simscape-analytical) and are found to perfectly match.
|
||||
|
||||
<a id="figure--fig:plant-simscape-analytical"></a>
|
||||
|
||||
{{< figure src="figs/plant_simscape_analytical.png" caption="<span class='figure-number'>Figure 4: </span>Bode plot of the transfer function from \\([F\_u, F\_v]\\) to \\([d\_u, d\_v]\\) as identified from the Simscape model and from an analytical model" >}}
|
||||
|
||||
|
||||
### Effect of the rotation speed {#effect-of-the-rotation-speed}
|
||||
|
||||
The transfer functions from \\([F\_u, F\_v]\\) to \\([d\_u, d\_v]\\) are identified for the following rotating speeds.
|
||||
|
||||
```matlab
|
||||
Ws = [0, 0.2, 0.7, 1.1]*w0; % Rotating Speeds [rad/s]
|
||||
```
|
||||
|
||||
```matlab
|
||||
Gs = {zeros(2, 2, length(Ws))};
|
||||
|
||||
for W_i = 1:length(Ws)
|
||||
W = Ws(W_i);
|
||||
|
||||
Gs(:, :, W_i) = {(1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
|
||||
-2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)]};
|
||||
end
|
||||
```
|
||||
|
||||
They are compared in Figures [Figure 5](#figure--fig:plant-compare-rotating-speed-direct) and [Figure 6](#figure--fig:plant-compare-rotating-speed-coupling).
|
||||
|
||||
<a id="figure--fig:plant-compare-rotating-speed-direct"></a>
|
||||
|
||||
{{< figure src="figs/plant_compare_rotating_speed_direct.png" caption="<span class='figure-number'>Figure 5: </span>Comparison of the transfer functions from \\([F\_u, F\_v]\\) to \\([d\_u, d\_v]\\) for several rotating speed - Direct Terms" >}}
|
||||
|
||||
<a id="figure--fig:plant-compare-rotating-speed-coupling"></a>
|
||||
|
||||
{{< figure src="figs/plant_compare_rotating_speed_coupling.png" caption="<span class='figure-number'>Figure 6: </span>Comparison of the transfer functions from \\([F\_u, F\_v]\\) to \\([d\_u, d\_v]\\) for several rotating speed - Coupling Terms" >}}
|
||||
|
||||
|
||||
## Problem with pure Integral Force Feedback {#problem-with-pure-integral-force-feedback}
|
||||
|
||||
<span class="org-target" id="org-target--sec-iff-pure-int"></span>
|
||||
|
||||
Force sensors are added in series with the two actuators (Figure [Figure 7](#figure--fig:system-iff)).
|
||||
|
||||
Two identical controllers \\(K\_F\\) are used to feedback each of the sensed force to its associated actuator.
|
||||
|
||||
<a id="figure--fig:system-iff"></a>
|
||||
|
||||
{{< figure src="system_iff.png" caption="<span class='figure-number'>Figure 7: </span>System with added Force Sensor in series with the actuators" >}}
|
||||
|
||||
|
||||
### Plant Parameters {#plant-parameters}
|
||||
|
||||
Let's define initial values for the model.
|
||||
|
||||
```matlab
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
```
|
||||
|
||||
```matlab
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
```
|
||||
|
||||
|
||||
### Equations {#equations}
|
||||
|
||||
The sensed forces are equal to:
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} f\_{u} \\\ f\_{v} \end{bmatrix} =
|
||||
\begin{bmatrix}
|
||||
1 & 0 \\\\
|
||||
0 & 1
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} F\_u \\\ F\_v \end{bmatrix} - (c s + k)
|
||||
\begin{bmatrix} d\_u \\\ d\_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
Which then gives:
|
||||
|
||||
<div class="important">
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} f\_{u} \\\ f\_{v} \end{bmatrix} =
|
||||
\bm{G}\_{f}
|
||||
\begin{bmatrix} F\_u \\\ F\_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} f\_{u} \\\ f\_{v} \end{bmatrix} =
|
||||
\frac{1}{G\_{fp}}
|
||||
\begin{bmatrix}
|
||||
G\_{fz} & -G\_{fc} \\\\
|
||||
G\_{fc} & G\_{fz}
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} F\_u \\\ F\_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\begin{align}
|
||||
G\_{fp} &= \left( \frac{s^2}{{\omega\_0}^2} + 2 \xi \frac{s}{\omega\_0} + 1 - \frac{{\Omega}^2}{{\omega\_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega\_0} \frac{s}{\omega\_0} \right)^2 \\\\
|
||||
G\_{fz} &= \left( \frac{s^2}{{\omega\_0}^2} - \frac{\Omega^2}{{\omega\_0}^2} \right) \left( \frac{s^2}{{\omega\_0}^2} + 2 \xi \frac{s}{\omega\_0} + 1 - \frac{{\Omega}^2}{{\omega\_0}^2} \right) + \left( 2 \frac{\Omega}{\omega\_0} \frac{s}{\omega\_0} \right)^2 \\\\
|
||||
G\_{fc} &= \left( 2 \xi \frac{s}{\omega\_0} + 1 \right) \left( 2 \frac{\Omega}{\omega\_0} \frac{s}{\omega\_0} \right)
|
||||
\end{align}
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
### Comparison of the Analytical Model and the Simscape Model {#comparison-of-the-analytical-model-and-the-simscape-model}
|
||||
|
||||
The rotation speed is set to \\(\Omega = 0.1 \omega\_0\\).
|
||||
|
||||
```matlab
|
||||
W = 0.1*w0; % [rad/s]
|
||||
```
|
||||
|
||||
```matlab
|
||||
open('rotating_frame.slx');
|
||||
```
|
||||
|
||||
And the transfer function from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) is identified using the Simscape model.
|
||||
|
||||
```matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
```
|
||||
|
||||
```matlab
|
||||
Giff = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
Giff.InputName = {'Fu', 'Fv'};
|
||||
Giff.OutputName = {'fu', 'fv'};
|
||||
```
|
||||
|
||||
The same transfer function from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) is written down from the analytical model.
|
||||
|
||||
```matlab
|
||||
Giff_th = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];
|
||||
```
|
||||
|
||||
The two are compared in Figure [Figure 8](#figure--fig:plant-iff-comp-simscape-analytical) and found to perfectly match.
|
||||
|
||||
<a id="figure--fig:plant-iff-comp-simscape-analytical"></a>
|
||||
|
||||
{{< figure src="figs/plant_iff_comp_simscape_analytical.png" caption="<span class='figure-number'>Figure 8: </span>Comparison of the transfer functions from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) between the Simscape model and the analytical one" >}}
|
||||
|
||||
|
||||
### Effect of the rotation speed {#effect-of-the-rotation-speed}
|
||||
|
||||
The transfer functions from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) are identified for the following rotating speeds.
|
||||
|
||||
```matlab
|
||||
Ws = [0, 0.2, 0.7]*w0; % Rotating Speeds [rad/s]
|
||||
```
|
||||
|
||||
```matlab
|
||||
Gsiff = {zeros(2, 2, length(Ws))};
|
||||
|
||||
for W_i = 1:length(Ws)
|
||||
W = Ws(W_i);
|
||||
|
||||
Gsiff(:, :, W_i) = {1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2]};
|
||||
end
|
||||
```
|
||||
|
||||
The obtained transfer functions are shown in Figure [Figure 9](#figure--fig:plant-iff-compare-rotating-speed).
|
||||
|
||||
<a id="figure--fig:plant-iff-compare-rotating-speed"></a>
|
||||
|
||||
{{< figure src="figs/plant_iff_compare_rotating_speed.png" caption="<span class='figure-number'>Figure 9: </span>Comparison of the transfer functions from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) for several rotating speed" >}}
|
||||
|
||||
|
||||
### Decentralized Integral Force Feedback {#decentralized-integral-force-feedback}
|
||||
|
||||
The decentralized IFF controller consists of pure integrators:
|
||||
|
||||
\begin{equation}
|
||||
\bm{K}\_{\text{IFF}}(s) = \frac{g}{s} \begin{bmatrix}
|
||||
1 & 0 \\\\
|
||||
0 & 1
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
The Root Locus (evolution of the poles of the closed loop system in the complex plane as a function of \\(g\\)) is shown in Figure [Figure 10](#figure--fig:root-locus-pure-iff).
|
||||
It is shown that for non-null rotating speed, one pole is bound to the right-half plane, and thus the closed loop system is unstable.
|
||||
|
||||
<a id="figure--fig:root-locus-pure-iff"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_pure_iff.png" caption="<span class='figure-number'>Figure 10: </span>Root Locus for the Decentralized Integral Force Feedback controller. Several rotating speed are shown." >}}
|
||||
|
||||
|
||||
## Integral Force Feedback with an High Pass Filter {#integral-force-feedback-with-an-high-pass-filter}
|
||||
|
||||
<span class="org-target" id="org-target--sec-iff-pseudo-int"></span>
|
||||
|
||||
|
||||
### Plant Parameters {#plant-parameters}
|
||||
|
||||
Let's define initial values for the model.
|
||||
|
||||
```matlab
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
```
|
||||
|
||||
```matlab
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
```
|
||||
|
||||
|
||||
### Modified Integral Force Feedback Controller {#modified-integral-force-feedback-controller}
|
||||
|
||||
Let's modify the initial Integral Force Feedback Controller ; instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used:
|
||||
|
||||
\begin{equation}
|
||||
K\_{\text{IFF}}(s) = g\frac{1}{\omega\_i + s} \begin{bmatrix}
|
||||
1 & 0 \\\\
|
||||
0 & 1
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
where \\(\omega\_i\\) characterize down to which frequency the signal is integrated.
|
||||
|
||||
Let's arbitrary choose the following control parameters:
|
||||
|
||||
```matlab
|
||||
g = 2;
|
||||
wi = 0.1*w0;
|
||||
```
|
||||
|
||||
And the following rotating speed.
|
||||
|
||||
```matlab
|
||||
Giff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];
|
||||
```
|
||||
|
||||
The obtained Loop Gain is shown in Figure [Figure 11](#figure--fig:loop-gain-modified-iff).
|
||||
|
||||
<a id="figure--fig:loop-gain-modified-iff"></a>
|
||||
|
||||
{{< figure src="figs/loop_gain_modified_iff.png" caption="<span class='figure-number'>Figure 11: </span>Loop Gain for the modified IFF controller" >}}
|
||||
|
||||
|
||||
### Root Locus {#root-locus}
|
||||
|
||||
As shown in the Root Locus plot (Figure [Figure 12](#figure--fig:root-locus-modified-iff)), for some value of the gain, the system remains stable.
|
||||
|
||||
<a id="figure--fig:root-locus-modified-iff"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_modified_iff.png" caption="<span class='figure-number'>Figure 12: </span>Root Locus for the modified IFF controller" >}}
|
||||
|
||||
<a id="figure--fig:root-locus-modified-iff-zoom"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_modified_iff_zoom.png" caption="<span class='figure-number'>Figure 13: </span>Root Locus for the modified IFF controller - Zoom" >}}
|
||||
|
||||
|
||||
### What is the optimal \\(\omega\_i\\) and \\(g\\)? {#what-is-the-optimal-omega-i-and-g}
|
||||
|
||||
In order to visualize the effect of \\(\omega\_i\\) on the attainable damping, the Root Locus is displayed in Figure [Figure 14](#figure--fig:root-locus-wi-modified-iff) for the following \\(\omega\_i\\):
|
||||
|
||||
```matlab
|
||||
wis = [0.01, 0.1, 0.5, 1]*w0; % [rad/s]
|
||||
```
|
||||
|
||||
<a id="figure--fig:root-locus-wi-modified-iff"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_wi_modified_iff.png" caption="<span class='figure-number'>Figure 14: </span>Root Locus for the modified IFF controller (zoomed plot on the left)" >}}
|
||||
|
||||
<a id="figure--fig:root-locus-wi-modified-iff-zoom"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_wi_modified_iff_zoom.png" caption="<span class='figure-number'>Figure 15: </span>Root Locus for the modified IFF controller (zoomed plot on the left)" >}}
|
||||
|
||||
For the controller
|
||||
|
||||
\begin{equation}
|
||||
K\_{\text{IFF}}(s) = g\frac{1}{\omega\_i + s} \begin{bmatrix}
|
||||
1 & 0 \\\\
|
||||
0 & 1
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
The gain at which the system becomes unstable is
|
||||
|
||||
\begin{equation}
|
||||
g\_\text{max} = \omega\_i \left( \frac{{\omega\_0}^2}{\Omega^2} - 1 \right) \label{eq:iff\_gmax}
|
||||
\end{equation}
|
||||
|
||||
While it seems that small \\(\omega\_i\\) do allow more damping to be added to the system (Figure [Figure 14](#figure--fig:root-locus-wi-modified-iff)), the control gains may be limited to small values due to \ref{eq:iff\_gmax} thus reducing the attainable damping.
|
||||
|
||||
There must be an optimum for \\(\omega\_i\\).
|
||||
To find the optimum, the gain that maximize the simultaneous damping of the mode is identified for a wide range of \\(\omega\_i\\) (Figure [Figure 16](#figure--fig:mod-iff-damping-wi)).
|
||||
|
||||
```matlab
|
||||
wis = logspace(-2, 1, 100)*w0; % [rad/s]
|
||||
|
||||
opt_xi = zeros(1, length(wis)); % Optimal simultaneous damping
|
||||
opt_gain = zeros(1, length(wis)); % Corresponding optimal gain
|
||||
|
||||
for wi_i = 1:length(wis)
|
||||
wi = wis(wi_i);
|
||||
Kiff = 1/(s + wi)*eye(2);
|
||||
|
||||
fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);
|
||||
|
||||
[g_opt, xi_opt] = fminsearch(fun, 0.5*wi*((w0/W)^2 - 1));
|
||||
opt_xi(wi_i) = 1/xi_opt;
|
||||
opt_gain(wi_i) = g_opt;
|
||||
end
|
||||
```
|
||||
|
||||
<a id="figure--fig:mod-iff-damping-wi"></a>
|
||||
|
||||
{{< figure src="figs/mod_iff_damping_wi.png" caption="<span class='figure-number'>Figure 16: </span>Simultaneous attainable damping of the closed loop poles as a function of \\(\omega\_i\\)" >}}
|
||||
|
||||
|
||||
## IFF with a stiffness in parallel with the force sensor {#iff-with-a-stiffness-in-parallel-with-the-force-sensor}
|
||||
|
||||
<span class="org-target" id="org-target--sec-iff-parallel-stiffness"></span>
|
||||
|
||||
|
||||
### Schematic {#schematic}
|
||||
|
||||
In this section additional springs in parallel with the force sensors are added to counteract the negative stiffness induced by the rotation.
|
||||
|
||||
<a id="figure--fig:system-parallel-springs"></a>
|
||||
|
||||
{{< figure src="system_parallel_springs.png" caption="<span class='figure-number'>Figure 17: </span>Studied system with additional springs in parallel with the actuators and force sensors" >}}
|
||||
|
||||
In order to keep the overall stiffness \\(k = k\_a + k\_p\\) constant, a scalar parameter \\(\alpha\\) (\\(0 \le \alpha < 1\\)) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor
|
||||
|
||||
\begin{equation}
|
||||
k\_p = \alpha k, \quad k\_a = (1 - \alpha) k
|
||||
\end{equation}
|
||||
|
||||
|
||||
### Equations {#equations}
|
||||
|
||||
<div class="important">
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} f\_u \\\ f\_v \end{bmatrix} =
|
||||
\bm{G}\_k
|
||||
\begin{bmatrix} F\_u \\\ F\_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} f\_u \\\ f\_v \end{bmatrix} =
|
||||
\frac{1}{G\_{kp}}
|
||||
\begin{bmatrix}
|
||||
G\_{kz} & -G\_{kc} \\\\
|
||||
G\_{kc} & G\_{kz}
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} F\_u \\\ F\_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
With:
|
||||
|
||||
\begin{align}
|
||||
G\_{kp} &= \left( \frac{s^2}{{\omega\_0}^2} + 2\xi \frac{s}{{\omega\_0}^2} + 1 - \frac{\Omega^2}{{\omega\_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega\_0}\frac{s}{\omega\_0} \right)^2 \\\\
|
||||
G\_{kz} &= \left( \frac{s^2}{{\omega\_0}^2} - \frac{\Omega^2}{{\omega\_0}^2} + \alpha \right) \left( \frac{s^2}{{\omega\_0}^2} + 2\xi \frac{s}{{\omega\_0}^2} + 1 - \frac{\Omega^2}{{\omega\_0}^2} \right) + \left( 2 \frac{\Omega}{\omega\_0}\frac{s}{\omega\_0} \right)^2 \\\\
|
||||
G\_{kc} &= \left( 2 \xi \frac{s}{\omega\_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega\_0}\frac{s}{\omega\_0} \right)
|
||||
\end{align}
|
||||
|
||||
</div>
|
||||
|
||||
If we compare \\(G\_{kz}\\) and \\(G\_{fz}\\), we see that the spring in parallel adds a term \\(\alpha\\).
|
||||
In order to have two complex conjugate zeros (instead of real zeros):
|
||||
|
||||
\begin{equation}
|
||||
\alpha > \frac{\Omega^2}{{\omega\_0}^2} \quad \Leftrightarrow \quad k\_p > m \Omega^2
|
||||
\end{equation}
|
||||
|
||||
|
||||
### Plant Parameters {#plant-parameters}
|
||||
|
||||
Let's define initial values for the model.
|
||||
|
||||
```matlab
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
```
|
||||
|
||||
```matlab
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
```
|
||||
|
||||
|
||||
### Comparison of the Analytical Model and the Simscape Model {#comparison-of-the-analytical-model-and-the-simscape-model}
|
||||
|
||||
The same transfer function from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) is written down from the analytical model.
|
||||
|
||||
```matlab
|
||||
W = 0.1*w0; % [rad/s]
|
||||
|
||||
kp = 1.5*m*W^2;
|
||||
cp = 0;
|
||||
```
|
||||
|
||||
```matlab
|
||||
open('rotating_frame.slx');
|
||||
```
|
||||
|
||||
```matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
Giff = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
Giff.InputName = {'Fu', 'Fv'};
|
||||
Giff.OutputName = {'fu', 'fv'};
|
||||
```
|
||||
|
||||
```matlab
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff_th = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2 ];
|
||||
Giff_th.InputName = {'Fu', 'Fv'};
|
||||
Giff_th.OutputName = {'fu', 'fv'};
|
||||
```
|
||||
|
||||
<a id="figure--fig:plant-iff-kp-comp-simscape-analytical"></a>
|
||||
|
||||
{{< figure src="figs/plant_iff_kp_comp_simscape_analytical.png" caption="<span class='figure-number'>Figure 18: </span>Comparison of the transfer functions from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) between the Simscape model and the analytical one" >}}
|
||||
|
||||
|
||||
### Effect of the parallel stiffness on the IFF plant {#effect-of-the-parallel-stiffness-on-the-iff-plant}
|
||||
|
||||
The rotation speed is set to \\(\Omega = 0.1 \omega\_0\\).
|
||||
|
||||
```matlab
|
||||
W = 0.1*w0; % [rad/s]
|
||||
```
|
||||
|
||||
And the IFF plant (transfer function from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\)) is identified in three different cases:
|
||||
|
||||
- without parallel stiffness
|
||||
- with a small parallel stiffness \\(k\_p < m \Omega^2\\)
|
||||
- with a large parallel stiffness \\(k\_p > m \Omega^2\\)
|
||||
|
||||
The results are shown in Figure [Figure 19](#figure--fig:plant-iff-kp).
|
||||
|
||||
One can see that for \\(k\_p > m \Omega^2\\), the systems shows alternating complex conjugate poles and zeros.
|
||||
|
||||
```matlab
|
||||
kp = 0;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
```
|
||||
|
||||
```matlab
|
||||
kp = 0.5*m*W^2;
|
||||
k = 1 - kp;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff_s = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
```
|
||||
|
||||
```matlab
|
||||
kp = 1.5*m*W^2;
|
||||
k = 1 - kp;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff_l = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
```
|
||||
|
||||
<a id="figure--fig:plant-iff-kp"></a>
|
||||
|
||||
{{< figure src="figs/plant_iff_kp.png" caption="<span class='figure-number'>Figure 19: </span>Transfer function from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) for \\(k\_p = 0\\), \\(k\_p < m \Omega^2\\) and \\(k\_p > m \Omega^2\\)" >}}
|
||||
|
||||
|
||||
### IFF when adding a spring in parallel {#iff-when-adding-a-spring-in-parallel}
|
||||
|
||||
In Figure [Figure 20](#figure--fig:root-locus-iff-kp) is displayed the Root Locus in the three considered cases with
|
||||
|
||||
\begin{equation}
|
||||
K\_{\text{IFF}} = \frac{g}{s} \begin{bmatrix}
|
||||
1 & 0 \\\\
|
||||
0 & 1
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
One can see that for \\(k\_p > m \Omega^2\\), the root locus stays in the left half of the complex plane and thus the control system is unconditionally stable.
|
||||
|
||||
Thus, decentralized IFF controller with pure integrators can be used if:
|
||||
|
||||
\begin{equation}
|
||||
k\_{p} > m \Omega^2
|
||||
\end{equation}
|
||||
|
||||
<a id="figure--fig:root-locus-iff-kp"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_iff_kp.png" caption="<span class='figure-number'>Figure 20: </span>Root Locus" >}}
|
||||
|
||||
<a id="figure--fig:root-locus-iff-kp-zoom"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_iff_kp_zoom.png" caption="<span class='figure-number'>Figure 21: </span>Root Locus" >}}
|
||||
|
||||
|
||||
### Effect of \\(k\_p\\) on the attainable damping {#effect-of-k-p-on-the-attainable-damping}
|
||||
|
||||
However, having large values of \\(k\_p\\) may decrease the attainable damping.
|
||||
|
||||
To study the second point, Root Locus plots for the following values of \\(k\_p\\) are shown in Figure [Figure 22](#figure--fig:root-locus-iff-kps).
|
||||
|
||||
```matlab
|
||||
kps = [2, 20, 40]*m*W^2;
|
||||
```
|
||||
|
||||
It is shown that large values of \\(k\_p\\) decreases the attainable damping.
|
||||
|
||||
<a id="figure--fig:root-locus-iff-kps"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_iff_kps.png" caption="<span class='figure-number'>Figure 22: </span>Root Locus plot" >}}
|
||||
|
||||
```matlab
|
||||
alphas = logspace(-2, 0, 100);
|
||||
|
||||
opt_xi = zeros(1, length(alphas)); % Optimal simultaneous damping
|
||||
opt_gain = zeros(1, length(alphas)); % Corresponding optimal gain
|
||||
|
||||
Kiff = 1/s*eye(2);
|
||||
|
||||
for alpha_i = 1:length(alphas)
|
||||
kp = alphas(alpha_i);
|
||||
k = 1 - alphas(alpha_i);
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
|
||||
fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);
|
||||
|
||||
[g_opt, xi_opt] = fminsearch(fun, 2);
|
||||
opt_xi(alpha_i) = 1/xi_opt;
|
||||
opt_gain(alpha_i) = g_opt;
|
||||
end
|
||||
```
|
||||
|
||||
<a id="figure--fig:opt-damp-alpha"></a>
|
||||
|
||||
{{< figure src="figs/opt_damp_alpha.png" caption="<span class='figure-number'>Figure 23: </span>Attainable damping ratio and corresponding controller gain for different parameter \\(\alpha\\)" >}}
|
||||
|
||||
|
||||
## Comparison {#comparison}
|
||||
|
||||
<span class="org-target" id="org-target--sec-comparison"></span>
|
||||
|
||||
Two modifications to adapt the IFF control strategy to rotating platforms have been proposed.
|
||||
These two methods are now compared in terms of added damping, closed-loop compliance and transmissibility.
|
||||
|
||||
|
||||
### Plant Parameters {#plant-parameters}
|
||||
|
||||
Let's define initial values for the model.
|
||||
|
||||
```matlab
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
```
|
||||
|
||||
```matlab
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
```
|
||||
|
||||
The rotating speed is set to \\(\Omega = 0.1 \omega\_0\\).
|
||||
|
||||
```matlab
|
||||
W = 0.1*w0;
|
||||
```
|
||||
|
||||
|
||||
### Root Locus {#root-locus}
|
||||
|
||||
IFF with High Pass Filter
|
||||
|
||||
```matlab
|
||||
wi = 0.1*w0; % [rad/s]
|
||||
|
||||
Giff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];
|
||||
```
|
||||
|
||||
IFF With parallel Stiffness
|
||||
|
||||
```matlab
|
||||
kp = 5*m*W^2;
|
||||
k = k - kp;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff_kp = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2 ];
|
||||
|
||||
k = k + kp;
|
||||
```
|
||||
|
||||
<a id="figure--fig:comp-root-locus"></a>
|
||||
|
||||
{{< figure src="figs/comp_root_locus.png" caption="<span class='figure-number'>Figure 24: </span>Root Locus plot - Comparison of IFF with additional high pass filter, IFF with additional parallel stiffness" >}}
|
||||
|
||||
|
||||
### Controllers - Optimal Gains {#controllers-optimal-gains}
|
||||
|
||||
In order to compare to three considered Active Damping techniques, gains that yield maximum damping of all the modes are computed for each case.
|
||||
|
||||
The obtained damping ratio and control are shown below.
|
||||
|
||||
| | Obtained \\(\xi\\) | Control Gain |
|
||||
|---------------------|--------------------|--------------|
|
||||
| Modified IFF | 0.83 | 1.99 |
|
||||
| IFF with \\(k\_p\\) | 0.83 | 2.02 |
|
||||
|
||||
|
||||
### Passive Damping - Critical Damping {#passive-damping-critical-damping}
|
||||
|
||||
\begin{equation}
|
||||
\xi = \frac{c}{2 \sqrt{km}}
|
||||
\end{equation}
|
||||
|
||||
Critical Damping corresponds to to \\(\xi = 1\\), and thus:
|
||||
|
||||
\begin{equation}
|
||||
c\_{\text{crit}} = 2 \sqrt{km}
|
||||
\end{equation}
|
||||
|
||||
```matlab
|
||||
c_opt = 2*sqrt(k*m);
|
||||
```
|
||||
|
||||
|
||||
### Transmissibility And Compliance {#transmissibility-and-compliance}
|
||||
|
||||
<span class="org-target" id="org-target--sec-comp-transmissibilty"></span>
|
||||
|
||||
```matlab
|
||||
open('rotating_frame.slx');
|
||||
```
|
||||
|
||||
```matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/dw'], 1, 'input'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/fd'], 1, 'input'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Meas'], 1, 'output'); io_i = io_i + 1;
|
||||
```
|
||||
|
||||
```matlab
|
||||
G_ol = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G_ol.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_ol.OutputName = {'Dx', 'Dy'};
|
||||
```
|
||||
|
||||
|
||||
#### Passive Damping {#passive-damping}
|
||||
|
||||
```matlab
|
||||
kp = 0;
|
||||
cp = 0;
|
||||
```
|
||||
|
||||
```matlab
|
||||
c_old = c;
|
||||
c = c_opt;
|
||||
```
|
||||
|
||||
```matlab
|
||||
G_pas = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G_pas.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_pas.OutputName = {'Dx', 'Dy'};
|
||||
```
|
||||
|
||||
```matlab
|
||||
c = c_old;
|
||||
```
|
||||
|
||||
```matlab
|
||||
Kiff = opt_gain_iff/(wi + s)*tf(eye(2));
|
||||
```
|
||||
|
||||
```matlab
|
||||
G_iff = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G_iff.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_iff.OutputName = {'Dx', 'Dy'};
|
||||
```
|
||||
|
||||
```matlab
|
||||
kp = 5*m*W^2;
|
||||
cp = 0.01;
|
||||
```
|
||||
|
||||
```matlab
|
||||
Kiff = opt_gain_kp/s*tf(eye(2));
|
||||
```
|
||||
|
||||
```matlab
|
||||
G_kp = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G_kp.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_kp.OutputName = {'Dx', 'Dy'};
|
||||
```
|
||||
|
||||
<a id="figure--fig:comp-transmissibility"></a>
|
||||
|
||||
{{< figure src="figs/comp_transmissibility.png" caption="<span class='figure-number'>Figure 25: </span>Comparison of the transmissibility" >}}
|
||||
|
||||
<a id="figure--fig:comp-compliance"></a>
|
||||
|
||||
{{< figure src="figs/comp_compliance.png" caption="<span class='figure-number'>Figure 26: </span>Comparison of the obtained Compliance" >}}
|
||||
|
||||
|
||||
## Notations {#notations}
|
||||
|
||||
<span class="org-target" id="org-target--sec-notations"></span>
|
||||
|
||||
| | Mathematical Notation | Matlab | Unit |
|
||||
|---------------------------------------|----------------------------------|---------------|---------|
|
||||
| Actuator Stiffness | \\(k\\) | `k` | N/m |
|
||||
| Actuator Damping | \\(c\\) | `c` | N/(m/s) |
|
||||
| Payload Mass | \\(m\\) | `m` | kg |
|
||||
| Damping Ratio | \\(\xi = \frac{c}{2\sqrt{km}}\\) | `xi` | |
|
||||
| Actuator Force | \\(\bm{F}, F\_u, F\_v\\) | `F` `Fu` `Fv` | N |
|
||||
| Force Sensor signal | \\(\bm{f}, f\_u, f\_v\\) | `f` `fu` `fv` | N |
|
||||
| Relative Displacement | \\(\bm{d}, d\_u, d\_v\\) | `d` `du` `dv` | m |
|
||||
| Resonance freq. when \\(\Omega = 0\\) | \\(\omega\_0\\) | `w0` | rad/s |
|
||||
| Rotation Speed | \\(\Omega = \dot{\theta}\\) | `W` | rad/s |
|
||||
| Low Pass Filter corner frequency | \\(\omega\_i\\) | `wi` | rad/s |
|
||||
|
||||
| | Mathematical Notation | Matlab | Unit |
|
||||
|------------------|-----------------------|--------|---------|
|
||||
| Laplace variable | \\(s\\) | `s` | |
|
||||
| Complex number | \\(j\\) | `j` | |
|
||||
| Frequency | \\(\omega\\) | `w` | [rad/s] |
|
||||
|
||||
<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>Dehaeze, T., and C. Collette. 2020. “Active Damping of Rotating Platforms Using Integral Force Feedback.” In <i>Proceedings of the International Conference on Modal Analysis Noise and Vibration Engineering (ISMA)</i>.</div>
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Dehaeze, Thomas. 2020. “Active Damping of Rotating Positioning Platforms.” Source Code on Zonodo. doi:<a href="https://doi.org/10.5281/zenodo.3894342">10.5281/zenodo.3894342</a>.</div>
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_3"></a>Dehaeze, Thomas, and Christophe Collette. 2021. “Active Damping of Rotating Platforms Using Integral Force Feedback.” <i>Engineering Research Express</i>. <a href="http://iopscience.iop.org/article/10.1088/2631-8695/abe803">http://iopscience.iop.org/article/10.1088/2631-8695/abe803</a>.</div>
|
||||
</div>
|
||||
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|
||||
+++
|
||||
title = "Active Damping of Rotating Platforms using Integral Force Feedback - Tikz Figures"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Configuration file is accessible [here]({{< relref "config.md" >}}).
|
||||
|
||||
|
||||
## X-Y Rotating Positioning Platform {#x-y-rotating-positioning-platform}
|
||||
|
||||
```latex
|
||||
\begin{tikzpicture}
|
||||
% Angle
|
||||
\def\thetau{25}
|
||||
|
||||
% Rotational Stage
|
||||
\draw[fill=black!60!white] (0, 0) circle (4.3);
|
||||
\draw[fill=black!40!white] (0, 0) circle (3.8);
|
||||
|
||||
% Label
|
||||
\node[anchor=north west, rotate=\thetau] at (-2.5, 2.5) {\small Rotating Stage};
|
||||
|
||||
% Rotating Scope
|
||||
\begin{scope}[rotate=\thetau]
|
||||
% Rotating Frame
|
||||
\draw[fill=black!20!white] (-2.6, -2.6) rectangle (2.6, 2.6);
|
||||
% Label
|
||||
\node[anchor=north west, rotate=\thetau] at (-2.6, 2.6) {\small Suspended Platform};
|
||||
|
||||
% Mass
|
||||
\draw[fill=white] (-1, -1) rectangle (1, 1);
|
||||
% Label
|
||||
\node[anchor=south west, rotate=\thetau] at (-1, -1) {\small Payload};
|
||||
|
||||
% Attached Points
|
||||
\node[] at (-1, 0){$\bullet$};
|
||||
\draw[] (-1, 0) -- ++(-0.2, 0) coordinate(cu);
|
||||
\draw[] ($(cu) + (0, -0.8)$) coordinate(actu) -- ($(cu) + (0, 0.8)$) coordinate(ku);
|
||||
\node[] at (0, -1){$\bullet$};
|
||||
\draw[] (0, -1) -- ++(0, -0.2) coordinate(cv);
|
||||
\draw[] ($(cv) + (-0.8, 0)$)coordinate(kv) -- ($(cv) + (0.8, 0)$) coordinate(actv);
|
||||
|
||||
% Spring and Actuator for U
|
||||
\draw[actuator={0.6}{0.2}] (actu) -- node[above=0.1, rotate=\thetau]{$F_u$} (actu-|-2.6,0);
|
||||
\draw[spring=0.2] (ku) -- node[above=0.1, rotate=\thetau]{$k$} (ku-|-2.6,0);
|
||||
\draw[damper={8}{8}] (cu) -- node[above left=0.2 and -0.1, rotate=\thetau]{$c$} (cu-|-2.6,0);
|
||||
|
||||
\draw[actuator={0.6}{0.2}] (actv) -- node[left, rotate=\thetau]{$F_v$} (actv|-0,-2.6);
|
||||
\draw[spring=0.2] (kv) -- node[left, rotate=\thetau]{$k$} (kv|-0,-2.6);
|
||||
\draw[damper={8}{8}] (cv) -- node[left=0.1, rotate=\thetau]{$c$} (cv|-0,-2.6);
|
||||
\end{scope}
|
||||
|
||||
% Inertial Frame
|
||||
\draw[->] (-4, -4) -- ++(2, 0) node[below]{$\vec{i}_x$};
|
||||
\draw[->] (-4, -4) -- ++(0, 2) node[left]{$\vec{i}_y$};
|
||||
\draw[fill, color=black] (-4, -4) circle (0.06);
|
||||
\node[draw, circle, inner sep=0pt, minimum size=0.3cm, label=left:$\vec{i}_z$] at (-4, -4){};
|
||||
|
||||
\draw[->] (0, 0) node[above left, rotate=\thetau]{$\vec{i}_w$} -- ++(\thetau:2) node[above, rotate=\thetau]{$\vec{i}_u$};
|
||||
\draw[->] (0, 0) -- ++(\thetau+90:2) node[left, rotate=\thetau]{$\vec{i}_v$};
|
||||
\draw[fill, color=black] (0,0) circle (0.06);
|
||||
\node[draw, circle, inner sep=0pt, minimum size=0.3cm] at (0, 0){};
|
||||
\draw[dashed] (0, 0) -- ++(2, 0);
|
||||
\draw[] (1.5, 0) arc (0:\thetau:1.5) node[midway, right]{$\theta$};
|
||||
|
||||
\draw[->] (3.5, 0) arc (0:40:3.5) node[midway, left]{$\Omega$};
|
||||
\end{tikzpicture}
|
||||
```
|
||||
|
||||
{{< figure src="system.png" >}}
|
||||
|
||||
|
||||
## X-Y Rotating Positioning Platform {#x-y-rotating-positioning-platform}
|
||||
|
||||
```latex
|
||||
\tikzset{block/.default={0.8cm}{0.8cm}}
|
||||
\tikzset{addb/.append style={scale=0.7}}
|
||||
\tikzset{node distance=0.6}
|
||||
|
||||
\begin{tikzpicture}
|
||||
\node[block={1.8cm}{2.2cm}] (G) {$\bm{G}_f$};
|
||||
|
||||
% Inputs of the controllers
|
||||
\coordinate[] (output1) at ($(G.south east)!0.75!(G.north east)$);
|
||||
\coordinate[] (output2) at ($(G.south east)!0.25!(G.north east)$);
|
||||
\coordinate[] (input1) at ($(G.south west)!0.75!(G.north west)$);
|
||||
\coordinate[] (input2) at ($(G.south west)!0.25!(G.north west)$);
|
||||
|
||||
\node[block, left=1.8 of input1] (K1) {$K_F$};
|
||||
\node[block] (K2) at ($(K1.east|-input2)+(0.6, 0)$) {$K_F$};
|
||||
|
||||
% Connections and labels
|
||||
\draw[->] (K1.east) -- (input1)node[above left]{$F_u$}node[below left]{$-$};
|
||||
\draw[->] (K2.east) -- (input2)node[above left]{$F_v$}node[below left]{$-$};
|
||||
|
||||
\draw[->] (output1) -- ++(0.8, 0) node[above left]{$f_u$};
|
||||
\draw[->] (output2) -- ++(0.8, 0) node[above left]{$f_v$};
|
||||
|
||||
\draw[->] ($(output1)+(0.2, 0)$)node[branch]{} -- ++(0, 1.2) -| ($(K1.west) + (-0.8, 0)$)coordinate(start) -- (K1.west);
|
||||
\draw[->] ($(output2)+(0.2, 0)$)node[branch]{} -- ++(0, -1.2) -| (start|-K2) -- (K2.west);
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(K1.north west) (K2.south east)}, inner sep=6pt, draw, dashed, fill=black!20!white] (K) {};
|
||||
\node[below left] at (K.north east) {$\bm{K}_F$};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
```
|
||||
|
||||
{{< figure src="control_diagram_iff.png" >}}
|
||||
|
||||
|
||||
## Decentralized Integral Force Feedback {#decentralized-integral-force-feedback}
|
||||
|
||||
```latex
|
||||
\begin{tikzpicture}
|
||||
% Angle
|
||||
\def\thetau{25}
|
||||
|
||||
% Rotational Stage
|
||||
\draw[fill=black!60!white] (0, 0) circle (4.3);
|
||||
\draw[fill=black!40!white] (0, 0) circle (3.8);
|
||||
|
||||
% Label
|
||||
\node[anchor=north west, rotate=\thetau] at (-2.5, 2.5) {\small Rotating Stage};
|
||||
|
||||
% Rotating Scope
|
||||
\begin{scope}[rotate=\thetau]
|
||||
% Rotating Frame
|
||||
\draw[fill=black!20!white] (-2.6, -2.6) rectangle (2.6, 2.6);
|
||||
% Label
|
||||
\node[anchor=north west, rotate=\thetau] at (-2.6, 2.6) {\small Suspended Platform};
|
||||
|
||||
% Mass
|
||||
\draw[fill=white] (-1, -1) rectangle (1, 1);
|
||||
% Label
|
||||
\node[anchor=south west, rotate=\thetau] at (-1, -1) {\small Payload};
|
||||
|
||||
% Attached Points
|
||||
\node[] at (-1, 0){$\bullet$};
|
||||
\draw[] (-1, 0) -- ++(-0.2, 0) coordinate(au);
|
||||
\node[] at (0, -1){$\bullet$};
|
||||
\draw[] (0, -1) -- ++(0, -0.2) coordinate(av);
|
||||
|
||||
% Force Sensors
|
||||
\draw[fill=white] ($(au) + (-0.2, -0.5)$) rectangle ($(au) + (0, 0.5)$);
|
||||
\draw[] ($(au) + (-0.2, -0.5)$)coordinate(actu) -- ($(au) + (0, 0.5)$);
|
||||
\draw[] ($(au) + (-0.2, 0.5)$)coordinate(ku) -- ($(au) + (0, -0.5)$);
|
||||
|
||||
\draw[fill=white] ($(av) + (-0.5, -0.2)$) rectangle ($(av) + (0.5, 0)$);
|
||||
\draw[] ($(av) + ( 0.5, -0.2)$)coordinate(actv) -- ($(av) + (-0.5, 0)$);
|
||||
\draw[] ($(av) + (-0.5, -0.2)$)coordinate(kv) -- ($(av) + ( 0.5, 0)$);
|
||||
|
||||
% Spring and Actuator for U
|
||||
\draw[actuator={0.6}{0.2}] (actu) -- coordinate[midway](actumid) (actu-|-2.6,0);
|
||||
\draw[spring=0.2] (ku) -- node[above=0.1, rotate=\thetau]{$k$} (ku-|-2.6,0);
|
||||
|
||||
% \draw[actuator={0.6}{0.2}] (actv) -- node[right, rotate=\thetau]{$F_v$} (actv|-0,-2.6);
|
||||
\draw[actuator={0.6}{0.2}] (actv) -- coordinate[midway](actvmid) (actv|-0,-2.6);
|
||||
\draw[spring=0.2] (kv) -- node[left, rotate=\thetau]{$k$} (kv|-0,-2.6);
|
||||
|
||||
\node[block={0.8cm}{0.6cm}, rotate=\thetau] (Ku) at ($(actumid) + (0, -1.2)$) {$K_{F}$};
|
||||
\draw[->] ($(au) + (-0.1, -0.5)$) |- (Ku.east) node[below right, rotate=\thetau]{$f_{u}$};
|
||||
\draw[->] (Ku.north) -- ($(actumid) + (0, -0.1)$) node[below left, rotate=\thetau]{$F_u$} node[below right, rotate=\thetau]{$-$};
|
||||
|
||||
\node[block={0.8cm}{0.6cm}, rotate=\thetau] (Kv) at ($(actvmid) + (1.2, 0)$) {$K_{F}$};
|
||||
\draw[->] ($(av) + (0.5, -0.1)$) -| (Kv.north) node[above right, rotate=\thetau]{$f_{v}$};
|
||||
\draw[->] (Kv.west) -- ($(actvmid) + (0.1, 0)$) node[below right, rotate=\thetau]{$F_v$} node[above right, rotate=\thetau]{$-$};
|
||||
\end{scope}
|
||||
|
||||
% Inertial Frame
|
||||
\draw[->] (-4, -4) -- ++(2, 0) node[below]{$\vec{i}_x$};
|
||||
\draw[->] (-4, -4) -- ++(0, 2) node[left]{$\vec{i}_y$};
|
||||
\draw[fill, color=black] (-4, -4) circle (0.06);
|
||||
\node[draw, circle, inner sep=0pt, minimum size=0.3cm, label=left:$\vec{i}_z$] at (-4, -4){};
|
||||
|
||||
\node[draw, circle, inner sep=0pt, minimum size=0.3cm] at (0, 0){};
|
||||
\draw[->] (0, 0) node[above left, rotate=\thetau]{$\vec{i}_w$} -- ++(\thetau:2) node[above, rotate=\thetau]{$\vec{i}_u$};
|
||||
\draw[->] (0, 0) -- ++(\thetau+90:2) node[left, rotate=\thetau]{$\vec{i}_v$};
|
||||
\draw[dashed] (0, 0) -- ++(2, 0);
|
||||
\draw[] (1.5, 0) arc (0:\thetau:1.5) node[midway, right]{$\theta$};
|
||||
\node[] at (0,0) {$\bullet$};
|
||||
|
||||
\draw[->] (3.5, 0) arc (0:40:3.5) node[midway, left]{$\Omega$};
|
||||
\end{tikzpicture}
|
||||
```
|
||||
|
||||
{{< figure src="system_iff.png" >}}
|
||||
|
||||
|
||||
## Springs in parallel {#springs-in-parallel}
|
||||
|
||||
```latex
|
||||
\begin{tikzpicture}
|
||||
% Angle
|
||||
\def\thetau{25}
|
||||
|
||||
% Rotational Stage
|
||||
\draw[fill=black!60!white] (0, 0) circle (4.3);
|
||||
\draw[fill=black!40!white] (0, 0) circle (3.8);
|
||||
|
||||
% Label
|
||||
\node[anchor=north west, rotate=\thetau] at (-2.5, 2.5) {\small Rotating Stage};
|
||||
|
||||
% Rotating Scope
|
||||
\begin{scope}[rotate=\thetau]
|
||||
% Rotating Frame
|
||||
\draw[fill=black!20!white] (-2.6, -2.6) rectangle (2.6, 2.6);
|
||||
% Label
|
||||
\node[anchor=north west, rotate=\thetau] at (-2.6, 2.6) {\small Suspended Platform};
|
||||
|
||||
% Mass
|
||||
\draw[fill=white] (-1, -1) rectangle (1, 1);
|
||||
% Label
|
||||
\node[anchor=south west, rotate=\thetau] at (-1, -1) {\small Payload};
|
||||
|
||||
% Attached Points
|
||||
\draw[] (-1, 0) -- ++(-0.2, 0) coordinate(au);
|
||||
\draw[] (0, -1) -- ++(0, -0.2) coordinate(av);
|
||||
|
||||
% Force Sensors
|
||||
\draw[fill=white] ($(au) + (-0.2, -0.5)$) rectangle ($(au) + (0, 0.5)$);
|
||||
\draw[] ($(au) + (-0.2, -0.5)$)coordinate(actu) -- ($(au) + (0, 0.5)$);
|
||||
\draw[] ($(au) + (-0.2, 0.5)$)coordinate(ku) -- ($(au) + (0, -0.5)$);
|
||||
\node[below=0.1, rotate=\thetau] at ($(au) + (-0.1, -0.5)$) {$f_{u}$}
|
||||
|
||||
\draw[fill=white] ($(av) + (-0.5, -0.2)$) rectangle ($(av) + (0.5, 0)$);
|
||||
\draw[] ($(av) + ( 0.5, -0.2)$)coordinate(actv) -- ($(av) + (-0.5, 0)$);
|
||||
\draw[] ($(av) + (-0.5, -0.2)$)coordinate(kv) -- ($(av) + ( 0.5, 0)$) ;
|
||||
\node[right=0.1, rotate=\thetau] at ($(av) + (0.5, -0.1)$) {$f_{v}$}
|
||||
|
||||
% Spring and Actuator for U
|
||||
\draw[actuator={0.6}{0.2}] (actu) -- node[below=0.1, rotate=\thetau]{$F_u$} (actu-|-2.6,0);
|
||||
\draw[spring=0.2] (ku) -- node[below=0.1, rotate=\thetau]{$k_a$} (ku-|-2.6,0);
|
||||
\draw[spring=0.2] (-1, 0.8) -- node[above=0.1, rotate=\thetau]{$k_p$} (-1, 0.8-|-2.6,0);
|
||||
|
||||
\draw[actuator={0.6}{0.2}] (actv) -- node[right=0.1, rotate=\thetau]{$F_v$} (actv|-0,-2.6);
|
||||
\draw[spring=0.2] (kv) -- node[right=0.1, rotate=\thetau]{$k_a$} (kv|-0,-2.6);
|
||||
\draw[spring=0.2] (-0.8, -1) -- node[left=0.1, rotate=\thetau]{$k_p$} (-0.8, -1|-0,-2.6);
|
||||
\end{scope}
|
||||
|
||||
% Inertial Frame
|
||||
\draw[->] (-4, -4) -- ++(2, 0) node[below]{$\vec{i}_x$};
|
||||
\draw[->] (-4, -4) -- ++(0, 2) node[left]{$\vec{i}_y$};
|
||||
\draw[fill, color=black] (-4, -4) circle (0.06);
|
||||
\node[draw, circle, inner sep=0pt, minimum size=0.3cm, label=left:$\vec{i}_z$] at (-4, -4){};
|
||||
|
||||
\node[draw, circle, inner sep=0pt, minimum size=0.3cm] at (0, 0){};
|
||||
\draw[->] (0, 0) node[above left, rotate=\thetau]{$\vec{i}_w$} -- ++(\thetau:2) node[above, rotate=\thetau]{$\vec{i}_u$};
|
||||
\draw[->] (0, 0) -- ++(\thetau+90:2) node[left, rotate=\thetau]{$\vec{i}_v$};
|
||||
\draw[dashed] (0, 0) -- ++(2, 0);
|
||||
\draw[] (1.5, 0) arc (0:\thetau:1.5) node[midway, right]{$\theta$};
|
||||
\node[] at (0,0) {$\bullet$};
|
||||
|
||||
\draw[->] (3.5, 0) arc (0:40:3.5) node[midway, left]{$\Omega$};
|
||||
\end{tikzpicture}
|
||||
```
|
||||
|
||||
{{< figure src="system_parallel_springs.png" >}}
|
||||
@@ -0,0 +1,765 @@
|
||||
+++
|
||||
title = "LaTeX Configuration for Tikz export"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
## Packages {#packages}
|
||||
|
||||
```latex
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[T1]{fontenc}
|
||||
|
||||
\usepackage[french, english]{babel} % Last language is main language
|
||||
|
||||
\usepackage{lmodern} % Latin Modern Font
|
||||
\usepackage{gensymb} % Generic symbols for both text and math mode
|
||||
|
||||
\usepackage{standalone} % Used to generate standalone Tikz
|
||||
|
||||
\usepackage{amsmath} % Main math Package
|
||||
\usepackage{mathtools} % Extension package to amsmath
|
||||
\usepackage{amsthm} % Typesetting theorems (AMS style)
|
||||
\usepackage{amsfonts} % More fonts from the AMS
|
||||
\usepackage{textcomp} % provide many text symbols
|
||||
\usepackage{steinmetz} % For phase symbol
|
||||
|
||||
\usepackage{xstring} % Utils to manipulate strings
|
||||
\usepackage{etoolbox} % Add basic if/then
|
||||
\usepackage{esvect} % Beautyfull vectors
|
||||
\usepackage{graphicx} % Enhanced support for graphics
|
||||
\usepackage{grffile} % Used by matlab2tikz
|
||||
|
||||
\usepackage{microtype} % typographic tuning
|
||||
\usepackage{setspace} % for line spacing, e.g. \onehalfspacing
|
||||
\usepackage{tabularx} % table features
|
||||
\usepackage{enumitem} % for simple list modifications
|
||||
\usepackage{booktabs} % better table support
|
||||
|
||||
\usepackage{stackengine} %
|
||||
|
||||
\usepackage[load-configurations=abbreviations]{siunitx} % SI units
|
||||
\sisetup{
|
||||
locale = US,
|
||||
detect-all,
|
||||
range-phrase=--,
|
||||
range-units=single
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Tikz related packages {#tikz-related-packages}
|
||||
|
||||
```latex
|
||||
\usepackage{tikz} % Tikz
|
||||
\usepackage{tikzscale} % Used to scale Tikz graphics
|
||||
\usepackage{adjustbox} % Used to proper positioning of tikz pictures
|
||||
\usepackage{circuitikz} % Draw electronic circuits
|
||||
\usepackage{pgfpages} % Needed to use notes
|
||||
\usepackage{pgfplots} % Used to plot functions
|
||||
```
|
||||
|
||||
|
||||
## Tikz Libraries {#tikz-libraries}
|
||||
|
||||
```latex
|
||||
\usetikzlibrary{arrows} % Arrow tip library
|
||||
\usetikzlibrary{arrows.meta} % Add some arrows
|
||||
\usetikzlibrary{calc} % The library allows advanced Coordinate Calculations
|
||||
\usetikzlibrary{intersections} % calculate intersections of paths
|
||||
\usetikzlibrary{matrix} %
|
||||
\usetikzlibrary{patterns} %
|
||||
\usetikzlibrary{shapes} % Defines circle and rectangle
|
||||
\usetikzlibrary{shapes.geometric} % Use for the shape diamond and isosceles triangle
|
||||
\usetikzlibrary{snakes} % snake=coil and snake=zigzag using segment amplitude=10pt
|
||||
\usetikzlibrary{positioning} % Additional options for placing nodes
|
||||
\usetikzlibrary{3d} % Plot 3D shapes
|
||||
\usetikzlibrary{spy} % Creating a magnified area
|
||||
\usetikzlibrary{decorations.text} % Used to make text follows a curve
|
||||
\usetikzlibrary{decorations.pathmorphing} % deformation of a path
|
||||
\usetikzlibrary{decorations.markings} % Used for spring and damper
|
||||
\usetikzlibrary{babel} % A tiny library that make the interaction with the babel package easier
|
||||
\usetikzlibrary{plotmarks} % This library defines a number of plot marks
|
||||
\usetikzlibrary{fit} % Used to make rectangle as nodes by specifying two points
|
||||
\usetikzlibrary{backgrounds} % Used to put things under others
|
||||
```
|
||||
|
||||
|
||||
## PGF Plot libraries and config {#pgf-plot-libraries-and-config}
|
||||
|
||||
```latex
|
||||
\usepgfplotslibrary{patchplots}
|
||||
\usepgfplotslibrary{groupplots}
|
||||
|
||||
\pgfplotsset{compat=newest}
|
||||
\pgfplotsset{plot coordinates/math parser=false}
|
||||
```
|
||||
|
||||
|
||||
## Setup size of figures {#setup-size-of-figures}
|
||||
|
||||
```latex
|
||||
\newlength{\fheight}
|
||||
\newlength{\fwidth}
|
||||
|
||||
\setlength{\fwidth}{85mm}
|
||||
\setlength{\fheight}{112mm}
|
||||
```
|
||||
|
||||
|
||||
## Setup Arrows style {#setup-arrows-style}
|
||||
|
||||
```latex
|
||||
\tikzset{>=Stealth}
|
||||
% Setup default Linewidth
|
||||
\tikzset{every path/.style={line width=1pt}}
|
||||
```
|
||||
|
||||
|
||||
## Colors {#colors}
|
||||
|
||||
```latex
|
||||
\usepackage{xcolor}% Color extension
|
||||
|
||||
\definecolor{colorblack}{rgb}{0, 0, 0}
|
||||
\definecolor{colorblue}{rgb}{0, 0.4470, 0.7410}
|
||||
\definecolor{colorred}{rgb}{0.8500, 0.3250, 0.0980}
|
||||
\definecolor{coloryellow}{rgb}{0.9290, 0.6940, 0.1250}
|
||||
\definecolor{colorpurple}{rgb}{0.4940, 0.1840, 0.5560}
|
||||
\definecolor{colorgreen}{rgb}{0.4660, 0.6740, 0.1880}
|
||||
\definecolor{colorcyan}{rgb}{0.3010, 0.7450, 0.9330}
|
||||
\definecolor{colorbordeau}{rgb}{0.6350, 0.0780, 0.1840}
|
||||
|
||||
% Main color
|
||||
\definecolor{maincolor}{RGB}{89, 9, 38}
|
||||
\definecolor{secondcolor}{RGB}{20, 9, 89}
|
||||
```
|
||||
|
||||
|
||||
## Control {#control}
|
||||
|
||||
|
||||
### Blocks {#blocks}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
block/.style n args={2}{%
|
||||
draw,
|
||||
fill=white,
|
||||
minimum width = #1,
|
||||
minimum height = #2,
|
||||
},
|
||||
block/.default={1.2cm}{1.0cm}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Branches {#branches}
|
||||
|
||||
```latex
|
||||
\tikzstyle{branch}=[fill,shape=circle,minimum size=4pt,inner sep=0pt]
|
||||
\tikzstyle{->top}=[-{Stealth[color=black, scale=0.8]}, draw=white, double=black, double distance=1pt, line width=1pt]
|
||||
\tikzstyle{<-top}=[{stealth[color=black, scale=0.8]}-, draw=white, double=black, double distance=1pt, line width=1pt]
|
||||
```
|
||||
|
||||
|
||||
### Hand Writen Style {#hand-writen-style}
|
||||
|
||||
Usefull for schematic plots
|
||||
|
||||
```latex
|
||||
\tikzstyle{handwriten}=[decorate,decoration={random steps,amplitude=0.1pt,segment length=0.8pt}]
|
||||
```
|
||||
|
||||
|
||||
### DAC {#dac}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
DAC/.style={%
|
||||
draw,
|
||||
signal,
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### ADC {#adc}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
ADC/.style={%
|
||||
draw,
|
||||
signal,
|
||||
signal to = west,
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Gain {#gain}
|
||||
|
||||
Maybe use `isosceles` instead of regular polygon?
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
gain right/.style={%
|
||||
draw,
|
||||
regular polygon,
|
||||
regular polygon sides = 3,
|
||||
inner sep = 2pt,
|
||||
shape border rotate=-90
|
||||
},
|
||||
gain left/.style={%
|
||||
draw,
|
||||
regular polygon,
|
||||
regular polygon sides = 3,
|
||||
inner sep = 2pt,
|
||||
shape border rotate=90
|
||||
},
|
||||
gain top/.style={%
|
||||
draw,
|
||||
regular polygon,
|
||||
regular polygon sides = 3,
|
||||
inner sep = 2pt,
|
||||
shape border rotate=0
|
||||
},
|
||||
gain bottom/.style={%
|
||||
draw,
|
||||
regular polygon,
|
||||
regular polygon sides = 3,
|
||||
inner sep = 2pt,
|
||||
shape border rotate=180
|
||||
},
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Add / Substract / Divide / Multiply block {#add-substract-divide-multiply-block}
|
||||
|
||||
```latex
|
||||
\tikzset{% Add block with Circled operations
|
||||
addc/.style n args={5}{%
|
||||
draw,
|
||||
fill=white,
|
||||
circle,
|
||||
outer sep = 0pt,
|
||||
inner sep = 0pt,
|
||||
minimum size = 2em,
|
||||
execute at begin node={\LARGE $#1$},
|
||||
append after command={\pgfextra{\let\mainnode=\tikzlastnode}
|
||||
\ifx#2\empty\else
|
||||
node[draw, circle, outer sep=6pt, inner sep=0pt, above left] at (\mainnode.west) {$#2$}%
|
||||
\fi
|
||||
\ifx#3\empty\else
|
||||
node[draw, circle, outer sep=6pt, inner sep=0pt, above right] at (\mainnode.north) {$#3$}%
|
||||
\fi
|
||||
\ifx#4\empty\else
|
||||
node[draw, circle, outer sep=6pt, inner sep=0pt, below right] at (\mainnode.east) {$#4$}%
|
||||
\fi
|
||||
\ifx#5\empty\else
|
||||
node[draw, circle, outer sep=6pt, inner sep=0pt, below left] at (\mainnode.south) {$#5$}%
|
||||
\fi
|
||||
}
|
||||
},
|
||||
addc/.default={+}{}{}{}{},
|
||||
}
|
||||
```
|
||||
|
||||
```latex
|
||||
\tikzset{% Add Block
|
||||
addb/.style n args={5}{%
|
||||
draw,
|
||||
fill=white,
|
||||
circle,
|
||||
outer sep = 0pt,
|
||||
inner sep = 0pt,
|
||||
minimum size = 2em,
|
||||
execute at begin node={\LARGE $#1$},
|
||||
append after command={\pgfextra{\let\mainnode=\tikzlastnode}
|
||||
\ifx#2\empty\else
|
||||
node[outer sep=2pt, inner sep=0pt, above left] at (\mainnode.west) {$#2$}%
|
||||
\fi
|
||||
\ifx#3\empty\else
|
||||
node[outer sep=2pt, inner sep=0pt, above right] at (\mainnode.north) {$#3$}%
|
||||
\fi
|
||||
\ifx#4\empty\else
|
||||
node[outer sep=2pt, inner sep=0pt, below right] at (\mainnode.east) {$#4$}%
|
||||
\fi
|
||||
\ifx#5\empty\else
|
||||
node[outer sep=2pt, inner sep=0pt, below left] at (\mainnode.south) {$#5$}%
|
||||
\fi
|
||||
}
|
||||
},
|
||||
addb/.default={+}{}{}{}{},
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Plots {#plots}
|
||||
|
||||
|
||||
### Grid {#grid}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{grid style={black}}
|
||||
\pgfplotsset{major grid style={black!30!white}}
|
||||
\pgfplotsset{minor grid style={black!10!white}}
|
||||
\pgfplotsset{xmajorgrids}
|
||||
\pgfplotsset{ymajorgrids}
|
||||
```
|
||||
|
||||
|
||||
### Lines {#lines}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{separate axis lines=false} % draw axis as rectangle and not as 4 lines
|
||||
\pgfplotsset{every outer x axis line/.append style={black}}
|
||||
\pgfplotsset{every outer y axis line/.append style={black}}
|
||||
\pgfplotsset{axis background/.style={fill=white}}
|
||||
\pgfplotsset{axis x line*=bottom} % solid line on the bottom with thin on the top
|
||||
\pgfplotsset{axis y line*=left} % solid line on the left with thin on the right
|
||||
```
|
||||
|
||||
|
||||
### Ticks {#ticks}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{every y tick label/.append style={font=\color{black}}}
|
||||
\pgfplotsset{every y tick/.append style={black}}
|
||||
\pgfplotsset{every x tick label/.append style={font=\color{black}}}
|
||||
\pgfplotsset{every x tick/.append style={black}}
|
||||
```
|
||||
|
||||
|
||||
### Size {#size}
|
||||
|
||||
If `scale only axis=false` (the default), pgfplots will try to produce the desired width including labels, titles and ticks.
|
||||
|
||||
```latex
|
||||
\pgfplotsset{scale only axis=true}
|
||||
```
|
||||
|
||||
|
||||
### Label {#label}
|
||||
|
||||
Used to align all of ylabel of one figure.
|
||||
|
||||
```latex
|
||||
\pgfplotsset{ylabel absolute}
|
||||
```
|
||||
|
||||
|
||||
### Legend {#legend}
|
||||
|
||||
```latex
|
||||
% https://tex.stackexchange.com/questions/54794/using-a-pgfplots-style-legend-in-a-plain-old-tikzpicture#54834
|
||||
|
||||
% argument #1: any options
|
||||
\newenvironment{customlegend}[1][]{%
|
||||
\begingroup
|
||||
% inits/clears the lists (which might be populated from previous
|
||||
% axes):
|
||||
\csname pgfplots@init@cleared@structures\endcsname
|
||||
\pgfplotsset{#1}%
|
||||
}{%
|
||||
% draws the legend:
|
||||
\csname pgfplots@createlegend\endcsname
|
||||
\endgroup
|
||||
}%
|
||||
|
||||
% makes \addlegendimage available (typically only available within an
|
||||
% axis environment):
|
||||
\def\addlegendimage{\csname pgfplots@addlegendimage\endcsname}
|
||||
|
||||
% definition to insert numbers
|
||||
% \pgfkeys{/pgfplots/number in legend/.style={%
|
||||
% /pgfplots/legend image code/.code={%
|
||||
% \node at (0.125,-0.0225){#1}; % <= changed x value
|
||||
% },%
|
||||
% },
|
||||
% }
|
||||
\pgfplotsset{
|
||||
every legend to name picture/.style={west}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Upper and Lower bounds {#upper-and-lower-bounds}
|
||||
|
||||
```latex
|
||||
\tikzstyle{upperbound}=[line cap=round, postaction={decorate,draw,decoration={border, segment length=0.2cm, amplitude=0.3cm, angle=60}}]
|
||||
\tikzstyle{lowerbound}=[line cap=round, postaction={decorate,draw,decoration={border, segment length=0.2cm, amplitude=0.3cm, angle=-60}}]
|
||||
```
|
||||
|
||||
And we add the corresdonding
|
||||
|
||||
```latex
|
||||
\pgfplotsset{
|
||||
/pgfplots/upperbound/.style 1 args={
|
||||
legend image code/.code={
|
||||
\draw[##1, upperbound]
|
||||
plot coordinates {
|
||||
(0cm,0cm)
|
||||
(0.6cm,0cm)
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Pole {#pole}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
pole/.style{%
|
||||
color=red,
|
||||
cross out,
|
||||
draw,
|
||||
inner sep=0pt,
|
||||
outer sep=0pt,
|
||||
minimum size=#1pt
|
||||
},
|
||||
pole/.default={4}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Zero {#zero}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
zero/.style{%
|
||||
color=red,
|
||||
circle,
|
||||
draw,
|
||||
inner sep=0pt,
|
||||
outer sep=0pt,
|
||||
minimum size=#1pt
|
||||
},
|
||||
zero/.default={4}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Mechanical {#mechanical}
|
||||
|
||||
|
||||
### Spring {#spring}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
spring/.style={%
|
||||
thick,
|
||||
decoration={
|
||||
zigzag,
|
||||
pre length = #1cm,
|
||||
post length = #1cm,
|
||||
segment length = 6
|
||||
},
|
||||
decorate
|
||||
},
|
||||
spring/.default={0.2}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Coil {#coil}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
coil/.style n args={2}{%
|
||||
thick,
|
||||
decoration={
|
||||
coil,
|
||||
pre length = #1cm,
|
||||
post length = #2cm,
|
||||
segment length = 4
|
||||
},
|
||||
decorate
|
||||
},
|
||||
coil/.default={0.3}{0.3}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Damper {#damper}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
damper/.style n args={2}{%
|
||||
thick,
|
||||
decoration={markings, mark connection node=dmp, mark=at position 0.5 with {
|
||||
\node (dmp) [thick,
|
||||
inner sep = 0pt,
|
||||
transform shape,
|
||||
rotate =-90,
|
||||
minimum width = #1pt,
|
||||
minimum height = #2pt,
|
||||
draw=none] {};
|
||||
\draw [thick] ($(dmp.north east)+(0.6*#2pt,0)$) -- (dmp.south east) -- (dmp.south west) -- ($(dmp.north west)+(0.6*#2pt,0)$);
|
||||
\draw [thick] ($(dmp.north)+(0,-0.3*#1pt)$) -- ($(dmp.north)+(0,0.3*#1pt)$);
|
||||
}
|
||||
},
|
||||
decorate
|
||||
},
|
||||
damper/.default={12}{3}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Actuator {#actuator}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
actuator/.style n args={2}{%
|
||||
thick,
|
||||
draw=none,
|
||||
decoration={
|
||||
markings,
|
||||
mark connection node=my node,
|
||||
mark=at position .5 with {
|
||||
\node [draw, inner sep=0pt, minimum width=#1cm, minimum height=#2cm,
|
||||
transform shape, fill=white] (my node) {};
|
||||
},
|
||||
mark=at position .0 with {
|
||||
\draw[<-] (0, 0) -- (my node);
|
||||
},
|
||||
mark=at position 1.0 with {
|
||||
\draw[<-] (0, 0) -- (my node);
|
||||
}
|
||||
},
|
||||
decorate
|
||||
},
|
||||
actuator/.default={0.5}{0.2}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Ground {#ground}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
ground/.style n args={2}{%
|
||||
fill,
|
||||
pattern = north east lines,
|
||||
draw = none,
|
||||
anchor = north,
|
||||
minimum width = #1cm,
|
||||
minimum height = #2cm,
|
||||
append after command={
|
||||
(\tikzlastnode.north west) edge (\tikzlastnode.north east)
|
||||
}
|
||||
},
|
||||
ground/.default={2.5}{0.3}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Force Sensor {#force-sensor}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
forcesensor/.style n args={2}{%
|
||||
rectangle,
|
||||
outer sep=0pt,
|
||||
inner sep=0pt,
|
||||
draw=black,
|
||||
fill=white!60!black,
|
||||
anchor=south,
|
||||
minimum width =#1cm,
|
||||
minimum height=#2cm,
|
||||
append after command={
|
||||
[every edge/.append style={
|
||||
thick,
|
||||
black,
|
||||
}]
|
||||
(\tikzlastnode.north west) edge (\tikzlastnode.south east)
|
||||
(\tikzlastnode.north east) edge (\tikzlastnode.south west)
|
||||
}
|
||||
},
|
||||
forcesensor/.default={2.0}{0.5}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Inertial Sensor {#inertial-sensor}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
inertialsensor/.style={%
|
||||
rectangle,
|
||||
outer sep=0pt,
|
||||
inner sep=0pt,
|
||||
draw=black,
|
||||
fill=white!60!black,
|
||||
anchor=south east,
|
||||
minimum size=#1cm,
|
||||
append after command={
|
||||
[every edge/.append style={
|
||||
thick,
|
||||
black,
|
||||
}]
|
||||
(\tikzlastnode.north west) edge (\tikzlastnode.south east)
|
||||
(\tikzlastnode.north east) edge (\tikzlastnode.south west)
|
||||
}
|
||||
},
|
||||
inertialsensor/.default={0.3}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Axis Rotator {#axis-rotator}
|
||||
|
||||
```latex
|
||||
\newcommand{\AxisRotator}[1][rotate=0]{%
|
||||
\tikz [x=0.1cm,y=0.30cm,-stealth,#1] \draw (0,0) arc (-150:150:1 and 1);%
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Cross {#cross}
|
||||
|
||||
```latex
|
||||
\tikzstyle{cross}=[path picture={
|
||||
\draw[black]
|
||||
(path picture bounding box.south east) -- (path picture bounding box.north west) (path picture bounding box.south west) -- (path picture bounding box.north east);
|
||||
}]
|
||||
|
||||
```
|
||||
|
||||
|
||||
### Piezoelectric actuator {#piezoelectric-actuator}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
piezo/.style n args={3}{%
|
||||
draw,
|
||||
rectangle,
|
||||
minimum width = #1cm,
|
||||
minimum height = #2cm,
|
||||
fill=blue!10!white,
|
||||
anchor=center,
|
||||
append after command={
|
||||
[every edge/.append style={
|
||||
thick,
|
||||
black,
|
||||
}]
|
||||
\foreach \i in {1,...,#3}{
|
||||
(${\i/(1+#3)}*(\tikzlastnode.north west)+{(1+#3-\i)/(1+#3)}*(\tikzlastnode.south west)+0.1*(#1,0)$) edge (${\i/(1+#3)}*(\tikzlastnode.north east)+{(1+#3-\i)/(1+#3)}*(\tikzlastnode.south east)-0.1*(#1,0)$)
|
||||
}
|
||||
}
|
||||
},
|
||||
piezo/.default={2}{4}{10}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Voice coil {#voice-coil}
|
||||
|
||||
```latex
|
||||
\def\voicecoil#1#2#3{
|
||||
% ======================
|
||||
% Parameters
|
||||
% ======================
|
||||
\def\voicecoilw{#1} % Total Width
|
||||
\def\voicecoilh{#2} % Total Height
|
||||
|
||||
\def\magnetw{\voicecoilw} % Width of the magnet
|
||||
\def\magneth{\voicecoilh/1.4} % Height of the magnet
|
||||
|
||||
\def\magnetwb{0.15*\magnetw} % Width of the borders of the magnet
|
||||
\def\magnetmw{0.15*\magnetw} % Width of the middle part of the magnet
|
||||
\def\magnetwg{0.5*\magnetw} % Width of the gap of the magnet
|
||||
|
||||
\def\magnethl{\magnetwb} % Height of the low part of the magnet
|
||||
\def\magnetmh{0.15*\magneth} % Height of the middle part of the magnet
|
||||
\def\magnethg{0.2*\magneth} % Height of the gap of the magnet
|
||||
% ======================
|
||||
|
||||
\begin{scope}[shift={(0.5*\voicecoilw, 0.5*\voicecoilh)}, rotate=#3, shift={(0, -0.5*\voicecoilh)}]
|
||||
% ======================
|
||||
% Magnet
|
||||
% ======================
|
||||
\draw[fill=white] (0, 0) -| ++(0.5*\magnetw, \magneth) -| ++(-0.5*\magnetw+0.5*\magnetwg, -\magnethg) -| (0.5*\magnetw-\magnetwb, \magnethl) -| (-0.5*\magnetw+\magnetwb, \magneth-\magnethg) -| (-0.5*\magnetwg, \magneth) -| (-0.5*\magnetw, 0) -- (cycle);
|
||||
\begin{scope}[shift={(0, \magnethl)}]
|
||||
\draw[fill=red] (-0.5*\magnetmw, 0) rectangle (0.5*\magnetmw, \magnetmh);
|
||||
\draw[fill=blue] (-0.5*\magnetmw, \magnetmh) rectangle (0.5*\magnetmw, 2*\magnetmh);
|
||||
% Top conductive Magnet
|
||||
\draw[fill=white] (-0.5*\magnetmw, 2*\magnetmh) -| (0.5*\magnetmw, -\magnethl+\magneth-\magnethg) -| ++(0.1, \magnethg) -| ++(-0.2-\magnetmw, -\magnethg) -| (-0.5*\magnetmw, \magnetmh);
|
||||
\end{scope}
|
||||
% ======================
|
||||
|
||||
% ======================
|
||||
% Coil
|
||||
% ======================
|
||||
\pgfmathsetmacro{\coilwidth}{0.5*0.5*\magnetmw+0.5*0.1+0.25*\magnetwg}%
|
||||
\draw[] ( \coilwidth, 0.5*\magneth) -- ++(0, 0.7*\magneth);
|
||||
\draw[] (-\coilwidth, 0.5*\magneth) -- ++(0, 0.7*\magneth);
|
||||
% Point on the coil
|
||||
\foreach \x in {0,1,...,9}
|
||||
{
|
||||
\node[circle,inner sep=0.6pt,fill] at ( \coilwidth, \x*0.7*\magneth/10+0.5*\magneth);
|
||||
\node[circle,inner sep=0.6pt,fill] at (-\coilwidth, \x*0.7*\magneth/10+0.5*\magneth);
|
||||
}
|
||||
\draw[fill=white] (-0.5*\magnetw, 1.2*\magneth) rectangle ++(\magnetw, \magnethg);
|
||||
% ======================
|
||||
|
||||
% ======================
|
||||
% Coordinates
|
||||
% ======================
|
||||
% Force
|
||||
\coordinate[] (vc_force) at (0, \magneth-0.5*\magnethg);
|
||||
% Coil
|
||||
\coordinate[] (vc_coil) at (0, \voicecoilh);
|
||||
% Magnet
|
||||
\coordinate[] (vc_magnet) at (0, 0);
|
||||
% Coil Wires
|
||||
\coordinate[] (vc_wire_one) at ( \coilwidth, 1.2*\magneth);
|
||||
\coordinate[] (vc_wire_two) at (-\coilwidth, 1.2*\magneth);
|
||||
% ======================
|
||||
\end{scope}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Optics {#optics}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
->-/.style={
|
||||
decoration={
|
||||
markings,
|
||||
mark = at position #1 with {\arrow{>}
|
||||
}
|
||||
},
|
||||
postaction={decorate}
|
||||
}
|
||||
}
|
||||
\tikzset{%
|
||||
-<-/.style={
|
||||
decoration={
|
||||
markings,
|
||||
mark = at position #1 with {\arrow{<}
|
||||
}
|
||||
},
|
||||
postaction={decorate}
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Misc {#misc}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
labelc/.style= {%
|
||||
draw,
|
||||
fill=white,
|
||||
shape=circle,
|
||||
inner sep=2pt,
|
||||
outer sep=6pt,
|
||||
}
|
||||
}
|
||||
```
|
||||
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|
||||
+++
|
||||
title = "Active Damping of Rotating Platforms using Integral Force Feedback"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
venue = "Engineering Research Express"
|
||||
year = 2021
|
||||
pubtype = "journal"
|
||||
doi = "10.1088/2631-8695/abe803"
|
||||
code = "https://git.tdehaeze.xyz/tdehaeze/dehaeze21_activ_dampin_rotat_platf_using"
|
||||
+++
|
||||
|
||||
> **Abstract**:
|
||||
>
|
||||
> This paper investigates the use of Integral Force Feedback (IFF) for the active damping of rotating mechanical systems.
|
||||
> Guaranteed stability, typical benefit of IFF, is lost as soon as the system is rotating due to gyroscopic effects.
|
||||
> To overcome this issue, two modifications of the classical IFF control scheme are proposed.
|
||||
> The first consists of slightly modifying the control law while the second consists of adding springs in parallel with the force sensors.
|
||||
> Conditions for stability and optimal parameters are derived.
|
||||
> The results reveal that, despite their different implementations, both modified IFF control scheme have almost identical damping authority on the suspension modes.
|
||||
|
||||
|
||||
## Journal Paper ([pdf](journal/dehaeze21_activ_dampin_rotat_platf_using.pdf)) {#journal-paper--pdf-journal-dehaeze21-activ-dampin-rotat-platf-using-dot-pdf}
|
||||
|
||||
The paper has been created using [Org Mode](https://orgmode.org/) (generating [LaTeX](https://www.latex-project.org/) code) under [Emacs](https://www.gnu.org/software/emacs/).
|
||||
|
||||
To cite this journal paper use the following bibtex code.
|
||||
|
||||
```bibtex
|
||||
@article{dehaeze21_activ_dampin_rotat_platf_using,
|
||||
author = {Thomas Dehaeze and Christophe Collette},
|
||||
title = {Active Damping of Rotating Platforms Using Integral Force
|
||||
Feedback},
|
||||
journal = {Engineering Research Express},
|
||||
year = 2021,
|
||||
doi = {10.1088/2631-8695/abe803},
|
||||
url = {https://doi.org/10.1088/2631-8695/abe803},
|
||||
month = {Feb},
|
||||
}
|
||||
```
|
||||
|
||||
You can also use the formatted citation below.
|
||||
|
||||
> Dehaeze, T., & Collette, C., Active damping of rotating platforms using integral force feedback, Engineering Research Express, (2021).
|
||||
|
||||
|
||||
## Matlab Scripts ([link]({{< relref "matlab/index.md" >}})) {#matlab-scripts--link-matlab-index-dot-md}
|
||||
|
||||
The Matlab scripts that permits to obtain all the results presented in the paper are accessible [here]({{< relref "matlab/index.md" >}}).
|
||||
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@@ -0,0 +1,959 @@
|
||||
+++
|
||||
title = "Active Damping of Rotating Platforms using Integral Force Feedback - Matlab Computation"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
<hr>
|
||||
<p>This report is also available as a <a href="./index.pdf">pdf</a>.</p>
|
||||
<hr>
|
||||
|
||||
This document gathers the Matlab code used to for the conference paper (<a href="#citeproc_bib_item_1">Dehaeze and Collette 2020</a>) and the journal paper (<a href="#citeproc_bib_item_3">Dehaeze and Collette 2021</a>).
|
||||
|
||||
It is structured in several sections:
|
||||
|
||||
- Section : presents a simple model of a rotating suspended platform that will be used throughout this study.
|
||||
- Section : explains how the unconditional stability of IFF is lost due to Gyroscopic effects induced by the rotation.
|
||||
- Section : suggests a simple modification of the control law such that damping can be added to the suspension modes in a robust way.
|
||||
- Section : proposes to add springs in parallel with the force sensors to regain the unconditional stability of IFF.
|
||||
- Section : compares both proposed modifications to the classical IFF in terms of damping authority and closed-loop system behavior.
|
||||
- Section : contains the notations used for both the Matlab code and the paper
|
||||
|
||||
The matlab code is accessible on [Zonodo](https://zenodo.org/record/3894343) and [Github](https://github.com/tdehaeze/dehaeze20_contr_stewa_platf) (<a href="#citeproc_bib_item_2">Dehaeze 2020</a>). It can also be download as a `.zip` file [here](https://git.tdehaeze.xyz/tdehaeze/dehaeze21_activ_dampin_rotat_platf_using/archive/master.zip).
|
||||
|
||||
To run the Matlab code, go in the `matlab` directory and run the following Matlab files corresponding to each section.
|
||||
|
||||
<div class="table-caption">
|
||||
<span class="table-number">Table 1:</span>
|
||||
Paper's sections and corresponding Matlab files
|
||||
</div>
|
||||
|
||||
| Sections | Matlab File |
|
||||
|----------|----------------------------|
|
||||
| Section | `s1_system_description.m` |
|
||||
| Section | `s2_iff_pure_int.m` |
|
||||
| Section | `s3_iff_hpf.m` |
|
||||
| Section | `s4_iff_kp.m` |
|
||||
| Section | `s5_act_damp_comparison.m` |
|
||||
|
||||
|
||||
## System Description and Analysis {#system-description-and-analysis}
|
||||
|
||||
<span class="org-target" id="org-target--sec-system-description"></span>
|
||||
|
||||
|
||||
### System description {#system-description}
|
||||
|
||||
The system consists of one 2 degree of freedom translation stage on top of a spindle (figure [Figure 1](#figure--fig:system)).
|
||||
|
||||
<a id="figure--fig:system"></a>
|
||||
|
||||
{{< figure src="figs-paper/system.png" caption="<span class='figure-number'>Figure 1: </span>Schematic of the studied system" >}}
|
||||
|
||||
The control inputs are the forces applied by the actuators of the translation stage (\\(F\_u\\) and \\(F\_v\\)).
|
||||
As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along \\(\vec{i}\_u\\) and \\(\vec{i}\_v\\).
|
||||
|
||||
|
||||
### Equations {#equations}
|
||||
|
||||
Based on the Figure [Figure 1](#figure--fig:system), the equations of motions are:
|
||||
|
||||
<div class="important">
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} d\_u \\\ d\_v \end{bmatrix} =
|
||||
\bm{G}\_d
|
||||
\begin{bmatrix} F\_u \\\ F\_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
Where \\(\bm{G}\_d\\) is a \\(2 \times 2\\) transfer function matrix.
|
||||
|
||||
\begin{equation}
|
||||
\bm{G}\_d = \frac{1}{k} \frac{1}{G\_{dp}}
|
||||
\begin{bmatrix}
|
||||
G\_{dz} & G\_{dc} \\\\
|
||||
-G\_{dc} & G\_{dz}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
With:
|
||||
|
||||
\begin{align}
|
||||
G\_{dp} &= \left( \frac{s^2}{{\omega\_0}^2} + 2 \xi \frac{s}{\omega\_0} + 1 - \frac{{\Omega}^2}{{\omega\_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega\_0} \frac{s}{\omega\_0} \right)^2 \\\\
|
||||
G\_{dz} &= \frac{s^2}{{\omega\_0}^2} + 2 \xi \frac{s}{\omega\_0} + 1 - \frac{{\Omega}^2}{{\omega\_0}^2} \\\\
|
||||
G\_{dc} &= 2 \frac{\Omega}{\omega\_0} \frac{s}{\omega\_0}
|
||||
\end{align}
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
### Numerical Values {#numerical-values}
|
||||
|
||||
Let's define initial values for the model.
|
||||
|
||||
```matlab
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
```
|
||||
|
||||
```matlab
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
```
|
||||
|
||||
|
||||
### Campbell Diagram {#campbell-diagram}
|
||||
|
||||
The Campbell Diagram displays the evolution of the real and imaginary parts of the system as a function of the rotating speed.
|
||||
|
||||
It is shown in Figures [Figure 2](#figure--fig:campbell-diagram-real) and [Figure 3](#figure--fig:campbell-diagram-imag), and one can see that the system becomes unstable for \\(\Omega > \omega\_0\\) (the real part of one of the poles becomes positive).
|
||||
|
||||
<a id="figure--fig:campbell-diagram-real"></a>
|
||||
|
||||
{{< figure src="figs/campbell_diagram_real.png" caption="<span class='figure-number'>Figure 2: </span>Campbell Diagram - Real Part" >}}
|
||||
|
||||
<a id="figure--fig:campbell-diagram-imag"></a>
|
||||
|
||||
{{< figure src="figs/campbell_diagram_imag.png" caption="<span class='figure-number'>Figure 3: </span>Campbell Diagram - Imaginary Part" >}}
|
||||
|
||||
|
||||
### Simscape Model {#simscape-model}
|
||||
|
||||
In order to validate all the equations of motion, a Simscape model of the same system has been developed.
|
||||
The dynamics of the system can be identified from the Simscape model and compare with the analytical model.
|
||||
|
||||
The rotating speed for the Simscape Model is defined.
|
||||
|
||||
```matlab
|
||||
W = 0.1; % Rotation Speed [rad/s]
|
||||
```
|
||||
|
||||
```matlab
|
||||
open('rotating_frame.slx');
|
||||
```
|
||||
|
||||
The transfer function from \\([F\_u, F\_v]\\) to \\([d\_u, d\_v]\\) is identified from the Simscape model.
|
||||
|
||||
```matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/G'], 2, 'openoutput'); io_i = io_i + 1;
|
||||
```
|
||||
|
||||
```matlab
|
||||
G = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G.InputName = {'Fu', 'Fv'};
|
||||
G.OutputName = {'du', 'dv'};
|
||||
```
|
||||
|
||||
The same transfer function from \\([F\_u, F\_v]\\) to \\([d\_u, d\_v]\\) is written down from the analytical model.
|
||||
|
||||
```matlab
|
||||
Gth = (1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
|
||||
-2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)];
|
||||
```
|
||||
|
||||
Both transfer functions are compared in Figure [Figure 4](#figure--fig:plant-simscape-analytical) and are found to perfectly match.
|
||||
|
||||
<a id="figure--fig:plant-simscape-analytical"></a>
|
||||
|
||||
{{< figure src="figs/plant_simscape_analytical.png" caption="<span class='figure-number'>Figure 4: </span>Bode plot of the transfer function from \\([F\_u, F\_v]\\) to \\([d\_u, d\_v]\\) as identified from the Simscape model and from an analytical model" >}}
|
||||
|
||||
|
||||
### Effect of the rotation speed {#effect-of-the-rotation-speed}
|
||||
|
||||
The transfer functions from \\([F\_u, F\_v]\\) to \\([d\_u, d\_v]\\) are identified for the following rotating speeds.
|
||||
|
||||
```matlab
|
||||
Ws = [0, 0.2, 0.7, 1.1]*w0; % Rotating Speeds [rad/s]
|
||||
```
|
||||
|
||||
```matlab
|
||||
Gs = {zeros(2, 2, length(Ws))};
|
||||
|
||||
for W_i = 1:length(Ws)
|
||||
W = Ws(W_i);
|
||||
|
||||
Gs(:, :, W_i) = {(1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
|
||||
-2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)]};
|
||||
end
|
||||
```
|
||||
|
||||
They are compared in Figures [Figure 5](#figure--fig:plant-compare-rotating-speed-direct) and [Figure 6](#figure--fig:plant-compare-rotating-speed-coupling).
|
||||
|
||||
<a id="figure--fig:plant-compare-rotating-speed-direct"></a>
|
||||
|
||||
{{< figure src="figs/plant_compare_rotating_speed_direct.png" caption="<span class='figure-number'>Figure 5: </span>Comparison of the transfer functions from \\([F\_u, F\_v]\\) to \\([d\_u, d\_v]\\) for several rotating speed - Direct Terms" >}}
|
||||
|
||||
<a id="figure--fig:plant-compare-rotating-speed-coupling"></a>
|
||||
|
||||
{{< figure src="figs/plant_compare_rotating_speed_coupling.png" caption="<span class='figure-number'>Figure 6: </span>Comparison of the transfer functions from \\([F\_u, F\_v]\\) to \\([d\_u, d\_v]\\) for several rotating speed - Coupling Terms" >}}
|
||||
|
||||
|
||||
## Problem with pure Integral Force Feedback {#problem-with-pure-integral-force-feedback}
|
||||
|
||||
<span class="org-target" id="org-target--sec-iff-pure-int"></span>
|
||||
|
||||
Force sensors are added in series with the two actuators (Figure [Figure 7](#figure--fig:system-iff)).
|
||||
|
||||
Two identical controllers \\(K\_F\\) are used to feedback each of the sensed force to its associated actuator.
|
||||
|
||||
<a id="figure--fig:system-iff"></a>
|
||||
|
||||
{{< figure src="figs-paper/system_iff.png" caption="<span class='figure-number'>Figure 7: </span>System with added Force Sensor in series with the actuators" >}}
|
||||
|
||||
|
||||
### Plant Parameters {#plant-parameters}
|
||||
|
||||
Let's define initial values for the model.
|
||||
|
||||
```matlab
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
```
|
||||
|
||||
```matlab
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
```
|
||||
|
||||
|
||||
### Equations {#equations}
|
||||
|
||||
The sensed forces are equal to:
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} f\_{u} \\\ f\_{v} \end{bmatrix} =
|
||||
\begin{bmatrix}
|
||||
1 & 0 \\\\
|
||||
0 & 1
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} F\_u \\\ F\_v \end{bmatrix} - (c s + k)
|
||||
\begin{bmatrix} d\_u \\\ d\_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
Which then gives:
|
||||
|
||||
<div class="important">
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} f\_{u} \\\ f\_{v} \end{bmatrix} =
|
||||
\bm{G}\_{f}
|
||||
\begin{bmatrix} F\_u \\\ F\_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} f\_{u} \\\ f\_{v} \end{bmatrix} =
|
||||
\frac{1}{G\_{fp}}
|
||||
\begin{bmatrix}
|
||||
G\_{fz} & -G\_{fc} \\\\
|
||||
G\_{fc} & G\_{fz}
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} F\_u \\\ F\_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\begin{align}
|
||||
G\_{fp} &= \left( \frac{s^2}{{\omega\_0}^2} + 2 \xi \frac{s}{\omega\_0} + 1 - \frac{{\Omega}^2}{{\omega\_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega\_0} \frac{s}{\omega\_0} \right)^2 \\\\
|
||||
G\_{fz} &= \left( \frac{s^2}{{\omega\_0}^2} - \frac{\Omega^2}{{\omega\_0}^2} \right) \left( \frac{s^2}{{\omega\_0}^2} + 2 \xi \frac{s}{\omega\_0} + 1 - \frac{{\Omega}^2}{{\omega\_0}^2} \right) + \left( 2 \frac{\Omega}{\omega\_0} \frac{s}{\omega\_0} \right)^2 \\\\
|
||||
G\_{fc} &= \left( 2 \xi \frac{s}{\omega\_0} + 1 \right) \left( 2 \frac{\Omega}{\omega\_0} \frac{s}{\omega\_0} \right)
|
||||
\end{align}
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
### Comparison of the Analytical Model and the Simscape Model {#comparison-of-the-analytical-model-and-the-simscape-model}
|
||||
|
||||
The rotation speed is set to \\(\Omega = 0.1 \omega\_0\\).
|
||||
|
||||
```matlab
|
||||
W = 0.1*w0; % [rad/s]
|
||||
```
|
||||
|
||||
```matlab
|
||||
open('rotating_frame.slx');
|
||||
```
|
||||
|
||||
And the transfer function from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) is identified using the Simscape model.
|
||||
|
||||
```matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
```
|
||||
|
||||
```matlab
|
||||
Giff = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
Giff.InputName = {'Fu', 'Fv'};
|
||||
Giff.OutputName = {'fu', 'fv'};
|
||||
```
|
||||
|
||||
The same transfer function from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) is written down from the analytical model.
|
||||
|
||||
```matlab
|
||||
Giff_th = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];
|
||||
```
|
||||
|
||||
The two are compared in Figure [Figure 8](#figure--fig:plant-iff-comp-simscape-analytical) and found to perfectly match.
|
||||
|
||||
<a id="figure--fig:plant-iff-comp-simscape-analytical"></a>
|
||||
|
||||
{{< figure src="figs/plant_iff_comp_simscape_analytical.png" caption="<span class='figure-number'>Figure 8: </span>Comparison of the transfer functions from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) between the Simscape model and the analytical one" >}}
|
||||
|
||||
|
||||
### Effect of the rotation speed {#effect-of-the-rotation-speed}
|
||||
|
||||
The transfer functions from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) are identified for the following rotating speeds.
|
||||
|
||||
```matlab
|
||||
Ws = [0, 0.2, 0.7]*w0; % Rotating Speeds [rad/s]
|
||||
```
|
||||
|
||||
```matlab
|
||||
Gsiff = {zeros(2, 2, length(Ws))};
|
||||
|
||||
for W_i = 1:length(Ws)
|
||||
W = Ws(W_i);
|
||||
|
||||
Gsiff(:, :, W_i) = {1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2]};
|
||||
end
|
||||
```
|
||||
|
||||
The obtained transfer functions are shown in Figure [Figure 9](#figure--fig:plant-iff-compare-rotating-speed).
|
||||
|
||||
<a id="figure--fig:plant-iff-compare-rotating-speed"></a>
|
||||
|
||||
{{< figure src="figs/plant_iff_compare_rotating_speed.png" caption="<span class='figure-number'>Figure 9: </span>Comparison of the transfer functions from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) for several rotating speed" >}}
|
||||
|
||||
|
||||
### Decentralized Integral Force Feedback {#decentralized-integral-force-feedback}
|
||||
|
||||
The decentralized IFF controller consists of pure integrators:
|
||||
|
||||
\begin{equation}
|
||||
\bm{K}\_{\text{IFF}}(s) = \frac{g}{s} \begin{bmatrix}
|
||||
1 & 0 \\\\
|
||||
0 & 1
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
The Root Locus (evolution of the poles of the closed loop system in the complex plane as a function of \\(g\\)) is shown in Figure [Figure 10](#figure--fig:root-locus-pure-iff).
|
||||
It is shown that for non-null rotating speed, one pole is bound to the right-half plane, and thus the closed loop system is unstable.
|
||||
|
||||
<a id="figure--fig:root-locus-pure-iff"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_pure_iff.png" caption="<span class='figure-number'>Figure 10: </span>Root Locus for the Decentralized Integral Force Feedback controller. Several rotating speed are shown." >}}
|
||||
|
||||
|
||||
## Integral Force Feedback with an High Pass Filter {#integral-force-feedback-with-an-high-pass-filter}
|
||||
|
||||
<span class="org-target" id="org-target--sec-iff-pseudo-int"></span>
|
||||
|
||||
|
||||
### Plant Parameters {#plant-parameters}
|
||||
|
||||
Let's define initial values for the model.
|
||||
|
||||
```matlab
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
```
|
||||
|
||||
```matlab
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
```
|
||||
|
||||
|
||||
### Modified Integral Force Feedback Controller {#modified-integral-force-feedback-controller}
|
||||
|
||||
Let's modify the initial Integral Force Feedback Controller ; instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used:
|
||||
|
||||
\begin{equation}
|
||||
K\_{\text{IFF}}(s) = g\frac{1}{\omega\_i + s} \begin{bmatrix}
|
||||
1 & 0 \\\\
|
||||
0 & 1
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
where \\(\omega\_i\\) characterize down to which frequency the signal is integrated.
|
||||
|
||||
Let's arbitrary choose the following control parameters:
|
||||
|
||||
```matlab
|
||||
g = 2;
|
||||
wi = 0.1*w0;
|
||||
```
|
||||
|
||||
And the following rotating speed.
|
||||
|
||||
```matlab
|
||||
Giff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];
|
||||
```
|
||||
|
||||
The obtained Loop Gain is shown in Figure [Figure 11](#figure--fig:loop-gain-modified-iff).
|
||||
|
||||
<a id="figure--fig:loop-gain-modified-iff"></a>
|
||||
|
||||
{{< figure src="figs/loop_gain_modified_iff.png" caption="<span class='figure-number'>Figure 11: </span>Loop Gain for the modified IFF controller" >}}
|
||||
|
||||
|
||||
### Root Locus {#root-locus}
|
||||
|
||||
As shown in the Root Locus plot (Figure [Figure 12](#figure--fig:root-locus-modified-iff)), for some value of the gain, the system remains stable.
|
||||
|
||||
<a id="figure--fig:root-locus-modified-iff"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_modified_iff.png" caption="<span class='figure-number'>Figure 12: </span>Root Locus for the modified IFF controller" >}}
|
||||
|
||||
<a id="figure--fig:root-locus-modified-iff-zoom"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_modified_iff_zoom.png" caption="<span class='figure-number'>Figure 13: </span>Root Locus for the modified IFF controller - Zoom" >}}
|
||||
|
||||
|
||||
### What is the optimal \\(\omega\_i\\) and \\(g\\)? {#what-is-the-optimal-omega-i-and-g}
|
||||
|
||||
In order to visualize the effect of \\(\omega\_i\\) on the attainable damping, the Root Locus is displayed in Figure [Figure 14](#figure--fig:root-locus-wi-modified-iff) for the following \\(\omega\_i\\):
|
||||
|
||||
```matlab
|
||||
wis = [0.01, 0.1, 0.5, 1]*w0; % [rad/s]
|
||||
```
|
||||
|
||||
<a id="figure--fig:root-locus-wi-modified-iff"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_wi_modified_iff.png" caption="<span class='figure-number'>Figure 14: </span>Root Locus for the modified IFF controller (zoomed plot on the left)" >}}
|
||||
|
||||
<a id="figure--fig:root-locus-wi-modified-iff-zoom"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_wi_modified_iff_zoom.png" caption="<span class='figure-number'>Figure 15: </span>Root Locus for the modified IFF controller (zoomed plot on the left)" >}}
|
||||
|
||||
For the controller
|
||||
|
||||
\begin{equation}
|
||||
K\_{\text{IFF}}(s) = g\frac{1}{\omega\_i + s} \begin{bmatrix}
|
||||
1 & 0 \\\\
|
||||
0 & 1
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
The gain at which the system becomes unstable is
|
||||
|
||||
\begin{equation}
|
||||
g\_\text{max} = \omega\_i \left( \frac{{\omega\_0}^2}{\Omega^2} - 1 \right) \label{eq:iff\_gmax}
|
||||
\end{equation}
|
||||
|
||||
While it seems that small \\(\omega\_i\\) do allow more damping to be added to the system (Figure [Figure 14](#figure--fig:root-locus-wi-modified-iff)), the control gains may be limited to small values due to \ref{eq:iff\_gmax} thus reducing the attainable damping.
|
||||
|
||||
There must be an optimum for \\(\omega\_i\\).
|
||||
To find the optimum, the gain that maximize the simultaneous damping of the mode is identified for a wide range of \\(\omega\_i\\) (Figure [Figure 16](#figure--fig:mod-iff-damping-wi)).
|
||||
|
||||
```matlab
|
||||
wis = logspace(-2, 1, 100)*w0; % [rad/s]
|
||||
|
||||
opt_xi = zeros(1, length(wis)); % Optimal simultaneous damping
|
||||
opt_gain = zeros(1, length(wis)); % Corresponding optimal gain
|
||||
|
||||
for wi_i = 1:length(wis)
|
||||
wi = wis(wi_i);
|
||||
Kiff = 1/(s + wi)*eye(2);
|
||||
|
||||
fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);
|
||||
|
||||
[g_opt, xi_opt] = fminsearch(fun, 0.5*wi*((w0/W)^2 - 1));
|
||||
opt_xi(wi_i) = 1/xi_opt;
|
||||
opt_gain(wi_i) = g_opt;
|
||||
end
|
||||
```
|
||||
|
||||
<a id="figure--fig:mod-iff-damping-wi"></a>
|
||||
|
||||
{{< figure src="figs/mod_iff_damping_wi.png" caption="<span class='figure-number'>Figure 16: </span>Simultaneous attainable damping of the closed loop poles as a function of \\(\omega\_i\\)" >}}
|
||||
|
||||
|
||||
## IFF with a stiffness in parallel with the force sensor {#iff-with-a-stiffness-in-parallel-with-the-force-sensor}
|
||||
|
||||
<span class="org-target" id="org-target--sec-iff-parallel-stiffness"></span>
|
||||
|
||||
|
||||
### Schematic {#schematic}
|
||||
|
||||
In this section additional springs in parallel with the force sensors are added to counteract the negative stiffness induced by the rotation.
|
||||
|
||||
<a id="figure--fig:system-parallel-springs"></a>
|
||||
|
||||
{{< figure src="figs-paper/system_parallel_springs.png" caption="<span class='figure-number'>Figure 17: </span>Studied system with additional springs in parallel with the actuators and force sensors" >}}
|
||||
|
||||
In order to keep the overall stiffness \\(k = k\_a + k\_p\\) constant, a scalar parameter \\(\alpha\\) (\\(0 \le \alpha < 1\\)) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor
|
||||
|
||||
\begin{equation}
|
||||
k\_p = \alpha k, \quad k\_a = (1 - \alpha) k
|
||||
\end{equation}
|
||||
|
||||
|
||||
### Equations {#equations}
|
||||
|
||||
<div class="important">
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} f\_u \\\ f\_v \end{bmatrix} =
|
||||
\bm{G}\_k
|
||||
\begin{bmatrix} F\_u \\\ F\_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} f\_u \\\ f\_v \end{bmatrix} =
|
||||
\frac{1}{G\_{kp}}
|
||||
\begin{bmatrix}
|
||||
G\_{kz} & -G\_{kc} \\\\
|
||||
G\_{kc} & G\_{kz}
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} F\_u \\\ F\_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
With:
|
||||
|
||||
\begin{align}
|
||||
G\_{kp} &= \left( \frac{s^2}{{\omega\_0}^2} + 2\xi \frac{s}{{\omega\_0}^2} + 1 - \frac{\Omega^2}{{\omega\_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega\_0}\frac{s}{\omega\_0} \right)^2 \\\\
|
||||
G\_{kz} &= \left( \frac{s^2}{{\omega\_0}^2} - \frac{\Omega^2}{{\omega\_0}^2} + \alpha \right) \left( \frac{s^2}{{\omega\_0}^2} + 2\xi \frac{s}{{\omega\_0}^2} + 1 - \frac{\Omega^2}{{\omega\_0}^2} \right) + \left( 2 \frac{\Omega}{\omega\_0}\frac{s}{\omega\_0} \right)^2 \\\\
|
||||
G\_{kc} &= \left( 2 \xi \frac{s}{\omega\_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega\_0}\frac{s}{\omega\_0} \right)
|
||||
\end{align}
|
||||
|
||||
</div>
|
||||
|
||||
If we compare \\(G\_{kz}\\) and \\(G\_{fz}\\), we see that the spring in parallel adds a term \\(\alpha\\).
|
||||
In order to have two complex conjugate zeros (instead of real zeros):
|
||||
|
||||
\begin{equation}
|
||||
\alpha > \frac{\Omega^2}{{\omega\_0}^2} \quad \Leftrightarrow \quad k\_p > m \Omega^2
|
||||
\end{equation}
|
||||
|
||||
|
||||
### Plant Parameters {#plant-parameters}
|
||||
|
||||
Let's define initial values for the model.
|
||||
|
||||
```matlab
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
```
|
||||
|
||||
```matlab
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
```
|
||||
|
||||
|
||||
### Comparison of the Analytical Model and the Simscape Model {#comparison-of-the-analytical-model-and-the-simscape-model}
|
||||
|
||||
The same transfer function from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) is written down from the analytical model.
|
||||
|
||||
```matlab
|
||||
W = 0.1*w0; % [rad/s]
|
||||
|
||||
kp = 1.5*m*W^2;
|
||||
cp = 0;
|
||||
```
|
||||
|
||||
```matlab
|
||||
open('rotating_frame.slx');
|
||||
```
|
||||
|
||||
```matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
Giff = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
Giff.InputName = {'Fu', 'Fv'};
|
||||
Giff.OutputName = {'fu', 'fv'};
|
||||
```
|
||||
|
||||
```matlab
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff_th = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2 ];
|
||||
Giff_th.InputName = {'Fu', 'Fv'};
|
||||
Giff_th.OutputName = {'fu', 'fv'};
|
||||
```
|
||||
|
||||
<a id="figure--fig:plant-iff-kp-comp-simscape-analytical"></a>
|
||||
|
||||
{{< figure src="figs/plant_iff_kp_comp_simscape_analytical.png" caption="<span class='figure-number'>Figure 18: </span>Comparison of the transfer functions from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) between the Simscape model and the analytical one" >}}
|
||||
|
||||
|
||||
### Effect of the parallel stiffness on the IFF plant {#effect-of-the-parallel-stiffness-on-the-iff-plant}
|
||||
|
||||
The rotation speed is set to \\(\Omega = 0.1 \omega\_0\\).
|
||||
|
||||
```matlab
|
||||
W = 0.1*w0; % [rad/s]
|
||||
```
|
||||
|
||||
And the IFF plant (transfer function from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\)) is identified in three different cases:
|
||||
|
||||
- without parallel stiffness
|
||||
- with a small parallel stiffness \\(k\_p < m \Omega^2\\)
|
||||
- with a large parallel stiffness \\(k\_p > m \Omega^2\\)
|
||||
|
||||
The results are shown in Figure [Figure 19](#figure--fig:plant-iff-kp).
|
||||
|
||||
One can see that for \\(k\_p > m \Omega^2\\), the systems shows alternating complex conjugate poles and zeros.
|
||||
|
||||
```matlab
|
||||
kp = 0;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
```
|
||||
|
||||
```matlab
|
||||
kp = 0.5*m*W^2;
|
||||
k = 1 - kp;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff_s = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
```
|
||||
|
||||
```matlab
|
||||
kp = 1.5*m*W^2;
|
||||
k = 1 - kp;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff_l = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
```
|
||||
|
||||
<a id="figure--fig:plant-iff-kp"></a>
|
||||
|
||||
{{< figure src="figs/plant_iff_kp.png" caption="<span class='figure-number'>Figure 19: </span>Transfer function from \\([F\_u, F\_v]\\) to \\([f\_u, f\_v]\\) for \\(k\_p = 0\\), \\(k\_p < m \Omega^2\\) and \\(k\_p > m \Omega^2\\)" >}}
|
||||
|
||||
|
||||
### IFF when adding a spring in parallel {#iff-when-adding-a-spring-in-parallel}
|
||||
|
||||
In Figure [Figure 20](#figure--fig:root-locus-iff-kp) is displayed the Root Locus in the three considered cases with
|
||||
|
||||
\begin{equation}
|
||||
K\_{\text{IFF}} = \frac{g}{s} \begin{bmatrix}
|
||||
1 & 0 \\\\
|
||||
0 & 1
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
One can see that for \\(k\_p > m \Omega^2\\), the root locus stays in the left half of the complex plane and thus the control system is unconditionally stable.
|
||||
|
||||
Thus, decentralized IFF controller with pure integrators can be used if:
|
||||
|
||||
\begin{equation}
|
||||
k\_{p} > m \Omega^2
|
||||
\end{equation}
|
||||
|
||||
<a id="figure--fig:root-locus-iff-kp"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_iff_kp.png" caption="<span class='figure-number'>Figure 20: </span>Root Locus" >}}
|
||||
|
||||
<a id="figure--fig:root-locus-iff-kp-zoom"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_iff_kp_zoom.png" caption="<span class='figure-number'>Figure 21: </span>Root Locus" >}}
|
||||
|
||||
|
||||
### Effect of \\(k\_p\\) on the attainable damping {#effect-of-k-p-on-the-attainable-damping}
|
||||
|
||||
However, having large values of \\(k\_p\\) may decrease the attainable damping.
|
||||
|
||||
To study the second point, Root Locus plots for the following values of \\(k\_p\\) are shown in Figure [Figure 22](#figure--fig:root-locus-iff-kps).
|
||||
|
||||
```matlab
|
||||
kps = [2, 20, 40]*m*W^2;
|
||||
```
|
||||
|
||||
It is shown that large values of \\(k\_p\\) decreases the attainable damping.
|
||||
|
||||
<a id="figure--fig:root-locus-iff-kps"></a>
|
||||
|
||||
{{< figure src="figs/root_locus_iff_kps.png" caption="<span class='figure-number'>Figure 22: </span>Root Locus plot" >}}
|
||||
|
||||
```matlab
|
||||
alphas = logspace(-2, 0, 100);
|
||||
|
||||
opt_xi = zeros(1, length(alphas)); % Optimal simultaneous damping
|
||||
opt_gain = zeros(1, length(alphas)); % Corresponding optimal gain
|
||||
|
||||
Kiff = 1/s*eye(2);
|
||||
|
||||
for alpha_i = 1:length(alphas)
|
||||
kp = alphas(alpha_i);
|
||||
k = 1 - alphas(alpha_i);
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
|
||||
fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);
|
||||
|
||||
[g_opt, xi_opt] = fminsearch(fun, 2);
|
||||
opt_xi(alpha_i) = 1/xi_opt;
|
||||
opt_gain(alpha_i) = g_opt;
|
||||
end
|
||||
```
|
||||
|
||||
<a id="figure--fig:opt-damp-alpha"></a>
|
||||
|
||||
{{< figure src="figs/opt_damp_alpha.png" caption="<span class='figure-number'>Figure 23: </span>Attainable damping ratio and corresponding controller gain for different parameter \\(\alpha\\)" >}}
|
||||
|
||||
|
||||
## Comparison {#comparison}
|
||||
|
||||
<span class="org-target" id="org-target--sec-comparison"></span>
|
||||
|
||||
Two modifications to adapt the IFF control strategy to rotating platforms have been proposed.
|
||||
These two methods are now compared in terms of added damping, closed-loop compliance and transmissibility.
|
||||
|
||||
|
||||
### Plant Parameters {#plant-parameters}
|
||||
|
||||
Let's define initial values for the model.
|
||||
|
||||
```matlab
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
```
|
||||
|
||||
```matlab
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
```
|
||||
|
||||
The rotating speed is set to \\(\Omega = 0.1 \omega\_0\\).
|
||||
|
||||
```matlab
|
||||
W = 0.1*w0;
|
||||
```
|
||||
|
||||
|
||||
### Root Locus {#root-locus}
|
||||
|
||||
IFF with High Pass Filter
|
||||
|
||||
```matlab
|
||||
wi = 0.1*w0; % [rad/s]
|
||||
|
||||
Giff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];
|
||||
```
|
||||
|
||||
IFF With parallel Stiffness
|
||||
|
||||
```matlab
|
||||
kp = 5*m*W^2;
|
||||
k = k - kp;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff_kp = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2 ];
|
||||
|
||||
k = k + kp;
|
||||
```
|
||||
|
||||
<a id="figure--fig:comp-root-locus"></a>
|
||||
|
||||
{{< figure src="figs/comp_root_locus.png" caption="<span class='figure-number'>Figure 24: </span>Root Locus plot - Comparison of IFF with additional high pass filter, IFF with additional parallel stiffness" >}}
|
||||
|
||||
|
||||
### Controllers - Optimal Gains {#controllers-optimal-gains}
|
||||
|
||||
In order to compare to three considered Active Damping techniques, gains that yield maximum damping of all the modes are computed for each case.
|
||||
|
||||
The obtained damping ratio and control are shown below.
|
||||
|
||||
| | Obtained \\(\xi\\) | Control Gain |
|
||||
|---------------------|--------------------|--------------|
|
||||
| Modified IFF | 0.83 | 1.99 |
|
||||
| IFF with \\(k\_p\\) | 0.83 | 2.02 |
|
||||
|
||||
|
||||
### Passive Damping - Critical Damping {#passive-damping-critical-damping}
|
||||
|
||||
\begin{equation}
|
||||
\xi = \frac{c}{2 \sqrt{km}}
|
||||
\end{equation}
|
||||
|
||||
Critical Damping corresponds to to \\(\xi = 1\\), and thus:
|
||||
|
||||
\begin{equation}
|
||||
c\_{\text{crit}} = 2 \sqrt{km}
|
||||
\end{equation}
|
||||
|
||||
```matlab
|
||||
c_opt = 2*sqrt(k*m);
|
||||
```
|
||||
|
||||
|
||||
### Transmissibility And Compliance {#transmissibility-and-compliance}
|
||||
|
||||
<span class="org-target" id="org-target--sec-comp-transmissibilty"></span>
|
||||
|
||||
```matlab
|
||||
open('rotating_frame.slx');
|
||||
```
|
||||
|
||||
```matlab
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/dw'], 1, 'input'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/fd'], 1, 'input'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Meas'], 1, 'output'); io_i = io_i + 1;
|
||||
```
|
||||
|
||||
```matlab
|
||||
G_ol = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G_ol.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_ol.OutputName = {'Dx', 'Dy'};
|
||||
```
|
||||
|
||||
|
||||
#### Passive Damping {#passive-damping}
|
||||
|
||||
```matlab
|
||||
kp = 0;
|
||||
cp = 0;
|
||||
```
|
||||
|
||||
```matlab
|
||||
c_old = c;
|
||||
c = c_opt;
|
||||
```
|
||||
|
||||
```matlab
|
||||
G_pas = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G_pas.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_pas.OutputName = {'Dx', 'Dy'};
|
||||
```
|
||||
|
||||
```matlab
|
||||
c = c_old;
|
||||
```
|
||||
|
||||
```matlab
|
||||
Kiff = opt_gain_iff/(wi + s)*tf(eye(2));
|
||||
```
|
||||
|
||||
```matlab
|
||||
G_iff = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G_iff.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_iff.OutputName = {'Dx', 'Dy'};
|
||||
```
|
||||
|
||||
```matlab
|
||||
kp = 5*m*W^2;
|
||||
cp = 0.01;
|
||||
```
|
||||
|
||||
```matlab
|
||||
Kiff = opt_gain_kp/s*tf(eye(2));
|
||||
```
|
||||
|
||||
```matlab
|
||||
G_kp = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G_kp.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_kp.OutputName = {'Dx', 'Dy'};
|
||||
```
|
||||
|
||||
<a id="figure--fig:comp-transmissibility"></a>
|
||||
|
||||
{{< figure src="figs/comp_transmissibility.png" caption="<span class='figure-number'>Figure 25: </span>Comparison of the transmissibility" >}}
|
||||
|
||||
<a id="figure--fig:comp-compliance"></a>
|
||||
|
||||
{{< figure src="figs/comp_compliance.png" caption="<span class='figure-number'>Figure 26: </span>Comparison of the obtained Compliance" >}}
|
||||
|
||||
|
||||
## Notations {#notations}
|
||||
|
||||
<span class="org-target" id="org-target--sec-notations"></span>
|
||||
|
||||
| | Mathematical Notation | Matlab | Unit |
|
||||
|---------------------------------------|----------------------------------|---------------|---------|
|
||||
| Actuator Stiffness | \\(k\\) | `k` | N/m |
|
||||
| Actuator Damping | \\(c\\) | `c` | N/(m/s) |
|
||||
| Payload Mass | \\(m\\) | `m` | kg |
|
||||
| Damping Ratio | \\(\xi = \frac{c}{2\sqrt{km}}\\) | `xi` | |
|
||||
| Actuator Force | \\(\bm{F}, F\_u, F\_v\\) | `F` `Fu` `Fv` | N |
|
||||
| Force Sensor signal | \\(\bm{f}, f\_u, f\_v\\) | `f` `fu` `fv` | N |
|
||||
| Relative Displacement | \\(\bm{d}, d\_u, d\_v\\) | `d` `du` `dv` | m |
|
||||
| Resonance freq. when \\(\Omega = 0\\) | \\(\omega\_0\\) | `w0` | rad/s |
|
||||
| Rotation Speed | \\(\Omega = \dot{\theta}\\) | `W` | rad/s |
|
||||
| Low Pass Filter corner frequency | \\(\omega\_i\\) | `wi` | rad/s |
|
||||
|
||||
| | Mathematical Notation | Matlab | Unit |
|
||||
|------------------|-----------------------|--------|---------|
|
||||
| Laplace variable | \\(s\\) | `s` | |
|
||||
| Complex number | \\(j\\) | `j` | |
|
||||
| Frequency | \\(\omega\\) | `w` | [rad/s] |
|
||||
|
||||
<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>Dehaeze, T., and C. Collette. 2020. “Active Damping of Rotating Platforms Using Integral Force Feedback.” In <i>Proceedings of the International Conference on Modal Analysis Noise and Vibration Engineering (ISMA)</i>.</div>
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Dehaeze, Thomas. 2020. “Active Damping of Rotating Positioning Platforms.” Source Code on Zonodo. doi:<a href="https://doi.org/10.5281/zenodo.3894342">10.5281/zenodo.3894342</a>.</div>
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_3"></a>Dehaeze, Thomas, and Christophe Collette. 2021. “Active Damping of Rotating Platforms Using Integral Force Feedback.” <i>Engineering Research Express</i>. <a href="http://iopscience.iop.org/article/10.1088/2631-8695/abe803">http://iopscience.iop.org/article/10.1088/2631-8695/abe803</a>.</div>
|
||||
</div>
|
||||
@@ -1,30 +0,0 @@
|
||||
+++
|
||||
title = "Mechatronics Approach for the Development of a Nano-Active-Stabilization-System"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
> **Abstract**
|
||||
>
|
||||
> With the growing number of fourth generation light sources, there is an increased need of fast positioning end-stations with nanometric precision.
|
||||
> Such systems are usually including dedicated control strategies, and many factors may limit their performances.
|
||||
> In order to design such complex systems in a predictive way, a mechatronic design approach also known as "model based design", may be utilized.
|
||||
> In this paper, we present how this mechatronic design approach was used for the development of a nano-hexapod for the ESRF ID31 beamline.
|
||||
> The chosen design approach consists of using models of the mechatronic system (including sensors, actuators and control strategies) to predict its behavior.
|
||||
> Based on this behavior and closed-loop simulations, the elements that are limiting the performances can be identified and re-designed accordingly.
|
||||
> This allows to make adequate choices concerning the design of the nano-hexapod and the overall mechatronic architecture early in the project and save precious time and resources.
|
||||
> Several test benches were used to validate the models and to gain confidence on the predictability of the final system's performances.
|
||||
> Measured nano-hexapod's dynamics was shown to be in very good agreement with the models.
|
||||
> Further tests should be done in order to confirm that the performances of the system match the predicted one.
|
||||
> The presented development approach is foreseen to be applied more frequently to future mechatronic system design at the ESRF.
|
||||
|
||||
|
||||
## Conference Paper [pdf](/ox-hugo/dehaeze21_mechatronics_approach_nass.pdf) {#conference-paper-pdf--dehaeze21-mechatronics-approach-nass-dot-pdf}
|
||||
|
||||
|
||||
## Code {#code}
|
||||
|
||||
[nass-mechatronics on Gitea](https://git.tdehaeze.xyz/tdehaeze/nass-mechatronics)
|
||||
|
||||
|
||||
## References {#references}
|
||||
@@ -0,0 +1,65 @@
|
||||
+++
|
||||
title = "Mechatronics Approach for the Development of a Nano-Active-Stabilization-System"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
venue = "MEDSI 2020"
|
||||
year = 2021
|
||||
pubtype = "conference"
|
||||
doi = "10.18429/JACoW-MEDSI2020-TUIO02"
|
||||
code = "https://git.tdehaeze.xyz/tdehaeze/dehaeze21_mechatronics_approach_nass"
|
||||
video = "https://www.youtube.com/watch?v=kaplQJoqqDg"
|
||||
+++
|
||||
|
||||
> **Abstract**:
|
||||
>
|
||||
> With the growing number of fourth generation light sources, there is an increased need of fast positioning end-stations with nanometric precision.
|
||||
> Such systems are usually including dedicated control strategies, and many factors may limit their performances.
|
||||
> In order to design such complex systems in a predictive way, a mechatronic design approach also known as "model based design", may be utilized.
|
||||
> In this paper, we present how this mechatronic design approach was used for the development of a nano-hexapod for the ESRF ID31 beamline.
|
||||
> The chosen design approach consists of using models of the mechatronic system (including sensors, actuators and control strategies) to predict its behavior.
|
||||
> Based on this behavior and closed-loop simulations, the elements that are limiting the performances can be identified and re-designed accordingly.
|
||||
> This allows to make adequate choices concerning the design of the nano-hexapod and the overall mechatronic architecture early in the project and save precious time and resources.
|
||||
> Several test benches were used to validate the models and to gain confidence on the predictability of the final system's performances.
|
||||
> Measured nano-hexapod's dynamics was shown to be in very good agreement with the models.
|
||||
> Further tests should be done in order to confirm that the performances of the system match the predicted one.
|
||||
> The presented development approach is foreseen to be applied more frequently to future mechatronic system design at the ESRF.
|
||||
|
||||
|
||||
## Conference Paper ([pdf](paper/dehaeze21_mechatronics_approach_nass.pdf)) {#conference-paper--pdf-paper-dehaeze21-mechatronics-approach-nass-dot-pdf}
|
||||
|
||||
|
||||
## Talk ([link](talk/dehaeze21_mechatronics_approach_nass_talk.pdf)) {#talk--link-talk-dehaeze21-mechatronics-approach-nass-talk-dot-pdf}
|
||||
|
||||
<iframe width="720"
|
||||
height="540"
|
||||
src="https://www.youtube.com/embed/kaplQJoqqDg"
|
||||
frameborder="0" allowfullscreen> </iframe>
|
||||
|
||||
|
||||
## Figures ([link]({{< relref "tikz/_index.md" >}})) {#figures--link-tikz-figures-dot-md}
|
||||
|
||||
All the figures in the paper are generated using either [TikZ](https://sourceforge.net/projects/pgf/) or [Inkscape](https://inkscape.org/). The code snippets that was used to generate the figures are accessible [here]({{< relref "tikz/_index.md" >}}).
|
||||
|
||||
|
||||
## Cite this work {#cite-this-work}
|
||||
|
||||
To cite this conference paper use the following bibTeX code.
|
||||
|
||||
```bibtex
|
||||
@inproceedings{dehaeze21_mechat_approac_devel_nano_activ_stabil_system,
|
||||
author = {Dehaeze, T. and Bonnefoy, J. and Collette, C.},
|
||||
title = {Mechatronics Approach for the Development of a
|
||||
Nano-Active-Stabilization-System},
|
||||
booktitle = {MEDSI'20},
|
||||
year = 2021,
|
||||
language = {english},
|
||||
publisher = {JACoW Publishing},
|
||||
series = {Mechanical Engineering Design of Synchrotron Radiation
|
||||
Equipment and Instrumentation},
|
||||
venue = {Chicago, USA},
|
||||
}
|
||||
```
|
||||
|
||||
You can also use the formatted citation below.
|
||||
|
||||
> Dehaeze, T., Bonnefoy, J., & Collette, C., Mechatronics approach for the development of a nano-active-stabilization-system, In MEDSI'20 (2021), JACoW Publishing.
|
||||
BIN
Binary file not shown.
BIN
Binary file not shown.
@@ -0,0 +1,363 @@
|
||||
+++
|
||||
title = "Tikz Figures"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
## Mechatronic Approach {#mechatronic-approach}
|
||||
|
||||
```latex
|
||||
\graphicspath{ {/home/thomas/Cloud/thesis/papers/dehaeze21_mechatronics_approach_nass/tikz/figs-tikz} }
|
||||
|
||||
\begin{tikzpicture}
|
||||
% Styles
|
||||
\tikzset{myblock/.style= {draw, thin, color=white!70!black, fill=white, text width=3cm, align=center, minimum height=1.4cm}};
|
||||
\tikzset{mylabel/.style= {anchor=north, below, font=\bfseries\small, color=black, text width=3cm, align=center}};
|
||||
\tikzset{mymodel/.style= {anchor=south, above, font=\small, color=black, text width=3cm, align=center}};
|
||||
\tikzset{mystep/.style= {->, ultra thick}};
|
||||
|
||||
% Blocks
|
||||
\node[draw, fill=lightblue, align=center, label={[mylabel, text width=8.0cm] Dynamical Models}, minimum height = 4.5cm, text width = 8.0cm] (model) at (0, 0) {};
|
||||
|
||||
\node[myblock, fill=lightgreen, label={[mylabel] Disturbances}, left = 3 of model.west] (dist) {};
|
||||
\node[myblock, fill=lightgreen, label={[mylabel] $\mu$ Station}, below = 2pt of dist] (mustation) {};
|
||||
\node[myblock, fill=lightgreen, label={[mylabel] $\nu$ Hexapod}, above = 2pt of dist] (nanohexapod) {};
|
||||
|
||||
\node[myblock, fill=lightyellow, label={[mylabel] Mech. Design}, above = 1 of model.north] (mechanical) {};
|
||||
\node[myblock, fill=lightyellow, label={[mylabel] Instrumentation}, left = 2pt of mechanical] (instrumentation) {};
|
||||
\node[myblock, fill=lightyellow, label={[mylabel] FEM}, right = 2pt of mechanical] (fem) {};
|
||||
|
||||
\node[myblock, fill=lightred, label={[mylabel] Test Benches}, right = 3 of model.east] (testbenches) {};
|
||||
\node[myblock, fill=lightred, label={[mylabel] Assembly}, above = 2pt of testbenches] (mounting) {};
|
||||
\node[myblock, fill=lightred, label={[mylabel] Implementation}, below = 2pt of testbenches] (implementation) {};
|
||||
|
||||
% Text
|
||||
\node[anchor=south, above, text width=8cm, align=left] at (model.south) {Extensive use of models for:\begin{itemize}[noitemsep,topsep=5pt]\item Extraction of transfer functions \\ \item Choice of appropriate control architecture \\ \item Tuning of control laws \\ \item Closed loop simulations \\ \item Noise budgets / Evaluation of performances \\ \item Sensibility to parameters / disturbances\end{itemize}\centerline{Models are at the core the mecatronic approach!}};
|
||||
|
||||
\node[mymodel] at (mustation.south) {Multiple stages \\ Complex dynamics};
|
||||
\node[mymodel] at (dist.south) {Ground motion \\ Position errors};
|
||||
\node[mymodel] at (nanohexapod.south) {Different concepts \\ Sensors, Actuators};
|
||||
|
||||
\node[mymodel] at (instrumentation.south) {Sensors, Actuators \\ Electronics};
|
||||
\node[mymodel] at (mechanical.south) {Proper integration \\ Ease of assembly};
|
||||
\node[mymodel] at (fem.south) {Optimize key parts: \\ Joints, Plates, APA};
|
||||
|
||||
\node[mymodel] at (mounting.south) {Struts \\ Nano-Hexapod};
|
||||
\node[mymodel] at (testbenches.south) {Instrumentation \\ APA, Struts};
|
||||
\node[mymodel] at (implementation.south) {Control tests \\ $\mu$ Station};
|
||||
|
||||
% Links
|
||||
\draw[->] (dist.east) -- node[above, midway]{{\small Measurements}} node[below,midway]{{\small Spectral Analysis}} (dist.east-|model.west);
|
||||
\draw[->] (mustation.east) -- node[above, midway]{{\small Measurements}} node[below, midway]{{\small CAD Model}} (mustation.east-|model.west);
|
||||
|
||||
\draw[->] ($(nanohexapod.east-|model.west)-(0, 0.15)$) -- node[below, midway]{{\small Optimization}} ($(nanohexapod.east)-(0, 0.15)$);
|
||||
\draw[<-] ($(nanohexapod.east-|model.west)+(0, 0.15)$) -- node[above, midway]{{\small Model}} ($(nanohexapod.east)+(0, 0.15)$);
|
||||
|
||||
\draw[->] ($(fem.south|-model.north)+(0.15, 0)$) -- node[right, midway]{{\small Specif.}} ($(fem.south)+(0.15,0)$);
|
||||
\draw[<-] ($(fem.south|-model.north)-(0.15, 0)$) -- node[left, midway,align=right]{{\small Super}\\{\small Element}} ($(fem.south)-(0.15,0)$);
|
||||
|
||||
\draw[->] ($(mechanical.south|-model.north)+(0.15, 0)$) -- node[right, midway]{{\small Specif.}} ($(mechanical.south)+(0.15,0)$);
|
||||
\draw[<-] ($(mechanical.south|-model.north)-(0.15, 0)$) -- node[left, midway,align=right]{{\small CAD}\\{\small model}} ($(mechanical.south)-(0.15,0)$);
|
||||
|
||||
\draw[->] ($(instrumentation.south|-model.north)+(0.15, 0)$) -- node[right, midway]{{\small Specif.}} ($(instrumentation.south)+(0.15,0)$);
|
||||
\draw[<-] ($(instrumentation.south|-model.north)-(0.15, 0)$) -- node[left, midway]{{\small Model}} ($(instrumentation.south)-(0.15,0)$);
|
||||
|
||||
\draw[->] ($(mounting.west-|model.east)+(0, 0.15)$) -- node[above, midway]{{\small Requirements}} ($(mounting.west)+(0, 0.15)$);
|
||||
\draw[<-] ($(mounting.west-|model.east)-(0, 0.15)$) -- node[below, midway]{{\small Model refinement}} ($(mounting.west)-(0, 0.15)$);
|
||||
|
||||
\draw[->] ($(testbenches.west-|model.east)+(0, 0.15)$) -- node[above, midway]{{\small Control Laws}} ($(testbenches.west)+(0, 0.15)$);
|
||||
\draw[<-] ($(testbenches.west-|model.east)-(0, 0.15)$) -- node[below, midway]{{\small Model refinement}} ($(testbenches.west)-(0, 0.15)$);
|
||||
|
||||
\draw[->] ($(implementation.west-|model.east)+(0, 0.15)$) -- node[above, midway]{{\small Control Laws}} ($(implementation.west)+(0, 0.15)$);
|
||||
\draw[<-] ($(implementation.west-|model.east)-(0, 0.15)$) -- node[below, midway]{{\small Model refinement}} ($(implementation.west)-(0, 0.15)$);
|
||||
|
||||
% Main steps
|
||||
\node[font=\bfseries, rotate=90, anchor=south, above] (conceptual_phase_node) at (dist.west) {1 - Conceptual Phase};
|
||||
\node[font=\bfseries, above] (detailed_phase_node) at (mechanical.north) {2 - Detail Design Phase};
|
||||
\node[font=\bfseries, rotate=-90, anchor=south, above] (implementation_phase_node) at (testbenches.east) {3 - Experimental Phase};
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(conceptual_phase_node.north|-nanohexapod.north) (mustation.south east)}, fill=lightgreen!50!white, draw, inner sep=2pt] (conceptual_phase) {};
|
||||
\node[fit={(detailed_phase_node.north-|instrumentation.west) (fem.south east)}, fill=lightyellow!50!white, draw, inner sep=2pt] (detailed_phase) {};
|
||||
\node[fit={(implementation_phase_node.north|-mounting.north) (implementation.south west)}, fill=lightred!50!white, draw, inner sep=2pt] (implementation_phase) {};
|
||||
% \node[above left] at (dob.south east) {DOB};
|
||||
\end{scope}
|
||||
|
||||
% Between main steps
|
||||
\draw[mystep, postaction={decorate,decoration={raise=1ex,text along path,text align=center,text={Concept Validation}}}] (conceptual_phase.north) to[out=90, in=180] (detailed_phase.west);
|
||||
\draw[mystep, postaction={decorate,decoration={raise=1ex,text along path,text align=center,text={Procurement}}}] (detailed_phase.east) to[out=0, in=90] (implementation_phase.north);
|
||||
|
||||
% % Inside Model
|
||||
% \node[inner sep=1pt, outer sep=6pt, anchor=north west, draw, fill=white, thin] (multibodymodel) at ($(model.north west) - (0, 0.5)$)
|
||||
% {\includegraphics[width=5.6cm]{simscape_nano_hexapod.png}};
|
||||
|
||||
% \node[inner sep=1pt, outer sep=6pt, anchor=south west, draw, fill=white, thin] (simscape) at (model.south west)
|
||||
% {\includegraphics[width=5.6cm]{simscape_picture.jpg}};
|
||||
|
||||
% % Feedback Model
|
||||
% \node[inner sep=3pt, outer sep=6pt, anchor=north east, draw, fill=white, thin] (simscape_sim) at ($(model.north east) - (0, 0.5)$)
|
||||
% {\includegraphics[width=3.6cm]{simscape_simulations.pdf}};
|
||||
|
||||
% % FeedBack
|
||||
% \node[inner sep=3pt, outer sep=6pt, anchor=south east, draw, fill=white, thin] (feedback) at (model.south east)
|
||||
% {\includegraphics[width=3.6cm]{classical_feedback_small.pdf}};
|
||||
\end{tikzpicture}
|
||||
```
|
||||
|
||||
{{< figure src="nass_mechatronics_approach.png" >}}
|
||||
|
||||
|
||||
## HAC-LAC Representation (two columns) {#hac-lac-representation--two-columns}
|
||||
|
||||
```latex
|
||||
\graphicspath{ {/home/thomas/Cloud/thesis/papers/dehaeze21_mechatronics_approach_nass/tikz/figs-tikz} }
|
||||
|
||||
\begin{tikzpicture}
|
||||
\node[inner sep=3pt, fill=white, draw] (plant) at (0, 0)
|
||||
{\includegraphics[width=4.5cm]{nass_concept_schematic.pdf}};
|
||||
|
||||
\coordinate[] (outputf) at ($(plant.south east)!0.75!(plant.north east)$);
|
||||
\coordinate[] (outputx) at ($(plant.south east)!0.25!(plant.north east)$);
|
||||
|
||||
\node[block, left=0.6 of plant] (amp) {Amplifier};
|
||||
\node[DAC, left=0.6 of amp] (dac) {DAC};
|
||||
\node[ADC] (adc) at ($(plant.north-|dac) + (0, 0.2)$) {ADC};
|
||||
\node[addb, left=0.6 of dac] (addu) {};
|
||||
\node[block, above=0.4 of addu] (Kiff) {$\bm{K}_{\mathcal{L}}$};
|
||||
\node[block, left=0.6 of addu] (Kl) {$\bm{K}_{\mathcal{X}}$};
|
||||
\node[block, left=0.6 of Kl] (J) {$\bm{J}$};
|
||||
\node[block, left=0.6 of J] (pos_error) {Pos. Err.};
|
||||
|
||||
\draw[->] (outputf) -- ++(0.2, 0)node[branch]{} |- (adc.east);
|
||||
\draw[->] (outputf) --node[midway, below]{$\bm{\tau}_m$} ++(0.8, 0);
|
||||
|
||||
\draw[->] (outputx) -- ++(0.2, 0)node[branch]{} |- ($(plant.south)+(0, -0.2)$) -| (pos_error.south);
|
||||
\draw[->] (outputx) --node[midway, above]{$\bm{\mathcal{X}}_m$} ++(0.8, 0);
|
||||
|
||||
\draw[->] (pos_error.east) -- node[midway, above]{$\bm{\epsilon}_{\mathcal{X}}$} (J.west);
|
||||
\draw[->] (J.east) -- node[midway, above]{$\bm{\epsilon}_{\mathcal{L}}$} (Kl.west);
|
||||
\draw[->] (Kl.east) -- node[midway, above]{$\bm{u}^\prime$} (addu.west);
|
||||
\draw[->] (addu.east) -- node[midway, above]{$\bm{u}$} (dac.west);
|
||||
\draw[->] (dac.east) -- (amp.west);
|
||||
\draw[->] (amp.east) -- (plant.west);
|
||||
\draw[->] (adc.west) -| (Kiff.north);
|
||||
\draw[->] (Kiff.south) -- (addu.north);
|
||||
\draw[<-] (pos_error.west) -- node[midway, above]{$\bm{r}_\mu$} ++(-0.8, 0);
|
||||
\end{tikzpicture}
|
||||
```
|
||||
|
||||
{{< figure src="nass_hac_lac_block_diagram.png" >}}
|
||||
|
||||
|
||||
## HAC-LAC alternative (one column) {#hac-lac-alternative--one-column}
|
||||
|
||||
```latex
|
||||
\graphicspath{ {/home/thomas/Cloud/thesis/papers/dehaeze21_mechatronics_approach_nass/tikz/figs-tikz} }
|
||||
|
||||
\begin{tikzpicture}
|
||||
% Plant
|
||||
\node[inner sep=3pt, fill=white, draw] (plant) at (0, 0)
|
||||
{\includegraphics[width=4cm]{nass_concept_schematic.pdf}};
|
||||
|
||||
% Plant outputs
|
||||
\coordinate[] (outputf) at ($(plant.south east)!0.8!(plant.north east)$);
|
||||
\coordinate[] (outputx) at ($(plant.south east)!0.2!(plant.north east)$);
|
||||
|
||||
% Blocks
|
||||
\node[addb, left=0.6 of plant] (addu) {};
|
||||
\node[block, above=0.4 of addu] (Kiff) {$\bm{K}_{\text{\tiny IFF}}$};
|
||||
\node[block, left=1.0 of addu] (Kl) {$\bm{K}_{\mathcal{L}}$};
|
||||
\node[block, left=0.6 of Kl] (J) {$\bm{J}$};
|
||||
\node[addb={+}{}{}{}{-}, left=0.6 of J] (pos_error) {};
|
||||
|
||||
% Lines
|
||||
\draw[->] (outputf) -- ++(0.2, 0)node[below]{$\bm{\tau}$} |- ($(plant.north)+(0, 0.2)$) -| (Kiff.north);
|
||||
\draw[->] (outputx) -- ++(0.6, 0)node[above]{$\bm{\mathcal{X}}$} |- ($(plant.south)+(0, -0.4)$) -| (pos_error.south);
|
||||
\draw[->] (pos_error.east) -- node[midway, above]{$\bm{\epsilon}_{\mathcal{X}}$} (J.west);
|
||||
\draw[->] (J.east) -- node[midway, above]{$\bm{\epsilon}_{\mathcal{L}}$} (Kl.west);
|
||||
\draw[->] (Kl.east) -- node[near start, above]{$\bm{u}^\prime$} (addu.west);
|
||||
\draw[->] (addu.east) -- node[midway, above]{$\bm{u}$} (plant.west);
|
||||
\draw[->] (Kiff.south) -- (addu.north);
|
||||
\draw[<-] (pos_error.west) -- node[midway, above]{$\bm{r}$} ++(-0.6, 0);
|
||||
|
||||
% Damped plant
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(plant.south-|Kiff.west) ($(plant.north east)+(0.2cm,0.2cm)$)}, fill=black!10!white, draw, dashed, inner sep=0.2cm] (damped_plant) {};
|
||||
\node[above right, align=left] at (damped_plant.south west) {\small Damped\\Plant};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
```
|
||||
|
||||
{{< figure src="nass_hac_lac_block_diagram_without_elec.png" >}}
|
||||
|
||||
|
||||
## Mass Spring Damper Model {#mass-spring-damper-model}
|
||||
|
||||
```latex
|
||||
\begin{tikzpicture}
|
||||
% ====================
|
||||
% Parameters
|
||||
% ====================
|
||||
\def\bracs{0.05} % Brace spacing vertically
|
||||
\def\brach{-12pt} % Brace shift horizontaly
|
||||
% ====================
|
||||
|
||||
% ====================
|
||||
% Ground
|
||||
% ====================
|
||||
\draw (-0.9, 0) -- (0.9, 0);
|
||||
\draw[dashed] (0.9, 0) -- ++(0.5, 0);
|
||||
\draw[->] (1.3, 0) -- ++(0, 0.4) node[right]{$w$};
|
||||
% ====================
|
||||
|
||||
% ====================
|
||||
% Granite
|
||||
\begin{scope}[shift={(0, 0)}]
|
||||
\draw[fill=white] (-0.9, 1.2) rectangle (0.9, 2.0) node[pos=0.5]{$\scriptstyle\text{granite}$};
|
||||
\draw[spring] (-0.7, 0) -- ++(0, 1.2);
|
||||
\draw[damper] ( 0, 0) -- ++(0, 1.2);
|
||||
|
||||
\draw[dashed] ( 0.9, 2.0) -- ++(2.0, 0) coordinate(xg);
|
||||
|
||||
% \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
||||
% (-0.9, \bracs) -- ++(0, 2.0) node[midway,rotate=90,anchor=south,yshift=10pt]{Granite};
|
||||
\end{scope}
|
||||
% ====================
|
||||
|
||||
% ====================
|
||||
% Stages
|
||||
\begin{scope}[shift={(0, 2.0)}]
|
||||
\draw[fill=white] (-0.9, 1.2) rectangle (0.9, 2.0) node[pos=0.5]{$\scriptstyle\mu\text{-station}$};
|
||||
\draw[spring] (-0.7, 0) -- ++(0, 1.2);
|
||||
\draw[damper] ( 0, 0) -- ++(0, 1.2);
|
||||
\draw[actuator] ( 0.7, 0) -- ++(0, 1.2) node[midway, right=0.1](ft){$f_t$};
|
||||
|
||||
% \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
||||
% (-0.9, \bracs) -- ++(0, 2.0) node[midway,rotate=90,anchor=south,yshift=10pt]{$\mu\text{-station}$};
|
||||
\end{scope}
|
||||
% ====================
|
||||
|
||||
|
||||
% ====================
|
||||
% NASS
|
||||
\begin{scope}[shift={(0, 4.0)}]
|
||||
\draw[fill=white] (-0.9, 1.5) rectangle (0.9, 2.3) node[pos=0.5]{$\scriptstyle\nu\text{-hexapod}$};
|
||||
\draw[dashed] (0.9, 2.3) -- ++(2.0, 0) coordinate(xnpos);
|
||||
|
||||
\draw[spring] (-0.7, 0) -- ++(0, 1.2) node[midway, left=0.1]{};
|
||||
\draw[damper] ( 0, 0) -- ++(0, 1.2) node[midway, left=0.2]{};
|
||||
\draw[actuator] ( 0.7, 0) -- ++(0, 1.2) coordinate[midway, below right=0.2 and 0.1](f);
|
||||
|
||||
\node[forcesensor={1.8}{0.3}] (fsensn) at (0, 1.2){};
|
||||
|
||||
% \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
||||
% (-0.9, \bracs) -- ++(0, 2.2) node[midway,rotate=90,anchor=south,yshift=10pt]{$\nu\text{-hexapod}$};
|
||||
\end{scope}
|
||||
% ====================
|
||||
|
||||
% ====================
|
||||
% Measured Displacement
|
||||
\draw[<->, dashed] ($(xg)+(-0.1, 0)$) node[above left](d){$d$} -- ($(xnpos)+(-0.1, 0)$);
|
||||
% ====================
|
||||
|
||||
% ====================
|
||||
% IFF Control
|
||||
\node[block={2em}{1.5em}, right=0.6 of fsensn] (iff) {$K_{\scriptscriptstyle IFF}$};
|
||||
\node[addb] (ctrladd) at (f-|iff) {};
|
||||
\node[block={2em}{1.5em}, below=0.6 of ctrladd] (ctrl) {$K_{X}$};
|
||||
|
||||
\draw[->] (fsensn.east) -- node[midway, above]{$\tau_m$} (iff.west);
|
||||
\draw[->] (iff.south) -- (ctrladd.north);
|
||||
\draw[->] (ctrladd.west) -- (f.east) node[above right]{$u$};
|
||||
\draw[->] (d.west) -| (ctrl.south);
|
||||
\draw[->] (ctrl.north) -- (ctrladd.south) node[below right]{$u^\prime$};
|
||||
% ====================
|
||||
\end{tikzpicture}
|
||||
```
|
||||
|
||||
{{< figure src="mass_spring_damper_hac_lac.png" >}}
|
||||
|
||||
|
||||
## Mass Spring Damper Model - Bis {#mass-spring-damper-model-bis}
|
||||
|
||||
```latex
|
||||
\begin{tikzpicture}
|
||||
% ====================
|
||||
% Parameters
|
||||
% ====================
|
||||
\def\bracs{0.05} % Brace spacing vertically
|
||||
\def\brach{-12pt} % Brace shift horizontaly
|
||||
% ====================
|
||||
|
||||
% ====================
|
||||
% Ground
|
||||
% ====================
|
||||
\draw (-0.9, 0) -- (0.9, 0);
|
||||
\draw[dashed] (0.9, 0) -- ++(0.5, 0);
|
||||
\draw[->] (1.3, 0) -- ++(0, 0.4) node[right]{$w$};
|
||||
% ====================
|
||||
|
||||
% ====================
|
||||
% Granite
|
||||
\begin{scope}[shift={(0, 0)}]
|
||||
\draw[fill=white] (-0.9, 1.2) rectangle (0.9, 2.0) node[pos=0.5]{$\scriptstyle\text{granite}$};
|
||||
\draw[spring] (-0.7, 0) -- ++(0, 1.2);
|
||||
\draw[damper] ( 0, 0) -- ++(0, 1.2);
|
||||
|
||||
\draw[dashed] ( 0.9, 2.0) -- ++(2.0, 0) coordinate(xg);
|
||||
|
||||
% \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
||||
% (-0.9, \bracs) -- ++(0, 2.0) node[midway,rotate=90,anchor=south,yshift=10pt]{Granite};
|
||||
\end{scope}
|
||||
% ====================
|
||||
|
||||
% ====================
|
||||
% Stages
|
||||
\begin{scope}[shift={(0, 2.0)}]
|
||||
\draw[fill=white] (-0.9, 1.2) rectangle (0.9, 2.0) node[pos=0.5]{$\scriptstyle\mu\text{-station}$};
|
||||
|
||||
\coordinate (mustation) at (0.9, 1.6);
|
||||
|
||||
\draw[spring] (-0.7, 0) -- ++(0, 1.2);
|
||||
\draw[damper] ( 0, 0) -- ++(0, 1.2);
|
||||
\draw[actuator] ( 0.7, 0) -- ++(0, 1.2) node[midway, right=0.1](ft){$f_t$};
|
||||
|
||||
% \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
||||
% (-0.9, \bracs) -- ++(0, 2.0) node[midway,rotate=90,anchor=south,yshift=10pt]{$\mu\text{-station}$};
|
||||
\end{scope}
|
||||
% ====================
|
||||
|
||||
|
||||
% ====================
|
||||
% NASS
|
||||
\begin{scope}[shift={(0, 4.0)}]
|
||||
\draw[fill=white] (-0.9, 1.2) rectangle (0.9, 2.0) node[pos=0.5]{$\scriptstyle\nu\text{-hexapod}$};
|
||||
\draw[dashed] (0.9, 2.0) -- ++(2.0, 0) coordinate(xnpos);
|
||||
|
||||
\draw[spring] (-0.7, 0) -- ++(0, 1.2) node[midway, left=0.1]{};
|
||||
\draw[damper] ( 0, 0) -- ++(0, 1.2) node[midway, left=0.2]{};
|
||||
\draw[actuator] ( 0.7, 0) -- ++(0, 1.2) coordinate[midway, right=0.1](f);
|
||||
|
||||
% \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] %
|
||||
% (-0.9, \bracs) -- ++(0, 2.2) node[midway,rotate=90,anchor=south,yshift=10pt]{$\nu\text{-hexapod}$};
|
||||
\end{scope}
|
||||
% ====================
|
||||
|
||||
% ====================
|
||||
% Measured Displacement
|
||||
\draw[<->, dashed] ($(xg)+(-0.1, 0)$) node[above left](d){$d$} -- ($(xnpos)+(-0.1, 0)$);
|
||||
% ====================
|
||||
|
||||
% ====================
|
||||
% IFF Control
|
||||
% \node[block={2em}{1.5em}, right=0.6 of fsensn] (iff) {$K_{\scriptscriptstyle IFF}$};
|
||||
% \node[addb] (ctrladd) at (f-|iff) {};
|
||||
\node[block={2em}{1.5em}, right=0.6 of mustation] (ctrl) {$K$};
|
||||
|
||||
% \draw[->] (fsensn.east) -- node[midway, above]{$\tau_m$} (iff.west);
|
||||
% \draw[->] (iff.south) -- (ctrladd.north);
|
||||
% \draw[->] (ctrladd.west) -- (f.east) node[above right]{$u$};
|
||||
\draw[->] (d.west) -| (ctrl.south);
|
||||
\draw[->] (ctrl.north) |- (f) node[above right]{$u$};
|
||||
% ====================
|
||||
\end{tikzpicture}
|
||||
```
|
||||
@@ -0,0 +1,775 @@
|
||||
+++
|
||||
title = "LaTeX Configuration for Tikz Figures"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
## Packages {#packages}
|
||||
|
||||
```latex
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[T1]{fontenc}
|
||||
|
||||
\usepackage[french, english]{babel} % Last language is main language
|
||||
|
||||
\usepackage{lmodern} % Latin Modern Font
|
||||
\usepackage{gensymb} % Generic symbols for both text and math mode
|
||||
|
||||
\usepackage{standalone} % Used to generate standalone Tikz
|
||||
|
||||
\usepackage{amsmath} % Main math Package
|
||||
\usepackage{mathtools} % Extension package to amsmath
|
||||
\usepackage{amsthm} % Typesetting theorems (AMS style)
|
||||
\usepackage{amsfonts} % More fonts from the AMS
|
||||
\usepackage{textcomp} % provide many text symbols
|
||||
\usepackage{steinmetz} % For phase symbol
|
||||
|
||||
\usepackage{xstring} % Utils to manipulate strings
|
||||
\usepackage{etoolbox} % Add basic if/then
|
||||
\usepackage{esvect} % Beautyfull vectors
|
||||
\usepackage{graphicx} % Enhanced support for graphics
|
||||
\usepackage{grffile} % Used by matlab2tikz
|
||||
|
||||
\usepackage{microtype} % typographic tuning
|
||||
\usepackage{setspace} % for line spacing, e.g. \onehalfspacing
|
||||
\usepackage{tabularx} % table features
|
||||
\usepackage{enumitem} % for simple list modifications
|
||||
\usepackage{booktabs} % better table support
|
||||
|
||||
\usepackage{stackengine} %
|
||||
|
||||
\usepackage[load-configurations=abbreviations]{siunitx} % SI units
|
||||
\sisetup{
|
||||
locale = US,
|
||||
detect-all,
|
||||
range-phrase=--,
|
||||
range-units=single
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Tikz related packages {#tikz-related-packages}
|
||||
|
||||
```latex
|
||||
\usepackage{tikz} % Tikz
|
||||
\usepackage{tikzscale} % Used to scale Tikz graphics
|
||||
\usepackage{adjustbox} % Used to proper positioning of tikz pictures
|
||||
\usepackage{circuitikz} % Draw electronic circuits
|
||||
\usepackage{pgfpages} % Needed to use notes
|
||||
\usepackage{pgfplots} % Used to plot functions
|
||||
```
|
||||
|
||||
|
||||
## Tikz Libraries {#tikz-libraries}
|
||||
|
||||
```latex
|
||||
\usetikzlibrary{arrows} % Arrow tip library
|
||||
\usetikzlibrary{arrows.meta} % Add some arrows
|
||||
\usetikzlibrary{calc} % The library allows advanced Coordinate Calculations
|
||||
\usetikzlibrary{intersections} % calculate intersections of paths
|
||||
\usetikzlibrary{matrix} %
|
||||
\usetikzlibrary{patterns} %
|
||||
\usetikzlibrary{shapes} % Defines circle and rectangle
|
||||
\usetikzlibrary{shapes.geometric} % Use for the shape diamond and isosceles triangle
|
||||
\usetikzlibrary{snakes} % snake=coil and snake=zigzag using segment amplitude=10pt
|
||||
\usetikzlibrary{positioning} % Additional options for placing nodes
|
||||
\usetikzlibrary{3d} % Plot 3D shapes
|
||||
\usetikzlibrary{spy} % Creating a magnified area
|
||||
\usetikzlibrary{decorations.text} % Used to make text follows a curve
|
||||
\usetikzlibrary{decorations.pathmorphing} % deformation of a path
|
||||
\usetikzlibrary{decorations.markings} % Used for spring and damper
|
||||
\usetikzlibrary{babel} % A tiny library that make the interaction with the babel package easier
|
||||
\usetikzlibrary{plotmarks} % This library defines a number of plot marks
|
||||
\usetikzlibrary{fit} % Used to make rectangle as nodes by specifying two points
|
||||
\usetikzlibrary{backgrounds} % Used to put things under others
|
||||
```
|
||||
|
||||
|
||||
## PGF Plot libraries and config {#pgf-plot-libraries-and-config}
|
||||
|
||||
```latex
|
||||
\usepgfplotslibrary{patchplots}
|
||||
\usepgfplotslibrary{groupplots}
|
||||
|
||||
\pgfplotsset{compat=newest}
|
||||
\pgfplotsset{plot coordinates/math parser=false}
|
||||
```
|
||||
|
||||
|
||||
## Setup Arrows style {#setup-arrows-style}
|
||||
|
||||
```latex
|
||||
\tikzset{>=Stealth}
|
||||
% Setup default Linewidth
|
||||
\tikzset{every path/.style={line width=1pt}}
|
||||
```
|
||||
|
||||
|
||||
## Colors {#colors}
|
||||
|
||||
```latex
|
||||
\usepackage{xcolor}% Color extension
|
||||
|
||||
\definecolor{colorblack}{rgb}{0, 0, 0}
|
||||
\definecolor{colorblue}{HTML}{0072bd}
|
||||
\definecolor{colorred}{HTML}{d95218}
|
||||
\definecolor{coloryellow}{HTML}{ecb01f}
|
||||
\definecolor{colorpurple}{HTML}{7d2e8e}
|
||||
\definecolor{colorgreen}{HTML}{77ab2f}
|
||||
|
||||
\definecolor{lightblue}{HTML}{dbf0ff}
|
||||
\definecolor{lightred}{HTML}{f9d9cb}
|
||||
\definecolor{lightyellow}{HTML}{faf0d1}
|
||||
\definecolor{lightpurple}{HTML}{efdcf4}
|
||||
\definecolor{lightgreen}{HTML}{e6f3d3}
|
||||
|
||||
% Main color
|
||||
\definecolor{maincolor}{RGB}{89, 9, 38}
|
||||
\definecolor{secondcolor}{RGB}{20, 9, 89}
|
||||
```
|
||||
|
||||
|
||||
## Control {#control}
|
||||
|
||||
|
||||
### Blocks {#blocks}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
block/.style n args={2}{%
|
||||
draw,
|
||||
fill=white,
|
||||
minimum width = #1,
|
||||
minimum height = #2,
|
||||
},
|
||||
block/.default={1.2cm}{1.0cm}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Branches {#branches}
|
||||
|
||||
```latex
|
||||
\tikzstyle{branch}=[fill,shape=circle,minimum size=4pt,inner sep=0pt]
|
||||
\tikzstyle{->top}=[-{Stealth[color=black, scale=0.8]}, draw=white, double=black, double distance=1pt, line width=1pt]
|
||||
\tikzstyle{<-top}=[{stealth[color=black, scale=0.8]}-, draw=white, double=black, double distance=1pt, line width=1pt]
|
||||
```
|
||||
|
||||
|
||||
### Hand Writen Style {#hand-writen-style}
|
||||
|
||||
Usefull for schematic plots
|
||||
|
||||
```latex
|
||||
\tikzstyle{handwriten}=[decorate,decoration={random steps,amplitude=0.1pt,segment length=0.8pt}]
|
||||
```
|
||||
|
||||
|
||||
### DAC {#dac}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
DAC/.style={%
|
||||
draw,
|
||||
signal,
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### ADC {#adc}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
ADC/.style={%
|
||||
draw,
|
||||
signal,
|
||||
signal to = west,
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Gain {#gain}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
gain right/.style={%
|
||||
draw,
|
||||
regular polygon,
|
||||
regular polygon sides = 3,
|
||||
inner sep = 2pt,
|
||||
shape border rotate=-90
|
||||
},
|
||||
gain left/.style={%
|
||||
draw,
|
||||
regular polygon,
|
||||
regular polygon sides = 3,
|
||||
inner sep = 2pt,
|
||||
shape border rotate=90
|
||||
},
|
||||
gain top/.style={%
|
||||
draw,
|
||||
regular polygon,
|
||||
regular polygon sides = 3,
|
||||
inner sep = 2pt,
|
||||
shape border rotate=0
|
||||
},
|
||||
gain bottom/.style={%
|
||||
draw,
|
||||
regular polygon,
|
||||
regular polygon sides = 3,
|
||||
inner sep = 2pt,
|
||||
shape border rotate=180
|
||||
},
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Add / Substract / Divide / Multiply block {#add-substract-divide-multiply-block}
|
||||
|
||||
```latex
|
||||
\tikzset{% Add block with Circled operations
|
||||
addc/.style n args={5}{%
|
||||
draw,
|
||||
fill=white,
|
||||
circle,
|
||||
outer sep = 0pt,
|
||||
inner sep = 0pt,
|
||||
minimum size = 2em,
|
||||
execute at begin node={\LARGE $#1$},
|
||||
append after command={\pgfextra{\let\mainnode=\tikzlastnode}
|
||||
\ifx#2\empty\else
|
||||
node[draw, circle, outer sep=6pt, inner sep=0pt, above left] at (\mainnode.west) {$#2$}%
|
||||
\fi
|
||||
\ifx#3\empty\else
|
||||
node[draw, circle, outer sep=6pt, inner sep=0pt, above right] at (\mainnode.north) {$#3$}%
|
||||
\fi
|
||||
\ifx#4\empty\else
|
||||
node[draw, circle, outer sep=6pt, inner sep=0pt, below right] at (\mainnode.east) {$#4$}%
|
||||
\fi
|
||||
\ifx#5\empty\else
|
||||
node[draw, circle, outer sep=6pt, inner sep=0pt, below left] at (\mainnode.south) {$#5$}%
|
||||
\fi
|
||||
}
|
||||
},
|
||||
addc/.default={+}{}{}{}{},
|
||||
}
|
||||
```
|
||||
|
||||
```latex
|
||||
\tikzset{% Add Block
|
||||
addb/.style n args={5}{%
|
||||
draw,
|
||||
fill=white,
|
||||
circle,
|
||||
outer sep = 0pt,
|
||||
inner sep = 0pt,
|
||||
minimum size = 2em,
|
||||
execute at begin node={\LARGE $#1$},
|
||||
append after command={\pgfextra{\let\mainnode=\tikzlastnode}
|
||||
\ifx#2\empty\else
|
||||
node[outer sep=2pt, inner sep=0pt, above left] at (\mainnode.west) {$#2$}%
|
||||
\fi
|
||||
\ifx#3\empty\else
|
||||
node[outer sep=2pt, inner sep=0pt, above right] at (\mainnode.north) {$#3$}%
|
||||
\fi
|
||||
\ifx#4\empty\else
|
||||
node[outer sep=2pt, inner sep=0pt, below right] at (\mainnode.east) {$#4$}%
|
||||
\fi
|
||||
\ifx#5\empty\else
|
||||
node[outer sep=2pt, inner sep=0pt, below left] at (\mainnode.south) {$#5$}%
|
||||
\fi
|
||||
}
|
||||
},
|
||||
addb/.default={+}{}{}{}{},
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Plots {#plots}
|
||||
|
||||
|
||||
### Default line caps {#default-line-caps}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{
|
||||
every axis plot/.append style={line join=round},
|
||||
every axis plot/.append style={line cap=round},
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Grid {#grid}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{grid style={black}}
|
||||
\pgfplotsset{major grid style={black!30!white}}
|
||||
\pgfplotsset{minor grid style={black!10!white}}
|
||||
\pgfplotsset{xmajorgrids}
|
||||
\pgfplotsset{ymajorgrids}
|
||||
```
|
||||
|
||||
|
||||
### Lines {#lines}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{separate axis lines=false} % draw axis as rectangle and not as 4 lines
|
||||
\pgfplotsset{every outer x axis line/.append style={black}}
|
||||
\pgfplotsset{every outer y axis line/.append style={black}}
|
||||
\pgfplotsset{axis background/.style={fill=white}}
|
||||
\pgfplotsset{axis x line*=bottom} % solid line on the bottom with thin on the top
|
||||
\pgfplotsset{axis y line*=left} % solid line on the left with thin on the right
|
||||
```
|
||||
|
||||
|
||||
### Ticks {#ticks}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{every y tick label/.append style={font=\color{black}}}
|
||||
\pgfplotsset{every y tick/.append style={black}}
|
||||
\pgfplotsset{every x tick label/.append style={font=\color{black}}}
|
||||
\pgfplotsset{every x tick/.append style={black}}
|
||||
```
|
||||
|
||||
|
||||
### Size {#size}
|
||||
|
||||
If `scale only axis=false` (the default), pgfplots will try to produce the desired width including labels, titles and ticks.
|
||||
|
||||
```latex
|
||||
\pgfplotsset{scale only axis=true}
|
||||
```
|
||||
|
||||
|
||||
### Label {#label}
|
||||
|
||||
Used to align all of ylabel of one figure.
|
||||
|
||||
```latex
|
||||
\pgfplotsset{ylabel absolute}
|
||||
```
|
||||
|
||||
|
||||
### Legend {#legend}
|
||||
|
||||
```latex
|
||||
% https://tex.stackexchange.com/questions/54794/using-a-pgfplots-style-legend-in-a-plain-old-tikzpicture#54834
|
||||
|
||||
% argument #1: any options
|
||||
\newenvironment{customlegend}[1][]{%
|
||||
\begingroup
|
||||
% inits/clears the lists (which might be populated from previous
|
||||
% axes):
|
||||
\csname pgfplots@init@cleared@structures\endcsname
|
||||
\pgfplotsset{#1}%
|
||||
}{%
|
||||
% draws the legend:
|
||||
\csname pgfplots@createlegend\endcsname
|
||||
\endgroup
|
||||
}%
|
||||
|
||||
% makes \addlegendimage available (typically only available within an
|
||||
% axis environment):
|
||||
\def\addlegendimage{\csname pgfplots@addlegendimage\endcsname}
|
||||
|
||||
% definition to insert numbers
|
||||
% \pgfkeys{/pgfplots/number in legend/.style={%
|
||||
% /pgfplots/legend image code/.code={%
|
||||
% \node at (0.125,-0.0225){#1}; % <= changed x value
|
||||
% },%
|
||||
% },
|
||||
% }
|
||||
\pgfplotsset{
|
||||
every legend to name picture/.style={west}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Upper and Lower bounds {#upper-and-lower-bounds}
|
||||
|
||||
```latex
|
||||
\pgfplotsset{upperbound}=[line cap=round, postaction={decorate,draw,decoration={border, segment length=0.2cm, amplitude=0.3cm, angle=60}}]
|
||||
\pgfplotsset{lowerbound}=[line cap=round, postaction={decorate,draw,decoration={border, segment length=0.2cm, amplitude=0.3cm, angle=-60}}]
|
||||
```
|
||||
|
||||
And we add the corresdonding
|
||||
|
||||
```latex
|
||||
\pgfplotsset{
|
||||
/pgfplots/upperbound/.style 1 args={
|
||||
legend image code/.code={
|
||||
\draw[##1, upperbound]
|
||||
plot coordinates {
|
||||
(0cm,0cm)
|
||||
(0.6cm,0cm)
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Pole {#pole}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
pole/.style{%
|
||||
color=red,
|
||||
cross out,
|
||||
draw,
|
||||
inner sep=0pt,
|
||||
outer sep=0pt,
|
||||
minimum size=#1pt
|
||||
},
|
||||
pole/.default={4}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Zero {#zero}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
zero/.style{%
|
||||
color=red,
|
||||
circle,
|
||||
draw,
|
||||
inner sep=0pt,
|
||||
outer sep=0pt,
|
||||
minimum size=#1pt
|
||||
},
|
||||
zero/.default={4}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Mechanical {#mechanical}
|
||||
|
||||
|
||||
### Spring {#spring}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
spring/.style={%
|
||||
thick,
|
||||
decoration={
|
||||
zigzag,
|
||||
pre length = #1cm,
|
||||
post length = #1cm,
|
||||
segment length = 6
|
||||
},
|
||||
decorate
|
||||
},
|
||||
spring/.default={0.2}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Coil {#coil}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
coil/.style n args={2}{%
|
||||
thick,
|
||||
decoration={
|
||||
coil,
|
||||
pre length = #1cm,
|
||||
post length = #2cm,
|
||||
segment length = 4
|
||||
},
|
||||
decorate
|
||||
},
|
||||
coil/.default={0.3}{0.3}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Damper {#damper}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
damper/.style n args={2}{%
|
||||
thick,
|
||||
decoration={markings, mark connection node=dmp, mark=at position 0.5 with {
|
||||
\node (dmp) [thick,
|
||||
inner sep = 0pt,
|
||||
transform shape,
|
||||
rotate =-90,
|
||||
minimum width = #1pt,
|
||||
minimum height = #2pt,
|
||||
draw=none] {};
|
||||
\draw [thick] ($(dmp.north east)+(0.6*#2pt,0)$) -- (dmp.south east) -- (dmp.south west) -- ($(dmp.north west)+(0.6*#2pt,0)$);
|
||||
\draw [thick] ($(dmp.north)+(0,-0.3*#1pt)$) -- ($(dmp.north)+(0,0.3*#1pt)$);
|
||||
}
|
||||
},
|
||||
decorate
|
||||
},
|
||||
damper/.default={12}{3}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Actuator {#actuator}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
actuator/.style n args={2}{%
|
||||
thick,
|
||||
draw=none,
|
||||
decoration={
|
||||
markings,
|
||||
mark connection node=my node,
|
||||
mark=at position .5 with {
|
||||
\node [draw, inner sep=0pt, minimum width=#1cm, minimum height=#2cm,
|
||||
transform shape, fill=white] (my node) {};
|
||||
},
|
||||
mark=at position .0 with {
|
||||
\draw[<-] (0, 0) -- (my node);
|
||||
},
|
||||
mark=at position 1.0 with {
|
||||
\draw[<-] (0, 0) -- (my node);
|
||||
}
|
||||
},
|
||||
decorate
|
||||
},
|
||||
actuator/.default={0.5}{0.2}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Ground {#ground}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
ground/.style n args={2}{%
|
||||
fill,
|
||||
pattern = north east lines,
|
||||
draw = none,
|
||||
anchor = north,
|
||||
minimum width = #1cm,
|
||||
minimum height = #2cm,
|
||||
append after command={
|
||||
(\tikzlastnode.north west) edge (\tikzlastnode.north east)
|
||||
}
|
||||
},
|
||||
ground/.default={2.5}{0.3}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Force Sensor {#force-sensor}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
forcesensor/.style n args={2}{%
|
||||
rectangle,
|
||||
outer sep=0pt,
|
||||
inner sep=0pt,
|
||||
draw=black,
|
||||
fill=white!60!black,
|
||||
anchor=south,
|
||||
minimum width =#1cm,
|
||||
minimum height=#2cm,
|
||||
append after command={
|
||||
[every edge/.append style={
|
||||
thick,
|
||||
black,
|
||||
}]
|
||||
(\tikzlastnode.north west) edge (\tikzlastnode.south east)
|
||||
(\tikzlastnode.north east) edge (\tikzlastnode.south west)
|
||||
}
|
||||
},
|
||||
forcesensor/.default={2.0}{0.5}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Inertial Sensor {#inertial-sensor}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
inertialsensor/.style={%
|
||||
rectangle,
|
||||
outer sep=0pt,
|
||||
inner sep=0pt,
|
||||
draw=black,
|
||||
fill=white!60!black,
|
||||
anchor=south east,
|
||||
minimum size=#1cm,
|
||||
append after command={
|
||||
[every edge/.append style={
|
||||
thick,
|
||||
black,
|
||||
}]
|
||||
(\tikzlastnode.north west) edge (\tikzlastnode.south east)
|
||||
(\tikzlastnode.north east) edge (\tikzlastnode.south west)
|
||||
}
|
||||
},
|
||||
inertialsensor/.default={0.3}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Cross {#cross}
|
||||
|
||||
```latex
|
||||
\tikzstyle{cross}=[path picture={
|
||||
\draw[black]
|
||||
(path picture bounding box.south east) -- (path picture bounding box.north west) (path picture bounding box.south west) -- (path picture bounding box.north east);
|
||||
}]
|
||||
|
||||
```
|
||||
|
||||
|
||||
### Piezoelectric actuator {#piezoelectric-actuator}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
piezo/.style n args={3}{%
|
||||
draw,
|
||||
rectangle,
|
||||
minimum width = #1cm,
|
||||
minimum height = #2cm,
|
||||
fill=blue!10!white,
|
||||
anchor=center,
|
||||
append after command={
|
||||
[every edge/.append style={
|
||||
thick,
|
||||
black,
|
||||
}]
|
||||
\foreach \i in {1,...,#3}{
|
||||
(${\i/(1+#3)}*(\tikzlastnode.north west)+{(1+#3-\i)/(1+#3)}*(\tikzlastnode.south west)+0.1*(#1,0)$) edge (${\i/(1+#3)}*(\tikzlastnode.north east)+{(1+#3-\i)/(1+#3)}*(\tikzlastnode.south east)-0.1*(#1,0)$)
|
||||
}
|
||||
}
|
||||
},
|
||||
piezo/.default={2}{4}{10}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Voice coil {#voice-coil}
|
||||
|
||||
```latex
|
||||
\def\voicecoil#1#2#3{
|
||||
% ======================
|
||||
% Parameters
|
||||
% ======================
|
||||
\def\voicecoilw{#1} % Total Width
|
||||
\def\voicecoilh{#2} % Total Height
|
||||
|
||||
\def\magnetw{\voicecoilw} % Width of the magnet
|
||||
\def\magneth{\voicecoilh/1.4} % Height of the magnet
|
||||
|
||||
\def\magnetwb{0.15*\magnetw} % Width of the borders of the magnet
|
||||
\def\magnetmw{0.15*\magnetw} % Width of the middle part of the magnet
|
||||
\def\magnetwg{0.5*\magnetw} % Width of the gap of the magnet
|
||||
|
||||
\def\magnethl{\magnetwb} % Height of the low part of the magnet
|
||||
\def\magnetmh{0.15*\magneth} % Height of the middle part of the magnet
|
||||
\def\magnethg{0.2*\magneth} % Height of the gap of the magnet
|
||||
% ======================
|
||||
|
||||
\begin{scope}[shift={(0.5*\voicecoilw, 0.5*\voicecoilh)}, rotate=#3, shift={(0, -0.5*\voicecoilh)}]
|
||||
% ======================
|
||||
% Magnet
|
||||
% ======================
|
||||
\draw[fill=white] (0, 0) -| ++(0.5*\magnetw, \magneth) -| ++(-0.5*\magnetw+0.5*\magnetwg, -\magnethg) -| (0.5*\magnetw-\magnetwb, \magnethl) -| (-0.5*\magnetw+\magnetwb, \magneth-\magnethg) -| (-0.5*\magnetwg, \magneth) -| (-0.5*\magnetw, 0) -- (cycle);
|
||||
\begin{scope}[shift={(0, \magnethl)}]
|
||||
\draw[fill=red] (-0.5*\magnetmw, 0) rectangle (0.5*\magnetmw, \magnetmh);
|
||||
\draw[fill=blue] (-0.5*\magnetmw, \magnetmh) rectangle (0.5*\magnetmw, 2*\magnetmh);
|
||||
% Top conductive Magnet
|
||||
\draw[fill=white] (-0.5*\magnetmw, 2*\magnetmh) -| (0.5*\magnetmw, -\magnethl+\magneth-\magnethg) -| ++(0.1, \magnethg) -| ++(-0.2-\magnetmw, -\magnethg) -| (-0.5*\magnetmw, \magnetmh);
|
||||
\end{scope}
|
||||
% ======================
|
||||
|
||||
% ======================
|
||||
% Coil
|
||||
% ======================
|
||||
\pgfmathsetmacro{\coilwidth}{0.5*0.5*\magnetmw+0.5*0.1+0.25*\magnetwg}%
|
||||
\draw[] ( \coilwidth, 0.5*\magneth) -- ++(0, 0.7*\magneth);
|
||||
\draw[] (-\coilwidth, 0.5*\magneth) -- ++(0, 0.7*\magneth);
|
||||
% Point on the coil
|
||||
\foreach \x in {0,1,...,9}
|
||||
{
|
||||
\node[circle,inner sep=0.6pt,fill] at ( \coilwidth, \x*0.7*\magneth/10+0.5*\magneth);
|
||||
\node[circle,inner sep=0.6pt,fill] at (-\coilwidth, \x*0.7*\magneth/10+0.5*\magneth);
|
||||
}
|
||||
\draw[fill=white] (-0.5*\magnetw, 1.2*\magneth) rectangle ++(\magnetw, \magnethg);
|
||||
% ======================
|
||||
|
||||
% ======================
|
||||
% Coordinates
|
||||
% ======================
|
||||
% Force
|
||||
\coordinate[] (vc_force) at (0, \magneth-0.5*\magnethg);
|
||||
% Coil
|
||||
\coordinate[] (vc_coil) at (0, \voicecoilh);
|
||||
% Magnet
|
||||
\coordinate[] (vc_magnet) at (0, 0);
|
||||
% Coil Wires
|
||||
\coordinate[] (vc_wire_one) at ( \coilwidth, 1.2*\magneth);
|
||||
\coordinate[] (vc_wire_two) at (-\coilwidth, 1.2*\magneth);
|
||||
% ======================
|
||||
\end{scope}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
### Axis Rotator {#axis-rotator}
|
||||
|
||||
```latex
|
||||
\newcommand{\AxisRotator}[1][rotate=0]{%
|
||||
\tikz [x=0.1cm,y=0.30cm,-stealth,#1] \draw (0,0) arc (-150:150:1 and 1);%
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Optics {#optics}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
->-/.style={
|
||||
decoration={
|
||||
markings,
|
||||
mark = at position #1 with {\arrow{>}
|
||||
}
|
||||
},
|
||||
postaction={decorate}
|
||||
}
|
||||
}
|
||||
\tikzset{%
|
||||
-<-/.style={
|
||||
decoration={
|
||||
markings,
|
||||
mark = at position #1 with {\arrow{<}
|
||||
}
|
||||
},
|
||||
postaction={decorate}
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## Misc {#misc}
|
||||
|
||||
```latex
|
||||
\tikzset{%
|
||||
labelc/.style= {%
|
||||
draw,
|
||||
fill=white,
|
||||
shape=circle,
|
||||
inner sep=2pt,
|
||||
outer sep=6pt,
|
||||
}
|
||||
}
|
||||
```
|
||||
|
||||
|
||||
## More Defaults specific to this paper {#more-defaults-specific-to-this-paper}
|
||||
|
||||
```latex
|
||||
\tikzset{block/.default={0.8cm}{0.8cm}}
|
||||
\tikzset{addb/.append style={scale=0.7}}
|
||||
\tikzset{node distance=0.6}
|
||||
```
|
||||
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@@ -79,15 +79,6 @@ note = "coming soon"
|
||||
|
||||
# ---------------------------------------------------------------- Conference Publications
|
||||
|
||||
[[publications]]
|
||||
key = "dehaeze22_fastj_uhv"
|
||||
type = "conference"
|
||||
authors = ["T. Dehaeze", "L. Ducotté"]
|
||||
title = "The Fastjack - A robust, UHV compatible and high performance linear actuator"
|
||||
venue = "euspen"
|
||||
year = 2022
|
||||
page = "dehaeze22_fast_jack"
|
||||
|
||||
[[publications]]
|
||||
key = "dehaeze21_mechat_approac_devel_nano_activ_stabil_system"
|
||||
type = "conference"
|
||||
|
||||
@@ -3,7 +3,7 @@
|
||||
|
||||
{{- $pdf := .Params.pdf -}}
|
||||
{{- if not $pdf }}
|
||||
{{- with .Resources.Match "paper/*.pdf" }}
|
||||
{{- with .Resources.Match "{paper,journal}/*.pdf" }}
|
||||
{{- range first 1 . }}
|
||||
{{- $pdf = .RelPermalink -}}
|
||||
{{- end }}
|
||||
|
||||
Some files were not shown because too many files have changed in this diff Show More
Reference in New Issue
Block a user