From 7bf7983835b4aa8e8e88d454869b4030fbfb97a7 Mon Sep 17 00:00:00 2001 From: Matteo Anedda <matteo.anedda@mailbox.tu-dresden.de> Date: Sun, 28 Nov 2021 14:15:17 +0100 Subject: [PATCH] ... --- CRE4E1A.tmp | 173 ++++++++++++++++++++++++++++ bibliography.bib | 160 +++++++++++++++++--------- images/IMG_20211124_153815.jpg | Bin 0 -> 3273511 bytes lst.tex | 3 + sections/einleitung.tex | 2 +- sections/fallbeispiel.tex | 53 +++++---- sections/grundlagen.tex | 63 ++++++----- sections/konzept.tex | 199 +++++++++++++++++++++++---------- sections/sota.tex | 86 ++++++++------ thesis.tex | 4 +- 10 files changed, 542 insertions(+), 201 deletions(-) create mode 100644 CRE4E1A.tmp create mode 100644 images/IMG_20211124_153815.jpg diff --git a/CRE4E1A.tmp b/CRE4E1A.tmp new file mode 100644 index 0000000..e2a26e8 --- /dev/null +++ b/CRE4E1A.tmp @@ -0,0 +1,173 @@ +% the following command is only required if the thesis is written in german +\RequirePackage[ngerman=ngerman-x-latest]{hyphsubst} + +% change to english for english theses +\documentclass[ngerman,BCOR=1.5cm,twoside]{tudscrreprt} + +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[ngerman]{babel} +\usepackage{isodate} +\usepackage{amssymb} +\usepackage{amsmath} + +\usepackage[ + style=numeric-comp, + backend=biber, + url=false, + doi=true, + isbn=false, + hyperref, +]{biblatex} +\addbibresource{bibliography.bib} +\AtEveryBibitem{% + \clearfield{note}% +} + +\renewbibmacro*{doi+eprint+url}{% + \printfield{doi}% + \newunit\newblock% + \iftoggle{bbx:eprint}{% + \usebibmacro{eprint}% + }{}% + \newunit\newblock% + \iffieldundef{doi}{% + \usebibmacro{url+urldate}}% + {}% + } + + + +\usepackage[hidelinks]{hyperref} % makes all links clickable but hides ugly boxes +\usepackage[capitalise,nameinlink,noabbrev]{cleveref} % automatically inserts Fig. X in the text with \cref{..} + +\usepackage[colorinlistoftodos,prependcaption,textsize=tiny]{todonotes} + +\usepackage{graphicx} +\graphicspath{ {./images/} } + +% if you need mathy stuff +\newtheorem{lem}{Lemma} +\crefname{lem}{Lemma}{Lemmas} +\newtheorem{thm}{Theorem} +\crefname{thm}{Theorem}{Theorems} +\newtheorem{defs}{Definition} +\crefname{defs}{Def.}{Defs.} + +\usepackage{blindtext} + +%\usepackage{tudscrsupervisor} % if you want to copy the sources of the task description into the thesis + +\usepackage{csquotes} + + + +\usepackage{caption} +\captionsetup{font=sf,labelfont=bf,labelsep=space} +\usepackage{floatrow} +\floatsetup{font=sf} +\floatsetup[table]{style=plaintop} +\captionsetup{singlelinecheck=off,format=hang,justification=raggedright} +\DeclareCaptionSubType[alph]{figure} +\DeclareCaptionSubType[alph]{table} +\captionsetup[subfloat]{labelformat=brace,list=off} + +\usepackage{booktabs} +\usepackage{array} +\usepackage{tabularx} +\usepackage{tabulary} +\usepackage{tabu} +\usepackage{longtable} + +\usepackage{quoting} + +\usepackage[babel]{microtype} + +\usepackage{xfrac} + +\usepackage{enumitem} +\setlist[itemize]{noitemsep} + +\usepackage{ellipsis} +\let\ellipsispunctuation\relax + +\input{lst.tex} +\usepackage{tikz} + +\usetikzlibrary{positioning,arrows,calc,math,angles,quotes,chains,shapes} +\newcommand{\R}{\mathbb{R}} +\newcommand{\N}{\mathbb{N}} +\usepackage{hyperref} +\usepackage{mathtools} + + +\usepackage{bera}% optional: just to have a nice mono-spaced font +\usepackage{listings} +\usepackage{xcolor} +\usepackage{lstautogobble} +\usepackage{dirtree} + +\begin{document} + +\faculty{Fakultät Informatik} +\department{} +\institute{Institut für Software- und Multimediatechnik} +\chair{Lehrstuhl für Softwaretechnologie} +\title{% + Automatisierte Konstruktion kollaborativer Multi-Roboter Arbeitsräume +} + +%% for a bachelor thesis +%\thesis{bachelor} +%\graduation[B.Sc.]{Bachelor of Science} + +% for a master thesis +\thesis{bachelor} +\graduation[B.Sc.]{Bachelor of Science} + +% for a diploma thesis +%\thesis{diploma} +%\graduation[Dipl.Inf.]{Diplom-Informatiker} + +\author{Matteo Anedda} +\emailaddress[]{matteo.anedda@mailbox.tu-dresden.de} +\matriculationnumber{4732423} +\matriculationyear{2017} +\dateofbirth{17.09.1997} +\placeofbirth{Riesa} +%\discipline{Distributed Systems Engineering} + +\course{Bachelor Informatik} + +\supervisor{% + Dipl.-Inf. Sebastian Ebert + \and Dipl.-Inf. Johannes Mey% +} + +\professor{Prof. Dr. rer. nat. habil. Uwe Aßmann} +\date{01.12.2021} +\maketitle +\newpage + +\tableofcontents +\listoffigures +\listoftables + +\input{sections/einleitung} +\input{sections/grundlagen.tex} +\input{sections/sota.tex} +\input{sections/konzept.tex} +\input{sections/fallbeispiel.tex} +\input{sections/implementierung.tex} +\input{sections/evaluation.tex} +% \input{sections/zusammenfassung.tex} + +\nocite{*} +\printbibliography[heading=bibintoc]\label{sec:bibliography}% + +%\appendix +%\input{sections/appendix} + +\confirmation + +\end{document} diff --git a/bibliography.bib b/bibliography.bib index ea3ddbf..0e904ce 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -8,20 +8,60 @@ url = {https://link.springer.com/article/10.1114/1.1326030}, keywords = {Biochemistry;Biological and Medical Physics;Biomedical Engineering;Biomedical Engineering and Bioengineering;Biomedicine;Biophysics;Classical Mechanics;general;Humans;Joints/physiology;Models, Biological;Movement/physiology;Rotation;Shoulder Joint/physiology}, pages = {1381--1392}, + pagination = {page}, volume = {28}, number = {11}, issn = {0090-6964}, journal = {Annals of biomedical engineering}, doi = {10.1114/1.1326030}, - file = {A Spherical Rotation Coordinate System for the Descripti:Attachments/A Spherical Rotation Coordinate System for the Descripti.pdf:application/pdf} + file = {A Spherical Rotation Coordinate System for the Descripti:Attachments/A Spherical Rotation Coordinate System for the Descripti.pdf:application/pdf}, + publisher = {Springer and {Kluwer Academic Publishers-Plenum Publishers}} } -@book{Deserno2004, +@incollection{Chitta2016, + author = {Chitta, Sachin}, + title = {MoveIt!: An Introduction}, + pages = {3--27}, + bookpagination = {page}, + volume = {625}, + publisher = {{Springer International Publishing}}, + isbn = {978-3-319-26052-5}, + series = {Studies in Computational Intelligence}, + editor = {Koubaa, Anis}, + booktitle = {Robot Operating System (ROS)}, + year = {2016}, + address = {Cham}, + doi = {10.1007/978-3-319-26054-9_1}, + file = {MoveIt An Introduction:Attachments/MoveIt An Introduction.pdf:application/pdf} +} + + +@article{DavidGossow2011, + abstract = {PDF | On Dec 1, 2011, David Gossow and others published Interactive Markers: 3-D User Interfaces for ROS Applications | Find, read and cite all the research you need on ResearchGate}, + author = {{David Gossow} and {Adam Leeper} and {Dave Hershberger} and {Matei T. Ciocarlie}}, + year = {2011}, + title = {Interactive Markers: 3-D User Interfaces for ROS Applications}, + url = {https://www.researchgate.net/profile/david-gossow/publication/220556155_interactive_markers_3-d_user_interfaces_for_ros_applications}, + pages = {14--15}, + pagination = {page}, + volume = {18}, + number = {4}, + issn = {1070-9932}, + journal = {IEEE Robotics {\&} Automation Magazine}, + doi = {10.1109/MRA.2011.943230}, + file = {Interactive Markers 3-D User Interfaces for ROS Applications:Attachments/Interactive Markers 3-D User Interfaces for ROS Applications.pdf:application/pdf}, + publisher = {{Institute of Electrical and Electronics Engineers}} +} + + +@article{Deserno2004, author = {Deserno, Markus}, year = {2004}, title = {How to generate equidistributed points on the surface of a sphere}, - url = {https://content.instructables.com/orig/fvd/crgp/ki8um30n/fvdcrgpki8um30n.pdf} + volume = {99}, + number = {2}, + journal = {If Polymerforshung (Ed.)} } @@ -29,6 +69,7 @@ author = {Forstenhausler, Marc and Wetner, Tim and Dietmayer, Klaus}, title = {Optimized Mobile Robot Positioning for better Utilization of the Workspace of an attached Manipulator}, pages = {2074--2079}, + bookpagination = {page}, publisher = {IEEE}, isbn = {978-1-7281-6794-7}, booktitle = {2020 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM)}, @@ -37,33 +78,55 @@ } +@article{frankaHB, + author = {{Franka Emika GmbH}}, + year = {2018}, + title = {Franka-Panda User Handbook}, + file = {Franka-Panda-UserGuide-eng:Attachments/Franka-Panda-UserGuide-eng.pdf:application/pdf} +} + + +@article{Holz2015, + author = {Holz, Dirk and Ichim, Alexandru E. and Tombari, Federico and Rusu, Radu B. and Behnke, Sven}, + year = {2015}, + title = {Registration with the Point Cloud Library: A Modular Framework for Aligning in 3-D}, + pages = {110--124}, + pagination = {page}, + volume = {22}, + number = {4}, + issn = {1070-9932}, + journal = {IEEE Robotics {\&} Automation Magazine}, + doi = {10.1109/MRA.2015.2432331} +} + + @article{Hornung2013, - abstract = {Three-dimensional models provide a volumetric representation of space which is important for a variety of robotic applications including flying robots and robots that are equipped with manipulators. In this paper, we present an open-source framework to generate volumetric 3D~environment models. Our mapping approach is based on octrees and uses probabilistic occupancy estimation. It explicitly represents not only occupied space, but also free and unknown areas. Furthermore, we propose an octree map compression method that keeps the 3D models compact. Our framework is available as an open-source C++ library and has already been successfully applied in several robotics projects. We present a series of experimental results carried out with real robots and on publicly available real-world datasets. The results demonstrate that our approach is able to update the representation efficiently and models the data consistently while keeping the memory requirement at a minimum.}, + abstract = {Request PDF | OctoMap: An efficient probabilistic 3D mapping framework based on octrees | Three-dimensional models provide a volumetric representation of space which is important for a variety of robotic applications including flying... | Find, read and cite all the research you need on ResearchGate}, author = {Hornung, Armin and Wurm, Kai M. and Bennewitz, Maren and Stachniss, Cyrill and Burgard, Wolfram}, year = {2013}, title = {OctoMap: an efficient probabilistic 3D mapping framework based on octrees}, - url = {https://link.springer.com/article/10.1007/s10514-012-9321-0}, - keywords = {Artificial Intelligence;Computer Imaging;Control;Mechatronics;Pattern Recognition and Graphics;Robotics;Robotics and Automation;Vision}, + url = {https://www.researchgate.net/publication/257523133_OctoMap_An_efficient_probabilistic_3D_mapping_framework_based_on_octrees}, pages = {189--206}, + pagination = {page}, volume = {34}, number = {3}, issn = {1573-7527}, journal = {Autonomous Robots}, doi = {10.1007/s10514-012-9321-0}, - file = {OctoMap an efficient probabilistic 3D mapping framework:Attachments/OctoMap an efficient probabilistic 3D mapping framework.pdf:application/pdf} + file = {Hornung, Wurm et al. 2013 - OctoMap an efficient probabilistic 3D:Attachments/Hornung, Wurm et al. 2013 - OctoMap an efficient probabilistic 3D.pdf:application/pdf}, + publisher = {Springer Verlag} } -@book{ITG2020, - year = {2020}, - title = {ISR 2020: 52th International Symposium on Robotics in conjunction with: automatica December 9 -- 10, 2020, Online-Event}, - url = {http://www.content-select.com/index.php?id=bib_view&ean=9783800754298}, - keywords = {Collaborative Robots;Human-robot-collaboration;Industrial Robots;Mobile Robots;Robot Technologies;Robotic;Robotics in Production}, - address = {Berlin}, - edition = {Neuerscheinung}, - publisher = {{VDE Verlag}}, - isbn = {9783800754298}, - editor = {ITG, V. D.E.} +@book{Koubaa.2016, + year = {2016}, + title = {Robot Operating System (ROS)}, + address = {Cham}, + publisher = {{Springer International Publishing}}, + isbn = {978-3-319-26052-5}, + series = {Studies in Computational Intelligence}, + editor = {Koubaa, Anis}, + doi = {10.1007/978-3-319-26054-9} } @@ -71,6 +134,7 @@ author = {Makhal, Abhijit and Goins, Alex K.}, title = {Reuleaux: Robot Base Placement by Reachability Analysis}, pages = {137--142}, + bookpagination = {page}, publisher = {IEEE}, isbn = {978-1-5386-4652-6}, booktitle = {2018 Second IEEE International Conference on Robotic Computing (IRC)}, @@ -86,12 +150,14 @@ title = {Multi-Robot Assembly Strategies and Metrics}, url = {https://www.researchgate.net/publication/322260778_Multi-Robot_Assembly_Strategies_and_Metrics}, pages = {1--32}, + pagination = {page}, volume = {51}, number = {1}, issn = {0360-0300}, journal = {ACM computing surveys}, doi = {10.1145/3150225}, - file = {Multi-Robot Assembly Strategies and Metrics:Attachments/Multi-Robot Assembly Strategies and Metrics.pdf:application/pdf} + file = {Multi-Robot Assembly Strategies and Metrics:Attachments/Multi-Robot Assembly Strategies and Metrics.pdf:application/pdf}, + publisher = {{Association for Computing Machinery}} } @@ -106,13 +172,20 @@ } -@proceedings{Porges2015, +@inproceedings{Porges2015, abstract = {The utility of a mobile manipulator largely depends on its kinematic structure and mounting point on the robot body. The reachable workspace of the robot can be obtained offline and modeled as a discretized map called Reachability map. A Capability map is obtained by including some quality measure for the local dexterity of the manipulator, which helps to identify good and bad regions for manipulation. Once the maps are obtained based on forward or inverse kinematic methods, they can be used for numerous analysis tasks such as robot kinematics and workspace quality assessment, robot mounting point analysis or redundancy and failure analysis. This paper covers basic aspects of the Reachability and Capability map generation and storage, and shows particular applications of the maps for space robotics.}, - year = {2015}, + author = {Porges, Oliver and Lampariello, Roberto and Artigas, Jordi and Wedler, Armin and Borst, Christoph and Roa, Maximo A.}, title = {Reachability and Dexterity: Analysis and Applications for Space Robotics}, - url = {http://robotics.estec.esa.int/ASTRA/Astra2015/index.html}, + url = {https://elib.dlr.de/97212/}, keywords = {dexterity;reachability;robotic manipulator}, - editor = {Porges, Oliver and Lampariello, Roberto and Artigas, Jordi and Wedler, Armin and Borst, Christoph and Roa, Maximo A.} + booktitle = {Workshop on Advanced Space Technologies for Robotics and Automation (ASTRA)}, + year = {2015} +} + + +@incollection{Quigley, + author = {Quigley, Morgan and Faust, Josh and Foote, Tully and Leibs, Jeremy and others}, + title = {ROS: an open-source Robot Operating System} } @@ -123,40 +196,34 @@ url = {https://link.springer.com/article/10.1007/BF03024331}, keywords = {general;Mathematical and Computational Engineering;Mathematical and Computational Physics;Mathematical Methods in Physics;Mathematics;Numerical and Computational Physics;Simulation;Theoretical}, pages = {5--11}, + pagination = {page}, volume = {19}, number = {1}, issn = {0343-6993}, journal = {The Mathematical Intelligencer}, doi = {10.1007/BF03024331}, - file = {Distributing many points on a sphere:Attachments/Distributing many points on a sphere.pdf:application/pdf} + file = {Distributing many points on a sphere:Attachments/Distributing many points on a sphere.pdf:application/pdf}, + publisher = {Springer and {Springer US}} } -@book{Siciliano2016, +@book{Siciliano.2016, year = {2016}, - title = {Springer Handbook of Robotics}, - price = {Book : circa EUR 306.90 (AT) (freier Preis), circa sfr 306.50 (freier Preis), circa EUR 298.53 (DE) (freier Preis)}, - keywords = {Basic Principles and Methods of Robotics;Biologically-Inspired Robots;Human-Robot Interaction;Industrial Robotics;Life-Like Robotics;Manipulation and Interfaces of Robots;Mobile and Distributed Robotics;Roboethics;Robot Structures;Robotics;Robotics Foundations;Sensing and Perception of Robots;Springer Handbook of Robotics}, - address = {Cham}, - edition = {2nd ed. 2017}, - publisher = {{Springer International Publishing} and Springer}, - isbn = {9783319325507}, + title = {Springer handbook of robotics}, + url = {http://lib.myilibrary.com?id=943278}, + address = {Berlin and Heidelberg}, + edition = {2nd edition}, + publisher = {Springer}, + isbn = {9783319325521}, editor = {Siciliano, Bruno and Khatib, Oussama} } -@misc{SphericalCoordinates, - year = {10/15/2021}, - title = {Spherical Coordinates -- from Wolfram MathWorld}, - url = {https://mathworld.wolfram.com/SphericalCoordinates.html}, - urldate = {10/15/2021} -} - - @inproceedings{Tao.662011682011, author = {Tao, Long and Liu, Zhigang}, title = {Optimization on multi-robot workcell layout in vertical plane}, pages = {744--749}, + bookpagination = {page}, publisher = {IEEE}, isbn = {978-1-4577-0268-6}, booktitle = {2011 IEEE International Conference on Information and Automation}, @@ -169,6 +236,7 @@ author = {Vahrenkamp, Nikolaus and Asfour, Tamim and Dillmann, Rudiger}, title = {Robot placement based on reachability inversion}, pages = {1970--1975}, + bookpagination = {page}, publisher = {IEEE}, isbn = {978-1-4673-5643-5}, booktitle = {2013 IEEE International Conference on Robotics and Automation}, @@ -177,22 +245,11 @@ } -@misc{Wikipedia2021, - abstract = {Roll-Nick-Gier-Winkel, englisch roll-pitch-yaw angle, sind spezielle Eulerwinkel (Lagewinkel), die zur Beschreibung der Ausrichtung eines Fahrzeugs im dreidimensionalen Raum herangezogen werden. Diese Art der Richtungsmessung und -bestimmung durch Drehratensensoren wurde zur Navigation im Luftverkehr eingef{\"u}hrt und wird inzwischen neben Luftfahrzeugen auch f{\"u}r Raum-, Land- und Wasserfahrzeuge verwendet.}, - editor = {Wikipedia}, - year = {2021}, - title = {Roll-Nick-Gier-Winkel}, - url = {https://de.wikipedia.org/w/index.php?title=Roll-Nick-Gier-Winkel&oldid=209197743}, - urldate = {7/11/2021}, - doi = {Page}, - file = {Roll-Nick-Gier-Winkel:Attachments/Roll-Nick-Gier-Winkel.pdf:application/pdf} -} - - @inproceedings{Zacharias2007, author = {Zacharias, Franziska and Borst, Christoph and Hirzinger, Gerd}, title = {Capturing robot workspace structure: representing robot capabilities}, pages = {3229--3236}, + bookpagination = {page}, publisher = {IEEE}, isbn = {978-1-4244-0911-2}, booktitle = {2007 IEEE/RSJ International Conference on Intelligent Robots and Systems}, @@ -205,6 +262,7 @@ author = {Zacharias, Franziska and Sepp, Wolfgang and Borst, Christoph and Hirzinger, Gerd}, title = {Using a model of the reachable workspace to position mobile manipulators for 3-d trajectories}, pages = {55--61}, + bookpagination = {page}, publisher = {IEEE}, isbn = {978-1-4244-4597-4}, booktitle = {2009 9th IEEE-RAS International Conference on Humanoid Robots}, diff --git a/images/IMG_20211124_153815.jpg b/images/IMG_20211124_153815.jpg new file mode 100644 index 0000000000000000000000000000000000000000..ee4170330af94b9e2dd18cd651c643f70cc1de99 GIT binary patch literal 3273511 zcmex=<NpH&0WUXCHwH#V1_nk3Mh1re4@F8`D>Bm<7<_#hv=|r|I2d>t85o!u7#J9Z zKm;R%&D+4hz{t<Q1XjVou!>Ot%;sZYVA#Yc1W^Mrk5P(&6{L-Uf#Cq7G@N~cQ3I-m ziGhL95UiJ-fq~%yqY;=b!oa|Aoe5$N69W?i6O%AhJtG4{dnN-5)XX;w49t4|d8q+8 zrNs>Xey#yNp}`7{W(Eev3=9km9#B<aIv_JSu{e>z$iT?Z%Fxit$V9==)XLbx%Fv8~ zfkB~Z7R+r-MNM<z?5YkWsJ#pf3?1qaHscoth6xM|AosEu7#SH%V1~Jwsc8ZWoIPzK zBSZ}&6EnjkMg}g3YnWC|VuG8$X%aJBeAgrvIQ!5fR=Ao|lh`5lf=r&o0cA5VFihfv zvIQ6zCUL>lUz#)lq!MH|)2&Gp;p~T#Cc)V+7QozZXkf4a8mFN6TEGMgTOo!8%rG`@ 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b/lst.tex @@ -46,6 +46,9 @@ \lstdefinelanguage{JSON}{ style=common-style, morecomment=[s]{<!--}{-->}, + morecomment=[l]{\%}, + float=tp, + floatplacement=tbp, literate= *{0}{{{\color{numb}0}}}{1} {1}{{{\color{numb}1}}}{1} diff --git a/sections/einleitung.tex b/sections/einleitung.tex index be2ebf4..ddfdd32 100644 --- a/sections/einleitung.tex +++ b/sections/einleitung.tex @@ -1,5 +1,5 @@ \chapter{Einleitung}\label{ch:introduction} -'Collaborative Multi-Robot Work-Cells' beschreibt ein Gebiet der Robotik, dessen Inhalt die Kooperation mehrerer robotischer Systeme hinsichtlich der Lösung einer spezifischen Aufgabe innerhalb einer räumlichen Domäne ist. Die zugrundeliegenden Algorithmen ermöglichen dabei die Bewältigung der Aufgaben ohne Kollisionen untereinander, sowie mit Hindernissen in der Umgebung. \par +'Kollaborative Multi-Roboter Arbeitsräume' beschreiben ein Gebiet der Robotik, dessen Inhalt die Kooperation mehrerer robotischer Systeme hinsichtlich der Lösung einer spezifischen Aufgabe innerhalb einer räumlichen Domäne ist. Die zugrundeliegenden Algorithmen ermöglichen dabei die Bewältigung der Aufgaben ohne Kollisionen untereinander, sowie mit Hindernissen in der Umgebung. \par Die Planung der Abläufe ist abhängig von der Beschaffenheit des Roboters, denn die Basis eines Roboters kann beweglich oder fest sein. Beispielsweise sind Roboterarme fest auf einer Platte montiert und somit fester Bestandteil der Umgebung. Ein Wechsel ihrer Position würde bedeuten, eine andere Stelle auf der Platte für die Montage zu präparieren, den Roboterarm zu demontieren und an anderer Stelle zu platzieren, was einen enormen Aufwand darstellt und den Erfolg der Handlung nicht garantiert. Daraus ist ersichtlich, dass die Positionierung unflexibler Roboter ein Problem aufweist, dessen Lösung direkten Einfluss hinsichtlich der zeitlicher Komponente, sowie der Bewältigung der spezifischen Aufgabe hat. Die Absolvierung dieser Aufgabe wird demzufolge durch effizientere Nutzung der Arbeitsumgebung besser umgesetzt, aufwändige Montagearbeiten während der Bearbeitung werden reduziert und der dadurch erschlossene Raum sowie die resultierende Zeit kann gewinnbringender genutzt werden. Im Anbetracht der stetig wachsenden Industrie und der sich daraus ergebenden Nachfrage nach effizienten automatisierten Produktionsverfahren, die den limitierten Arbeitsraum einer Produktionsstätte optimal nutzen, ist die Platzierung der Roboter ein entscheidender erster Schritt zur Optimierung des gesamten Arbeitsprozesses. \section{Zielsetzung der wissenschaftlichen Arbeit} diff --git a/sections/fallbeispiel.tex b/sections/fallbeispiel.tex index bebb4d0..9943b30 100644 --- a/sections/fallbeispiel.tex +++ b/sections/fallbeispiel.tex @@ -1,5 +1,5 @@ \chapter{Fallbeispiel}\label{ch:caseStudy} -Inhalt dieses Kapitels ist die Demonstration der Funktionsweise einzelner Elemente der Implementierung und dessen Synergie mit MoveIt spezifischen Komponenten, sowie der Integration des franka\_ros Pakets \footnote{\url{http://wiki.ros.org/franka_ros}} anhand eines Beispiels. Dieses Paket stellt hierbei die sowohl grafische, als auch funktionelle Beschreibung der Bestandteile des gleichnamigen robotischen Systems zur Verfügung, welches primär Objekt der Ausführung ist. +Inhalt dieses Kapitels ist die Demonstration der Funktionsweise einzelner Elemente der Implementierung und dessen Synergie mit \emph{MoveIt!} spezifischen Komponenten, sowie der Integration des franka\_ros Pakets \footnote{\url{http://wiki.ros.org/franka_ros}} anhand eines Beispiels. Dieses Paket stellt hierbei die sowohl grafische, als auch funktionelle Beschreibung der Bestandteile des gleichnamigen robotischen Systems zur Verfügung, welches primär Objekt der Ausführung ist. \section{Ausgangspunkt kinematischer Operationen} MoveIt bietet über die move\_group Schnittstelle eine Möglichkeit die Kommunikation mittels spezieller Nachrichten und somit der Planung und Exekution kinematischer Operationen durch den Roboter. Dies bedingt eine Konfigurationsdatei, welche das robotische System konkretisiert, indem beispielsweise Kontrollelemente wie Endeffektoren als Gruppen definiert oder Adjazenzen der einzelnen Festkörper in Form einer Kollisionsmatrix erfasst werden oder weitere Nodes initialisiert werden. Die Konfigurationsdatei stellt dadurch den Ausgangspunkt aller operativen Aspekte des Fallbeispiels dar und ist daher sowohl für einen Roboter, als auch für mehrere robotische Systeme in der Implementierung enthalten. Diese Konfigurationen wurden mittels \emph{moveit setup assistant (MSA)} generiert. @@ -11,33 +11,30 @@ Die \emph{move\_group} Schnittstelle bietet keine Möglichkeit zur Deklaration e Die Kalkulation valider Positionen für robotische Systeme erfordert die roboterzentrierte Inspektion des Arbeitsraums und eine Aufgabenbeschreibung, welche für $i \in \N_{>0}$ Endsysteme mindestens $i$ Operationsketten aufweist. Die diesbezüglich vorzunehmende Arbeitsraumanalyse und dessen Inversion ist aufgabenunabhängig, erfolgt daher a priori und kann auf alle Aufgabenbeschreibung angewendet werden, die vom Ausgangssystem der Analyse, wie beispielsweise dem \emph{Panda} Roboter der Franka Emika GmbH, operiert werden sollen. Die Aufgabenbeschreibung kann zum Ermittlungszeitpunkt vorgenommen werden oder schon persistiert vorliegen. \subsection{Aufgabenkonstruktion} -Die Aufgabenbeschreibung in Form einer Operationskette erfolgt über interaktive Marker, die jeweils die Abstell- beziehungsweise Greifposition eines Primitives darstellen. Jede dieser Positionen erfordert ein zusätzliches Kollisionsobjekt als Operationsoberfläche \emph{Support\_Surface}, welche dem robotischen System über das spezifische Nachrichten kommuniziert wird und ohne dessen eine Exekution nicht erfolgt. Dieser redundante Aufwand wird im Algorithmus berücksichtigt und ist implizit durch die Position und Dimension des Kettenglieds definiert. Demzufolge ist beispielsweise eine Differenzierung der zu erzeugenden Kollisionsobjekte, durch Platzierung zusätzlicher Abstelltische, kein Aspekt der Aufgabenbeschreibung. Ausgangspunkt einer Operationskette und dessen Teilketten ist der Startknoten, welcher durch seine Identität $id=0$ und der Farbe gekennzeichnet ist. Es ist stets möglich, einen weiteren Knoten zu generieren, der als weiteres Kettenglied farblich und schriftlich gekennzeichnet ist. Die Schnittoption im Menü des Startknotens dupliziert diesen und ist ab einer Operationskette beziehungsweise Teilkette der Länge zwei möglich. Eine valide Aufgabenbeschreibung ist nur formulierbar und wird persistiert, wenn mindestens zwei Teilketten existieren die jeweils mindestens zwei Glieder enthalten, da dies die Notwendigkeit zweier robotischer Systeme und das Vorhandensein einer Greif und Abstellposition impliziert. Die Wiederherstellung des Ausgangszustands ist ebenfalls durch einen zusätzlichen Menüpunkt möglich, dies eliminiert alle Glieder bis auf den Startknoten. Anlog dupliziert die Kooperationsoption das jeweils letzte Glied der Operationskette beziehungsweise Teilkette und visualisiert diese Schnittmenge ebenfalls schriftlich und farblich. Dies impliziert, dass eine Operationskette aus Schnitt- und Kooperationskomponenten bestehen kann, was für deren Anwendung auf $n \in \N_{>2}$ robotischen Systemen relevant ist. - - - - -%\begin{figure} -%\ffigbox[\FBwidth]% -% {\begin{subfloatrow}% -% \ffigbox[\FBwidth]% -% {\fbox{\includegraphics[width=0.35\textwidth, height = 6cm]{images/task1.png}}}% -% {\caption{Definition eines neuen Kettenglieds}} -% \ffigbox[\FBwidth]% -% {\fbox{\includegraphics[width=0.35\textwidth, height =6cm]{images/task2.png}}}% -% {\caption{Option zur Erstellung von Kooperationsgliedern}} -% \end{subfloatrow}% -% {\caption{}} -%\ffigbox[\FBwidth]% -% \begin{subfloatrow}% -% \ffigbox[\FBwidth]% -% {\fbox{\includegraphics[width=0.35\textwidth, height =6cm]{images/task4.png}}}% -% {\caption{Option zur Spaltung der Operationskette}} -% \ffigbox[\FBwidth]% -% {\fbox{\includegraphics[width=0.35\textwidth, height =6cm]{images/task3.png}}}% -% {\caption{Generierung der Aufgabenbeschreibung anhand einer validen Operationskette }} -% \end{subfloatrow}}% -% {\caption{Generierung der Aufgabenbeschreibung anhand einer validen Operationskette }} -%\end{figure} +Die Aufgabenbeschreibung in Form einer Operationskette erfolgt über interaktive Marker, die jeweils die Abstell- beziehungsweise Greifposition eines Primitives darstellen. Jede dieser Positionen erfordert ein zusätzliches Kollisionsobjekt als Operationsoberfläche \emph{Support\_Surface}, welche dem robotischen System über spezifische Nachrichten kommuniziert wird und ohne dessen eine Exekution nicht erfolgt. Dieser redundante Aufwand wird im Algorithmus berücksichtigt und ist implizit durch die Position und Dimension des Kettenglieds definiert. Demzufolge ist beispielsweise eine Differenzierung der zu erzeugenden Kollisionsobjekte, durch Platzierung zusätzlicher Abstelltische, kein Aspekt der Aufgabenbeschreibung. Ausgangspunkt einer Operationskette und dessen Teilketten ist der Startknoten, welcher durch seine Identität $id=0$ und der Farbe gekennzeichnet ist. Es ist stets möglich, einen weiteren Knoten zu generieren, der als weiteres Kettenglied farblich und schriftlich gekennzeichnet ist. Die Schnittoption im Menü des Startknotens dupliziert diesen und ist ab einer Operationskette beziehungsweise Teilkette der Länge zwei möglich. Eine valide Aufgabenbeschreibung ist nur formulierbar und wird persistiert, wenn mindestens zwei Teilketten existieren die jeweils mindestens zwei Glieder enthalten, da dies die Notwendigkeit zweier robotischer Systeme und das Vorhandensein einer Greif und Abstellposition impliziert. Die Wiederherstellung des Ausgangszustands ist ebenfalls durch einen zusätzlichen Menüpunkt möglich, dies eliminiert alle Glieder bis auf den Startknoten. Anlog dupliziert die Kooperationsoption das jeweils letzte Glied der Operationskette beziehungsweise Teilkette und visualisiert diese Schnittmenge ebenfalls schriftlich und farblich. Dies impliziert, dass eine Operationskette aus Schnitt- und Kooperationskomponenten bestehen kann, was für deren Anwendung auf $n \in \N_{>2}$ robotischen Systemen relevant ist. + + +\begin{figure}[h!] +\ffigbox[\FBwidth]% + {\begin{subfloatrow}[2] + \ffigbox[\FBwidth]% + {\fbox{\includegraphics[width=0.35\textwidth, height = 6cm]{images/task1.png}}}% + {\caption{Definition eines neuen Kettenglieds}\label{fig:bild1}}% + \ffigbox[\FBwidth]% + {\fbox{\includegraphics[width=0.35\textwidth, height = 6cm]{images/task2.png}}}% + {\caption{Option zur Erstellung von Kooperationsgliedern} \label{fig:bild2}} + \end{subfloatrow} + \begin{subfloatrow}[2] + \ffigbox[\FBwidth]% + {\fbox{\includegraphics[width=0.35\textwidth, height = 6cm]{images/task3.png}}}% + {\caption{Definition eines neuen Kettenglieds}\label{fig:bild3}}% + \ffigbox[\FBwidth]% + {\fbox{\includegraphics[width=0.35\textwidth, height = 6cm]{images/task4.png}}}% + {\caption{Option zur Erstellung von Kooperationsgliedern} \label{fig:bild4}} + \end{subfloatrow}} + {\caption{ Darstellung der Mengen $\protect P_{total}$ und $\protect OR_{total}$. Ein roter Marker repräsentiert den Vektoren, den der Endeffektor einnimmt. Der grüne Pfeil zeigt in die Richtung, in welche die Finger des Endeffektors ausgerichtet sind.} +\label{fig:fafesf}}% +\end{figure} \section{Positionsanalyse} Der Algorithmus zur Ermittlung von Positionen robotischer Systeme bezüglich einer Aufgabenbeschreibungen erfordert diese und zusätzlich den invertierten Arbeitsraum als Parameter. Alle Resultate sind Transparent über das graphische Programm RViz einsehbar, dabei kann der Umfang der visualisierten Informationen über Checkboxen reguliert werden. Die Metrik $D_{Reach}(Voxel)(TK)$ aus \autoref{eq:29}, welche die differenzierte Voxelisierung kalkuliert, visualisiert die Wertigkeit eines Voxels. Analog zur Menge $P_{Base}$ jedes Kettenglieds und deren Differenzierung in die jeweilige Operationskette beziehungsweise Teilkette ist zusätzlich die Projektion dieser Werte auf die XY Ebene des kartesischen Koordinatensystems eine mögliche visuelle Modalität. Der maximale Index je Teilkette und Voxel ist ebenfalls einsehbar und wird mit zusätzlichen Informationen, wie dem Namen der Aufgabenbeschreibung, persistiert. diff --git a/sections/grundlagen.tex b/sections/grundlagen.tex index 8635000..488aa62 100644 --- a/sections/grundlagen.tex +++ b/sections/grundlagen.tex @@ -3,21 +3,22 @@ Wesentliche Konzepte dieser Bachelorarbeit fundieren auf verschiedenen wissensch \section{Räumliche Partitionierung} -Die räumliche Partitionierung beschreibt das Unterteilen eines kartesischen Raumes in kleinere, disjunkte Teilräume und deren Indexierung und Speicherung in entsprechenden Datenstrukturen. Die Implementierung einer solcher Datenstruktur ist oft eine erweiterte Baumstruktur mit speziellen Eigenschaften, die dessen Einsatz im 3 dimensionalen Raum begünstigen. Beispielsweise sind solche Bäume aufgrund dessen Kompaktheit für die Aufbewahrung großer Datenmengen geeignet oder ermöglichen eine schnelle Suche unter dessen Blättern durch ihre Implementierung \cite{Hornung2013}. In der Literatur wird die Unterteilung eines quadratischen Raumes in Würfel fester Größe \emph{Voxelisierung} genannt \cite{Makhal2018} \cite{Zacharias2007}, wobei ein Voxels per Definition ein Pixel im 3-dimensionalen Raum ist. Fortan werden disjunkte Teilräume eines kartesischen Raums \emph{Voxel} genannt, die einen Container darstellen, dessen Inhalt eine Menge aus $n \in \N$ Vektoren $v \in \R^{3}$ ist. - +\label{sec:partitionierung} +Die räumliche Partitionierung beschreibt das Unterteilen eines kartesischen Raumes in kleinere, disjunkte Teilräume und deren Indexierung und Speicherung in entsprechenden Datenstrukturen. Die Implementierung einer solcher Datenstruktur ist oft eine erweiterte Baumstruktur mit speziellen Eigenschaften, die dessen Einsatz im 3 dimensionalen Raum begünstigen \cite{Hornung2013}. Beispielsweise sind solche Bäume aufgrund dessen Kompaktheit für die Aufbewahrung großer Datenmengen geeignet oder ermöglichen eine schnelle Suche unter dessen Blättern durch ihre Implementierung. In der Literatur wird die Unterteilung eines quadratischen Raumes in Volumen fester Größe \emph{Voxelisierung} genannt \cite{Makhal2018} \cite{Zacharias2007} \cite{Porges2015}. Diese Volumen beziehungsweise \emph{Voxel} definieren disjunkte Teilräume des kartesischen Raums. \section{Diskretisierung eines Objektes} +\label{sec:diskretisierung} Die Diskretisierung ist bekannt als ein Prozess der Zerlegung einer kontinuierliche Oberfläche durch Abtastung in ihre diskreten Teilbereiche oder Punkte, welche im Kontext dieser Arbeit durch Vektoren beschrieben sind. Anwendung findet dieses Verfahren Beispielsweise im Gebiet der Netzwerktechnologie, in der ein analoges Signal durch räumliche und zeitliche Diskretisierung in ein digitales Signal überführt wird. Angewendet auf 3-dimensionale Objekte entsteht durch Diskretisierung ein Gitter aus Vektoren, welche das zugrundeliegende Objekt approximieren \cite{Porges2015}. \section{Robotische Systeme} -Robotische Systeme setzen sich aus einer Menge an Festkörpern und deren Verknüpfungen durch verschiedene Gelenktypen zusammen. Ein solches System spannt eine Baumstruktur auf, dessen Wurzel durch eine fixe Basis definiert ist. Ein solcher kinematischer Baum ist verzweigt, sobald ein Festkörper Verknüpfungen zu mehreren Festkörpern aufweist. Solche kinematischen Strukturen können auch als verzweigte beziehungsweise unverzweigte kinematische Kette denotiert werden. Bildet der Baum keinen Kreis, so ist er eine offene kinematische Kette beziehungsweise ein offenes robotisches System. Diese robotischen Mechanismen definieren eine offene kinematische Kette, dessen letztes Glied als Endeffektor deklariert ist, während der erste Festkörper die Basis des Roboters darstellt \cite[46ff.]{Siciliano2016}. Roboterarme sind Beispiele für offene robotische Systeme beziehungsweise offene kinematischer Ketten und Bezugspunkt dieser wissenschaftlichen Arbeit. Nähere Informationen über die Kopplung spezifischer Festkörper werden im Kontext dieser Arbeit in der zugehörigen URDF beziehungsweise XACRO Datei des Roboters dokumentiert. Diese sind XML Erweiterungen zur Beschreibung von Robotern und bieten als solche einen entsprechenden erweiterten Funktionsumfang. Demzufolge können zusätzliche Informationen, in Form von \emph{tags}, Limitierungen bezüglicher des Bewegungsumfangs eines Festkörpers spezifizieren. \par -Ein Roboterarm kann anhand seiner Komponenten durch das Inkludieren einzelner URDF Dateien modelliert werden. Neben der Möglichkeit, ein robotisches System eigens zu konstruieren, werden fertige Systeme von offizieller Seite \cite{urdf_Examples_ros} bereitgestellt, wodurch ein Einblick in umfangreichere URDF-Dateien ermöglicht wird und der Konstruktionsaufwand entfällt. Beispielsweise ist das Referenzsystem \emph{Panda} der Franka Emika GmbH öffentlich zugänglich \cite{franka_ros}. +Robotische Systeme setzen sich aus einer Menge an Festkörpern und deren Verknüpfungen durch verschiedene Gelenktypen zusammen. Ein solches System spannt eine Baumstruktur auf, dessen Wurzel durch eine fixe Basis definiert ist. Ein solcher kinematischer Baum ist verzweigt, sobald ein Festkörper Verknüpfungen zu mehreren Festkörpern aufweist. Solche kinematischen Strukturen können auch als verzweigte beziehungsweise unverzweigte kinematische Kette denotiert werden. Bildet der Baum keinen Kreis, so ist er eine offene kinematische Kette beziehungsweise ein offenes robotisches System, dessen letztes Glied als Endeffektor deklariert ist, während der erste Festkörper die Basis des Roboters darstellt \cite[46ff.]{Siciliano2016}. Roboterarme sind Beispiele für offene robotische Systeme beziehungsweise offene kinematischer Ketten und Bezugspunkt dieser wissenschaftlichen Arbeit. Nähere Informationen über die Kopplung spezifischer Festkörper werden im Kontext dieser Arbeit in der zugehörigen URDF \footnote{\url{http://wiki.ros.org/urdf}} beziehungsweise XACRO \footnote{\url{http://wiki.ros.org/xacro}} Datei des Roboters dokumentiert. Innerhalb dieser Dateien können zusätzliche Informationen in Form von \emph{tags}, beispielsweise Limitierungen bezüglicher des Bewegungsumfangs eines Festkörpers, spezifizieren werden. \par +Ein Roboterarm kann anhand seiner Komponenten durch das Inkludieren einzelner URDF Dateien modelliert werden. Neben der Möglichkeit, ein robotisches System eigens zu konstruieren, sind fertige Systeme öffentlich \footnote{\url{http://wiki.ros.org/urdf/Examples}} zugänglich. So ist es beispielsweise auch die Roboterbeschreibung des Referenzsystems dieser wissenschaftlichen Arbeit, des Roboterarms \emph{Panda} der Franka Emika GmbH \footnote{\url{http://wiki.ros.org/franka_ros}}. \section{Kinematische Grundlagen} -Nachdem bereits die Klassifizierung eines robotischen Systems anhand seiner Baumstruktur eruiert wurde, werden in diesem Abschnitt weitere kinematischen Aspekte näher erläutert. Grundlegende Methodiken dieser wissenschaftlichen Arbeit fundieren auf Konzepten der Kinematik, welche als Teilbereich der Physik die Bewegung von Festkörpern in robotischen Systemen, abstrahierend von den daraus resultierenden Kräften und Momenten, umfasst \cite[9f.]{Siciliano2016}. Ein wesentlicher Aspekt ist hierbei die Repräsentation der Position und Orientierung von Körpern im 3-dimensionalen Raum und deren Relation zueinander, sowie die Beschreibung der Geometrie robotischer Mechanismen. Termini wie \emph{Pose} oder Referenzkoordinatensystem, in dieser Arbeit \emph{Frame}, sind existenzieller Bestandteil späterer Ausführungen und werden daher in diesem Kapitel näher erläutert. +Nachdem bereits die Klassifizierung eines robotischen Systems anhand seiner Baumstruktur eruiert wurde, werden in diesem Abschnitt weitere kinematischen Aspekte näher erläutert. Grundlegende Methodiken dieser wissenschaftlichen Arbeit fundieren auf Konzepten der Kinematik, welche als Teilbereich der Physik die Bewegung von Festkörpern in robotischen Systemen, abstrahierend von den daraus resultierenden Kräften und Momenten, umfasst \cite[9f.]{Siciliano2016}. Ein wesentlicher Aspekt ist hierbei die Repräsentation der Position und Orientierung von Körpern im 3-dimensionalen Raum und deren Relation zueinander, sowie die Beschreibung der Geometrie robotischer Mechanismen. \subsection{Pose, Position und Orientierung} -Die Kombination aus Position und Orientierung eines Festkörpers relativ zu einem Frame definiert eine \emph{Pose}. Frames bestehen aus dessen Koordinatenursprung und orthogonalen Basisvektoren $(v_{0}, v_{1}, v_{2})$. Beispielsweise beinhaltet das bekannte Referenzkoordinatensystem $Frame_{0}$ den Ursprung $(0,0,0)^{T}$ und Basisvektoren $(0,0,1)^{T}, (0,1,0)^{T}, (1,0,0)^{T}$, auf Grundlage dessen eine Pose $^{0}Pose$ relativ zu $Frame_{0}$ definiert und repräsentiert werden kann. Ein robotisches System ist durch die Definition der $n \in \N$ Frames für jedes seiner Festkörper in der URDF Datei konkretisiert \cite[9ff.]{Siciliano2016}. +Die Kombination aus Position und Orientierung eines Festkörpers relativ zu einem Frame definiert eine \emph{Pose}. Frames bestehen aus dessen Koordinatenursprung und orthogonalen Basisvektoren $(v_{0}, v_{1}, v_{2})$. Beispielsweise beinhaltet das bekannte Referenzkoordinatensystem $Frame_{0}$ den Ursprung $(0,0,0)^{T}$ und Basisvektoren $(0,0,1)^{T}, (0,1,0)^{T}, (1,0,0)^{T}$ auf deren Grundlage eine Pose $^{0}Pose$ relativ zu $Frame_{0}$ definiert und repräsentiert werden kann. Ein robotisches System ist durch die Definition der $n \in \N$ Frames für jedes seiner Festkörper in der URDF Datei konkretisiert \cite[9ff.]{Siciliano2016}. \subsubsection{Position} Der Koordinatenursprung eines Frames $i$ relativ zum Frame $j$ wird durch den Vektor $^{j}p_{i} \in \R^{3}$ gekennzeichnet. @@ -33,11 +34,11 @@ Der Koordinatenursprung eines Frames $i$ relativ zum Frame $j$ wird durch den Ve Aus \autoref{eq:1} ist ersichtlich, dass die Elemente des Koordinatenursprungs von Frame $i$ auf Koordinaten relativ zu $j$ referenzieren. Diese Abhängigkeit besteht analog zwischen Frames eines Roboters, welche jeweils relativ zum Vorgänger Frame der kinematischen Kette definiert sind. \subsubsection{Orientierung} -Eine Orientierung wird durch Rotationen $R$ dargestellt, die auf einen Körper angewendet werden können. Im Forschungsgebiet Robotik und der Computergrafik existieren verschiedene Ansätze zur mathematischen Beschreibung einer Rotation, welche sich konzeptuell unterscheiden. Beispielsweise kann eine Rotation durch Kompositionen aus Rotations-Matrizen oder der Anwendung komplexer Zahlen realisiert werden. Die spezielle RPY-Rotation\footnote{\url{https://de.wikipedia.org/w/index.php?title=Roll-Nick-Gier-Winkel&oldid=209197743}}, welche auf den Rotationswinkeln eines Körpers hinsichtlich seiner Achsen basiert, wird in \autoref{fig:1} anhand eines Flugzeugs visualisiert und ist Bestandteil späterer Ausführungen. +Eine Orientierung wird durch Rotationen $R$ dargestellt, die auf einen Körper angewendet werden können. Im Forschungsgebiet Robotik und der Computergrafik existieren verschiedene Ansätze zur mathematischen Beschreibung einer Rotation, welche sich konzeptuell unterscheiden. Beispielsweise kann eine Rotation durch Kompositionen aus Rotations-Matrizen oder der Anwendung komplexer Zahlen realisiert werden. Die spezielle RPY-Rotation\footnote{\url{https://de.wikipedia.org/w/index.php?title=Roll-Nick-Gier-Winkel&oldid=209197743}\label{RPY}}, welche auf den Rotationswinkeln eines Körpers hinsichtlich seiner Achsen basiert, wird in \autoref{fig:1} anhand eines Flugzeugs visualisiert und ist Bestandteil späterer Ausführungen. \begin{figure}[ht] \includegraphics[width = 4cm, height = 4cm]{images/RPY.png} -\caption{Roll-Pitch-Yaw beziehungsweise Roll-, Nick- und Gier- Winkel an einem Flugzeug demonstriert.} +\caption{Roll-Pitch-Yaw beziehungsweise Roll-, Nick- und Gier- Winkel an einem Flugzeug demonstriert.\footref{RPY}} \label{fig:1} \end{figure} @@ -58,11 +59,11 @@ Die Menge aller Rotations-Matrizen $R \in \R^{3 \times 3}$ bilden mit der Matrix Die Komposition von Rotationen verschiedener Frames wird in \autoref{eq:2} veranschaulicht. Dabei ist die Einhaltung der Reihenfolge bezüglich der Multiplikation, aufgrund der mangelnden Kommutativität der Gruppe SO(3), erforderlich. Durch dessen Eigenschaft der Geschlossenheit, ist das Resultat dieser Multiplikation ebenfalls $\in SO(3)$ \cite[11f.]{Siciliano2016}. \begin{equation} \label{eq:2} -^{k}R_{i} = ^{k}R_{j} \times ^{j}R_{i} +^{k}R_{i} = \prescript{k}{}{R}_{j} \times \prescript{j}{}{R}_{i} \end{equation} \subsubsection{homogene Transformation} -Eine homogene Matrix $T \in \R^{4\times4}$, bestehend aus einer Rotationsmatrix $R \in SO(3)$ und einem Vektor $p \in \R ^{3}$, beschreibt eine Transformations-Matrix, siehe \autoref{eq:3}. Diese ermöglicht die synchrone Ausführung von Translation und Rotation, wobei eine Translation die Verschiebung eines beliebigen Vektors beschreibt, ohne sich auf dessen Orientierung auszuwirken. Die aus Frame Rotationen $^{j}R_{i}$ und Koordinatenursprung $^{j}p_{i}$ bestehende Transformationsmatrizen $^{j}T_{i}$ eignet sich am besten zur Beschreibung einer Pose $i$ und ermöglichen zusätzlich die Transformation eines Vektors $^{i}u$ in den Vektor $^{j}u$, wie \autoref{eq:4} zeigt. +Eine homogene Matrix $T \in \R^{4\times4}$, bestehend aus einer Rotationsmatrix $R \in SO(3)$ und einem Vektor $p \in \R ^{3}$, beschreibt eine Transformations-Matrix, siehe \autoref{eq:3}. Diese ermöglicht die synchrone Ausführung von Translation und Rotation, wobei eine Translation die Verschiebung eines beliebigen Vektors beschreibt, ohne sich auf dessen Orientierung auszuwirken. Die aus Frame Rotationen $^{j}R_{i}$ und Koordinatenursprung $^{j}p_{i}$ bestehende Transformations-Matrix $^{j}T_{i}$ eignet sich am besten zur Beschreibung einer Pose $i$ und ermöglichen zusätzlich die Transformation eines Vektors $^{i}u$ in den Vektor $^{j}u$, wie \autoref{eq:4} zeigt. \begin{equation} \label{eq:3} @@ -84,7 +85,7 @@ T &= \begin{pmatrix} ^{j}u \\ 1 \\ \end{pmatrix} -= ^{j}T_{i} \times \begin{pmatrix} += \prescript{j}{}{T}_{i} \times \begin{pmatrix} ^{i}u \\ 1 \\ \end{pmatrix} @@ -104,7 +105,7 @@ Transformationsmatrizen $T$ bilden mit der Matrixmultiplikation eine nicht kommu Die Komposition aus Transformationen verschiedener Frames wird in Gleichung \autoref{fig:5} veranschaulicht. Dabei ist die Einhaltung der Reihenfolge bezüglich der Multiplikation, analog zur \autoref{eq:2} Gruppe $SO(3)$, erforderlich. Durch die Eigenschaft der Geschlossenheit der Gruppe SE(3), ist das Resultat dieser Multiplikation ebenfalls $\in SE(3)$ \cite[14f.]{Siciliano2016}. \begin{equation}\label{eq:5} -^{k}T_{i} = ^{k}T_{j} \times ^{j}T_{i} +^{k}T_{i} = \prescript{k}{}{T}_{j} \times \prescript{j}{}{T}_{i} \end{equation} Für die kinematische Kette eines robotischen Systems bestehenden aus $n \in \N$ Festkörpern, wobei $Frame_{n} = Frame_{Endeff}$ das Frame des Endeffektors ist, gilt \autoref{eq:6}, welche zusätzlich ein Beispiel für $n=8$ zeigt. Im Kontext dieser Arbeit bedeutet diese Gleichung, dass für jede vom Roboter erreichbare Endeffektor-Pose relativ zu $Frame_{0}$ eine solche Kette aus Transformationen existiert, welche jeweils die Pose eines Festkörpers des Roboters repräsentieren \cite[26f.]{Siciliano2016}. @@ -113,7 +114,7 @@ Für die kinematische Kette eines robotischen Systems bestehenden aus $n \in \N$ \label{eq:6} \begin{split} ^{0}T_{n} &= \prod_{i=0}^{n}{^{n-i}T_{i}} \\ -^{0}T_{8} &= ^{0}T_{1} \times ^{1}T_{2} \times ^{2}T_{3} \times ^{3}T_{4} \times ^{4}T_{5} \times ^{5}T_{6} \times ^{6}T_{7} \times ^{7}T_{8} +^{0}T_{8} &= \prescript{0}{}{T}_{1} \times \prescript{1}{}{T}_{2} \times \prescript{2}{}{T}_{3} \times \prescript{3}{}{T}_{4} \times \prescript{4}{}{T}_{5} \times \prescript{5}{}{T}_{6} \times \prescript{6}{}{T}_{7} \times \prescript{7}{}{T}_{8} \end{split} \end{equation} @@ -127,12 +128,7 @@ Für die kinematische Kette eines robotischen Systems bestehenden aus $n \in \N$ \section{Arbeitsbereich eines robotischen Systems} -Der Arbeitsbereich 'Workspace', beschreibt die Kapazitäten eins robotische Systems bezüglich der Interaktion mit dessen Umfeld. Die Darstellung des Arbeitsbereiches erfolgt durch 'reachability maps' oder 'capability maps', welche die Erreichbarkeit der Vektoren $v$ im Umfeld des Roboters durch den Endeffektor charakterisiert. -In den Publikationen werden diese Darstellungen gleichgesetzt. Dieses Vorgehen hat auch Bestand in dieser wissenschaftlichen Arbeit, demnach werden fortan reachability maps betrachtet, welche zusätzliche Eigenschaften der capability map besitzt. - -\subsection{reachability map} -Reachability maps visualisieren die Erreichbarkeit durch den Endeffektor, ohne weitere Angaben hinsichtlich der Qualität. Demnach ist beispielsweise ein Objekt erreichbar, was aber nicht eine erfolgreiche Exekution impliziert, da der Endeffektor Möglicherweise eine der definierten Greifposen nicht einnehmen kann. Demnach enthält eine reachability map die Information darüber, dass der Endeffektor die Objektposition erreicht, aber nicht welche Posen Operiert werden können. -Diese \autoref{fig:panda} zeigt eine solche Darstellung im Manual\footnote{\url{https://www.generationrobots.com/media/panda-franka-emika-datasheet.pdf}} des Referenzroboters 'Panda' aus verschiedenen Perspektiven, um den Arbeitsbereich des Roboters in mehreren Dimensionen zu visualisieren. +Der Arbeitsbereich \emph{Workspace}, beschreibt die Kapazitäten eins robotische Systems bezüglich der Interaktion mit dessen Umfeld. \autoref{fig:panda} aus dem Manual \cite{frankaHB} des Referenzroboters 'Panda' zeigt dessen Arbeitsbereich schematisch aus verschiedenen Perspektiven. Dabei erfolgt keine detaillierte Charakterisierung eines Bereiches, denn Eigenkollisionen des Roboters werden beispielsweise nicht erfasst beziehungsweise nicht gekennzeichnet. In dieser wissenschaftlichen Arbeit ist daher die Ermittlung spezieller Datenstrukturen erforderlich, um den Arbeitsbereich des Referenzroboters differenziert zu inspizieren. Zacharias et al. \cite{Zacharias2007} \cite{Zacharias2009} definieren hierzu 'reachability maps' und 'capability maps', welche die Erreichbarkeit der Vektoren $v$ im Umfeld des Roboters durch den Endeffektor klassifizieren. \ffigbox[\FBwidth]% {\begin{subfloatrow}% @@ -145,16 +141,17 @@ Diese \autoref{fig:panda} zeigt eine solche Darstellung im Manual\footnote{\url{ \end{subfloatrow}}% {\caption{Illustration des Arbeitsraums eines Panda Roboters der Franka Emika GmbH aus unterschiedlichen Perspektiven. Alle Angaben sind in Millimeter bemessen.}\label{fig:panda}}% +\subsection{reachability map} +Die reachability map ist eine Datenstruktur, welche die Erreichbarkeit eines Vektors $v$ anhand aller Orientierungen charakterisiert, die der Endeffektor einnehmen kann. Sie enthält eine Menge aus Tupeln, welche die Pose des Endeffektors sowie der Information bezüglich deren Erreichbarkeit in Form eines Wahrheitswertes spezifiziert. Diese Informationen können zusätzlich durch die Integration einer Metrik genutzt werden, um beispielsweise den Arbeitsbereich differenziert 3 dimensional Darzustellen, oder um mit den Posen weitere Berechnungen zu führen \cite{Zacharias2009}. Aufgrund des Aufbaus einer reachability map, eignet sich diese als Grundlage für spätere Verfahren und wird in dieser wissenschaftlichen Arbeit genutzt. \subsection{capability map} -In capability maps werden zusätzliche Informationen über die Erreichbarkeit der Bereiche im Umfeld des Roboters mit Farben gekennzeichnet. Die Färbung basiert auf der Qualifizierung, welche durch die berechenbaren Endeffektor Posen im Bereich bestimmt wird. Beispielsweise ist die Farbe rot ein Indikator für schwer vom Roboter zugängliche Bereiche, während blau leichte Greifbarkeit suggeriert. -%\cite{Reuleaux} \cite{zacharias2008positioning} +Capability maps sind Erweiterungen der reachability map, welche aufbauend auf den Inhalt einer reachability map weitere Metriken implementieren. Die Approximation der erfolgreich terminierten kinematischen Kalkulationen für einen Vektor $v$ durch Primitive, beispielsweise Kegeln oder Zylindern, ist der Grundlegende Ansatz dieser Datenstruktur. Demnach kann die Erreichbarkeit eines Vektors darin gemessen werden, welches Primitiv seine erfolgreichen Orientierungen approximiert. Diese Informationen können analog zur reachability map Visualisiert und persistiert werden \cite{Zacharias2007}. -\section{ROS und das MoveIt} -ROS \footnote{\url{https://www.ros.org/}} ist ein der Öffentlichkeit frei zugängliches Framework, welches auf Anwendungen bezüglich robotischer Systeme spezialisiert ist. Es ermöglicht beispielsweise die Kommunikation zwischen Robotern, Erfassung sensorischer Daten und deren Visualisierung in zusätzlichen Programmen. Dies wird durch die Initialisierung von Nodes realisiert, welche Informationen hinsichtlich eines spezifischen Themas publizieren oder durch das abonnieren des Themas empfangen. Pakete innerhalb eines Arbeitsraums im ROS Kontext Koordinieren die zugrundeliegenden Quelltexte der Nodes und dienen der Abstraktion implementierter Algorithmen. +\section{Das ROS Framework und MoveIt} +ROS \cite{Quigley} ist ein auf robotische Systeme spezialisiertes open-source Framework, mit dem Ziel, die Implementierung mit Robotern zu vereinheitlichen. Exekutierende Prozesse beziehungsweise software Module im ROS Kontext sind \emph{Nodes}. Diese Kommunizieren über spezifische Nachrichten, welche beispielsweise Listen, primitive Datentypen oder weitere Nachrichten enthalten können mit anderen Nodes. Dies erfolgt über das \emph{publisher-subscriber} Prinzip, wobei Nachrichten bezüglich eines Themas \emph{Topic} publiziert werden, welche von anderen Nodes durch das Abonnieren des Themas empfangen werden. ROS Pakete \emph{packages} realisieren die kooperatives Arbeiten, indem die Nodes größerer Projekte koordiniert werden können, so kann ein Paket beispielsweise die graphische Visualisierung eines Prozesses beinhalten, während ein anderes Paket andere Aspekte des Projekts, wie die Ausführung von Algorithmen, beinhalten. Grundsätzlich ist ein ROS package ein Verzeichnis, indem eine XML Datei die Abhängigkeiten zwischen enthaltenen Nodes und genutzten Bibliotheken dokumentiert. \subsection{Visualisierung in ROS} -\autoref{fig:RViz} zeigt RViz als ein ROS Node zur Visualisierung von Informationen im 3 dimensionalen Raum und wird in dieser Arbeit zur Illustration spezifischer Aspekte der Implementierung genutzt. Die Orientierung innerhalb der grafischen Benutzeroberfläche erfolgt anhand eines Gitters auf der horizontalen XY Ebene, dessen Kanten jeweils einen Meter voneinander distanziert sind. +Während die textuelle Visualisierung primitiver Datentypen oder des Inhalts ROS spezifischer Nachrichten über die Kommandozeile möglich sind, ermöglicht das \emph{publisher-subscriber} Prinzip die Implementierung separater Prozesse, welche Informationen anhand des Abonnierens einer topic 3 dimensional repräsentieren. Ein Beispiel hierfür ist \emph{RViz} als fester Bestandteil von ROS. RViz ist durch die Integration eigener Module erweiterbar und ermöglicht die 3 dimensionale Visualisierung von Frames, robotischer Systeme oder weiteren Objekten in Form von \emph{Meshes} \cite{Quigley}. \autoref{fig:RViz} zeigt eine leere Szene in RViz und dessen grafischer Benutzeroberfläche, wobei das dargestellte Gitters auf der horizontalen XY Ebene, dessen Kanten jeweils einen Meter voneinander distanziert sind, zur Orientierung innerhalb der Szene dient. \begin{figure}[ht] \includegraphics[width = 7cm, height = 5cm]{images/RViz.png} @@ -166,13 +163,24 @@ ROS \footnote{\url{https://www.ros.org/}} ist ein der Öffentlichkeit frei zugä Primitive wie Würfel, Pfeile oder Sphären sind simple Marker Elemente, welche anhand deren Position, Orientierung, Farbe und Dimension definiert sind und in RViz durch das Abhören des zugrundeliegenden Themas visualisiert werden, um die Resultate der zugrundeliegenden Implementierung zu inspizieren beziehungsweise analysieren. \subsubsection{Interaktive Marker Elemente} -Interaktive Marker Elemente erweitern die bereits präzisierten Marker um die Interaktionsmöglichkeit durch den Nutzer, wodurch Translationen und Rotationen auf beliebigen Achsen des Markers durch den Mauszeiger realisierbar sind. Zusätzlich kann jeder Marker ein eigenes Menü mit Untermenüs initialisieren, durch dessen Einträge zusätzliche anwendungsspezifische Algorithmen implementiert werden können. Diese Modifikationen werden durch zugrundeliegende visuelle Kontrollelemente des Markers und deren Kommunikation mit dem \emph{Interaktive Marker Server}\footnote{\url{http://wiki.ros.org/interactive_markers}} realisiert. Jede Interaktion mit einer Instanz eines interaktiven Markers wird von RViz publiziert. Der Server reagiert auf diese Nachricht mit der aktualisierten Position oder Orientierung des Markers. Zusätzlich ist die Implementierung weiterer Funktionen möglich, die als Reaktion auf eine Handlung in RViz vom Server ausgeführt werden sollen. +Interaktive Marker Elemente erweitern die bereits präzisierten Marker um die Interaktionsmöglichkeit durch den Nutzer, wodurch Translationen und Rotationen auf beliebigen Achsen des Markers durch den Mauszeiger realisierbar sind. Zusätzlich kann jeder Marker ein eigenes Menü mit Untermenüs initialisieren, durch dessen Einträge zusätzliche anwendungsspezifische Algorithmen implementiert werden können. Diese Modifikationen werden durch zugrundeliegende visuelle Kontrollelemente des Markers und deren Kommunikation mit dem \emph{Interactive Marker Server} \cite{DavidGossow2011} realisiert. Die Visualisierung dieser Marker mit ihren Kontrollelementen in RViz zeigt \autoref{fig:IM1}, wobei die Translation und Rotation durch Interaktion mit den Pfeilen beziehungsweise Kreisbögen innerhalb der grafischen Benutzeroberfläche von RViz erfolgt. \autoref{fig:IM2} zeigt die Kommunikation zwischen dem RViz Prozess und dem interactive Marker Server. Dabei wird jede Interaktion mit einer Instanz des interaktiven Markers von RViz publiziert. Der Server reagiert auf diese Nachricht mit der aktualisierten Position oder Orientierung des Markers. Zusätzlich ist die Implementierung benutzerdefinierter Funktionen möglich, die als Reaktion auf eine Handlung in RViz vom Server ausgeführt werden sollen. \begin{figure} \ffigbox[\FBwidth]% {\begin{subfloatrow}% \ffigbox[\FBwidth]% - {\fbox{\includegraphics[width=0.47\textwidth, height = 5.5cm]{images/IM.png}}}% + {\fbox{ + \begin{tikzpicture} [ >=stealth'] + \node[rectangle, draw=black, thick, minimum height=3em, align=center] (IMS) {Interacte Marker Server}; + \node[rectangle, draw=black, thick, minimum height=3em, align=center] (R) [below = 3.7cm]{RViz}; + \path (IMS.south) -- (IMS.south east) coordinate[pos=0.092] (a1); + \path (IMS.south) -- (IMS.south west) coordinate[pos=0.092] (a2); + \path (R.north) -- (R.north east) coordinate[pos=0.4] (b1); + \path (R.north) -- (R.north west) coordinate[pos=0.4] (b2); + \draw[->] (a1) -- (b1); + \draw[<-] (a2) -- (b2); + \end{tikzpicture}\label{fig:IM1} + }}% {\caption{Kommunikation zwischen Server und RViz}} \ffigbox[\FBwidth]% {\fbox{\includegraphics[width=0.47\textwidth, height =5.5cm]{images/IM2.png}}}% @@ -182,7 +190,8 @@ Interaktive Marker Elemente erweitern die bereits präzisierten Marker um die In \end{figure} \subsection{kinematische Operationen in ROS} -In dieser wissenschaftlichen Arbeit erfolgen Berechnungen der Kinematik, Kollision und Bewegungsplanung ausschließlich mit der ebenfalls öffentlich frei zugänglichen MoveIt \footnote{\url{https://moveit.ros.org/}} Software. Diese definiert den \emph{move\_group} Node, welcher durch Integration zusätzlicher Nodes und Komponenten dessen vollständigen Funktionsumfang erhält. Beispielsweise werden diese zusätzlichen Nodes innerhalb einer Konfigurationsdatei initiiert, welche gleichermaßen die URDF Datei der zugrundeliegenden robotischen Systeme spezifiziert. Die Generation solcher Konfigurationsdateien erfolgt a priori über den Node des Moveit Assistenten \emph{MSA}. +In dieser wissenschaftlichen Arbeit erfolgen Berechnungen bezüglich der Kinematik, Kollision und Bewegungsplanung ausschließlich mit der open source Software \emph{MoveIt!} \cite{Chitta2016}. Diese definiert den \emph{move\_group} Prozess als Zentrum der modularen Architektur, welcher durch Integration zusätzlicher Prozesse beziehungsweise deren \emph{topics} und Nachrichten beispielsweise die Exekution kinematischer Operationen erlaubt. Dieser zentrale Prozess wird in einem Konfigurationspaket initialisiert, welches zuvor innerhalb des \emph{MoveIt! Setup Assistant (MSA)} konstruiert wird. Dieser spezifiziert beispielsweise die SRDF \footnote{\url{http://wiki.ros.org/srdf}} des zugrundeliegenden robotischen Systems und enthält weitere Konfigurationen zu dessen Visualisierung und Simulation. \section{Anforderungen automatischer kollaborativer Multi-Roboter Arbeitsräume} +\label{sec:anforderungen} Marvel et al. adressieren die Skalierbarkeit, Synchronisierung der Roboter, Verteilung der spezifischen Aufgaben und Kollisionsvermeidung als grundlegende Aspekte der Planung eines Multi-Roboter Systems \cite{Marvel2018}. Diese Aspekte stehen in Wechselwirkung zueinander, denn die Kollisionsdetektion ist beispielsweise vereinfacht, wenn die Exekution der Aufgaben sequentiell erfolgt und daher nur ein Roboter zeitgleich Bewegungen ausführt, dies impliziert zugleich eine weniger komplexe Aufgabenverteilung. Erfolgen die Handlungen der Roboter synchron, so ist die Verteilung der der Aufgaben essenziell für eine erfolgreiche Terminierung und dessen Ausführungszeit und bedingt eine komplexere Kollisionsdetektion. Änderungen der Szenerie, Beispielsweise durch die Hinzunahme oder Abnahme eines Roboters, haben ebenfalls Auswirkungen auf die Ausführung beziehungsweise Kalkulationszeit, welche die Skalierbarkeit des Arbeitsraums konkretisiert. Diese Aspekte sind zusätzlich abhängig von deren Position in der Domäne, welche die Generierung automatischer Multi-Roboter Arbeitsräume ermöglicht und daher der Hauptaspekt dieser wissenschaftlichen Arbeit ist. Also eigentlich kann ich hier alles aus Evaluation rein kopieren, das klingt zumindest besser. diff --git a/sections/konzept.tex b/sections/konzept.tex index 745b5b1..098c137 100644 --- a/sections/konzept.tex +++ b/sections/konzept.tex @@ -1,5 +1,5 @@ -\chapter{Konzept zur Realisierung Mulit-Roboter Arbeitsplätze}\label{ch:concept} -Wie im vorherigen Kapitel beschrieben, charakterisieren wissenschaftliche Arbeiten die Analyse des Arbeitsraums einzelner Roboter, sowie die Kenntnis bezüglich auszuführender Aufgaben als notwendige Voraussetzungen, um anhand dieser Informationen sukzessiv die optimale Position des Roboters zu bestimmen. Dieses Kapitel dokumentiert die konzeptuellen Entscheidungen bezüglich gewählten Verfahren dieser Bachelorarbeit, basierend auf den zuvor beschriebenen Methodiken zur Arbeitsraumanalyse, dessen Invertierung und anschließender Auswertung. Das Konzept zur Definition spezifischer Aufgaben, sowie deren intuitive Generierung und Darstellung mittels grafischer Programme ist ebenfalls dokumentiert und vervollständigt somit den Algorithmus zur Kalkulation valider Roboterbasen. Die grundlegende Struktur persistierter Daten im YAML Format sind ebenfalls Bestandteil des Kapitels. Anhand dieser Realisierung erfolgt eine Erweiterung der Erkenntnisse vorangegangener wissenschaftlicher Arbeiten um die Anwendung auf mehrere robotische Systeme sowie der Konkretisierung spezifischer Aufgaben. Diese Erweiterung ermöglicht die automatisiert Schöpfung kollaborative Multi-Roboter Arbeitsräume. +\chapter{Konzept zur Realisierung Multi-Roboter Arbeitsräume}\label{ch:concept} +Wie im vorherigen Kapitel beschrieben, charakterisieren wissenschaftliche Arbeiten die Analyse des Arbeitsraums einzelner Roboter, sowie die Kenntnis bezüglich auszuführender Aufgaben als notwendige Voraussetzungen, um anhand dieser Informationen sukzessiv die optimale Position des Roboters zu bestimmen. Dieses Kapitel dokumentiert die konzeptuellen Entscheidungen bezüglich gewählten Verfahren dieser Bachelorarbeit, basierend auf den zuvor beschriebenen Methodiken zur Arbeitsraumanalyse, dessen Invertierung und anschließender Auswertung. Das Konzept zur Definition spezifischer Aufgaben, sowie deren intuitive Generierung und Darstellung mittels grafischer Programme ist ebenfalls dokumentiert und vervollständigt somit den Algorithmus zur Kalkulation valider Roboterbasen. Die grundlegende Struktur persistierter Daten im YAML Format sind ebenfalls Bestandteil des Kapitels. Anhand dieser Realisierung erfolgt eine Erweiterung der Erkenntnisse vorangegangener wissenschaftlicher Arbeiten um die Anwendung auf mehrere robotische Systeme sowie der Konkretisierung spezifischer Aufgaben. Diese Erweiterung ermöglicht die automatisiert Konstruktion kollaborative Multi-Roboter Arbeitsräume. \newcommand\pgfmathsinandcos[3]{% \pgfmathsetmacro#1{sin(#3)}% @@ -35,33 +35,41 @@ Wie im vorherigen Kapitel beschrieben, charakterisieren wissenschaftliche Arbeit } \section{Arbeitsraumanalyse robotischer Systeme} -Die Population valider Transformationen $^{0}T_{Endeff}$, welche Resultat der Vereinigung aus $OR_{total}$ und $P_{total}$ sind, ist Bestandteil der Arbeitsraumanalyse. Dabei ist den Formulierungen zur Generierung aller Orientierungen $OR_{total}$ zu entnehmen, dass diese auf den Polarkoordinaten $^{0}s \in S^{2}$ einer Sphäre basieren, welche durch ihren Ursprung und Radius definiert ist. Der implementiere Algorithmus zur Ermittlung aller Transformationen $^{0}T_{Endeff}$ kombiniert diese Mengen, indem jeder Vektor $^{0}p \in P_{total}$ den Ursprung einer Sphäre darstellt, und demzufolge Ausgangspunkt der Kalkulation seiner Orientierungen ist. +Das Generieren valider Transformationen $^{0}T_{Endeff}$, welche Resultat der Vereinigung aus $OR_{total}$ und $P_{total}$ sind, ist Bestandteil der Arbeitsraumanalyse. Dabei ist den Formulierungen zur Generierung aller Orientierungen $OR_{total}$ zu entnehmen, dass diese auf den Polarkoordinaten $^{0}s \in S^{2}$ einer Sphäre basieren, welche durch ihren Ursprung und Radius definiert ist. Der implementiere Algorithmus zur Ermittlung aller Transformationen $^{0}T_{Endeff}$ kombiniert diese Mengen, indem jeder Vektor $^{0}p \in P_{total}$ den Ursprung einer Sphäre darstellt, und demzufolge Ausgangspunkt der Kalkulation seiner Orientierungen ist. \subsection{Erfassung der Orientierungen und Positionen} -Basierend auf der Arbeit von Zacharias et al., welche die Erfassung der Menge $P_{total}$ anhand von Zufallsverfahren als unzureichend bezüglich der Repräsentation des Umfelds robotischer Systeme klassifiziert, wird in der Implementierung stattdessen die kubische Diskretisierung angewandt \cite{Zacharias2007}. Demzufolge unterliegt die Menge $P_{total}$ den Größen $q, a \in \R_{+}$, welche das Intervall des entstehenden Würfels beziehungsweise den Abstand zwischen den Vektoren $^{0}p \in P_{total}$ definieren.\par -Die Erfassung aller Orientierungen $OR_{total}$ basiert ebenfalls nicht auf Pseudozufall, da die anschließenden kinematischen Berechnungen den zeitaufwändigsten Faktor der Arbeitsraumanalyse darstellen, folglich ist eine Implementierung, dessen Resultat von der kumulativen Betrachtung einer enormen Anzahl zufällig generierter sphärischer Koordinaten abhängt, gleichermaßen unzulänglich. Dies impliziert den Einsatz der sphärischen Diskretisierungen anhand gleichmäßiger beziehungsweise ungleichmäßiger Verteilungen. \par +Basierend auf der Arbeit von Zacharias et al. \cite{Zacharias2007}, welche die Erfassung der Menge $P_{total}$ anhand von Zufallsverfahren als unzureichend bezüglich der Repräsentation des Umfelds robotischer Systeme klassifiziert, wird in der Implementierung stattdessen die kubische Diskretisierung angewandt. Demzufolge unterliegt die Menge $P_{total}$ den Größen $q, a \in \R_{+}$, welche das Intervall des entstehenden Würfels beziehungsweise den Abstand zwischen den Vektoren $^{0}p \in P_{total}$ definieren.\par +Die Erfassung aller Orientierungen $OR_{total}$ basiert ebenfalls nicht auf Pseudozufall, da die anschließenden kinematischen Berechnungen den zeitaufwändigsten Faktor der Arbeitsraumanalyse darstellen, folglich ist eine Implementierung, dessen Resultat von der kumulativen Betrachtung einer enormen Anzahl zufällig generierter sphärischer Koordinaten abhängt, gleichermaßen unzulänglich. Dies impliziert den Einsatz der sphärischen Diskretisierungen anhand gleichmäßiger beziehungsweise ungleichmäßiger Verteilungen. +Durch die Wahl des festen Abstands $a$ bezüglich der kubischen Diskretisierung und Definition der Vektoren $^{0}p \in P_{total}$ als Zentrum einer Sphäre, wird der Radius $r \in \R_{+}$ der Sphäre nach \autoref{eq:23} kalkuliert. \begin{equation} \label{eq:23} r = \frac {a}{2} \end{equation} -Durch die Wahl des festen Abstands $a$ bezüglich der kubischen Diskretisierung und Definition der Vektoren $^{0}p \in P_{total}$ als Zentrum einer Sphäre, wird der Radius $r \in \R_{+}$ der Sphäre nach \autoref{eq:23} kalkuliert. -Zusätzlich empfiehlt sich die vollständige Auslagerung der Berechnungen auf performante Rechner, um diese mit mehreren Rechenkernen zu operieren. Beispielsweise empfiehlt sich für diese Arbeit das Hochleistungsrechenzentrum der TU-Dresden. \autoref{lst:1} zeigt die Struktur der YAML-Datei, welche die RM repräsentiert. -\begin{lstlisting}[language=JSON,firstnumber=1,float, caption={Struktur der RM als Resultat der Arbeitsraumanalyse},captionpos=t, label={lst:1}] -Referenzsystem: "Emika Franka 'Panda'" -Map Name: "RM_\$(Intervall)_\$(Auflösung)" -Auflösung: float +Zusätzlich empfiehlt sich die vollständige Auslagerung der Berechnungen auf performante Rechner, um diese mit mehreren Rechenkernen zu operieren. Beispielsweise empfiehlt sich für diese Arbeit das Hochleistungsrechenzentrum der TU-Dresden. \par +\autoref{lst:1} zeigt die Struktur der YAML-Datei, welche eine reachability map schematisch repräsentiert. +\begin{lstlisting}[language=JSON,firstnumber=1,float, caption={Struktur der reachability map},captionpos=t, label={lst:1}] +Referenzsystem: str % "Emika Franka 'Panda'" +Map_Name: str % : "RM_\$(Intervall)_\$(Aufloesung)" +Aufloesung: float Intervall: float -Endeffektor-Pose: [ "x y z q_x q_y q_z q_w"] -IK Lösung: [true | false] +Threads: int +Zeit: float +OR_total: int +Endeffektor_Pose: [str] % "x y z q_x q_y q_z q_w" +% - "-1.250 -1.250 -1.250 0.000 0.831 0.000 0.556" +% - "-1.250 -1.250 -1.250 -0.831 -0.000 0.556 -0.000" +IK_Loesung: [bool] +% - false +% - false \end{lstlisting} -Die Felder 'Auflösung' und 'Intervall' entstammen den Parametern der kubischen Diskretisierung und definieren den Namen der Datei. Das Feld 'Endeffektor-Pose' enthält eine Sequenz aus String-Repräsentation aller Endeffektor-Posen. Diese bestehen aus Position und Orientierung, wobei die Orientierung als Quaternion notiert ist, was dem Präfix q\_ zu entnehmen ist. Das Ergebnis der kinematischen Funktion aus \autoref{eq:7} als Informationen bezüglich der Erreichbarkeit der Pose ist als Wahrheitswert/bool in der Sequenz des Feldes 'IK Lösung' dokumentiert. +Die Felder \texttt{Aufloesung} und \texttt{Intervall} entstammen den Parametern der kubischen Diskretisierung und definieren den Namen der Datei, sowie das Feld \texttt{Map\_Name}. \texttt{Endeffektor\_Pose} enthält eine Sequenz aus String-Repräsentation aller Endeffektor-Posen. Diese bestehen aus Position und Orientierung, wobei die Orientierung als Quaternion notiert ist, was dem Präfix q\_ zu entnehmen ist. \texttt{Zeit} enthält die Zeitspanne, in der alle kinematischen Berechnungen erfolgen, in Sekunden. Das Ergebnis der kinematischen Funktion aus \autoref{eq:7} als Informationen bezüglich der Erreichbarkeit der Pose ist als Wahrheitswert in der Sequenz des Feldes \texttt{IK\_Loesung} dokumentiert. Der Name des durchführenden robotischen Systems, die Kardinalität von $OR_{total}$ und die Anzahl der angewandten Rechenkernen werden in den Feldern \texttt{Referenzsystem}, \texttt{OR\_total} beziehungsweise \texttt{Threads} erfasst. \subsection{Komplexität} -Derart zeitaufwändigen Verfahren erfordern die Betrachtung der Komplexität, um anhand dessen die Anzahl aller auszuführenden Operationen und somit dessen maximalen Zeitaufwand zu prognostizieren. Die \autoref{eq:24} formulierte Einordnung der zeitlichen Komplexität erfolgt durch die Betrachtung der Kardinalitäten der jeweiligen Mengen. +Derart zeitaufwändigen Verfahren erfordern die Betrachtung der Komplexität, um anhand dessen die Anzahl aller auszuführenden Operationen und somit dessen maximalen Zeitaufwand abzuschätzen. Die \autoref{eq:24} formulierte Einordnung der zeitlichen Komplexität erfolgt durch die Betrachtung der Kardinalitäten der jeweiligen Mengen. \begin{equation} \label{eq:24} @@ -75,7 +83,7 @@ Derart zeitaufwändigen Verfahren erfordern die Betrachtung der Komplexität, um Davon ist abzuleiten, dass die Wahl der Faktoren $q, a \in \R_{+}$, sowie die Menge sphärischer Vektoren $S^{2}$ und der zusätzlich betrachteten Roll Winkel, welche Grundlegende Parameter der kubischen Diskretisierung beziehungsweise sphärischen Diskretisierung darstellen, die Komplexität kubisch oder linear beeinflussen. Basierend auf den folgenden Überlegungen und daraus resultierenden Vereinfachungen, kann die Auswirkung der sphärischen Komponente auf die Komplexität dezimiert werden. \subsubsection{Vereinfachung des Roll Segments} -Um die Komplexität zu beschränken, wird beruhend auf der Bauart eines Roboterarms die Annahme getroffen, dass der Endeffektor zu jeder Zeit eine vollständige Drehung um seine longitudinal Achse vollziehen kann. Dies dezimiert die Kardinalität $\vert OR_{total} \vert$ um das Roll Segment und stellt ein Ersparnis hinsichtlich einer Dimension an Berechnungen während der Analyse dar. Daraus resultiert die Menge $OR_{total}^{/R}$, dessen Kardinalität \autoref{eq:25} zeigt. +Um die Komplexität zu beschränken, wird beruhend auf der Bauart eines Roboterarms die Annahme getroffen, dass der Endeffektor zu jeder Zeit eine vollständige Drehung um seine longitudinal Achse vollziehen kann. Dies reduziert die Kardinalität $\vert OR_{total} \vert$ um das Roll Segment und stellt ein Ersparnis hinsichtlich einer Dimension an Berechnungen während der Analyse dar. Daraus resultiert die Menge $OR_{total}^{/R}$, dessen Kardinalität \autoref{eq:25} zeigt. \begin{equation} \begin{split} @@ -116,7 +124,6 @@ Aus den Vereinfachungen bezüglich verschiedener Aspekte der Menge $OR_{total}$ Aus vorangegangenen Kapiteln ist bekannt, dass die sphärische Diskretisierung durch Abtastung der Winkelintervalle oder der Verteilung von $n \in \N $ Vektoren $^{0}s$ auf einer Sphäre, erfolgt. Das Abtasten der Winkelintervalle basierend auf einer festen Auflösung generiert ein ungleichmäßiges sphärisches Gitter, wodurch die Vektoren $^{0}s$ nicht äquidistant verteilt sind. Zusätzlich ist aus \autoref{fig:7} ersichtlich, dass die Distanz der Vektoren an den Polen marginal ist. Diese Beobachtungen implizieren, dass die resultierende Menge $OR_{total}$ durch dessen hohes Maß an ähnlichen und uneinheitlichen Orientierungen, für die Bewertung der Erreichbarkeit nicht geeignet ist. Demnach ist die äquidistante Verteilung der Vektoren $^{0}s$ für eine qualitative Repräsentation der Erreichbarkeit eines Vektors $^{0}v_{Endeff}$, sowie der Dezimierung des zeitlichen Aufwands aller kinematischen Berechnungen, maßgeblich ist. \begin{figure} -\label{fig:7} \centering \begin{tikzpicture} % "THE GLOBE" showcase \def\R{2.5} % sphere radius @@ -126,76 +133,152 @@ Aus vorangegangenen Kapiteln ist bekannt, dass die sphärische Diskretisierung d \foreach \t in {-5,-35,...,-175} { \DrawLongitudeCircle[\R]{\t} } \end{tikzpicture} \caption{Gitter aus nicht äquidistanten Vektoren einer sphärischen Diskretisierung. Die Distanz zwischen Vektoren an den Polen ist vernachlässigbar gering.} +\label{fig:7} \end{figure} \subsubsection{Gleichmäßige Verteilung} -Methoden wie die Fibonacci- oder Spiral- Verteilungen bieten minimale Abweichungen, empfehlen sich daher für die Orientierungsgenerierung und bilden eine im Kontext dieser wissenschaftlichen Arbeit angemessene Repräsentation der Erreichbarkeit des Vektors $^{0}p_{Endeff}$. Sie gehören neben der ungleichmäßigen Diskretisierung zu den von Mikhal et al. angewendeten Methoden zur Arbeitsraumanalyse eines robotischen Systems \cite{Makhal2018}. Deserno illustriert in seiner Arbeit zur äquidistanten sphärischen Diskretisierung einen Algorithmus zur gleichmäßigen Verteilung vor, welche bisher nicht zum Zweck der Roboterplatzierung genutzt wurde. Die dokumentierte Pseudoimplementierung wurde für diese wissenschaftliche Arbeit übernommen \cite{Deserno2004}. +Methoden wie die Fibonacci- oder Spiral- Verteilungen bieten minimale Abweichungen, empfehlen sich daher für die Orientierungsgenerierung und bilden eine im Kontext dieser wissenschaftlichen Arbeit angemessene Repräsentation der Erreichbarkeit des Vektors $^{0}p_{Endeff}$. Sie gehören neben der ungleichmäßigen Diskretisierung zu den von Mikhal et al. angewendeten Methoden zur Arbeitsraumanalyse eines robotischen Systems \cite{Makhal2018}. Deserno \cite{Deserno2004} illustriert in seiner Arbeit zur äquidistanten sphärischen Diskretisierung einen Algorithmus zur gleichmäßigen Verteilung vor, welche bisher nicht zum Zweck der Roboterplatzierung genutzt wurde. Der dokumentierte Algorithmus wurde für diese wissenschaftliche Arbeit übernommen. \section{Invertierte Arbeitsräume} -Ziel dieses Abschnitts ist die Präzisierung des grundlegenden Ansatzes, dessen sich hinsichtlich der Eruierung optimaler Positionen bezüglich der Platzierung robotischer Systeme bedient wird. Dies beinhaltet die Anwendung der in den Grundlagen aufgeführten kinematischen Sachverhalte, sowie der Voraussetzung einer Pose $a \in A$ des zu operierenden Objekts, wobei $A \subset SE(3)$ die Menge aller Aufgabenposen relativ zum $Frame_{0}$ ist. +\label{sec:IA} +Ziel dieses Abschnitts ist die Präzisierung des grundlegenden Ansatzes, dessen sich hinsichtlich der Berechnung optimaler Positionen bezüglich der Platzierung robotischer Systeme bedient wird. Dies beinhaltet die Anwendung der in den Grundlagen aufgeführten kinematischen Sachverhalte, sowie der Voraussetzung einer Pose $t \in A$ des zu operierenden Objekts, wobei $A \subset SE(3)$ die Menge aller Aufgabenposen relativ zum $Frame_{0}$ ist. -Eine persistierte $RM$ beinhaltet eine Liste aller Vektoren $v \in P_{total}$ des kubisch diskretisierten Roboterumfeldes mit dessen zugehörigen Orientierungen $R \in OR_{total}$, zuzüglich des Ergebnisses der Funktion $f_{kin}$ aus \autoref{eq:7}. Jede positiv terminierte kinematische Berechnung klassifiziert die operierte Transformation als valide Endeffektor Pose. Für jede dieser validen Transformationen gilt nach \autoref{eq:6}, dass eine kinematische Kette jener Form existiert. +Eine persistierte reachability map beinhaltet eine Liste aller Vektoren $v \in P_{total}$ des kubisch diskretisierten Roboterumfeldes mit dessen zugehörigen Orientierungen $R \in OR_{total}$, zuzüglich des Ergebnisses der Funktion $f_{kin}$ aus \autoref{eq:7}. Jede positiv terminierte kinematische Berechnung klassifiziert die operierte Transformation als valide Endeffektor Pose. Für jede dieser validen Transformationen gilt nach \autoref{eq:6}, dass eine kinematische Kette jener Form existiert. -Die Roboterbasis kann der Pose des ersten Festkörpers in Form der Transformation $^{0}T_{Base}$ eines robotischen Systems entnommen werden, wie die Umformungen in \autoref{eq:28} sukzessiv zeigen. Dabei ist in diesem Beispiel die Transformation von $Frame_{0}$ zu $Frame_{1} = Frame_{Base}$ die Identität 1, da der Roboter keine Erhöhung durch einen Standfuß aufweist, sondern seine Basis $^{0}T_{Base}$ auf dem fixen $Frame_{0}$ des kinematischen Baumes liegt. Sollte dem nicht so sein, ist eine zusätzliche Berechnung erforderlich. +Die Roboterbasis kann der Pose des ersten Festkörpers in Form der Transformation $^{0}T_{Base}$ eines robotischen Systems entnommen werden, wie die Umformungen in \autoref{eq:28} sukzessiv zeigen. Zum besseren Verständnis erfolgt diese Berechnung für eine kinematische Kette aus $n = 8$ Gliedern. Dabei ist in diesem Beispiel die Transformation von $Frame_{0}$ zu $Frame_{1} = Frame_{Base}$ die Identität 1, da der Roboter keine Erhöhung durch einen Standfuß aufweist, sondern seine Basis $^{0}T_{Base}$ auf dem fixen $Frame_{0}$ des kinematischen Baumes liegt. Sollte dem nicht so sein, ist eine zusätzliche Berechnung erforderlich. -\begin{equation} +\begin{equation*} \label{eq:28} -\begin{split} -^{0}T_{Base} &= a \times ^{0}T_{Endeff} \\ -^{0}T_{Base} &= a \times ^{0}T_{1} \times ... ^{Endeff-1}T_{Endeff} \\ -^{0}T_{Base} &= a \times (^{0}T_{1} \times ... ^{Endeff-1}T_{Endeff})^{-1} \\ -^{0}T_{Base} &= ^{0}T_{Endeff} \times ^{Endeff}T_{Endeff-1} \times ... \times ^{Base}T_{0} \\ -^{0}T_{Base} &= ^{0}T_{Endeff} \times ^{Endeff}T_{Endeff-1} \times ... \times ^{2}T_{Base} \times 1 \\ -^{0}T_{Base} &= ^{0}T_{Endeff} \times ^{Endeff}T_{Endeff-1} \times ... \times ^{2}T_{Base} \\ -^{0}T_{Base} &= ^{0}T_{Endeff} \times ^{Endeff}T_{Base} \\ -^{0}T_{Base} &= ^{0}T_{Base} -\end{split} -\end{equation} +\begin{aligned} +^{0}T_{Base} &= t \times ^{0}T_{Endeff} \\ +^{0}T_{Base} &= t \times (\prescript{0}{}{T}_{1} \times \prescript{1}{}{T}_{2} \times \prescript{2}{}{T}_{3} \times \prescript{3}{}{T}_{4} \times \prescript{4}{}{T}_{5} \times \prescript{5}{}{T}_{6} \times \prescript{6}{}{T}_{7} \times \prescript{7}{}{T}_{Endeff}) &\text{\autoref{eq:6}} \\ +^{0}T_{Base} &= \prescript{0}{}{T}_{Endeff} \times (\prescript{0}{}{T}_{1} \times \prescript{1}{}{T}_{2} \times \prescript{2}{}{T}_{3} \times \prescript{3}{}{T}_{4} \times \prescript{4}{}{T}_{5} \times \prescript{5}{}{T}_{6} \times \prescript{6}{}{T}_{7} \times \prescript{7}{}{T}_{Endeff})^{-1} &\text{t = $\prescript{0}{}{T}_{Endeff}$}\\ +^{0}T_{Base} &= \prescript{0}{}{T}_{Endeff} \times (\prescript{Endeff}{}{T}_{7} \times \prescript{7}{}{T}_{6} \times \prescript{6}{}{T}_{5} \times \prescript{5}{}{T}_{4} \times \prescript{4}{}{T}_{3} \times \prescript{3}{}{T}_{2} \times \prescript{2}{}{T}_{1} \times \prescript{1}{}{T}_{0}) &\text{\autoref{eq:SE}}\\ +^{0}T_{Base} &= \prescript{0}{}{T}_{Endeff} \times (\prescript{Endeff}{}{T}_{7} \times \prescript{7}{}{T}_{6} \times \prescript{6}{}{T}_{5} \times \prescript{5}{}{T}_{4} \times \prescript{4}{}{T}_{3} \times \prescript{3}{}{T}_{2} \times \prescript{2}{}{T}_{Base} \times 1) &\text{\autoref{sec:IA}}\\ +^{0}T_{Base} &= \prescript{0}{}{T}_{Endeff} \times (\prescript{Endeff}{}{T}_{7} \times \prescript{7}{}{T}_{6} \times \prescript{6}{}{T}_{5} \times \prescript{5}{}{T}_{4} \times \prescript{4}{}{T}_{3} \times \prescript{3}{}{T}_{2} \times \prescript{2}{}{T}_{Base}) \\ +^{0}T_{Base} &= \prescript{0}{}{T}_{Endeff} \times ^{Endeff}T_{Base} &\text{\autoref{eq:5}} \\ +^{0}T_{Base} &= ^{0}T_{Base} &\text{\autoref{eq:5}} +\end{aligned} +\end{equation*} -Die Metrik aus \autoref{eq:18} wird dabei zu den inversen Transformationen $^{0}T_{Base}$ portiert und als Menge $IRM$ in eine YAML Datei persistiert, dessen Struktur der \autoref{lst:2} zeigt, wobei die Felder 'Basis-Pose' und 'Metrik' nicht aus der zugrundeliegenden $RM$ übernommen wurden. Das Ergebnis dieses Algorithmus kann bei Formeln der Form \autoref{eq:20} Angewendet werden, um die Roboter Positionierung hinsichtlich der Definition einer spezifischen Aufgabe zu errechnen. - -\begin{lstlisting}[language=json,firstnumber=1, caption={In 'Metrik' werden alle kalkulierten Werte der Metrik persistiert, während 'Basis-Pose' alle invertierten Endeffektor Posen enthält},captionpos=t, float, label={lst:2}] -Referenzsystem: "Emika Franka 'Panda'" -Map Name: "IRM_\$(Intervall)_\$(Auflösung)" -Auflösung: float -Intervall: float -Basis-Pose: [ "x y z q_x q_y q_z q_w"] +Dazu wird die Menge $IRM$ aus \autoref{eq:19} in eine YAML Datei persistiert, dessen Struktur der \autoref{lst:2} schematisch zeigt, wobei die Felder \texttt{Basis\_Pose} und \texttt{Metrik} aus der zugrundeliegenden reachability map kalkuliert werden. Die Felder \texttt{Referenzsystem}, \texttt{Aufloesung} und \texttt{Intervall} werden aus der reachability map übernommen, und bilden analog den Namen \texttt{Map\_Name} der Datei. \texttt{Zeit} dokumentiert die Zeit zur Kalkulation aller Posen $^{Endeff}T_{0}$. Das Ergebnis dieses Algorithmus kann bei Formeln der Form \autoref{eq:20} angewendet werden, um die Roboter Positionierung hinsichtlich der Definition einer spezifischen Aufgabe zu errechnen. + +\begin{lstlisting}[language=JSON,firstnumber=1, caption={ Struktur der inverse reachability map},captionpos=t, float, label={lst:2}] +Referenzsystem: str % $(Referenzsystem) aus reachability map +Map_Name: str % "IRM_\$(Intervall)_\$(Aufloesung)" +Aufloesung: float % $(Aufloesung) aus reachability map +Intervall: float % $(Intervall) aus reachability map +Zeit: float +Basis_Pose: [str] % ["x y z q_x q_y q_z q_w"] +% - "0.518 -0.250 -0.597 0.831 0.000 -0.555 -0.000" +% - "0.242 -0.775 0.170 0.849 -0.490 -0.169 0.098" Metrik: [float] +% - 0.375 +% - 0.375 \end{lstlisting} \section{Definition spezifischer Aufgaben} Das Konzept zur Planung und Erstellung spezifischer Aufgaben, sowie deren Realisierung innerhalb einer Szene und Visualisierung in einer geeigneten graphischen Umgebung wie beispielsweise RViz, ist Inhalt dieses Abschnittes. Dabei besteht eine Szene aus Kollisionsobjekten, welche Hindernisse oder Primitive definieren, die das robotische System umgehen beziehungsweise mit denen es Interagieren muss. Die Planung und Ausführung einer Aufgaben erfolgt über spezifische Nachrichten der move\_group Schnittstelle, welche dem Roboter die auszuführende Operation detailliert kommunizieren, Primitive in der Szene spezifizieren und Posen festgelegt, die vor und nach der Handlung angenommen werden sollen. -Aktuell bietet MoveIt die Möglichkeit, beliebig Kollisionsobjekte in der Szene zu generieren und persistieren, aber keinen Algorithmus zur Aufgabenbeschreibung, welcher die Reihenfolge konkretisiert, in der die Kommunikation mit den robotischen Systemen stattfindet. Diese fehlende Funktion wird mittels eines Algorithmus ergänzt, welcher die Interaktionen, wie beispielsweise die Translation und Rotation eines Kollisionsobjektes, durch den Nutzer und der grafischen Bedienoberfläche realisiert und anhand zusätzliche Optionen die Dokumentation von Aufgabenbeschreibungen ermöglicht. Dabei empfehlen sich Interaktive Marker durch ihren Funktionsumfang zur Repräsentation der Kollisionsobjekte, und bieten weitere Optionen zur Aufgabenbeschreibung in ihren Menüs. -Eine valide Aufgabenbeschreibung kann ebenfalls in YAML Dateien persistiert werden, dessen Struktur der \autoref{lst:3} veranschaulicht. +Aktuell bietet MoveIt die Möglichkeit, beliebig Kollisionsobjekte in der Szene zu generieren und persistieren, aber keinen Algorithmus zur Aufgabenbeschreibung, welcher die Reihenfolge konkretisiert, in der die Kommunikation mit den robotischen Systemen stattfindet. Diese fehlende Funktion wird mittels eines Algorithmus ergänzt, welcher die Interaktionen, wie beispielsweise die Translation und Rotation eines Kollisionsobjektes, durch den Nutzer und der grafischen Bedienoberfläche realisiert und anhand zusätzlicher Optionen die Dokumentation von Aufgabenbeschreibungen ermöglicht. Dabei empfehlen sich Interaktive Marker durch ihren Funktionsumfang zur Repräsentation der Kollisionsobjekte, und bieten weitere Optionen zur Aufgabenbeschreibung in ihren Menüs. +Eine valide Aufgabenbeschreibung kann ebenfalls in YAML Dateien persistiert werden, dessen Struktur der \autoref{lst:3} veranschaulicht. Der Nutzer hat die Möglichkeit, einen Titel der Aufgabe zu definieren, welche im Feld \texttt{Name der Aufgabe} erfasst wird. Das Feld \texttt{Aufgabenbeschreibung} ist eine Sequenz beziehungsweise Liste aus Objekten, dessen Inhalt eine weitere Sequenz aus Aufgabenpositionen in Form einer Kette ist, welche das jeweilige robotische System bearbeitet. Die Orientierung der Aufgabe ist nicht auf das zugrundeliegende Objekt bezogen, sondern dokumentiert die Rotation des Endeffektors um das Objekt beispielsweise zu greifen oder abzustellen. Demnach werden diese Rotationen pro Aufgabenposition nach den Algorithmen in \autoref{sec:ORS} generiert. -\begin{lstlisting}[language=JSON,firstnumber=1, float, caption={Der Nutzer hat die Möglichkeit, einen Titel der Aufgabe zu definieren. Das Feld 'Aufgabenbeschreibung' ist eine Sequenz beziehungsweise Liste aus Objekten, dessen Inhalt eine weitere Sequenz aus Aufgaben-Posen in Form einer Kette ist, welche das jeweilige robotische System operieren muss.},captionpos=t, label={lst:3}] -Name der Aufgabe: String +\begin{lstlisting}[language=JSON,firstnumber=1, float, caption={Struktur einer Aufgabenbeschreibung},captionpos=t, label={lst:3}] +Name_der_Aufgabe: str Aufgabenbeschreibung: - Objekt : [ "x y z q_x q_y q_z q_w"] + Objekt : [str] % "x y z" \end{lstlisting} \subsection{Operationsketten} -Aufgaben, wie das sequentielle Greifen und Abstellen eines Objektes gehören zur Aufgabengruppe \emph{Pick and Place}, welche in dieser Arbeit hauptsächlich untersucht werden. Diese Aufgaben werden mittels der Positionen konkretisiert, an denen eine Operation erfolgt. Diese sukzessiv zu operierende Menge an Positionen bildet eine Operationskette $K$, dessen Kettenglieder von robotischen Systemen abwechselnd iteriert werden. Die Menge aller Aufgaben, die Operationsketten abbilden, unterliegen der Partitionierung von $K$. So kann eine Kette bestehend aus $k \in \N_{>1}$ Partitionen von $k$ Robotern unabhängig voneinander oder kooperativ operiert werden. +Aufgaben, wie das sequentielle Greifen und Abstellen eines Objektes gehören zur Aufgabengruppe \emph{Pick and Place}, welche in dieser Arbeit hauptsächlich untersucht werden. Diese Aufgaben werden mittels der Positionen konkretisiert, an denen eine Operation erfolgt. Diese sukzessiv zu operierende Menge an Positionen bildet eine Operationskette $K$. Die Menge aller Aufgaben, die Operationsketten abbilden, unterliegen der Partitionierung von $K$. So kann eine Kette bestehend aus $k \in \N_{>1}$ Partitionen von $k$ Robotern unabhängig voneinander oder kooperativ operiert werden. \autoref{fig:OK} illustriert eine solche Operationskette mit $n = 5$ Kettengliedern. Das jeweils erste Glied einer Partition ist grün markiert, während Folgeglieder blau markiert sind. Jedes Glied ist aufsteigend nummeriert, wodurch die Reihenfolge impliziert wird, in der ein Objekt gegriffen beziehungsweise abgestellt werden soll. Die Färbung der Pfeile visualisiert den Roboter, der dieser Partition zugewiesen ist und diese ausführt. + +\begin{figure}[h!] +\centering +\begin{tikzpicture} [ >=stealth'] + \node[rectangle, draw=green!60, fill=green!5, thick, minimum size=3em, align=center] (a) {0}; + \node[rectangle, draw=blue!60, fill=blue!5, thick, minimum size=3em, align=center] (aa) [above right= of a]{1}; + \node[rectangle, draw=blue!60, fill=blue!5, thick, minimum size=3em, align=center] (aaa) [below right= of aa]{2}; + \node[rectangle, draw=blue!60, fill=blue!5, thick, minimum size=3em, align=center] (aaaa) [above right= of aaa]{3}; + \node[rectangle, draw=blue!60, fill=blue!5, thick, minimum size=3em, align=center] (aaaaa) [below right= of aaaa]{4}; + \draw[->] (a) -- (aa); + \draw[->] (aa) -- (aaa); + \draw[->] (aaa) -- (aaaa); + \draw[->] (aaaa) -- (aaaaa); +\end{tikzpicture} +\caption{Zwei dimensionale Darstellung einer Operationskette} +\label{fig:OK} +\end{figure} \subsubsection{Gespaltene Operationskette} Gespaltene Operationsketten partitionieren $K$ in $i \in \N_{>0}$ disjunkte Teilketten $TK \subset K $, welche $i$ Roboter isoliert voneinander bearbeiten. -Diese Aufgaben sind implizit durch die enthaltenen Teilketten im Positionsfeld der Aufgabenbeschreibung gegeben, indem alle enthaltenen Positionen disjunkt sind, das heißt es gibt keine Sequenz eines Objektes, die sich als Posen eines anderen Objektes wiederholen. +Diese Aufgaben sind implizit durch die enthaltenen Teilketten im Positionsfeld der Aufgabenbeschreibung gegeben, indem alle enthaltenen Positionen disjunkt sind, das heißt es gibt keine Sequenz eines Objektes, die sich als Position eines anderen Objektes wiederholen. \autoref{fig:hallo} zeigt eine gespaltene Operationskette mit zwei Teilkette. Die grünen Kettenglieder markieren jeweils den Start, an dem sich ein Kollisionsobjekt befindet. Die Anordnung und Färbung der Pfeile sowie die Nummerierung impliziert, dass zwei robotische Systeme diese Aufgabe getrennt voneinander ausführen. + +\begin{figure}[h!] +\centering +\begin{tikzpicture} [ >=stealth'] + \node[rectangle, draw=green!60, fill=green!5, thick, minimum size=3em, align=center] (a) {0}; + \node[rectangle, draw=blue!60, fill=blue!5, thick, minimum size=3em, align=center] (aa) [above right= of a]{1}; + \node[rectangle, draw=green!60, fill=green!5, thick, minimum size=3em, align=center] (aaa) [below right= of aa]{2}; + \node[rectangle, draw=blue!60, fill=blue!5, thick, minimum size=3em, align=center] (aaaa) [above right= of aaa]{3}; + \draw[->, draw=cyan!60] (a) -- (aa); + \draw[->, draw=magenta!60] (aaa) -- (aaaa); +\end{tikzpicture} +\caption{Zwei dimensionale Darstellung einer Operationskette bestehend aus disjunkten Teilketten} +\label{fig:hallo} +\end{figure} \subsubsection{Kooperative Operationskette} -Kooperative Ketten partitionieren $K$ in $i$ Teilketten $TK_{0}... TK_{i-1} \subset K $, deren Schnittmenge das jeweils letzte und erste Glied von $TK_{j}$ beziehungsweise $TK_{j+1}$, für $0 \leq j < i$, ist. Diese Einteilung impliziert, dass die Position innerhalb der Abstellnachricht des Vorgängerroboters ebenfalls Inhalt der Greifnachricht des Roboters $i$ ist. Diese Logik ist implizit aus der Sequenz eines Objekts entnehmbar, indem eine Pose wiederholt vorkommt. +Kooperative Ketten partitionieren $K$ in $i$ Teilketten $TK_{0}... TK_{i-1} \subset K $, deren Schnittmenge das jeweils letzte und erste Glied von $TK_{j}$ beziehungsweise $TK_{j+1}$, für $0 \leq j < i$, ist. Diese Einteilung impliziert, dass die Position innerhalb der Abstellnachricht des Vorgängerroboters ebenfalls Inhalt der Greifnachricht des Roboters $i$ ist. Diese Logik ist implizit aus der Sequenz eines Objekts im \autoref{lst:3} entnehmbar, indem eine Position wiederholt vorkommt. \autoref{fig:koop} Visualisiert das Kettenglied, welches zugleich End-, als auch Anfangsglied zweier Teilketten ist, farblich rot. Die Position ist demzufolge wiederholt, aufeinander folgend in der Aufgabenbeschreibung notiert. -\subsection{Trajektorien} -In dieser Aufgabengruppe bilden alle Positionen, die durch den Endeffektor operiert werden sollen, eine 3-dimensionale Kurve. Kurvenimplementierungen, wie Beispielsweise die Bezierkurve, können für diese Aufgaben genutzt werden, indem interaktive Marker die Kontrollpunkte der Kurve realisieren. Durch diesen Prozess entstehen wohlgeordnete Mengen aus $^{0}T_{Endeff}$ Transformationen die ähnlich der Operationsketten von mehreren Robotern bearbeitet werden können. +\begin{figure}[h!] +\centering +\begin{tikzpicture} [ >=stealth'] + \node[rectangle, draw=green!60, fill=green!5, thick, minimum size=3em, align=center] (a) {0}; + \node[rectangle, draw=blue!60, fill=blue!5, thick, minimum size=3em, align=center] (aa) [above right= of a]{1}; + \node[rectangle, draw=red!60, fill=red!5, thick, minimum size=3em, align=center] (aaa) [below right= of aa]{2}; + \node[rectangle, draw=blue!60, fill=blue!5, thick, minimum size=3em, align=center] (aaaa) [above right= of aaa]{3}; + \node[rectangle, draw=blue!60, fill=blue!5, thick, minimum size=3em, align=center] (aaaaa) [below right= of aaaa]{4}; + \draw[->, draw=cyan!60] (a) -- (aa); + \draw[->, draw=cyan!60] (aa) -- (aaa); + \draw[->, draw=magenta!60] (aaa) -- (aaaa); + \draw[->, draw=magenta!60] (aaaa) -- (aaaaa); +\end{tikzpicture} +\caption{Zwei dimensionale Darstellung einer kooperativen Operationskette} +\label{fig:koop} +\end{figure} \section{Positionsanalyse} Aus den persistierten Informationen der Arbeitsraumanalyse, dem Ansatz der invertierten Arbeitsräume und dessen Vereinigung mit den Aufgabenbeschreibungen resultiert die Menge der Basispositionen $P_{Base}$ kombiniert mit der Metrik $D_{Reach}$ aus \autoref{eq:18} für jeden Vektor $^{0}p_{Base} \in ^{0}P_{Base}$. -Der beschriebene Algorithmus zur Ermittlung der optimalen Position sieht vor, anhand der Voxelisierung einen Index für jeden Voxel zu kalkulieren und daraus Rückschlüsse bezüglich der Positionierung robotischer Systeme zu ziehen. Nachteil dieses Vorgangs ist die Intransparenz der Menge $P_{Base}$ bezüglich der Vektor $^{0}p_{Base}$, dessen Herkunft hinsichtlich der Aufgabenbeschreibung nicht ersichtlich ist. Daraus ist abzuleiten, dass die Partitionierung der Operationskette keinen Einfluss auf den Algorithmus hat, wodurch dessen Eignung für die Anwendung bezüglich mehrerer robotischer Systeme entfällt. Ein weiterer Nachteil ergibt sich implizit aus der Metrik $D_{Voxel}$ aus \autoref{eq:22}, denn diese bewertet einen Voxel, ohne dessen Inhalt bezüglich der Aufgabenbeschreibung zu gewichten. So werden beispielsweise Voxel mit hoher Kardinalität besser für die Aufgabe gewertet, obwohl dessen Inhalt bezüglich spezifischer Aufgabenpositionen mangelhaft ist. Basierend auf diesen Beobachtungen ist eine abstrahierende Betrachtung hinsichtlich der Kettenstruktur innerhalb der Aufgabenbeschreibung zielführender. +Der in \autoref{sec:positioning} beschriebene Algorithmus zur Ermittlung der optimalen Position sieht vor, anhand der Voxelisierung einen Index für jeden Voxel zu kalkulieren und daraus Rückschlüsse bezüglich der Positionierung robotischer Systeme zu ziehen. Nachteile dieses Vorgangs sind: -\subsection{differenzierte Voxelisierung} -Der in diesem Abschnitt dokumentierte Algorithmus stellt einen optimierten Ansatz zur Positionierung robotischer Systeme dar, indem die Operationsketten bezüglich einer Aufgabenbeschreibung in den Prozess integriert werden. Während der differenzierten Voxelisierung wird jedes Kettenglied $^{0}k \in K$ einzeln betrachtet und die Metrik $D_{Reach}(Voxel)(TK)$ pro Voxel und Kettenlied kalkuliert. Dieser Vorgang hat den Vorteil, die Voxel separat anhand der Teilketten $TK \subset K $ zu betrachten und anhand dieser Zuteilung jedes Objekt einem robotischen System zuzuordnen. Für jeden Voxel lässt sich so ein Index ermitteln, der den Ursprung der Vektoren $^{0}p_{Base} \in ^{0}P_{Base}$ im Voxel berücksichtigt und gleichermaßen nach dessen Inhalt bewertet. +\begin{enumerate} + \item Intransparenz der Menge $P_{Base}$ bezüglich der Vektor $^{0}p_{Base}$ \label{enumi:Intransparenz} + \item Metrik $D_{Voxel}$ ist nicht aussagekräftig für getrennte Aufgabenbeschreibungen\label{enumi:Metrik} +\end{enumerate} -\begin{equation} -\label{eq:29} -D_{Reach}(Voxel)(TK) = \frac{\displaystyle\sum_{i=0}^{\vert TK \vert} D_{Reach_{i}}^{average}(Voxel)}{\vert TK \vert} -\end{equation} +Der in \autoref{enumi:Intransparenz} beschriebene Aspekt bezieht sich auf die Vektoren $^{0}p_{Base} \in P_{Base}$, dessen Herkunft hinsichtlich der Aufgabenbeschreibung nicht ersichtlich ist. Daraus ist abzuleiten, dass die Partitionierung der Operationskette keinen Einfluss auf den Algorithmus hat, wodurch dessen Eignung für die Anwendung bezüglich mehrerer robotischer Systeme entfällt. Daraus ergibt sich implizit \autoref{enumi:Metrik}, denn die Metrik $D_{Voxel}$ aus \autoref{eq:22} bewertet einen Voxel, ohne dessen Inhalt bezüglich der Aufgabenbeschreibung zu gewichten. So werden beispielsweise Voxel mit hoher Kardinalität besser für die Aufgabe gewertet, obwohl dessen Inhalt bezüglich spezifischer Aufgabenpositionen mangelhaft ist. Basierend auf diesen Beobachtungen ist eine abstrahierende Betrachtung hinsichtlich der Kettenstruktur innerhalb der Aufgabenbeschreibung zielführender. +\subsection{differenzierte Voxelisierung} +Die beschriebenen Nachteile in \autoref{enumi:Intransparenz} und \autoref{enumi:Metrik} zeigen, dass der Algorithmus aus \autoref{sec:positioning} nicht auf mehrere robotische Systeme anwendbar ist. Diese Erkenntnis erfordert die Implementierung eines optimierten Ansatzes der Voxelisierung, indem die Operationsketten bezüglich einer Aufgabenbeschreibung in den Prozess integriert werden. Während der differenzierten Voxelisierung wird jede Teilkette $TK \subset OK$ einzeln in der \autoref{eq:28} betrachtet. +Diese Betrachtung hat folgende Vorteile: + +\begin{enumerate} + \item Filterung der Voxel nach Teilkette \label{enumi:filter} + \item Kettenglieder anderer Teilketten haben keinen Einfluss auf die Bewertung des Voxels \label{enumi:einfluss} +\end{enumerate} + +Durch die Dokumentation der zugrundeliegenden Multiplikation jedes Kettenglieds einer Teilkette $^{0}k \in TK$, dessen Resultat eine Basisposition $^{0}p_{Base} \in P_{Base}$ ist, können durch eine Iteration ermittelt werden, welches Voxel Basisposition aller Kettenglieder einer Teilkette enthält. Dieser implementierte Filter gleicht der Definition einer Schnittmenge aus allen Basisposition pro Kettenglied einer Teilkette. Die Division des arithmetischen Mittels aus \autoref{eq:21}, angewendet auf diese gefilterten Voxel, mit der Kardinalität der Teilkette ergeben die Bewertung eines Voxels bezüglich der Teilkette. Diese differenzierte Betrachtung verhindert, dass Kettenglieder anderer Teilketten den Wert eines Voxels beeinflussen. + +\section{Platzierung der robotischen Systeme} +Die Platzierung der robotischen Systeme erfolgt innerhalb der gewählten Domäne. Diese ist beispielsweise die Oberfläche eines oder mehrerer Tische, wie es im Ceti der Fall ist. Diese Oberfläche wird durch einen Bereich auf der XY Ebene imitiert, dessen Dimensionen angepasst werden können. Alle Voxel der Schnittmenge einer Teilkette werden dahingehend inspiziert, ob deren Basispositionen die gewählte Domäne schneiden. Anschließend erfolgt die Wahl des Voxels mit der maximalen Bewertung und die Basisposition wird abschließend festgelegt. Dieser Vorgang definiert die Zuteilung robotischer Systeme auf die Teilketten einer Aufgabenbeschreibung und wird als YAML Datei persistiert. \autoref{lst:Goal} zeigt schematisch die Verteilung der robotischen Syteme auf die Kollisionsobjekte. Die \texttt{Roboter\_Anzahl} entspricht der Anzahl aller Teilketten und spezifiziert die Anzahl der robotischen Systeme. Die während der differenzierten Voxelisierung vergangene Zeit wird im entsprechenden Feld erfasst. Die Basispositionen aller Roboter werden im \texttt{Basispositionen} Feld dokumentiert. Alle zusätzlichen Felder sind aus der zugrundeliegenden inverse reachability map, beziehungsweise der Aufgabenbeschreibung portiert. + +\begin{lstlisting}[language=JSON,firstnumber=1, caption={ Struktur der Positionsanalyse},captionpos=t, label={lst:Goal}] +Referenzsystem: str % "$(Referenzsystem)" aus inverse reachability map +Name_der_Aufgabe: str % "$(Name_der_Aufgabe)" aus Aufgabenbeschreibung +Roboter_Anzahl: int +Zeit: float +Move_Groups: + objekt: + panda_arm: [str] % "$(Aufgabenbeschreibung)" aufgetilt auf die Roboterarme +Basispositionen: + panda_arm: str % "x y z" +\end{lstlisting} \ No newline at end of file diff --git a/sections/sota.tex b/sections/sota.tex index eb57449..572be70 100644 --- a/sections/sota.tex +++ b/sections/sota.tex @@ -1,5 +1,5 @@ \chapter{Erkenntnisse zur Platzierung robotischer Systeme} \label{ch:sota} -Den dargelegten Grundlagen beziehungsweise dessen Abschnitt zur Formulierung der Anforderungen ist zu entnehmen, dass die Platzierung robotischer Systeme innerhalb einer Domäne ein fundamentaler Bestandteil zur automatischen Schaffung kollaborative Multi-Roboter Arbeitsräume ist. Die Einhaltung der physikalischen Beschränkungen bezüglich genutzter Roboter ist hierbei unerlässlich, um hinsichtlich ihrer Aufgabe und Position ein zufriedenstellendes Ergebnis zu erzielen. Einen Ansatz zur Erfassung dieser robotischer Grenzen konkretisieren Zacharias et al. in deren wissenschaftlichen Arbeit und kombiniert diesen in folgenden Publikationen mit zusätzlichen Aspekten der Kinematik, um einen Algorithmus zu formulieren, welcher sukzessiv die Position eines Roboters hinsichtlich seiner Grenzen bestimmt \cite{Zacharias2009}. Dieses Verfahren ist in die Komponenten +Den Anforderungen aus \autoref{sec:anforderungen} ist zu entnehmen, dass die Platzierung robotischer Systeme innerhalb einer Domäne ein fundamentaler Bestandteil zur automatischen Schaffung kollaborative Multi-Roboter Arbeitsräume ist. Die Einhaltung der physikalischen Beschränkungen bezüglich genutzter Roboter ist hierbei unerlässlich, um hinsichtlich ihrer Aufgabe und Position ein zufriedenstellendes Ergebnis zu erzielen. Einen Ansatz zur Erfassung dieser robotischer Grenzen konkretisieren Zacharias et al. \cite{Zacharias2007} und kombiniert diesen in folgenden Publikationen \cite{Zacharias2009} mit zusätzlichen Aspekten der Kinematik, um einen Algorithmus zu formulieren, welcher sukzessiv die Position eines Roboters hinsichtlich seiner Grenzen bestimmt. Dieses Verfahren ist in die Komponenten \emph{Arbeitsraumanalyse}, \emph{Arbeitsrauminvertierung} und \emph{Positionsanalyse} unterteilt, welche somit die Gliederung dieses Kapitels definieren, dessen Inhalt die Auseinandersetzung mit deren Methodiken ist. Die daraus gewonnenen Erkenntnisse werden in der konzeptuellen Ausarbeitung und Implementierung genutzt. \autoref{fig:4} zeigt die Reihenfolge der Komponenten im Algorithmus. \begin{figure}[h!] @@ -14,43 +14,56 @@ Den dargelegten Grundlagen beziehungsweise dessen Abschnitt zur Formulierung der \end{figure} \section{Arbeitsraumanalyse} -Vahrenkamp et al. beschreiben in ihrer Publikation die Analyse des Arbeitsraums eines robotischen Systems als Ausgangspunkt zur Berechnung dessen optimaler Positionierung hinsichtlich einer zuvor definierten Aufgabe \cite{Vahrenkamp2013}. Dazu dokumentieren Zacharias et al. in ihrer wissenschaftlichen Arbeit einen Ansatz, dessen Inhalt es ist, durch wiederholtes Anwenden kinematischer Algorithmen $f_{kin}$, gezeigt in \autoref{eq:7}, das unmittelbare Umfeld des Roboters zu erfassen und dieses Anhand einer Metrik zu charakterisieren. Den Berechnungen liegt eine feste Menge generierter Endeffektor-Posen $T \subset SE(3)$ in Form der Transformationen $^{0}T_{Endeff} \in T$ relativ zum $Frame_{0}$ zugrunde. Alle Transformation $\in T$ zuzüglich der Information über dessen Erreichbarkeit bilden die \emph{reachability map (RM)}, welche persistiert wird \autoref{eq:8}. +Vahrenkamp et al. beschreiben in ihrer Publikation die Analyse des Arbeitsraums eines robotischen Systems als Ausgangspunkt zur Berechnung dessen optimaler Positionierung hinsichtlich einer zuvor definierten Aufgabe \cite{Vahrenkamp2013}. Dazu dokumentieren Zacharias et al. in ihrer wissenschaftlichen Arbeit einen Ansatz, dessen Inhalt es ist, durch wiederholtes Anwenden kinematischer Algorithmen das unmittelbare Umfeld des Roboters zu erfassen und dieses Anhand einer Metrik zu charakterisieren. \autoref{eq:7} definiert diese Berechnungen diskret durch die Funktion $f_{kin}$. \begin{equation}\label{eq:7} -f_{kin}:T \to [0 | 1] + \begin{split} + f_{kin}&: T \to \{0 , 1\} \\ + f_{kin}(^{0}T_{Endeff})&= + \begin{cases*} + 1 & gdw. die Pose erreichbar ist \\ + 0 & sonst + \end{cases*} + \end{split} \end{equation} +Den Berechnungen liegt eine feste Menge generierter Endeffektor-Posen $T \subset SE(3)$ in Form der Transformationen $^{0}T_{Endeff} \in T$ relativ zum $Frame_{0}$ zugrunde und ihr Resultat ist bezogen auf die Terminierung ein Wahrheitswert. Alle Transformation zuzüglich der Information über dessen Erreichbarkeit bilden die \emph{reachability map (RM)}, welche in \autoref{eq:8} notiert ist. + + + \begin{equation}\label{eq:8} RM:= \{(t , f_{kin}(t)) | t \in T \} \end{equation} - +\begin{figure}[h!] \ffigbox[\FBwidth]% {\begin{subfloatrow}% \ffigbox[\FBwidth]% {\fbox{\includegraphics[width=0.47\textwidth, height = 5.5cm]{images/P_total.png}}}% - {\caption{Menge aller Positionen}\label{fig:PT}}% + {\caption{Menge aller Positionen $\protect P_{total}$}\label{fig:PT}}% \ffigbox[\FBwidth]% {\fbox{\includegraphics[width=0.47\textwidth, height =5.5cm]{images/OR_total.png}}}% - {\caption{Menge aller Orientierungen} \label{fig:OR}} + {\caption{Menge aller Orientierungen $\protect OR_{total}$} \label{fig:OR}} \end{subfloatrow}}% - {\caption{Abstrahierte Darstellung der Mengen $\protect P_{total}$ und $\protect OR_{total}$. Ein roter Marker repräsentiert den Vektoren, den der Endeffektor einnimmt. Der grüne Pfeil zeigt in die Richtung, in welche die Finger des Endeffektors ausgerichtet sind.} + {\caption{ Darstellung der Mengen $\protect P_{total}$ und $\protect OR_{total}$. Ein roter Marker repräsentiert den Vektoren, den der Endeffektor einnimmt. Der grüne Pfeil zeigt in die Richtung, in welche die Finger des Endeffektors ausgerichtet sind.} \label{fig:5}}% +\end{figure} -\label{eq:10} zeigt, dass ein Paar aus einem Vektor $^{0}v \in \R^{3}$ und einer Rotation $^{0}R \in SO(3)$ die Transformation $^{0}T_{Endeff}$ als Pose des Endeffektors bildet. Daraus ist abzuleiten, dass die Zusammenstellung einer Menge $P_{total}$ aus $^{0}v$ und die Generierung dessen zugehöriger Rotationen $^{0}R$, welche die Menge aller Orientierungen $OR_{total}$ definiert, abstrahiert erfolgt. Das Kreuzprodukt dieser Mengen beschreibt alle Endeffektor Posen $\in T$ als Grundlage anschließender kinematischer Berechnungen $f_{kin}$. \autoref{eq:9} visualisiert diese Mengen. -\begin{equation} -\label{eq:9} -T:= \{ P_{total} \times OR_{total} \} -\end{equation} +\autoref{eq:10} zeigt, dass ein Paar aus einem Vektor $^{0}v \in \R^{3}$ und einer Rotation $^{0}R \in SO(3)$ die Transformation $^{0}T_{Endeff}$ als Pose des Endeffektors bildet. Daraus ist abzuleiten, dass die Zusammenstellung einer Menge $P_{total}$ aus $^{0}v$ und die Generierung dessen zugehöriger Rotationen $^{0}R$, welche die Menge aller Orientierungen $OR_{total}$ definiert, abstrahiert erfolgt. \autoref{eq:9} beschreibt $T$ als das Kreuzprodukt dieser Mengen. \begin{equation} \label{eq:10} ^{0}T_{Endeff}:= \{(v,R) | v \in P_{total}, R \in OR_{total} \} \in T \end{equation} +\begin{equation} +\label{eq:9} +T:= \{ P_{total} \times OR_{total} \} +\end{equation} + \subsection{Menge aller Positionen} -Dieser Abschnitt umfasst Methodiken zur Population der Menge $^{0}P_{total}$. Die Wahl der Vektoren $^{0}v \in ^{0}P_{total}$ erfordert das bestimmen eines Intervalls $[-q,q] : q \in \R_{+}$, welches den tatsächlichen Arbeitsraum des robotischen Systems überschätzt. Anschließend erfolgt ein Algorithmus zur Generierung der Elemente $^{0}x, ^{0}y, ^{0}z \in [-q,q]$, die den Vektor $^{0}v$ definieren. Dieser Sachverhalt ist in \autoref{eq:11} dargestellt. +Dieser Abschnitt umfasst Methodiken zur Population der Menge $^{0}P_{total}$. Die Wahl der Vektoren $^{0}v \in ^{0}P_{total}$ erfordert das Bestimmen eines Intervalls $[-q,q] : q \in \R_{+}$ für jede Dimension anlog zu den in \autoref{fig:panda} gezeigten Perspektiven, welche den tatsächlichen Arbeitsraum des robotischen Systems überschätzt. Anschließend erfolgt ein Algorithmus zur Generierung der Elemente $^{0}x, ^{0}y, ^{0}z \in [-q,q]$, die den Vektor $^{0}v$ definieren. Dieser Sachverhalt ist in \autoref{eq:11} vereinfacht dargestellt. \begin{equation} \label{eq:11} @@ -58,10 +71,15 @@ P_{total}:= \{(x, y, z)^{T} | q \in \R_{+}; x, y ,z \in [-q,q] \} \end{equation} \subsubsection{Positionen durch Zufallsverfahren} -Dieses Verfahren bestimmt die Verteilung der Vektoren $^{0}v$, welche keiner räumlichen oder anderweitigen Abhängigkeit, mit Ausnahme des festen Intervalls $[-q,q]$, unterliegen. +Dieses Verfahren bestimmt die Verteilung der Vektoren $^{0}v$, welche keiner räumlichen oder anderweitigen Abhängigkeit, mit Ausnahme der festen Intervalle $[-q,q]$, unterliegen. Die Initialisierung erfolgt durch Angabe der $n \in \N_{>0}$ zu generierenden Vektoren, wodurch die Kardinalität von $^{0}P_{total}$ durch die Zahl $n$ bestimmt ist. + +\begin{equation} +\label{eq:kadi} +\vert ^{0}P_{total} \vert = n +\end{equation} \subsubsection{Positionen durch kubische Diskretisierung} -Die kubische Diskretisierung des Arbeitsraums robotischer Systeme stellt ein Verfahren zur Bestimmung der Vektoren $^{0}v$ dar, welche durch Angabe einer festen Auflösung $ a \in \R_{+}$ das Umfeld des Roboters in ein reguläres Gitter der Grenzen $[-q,q]$ unterteilt. Der daraus resultierende diskretisierte Würfel stellt eine gleichmäßige Verteilung der Vektoren $^{0}v$ im Abstand $a$ innerhalb des Intervalls $[-q,q]$ dar, welcher in \autoref{fig:PT} anhand roter Marker illustriert ist. Wie \autoref{eq:12} zeigt, ist die Kardinalität der Menge $^{0}P_{total}$ implizit durch das Intervall $[-q,q]$ und der Auflösung $a$ gegeben. +Die kubische Diskretisierung des Arbeitsraums robotischer Systeme stellt ein Verfahren zur Bestimmung der Vektoren $^{0}v$ dar, welche durch Angabe einer festen Auflösung $ a \in \R_{+}$ pro Dimension das Umfeld des Roboters in ein reguläres Gitter der Grenzen $[-q,q]$ unterteilt. Dem \autoref{sec:diskretisierung} ist zu entnehmen, dass die resultierende Menge $^{0}P_{total}$ durch die gleichmäßige Verteilung der Vektoren $^{0}v$ im Abstand $a$ innerhalb des Intervalls $[-q,q]$ einen Würfel approximiert, welcher in \autoref{fig:PT} anhand roter Marker illustriert ist. Wie \autoref{eq:12} vereinfacht zeigt, ist die Kardinalität der Menge $^{0}P_{total}$ implizit durch das Intervall $[-q,q]$ und der Auflösung $a$ gegeben. \begin{equation} \label{eq:12} @@ -69,7 +87,8 @@ Die kubische Diskretisierung des Arbeitsraums robotischer Systeme stellt ein Ver \end{equation} \subsection{Menge aller Orientierungen} -Die Menge $^{0}OR_{total}$ beschreibt alle möglichen Orientierungen, die für jeden Vektor generiert und operiert werden. Ein grundlegendes Verfahren hierzu ist die sphärische Diskretisierung, welche die Approximation einer Sphäre durch Vektoren $^{0}s \in \R^{3}$ um ein Zentrum $^{0}p \in \R^{3}$ mit festem Abstand $r \in \R_{+}$, beschreibt und in \autoref(fig:OR) durch den Ursprung eines Grünen Pfeils angedeutet ist. Ziel dieses Prozesses ist die Entwicklung aller Orientierungen für einen Vektor $^{0}p$ aus den sphärischen Polarkoordinaten , die $^{0}s \in S^{2}$ auf der erzeugten Sphäre definieren. Die Menge $S^{2}$ aller Vektoren auf einer Sphäre ist in aus \autoref{eq:13} formuliert. Hierzu beschreiben Azimut $\theta \in [ \, 0,2 \pi \,)$ und Colatitude $\phi \in [ \, 0,\pi \,]$ den Winkel auf der XY Ebene des kartesischen Koordinatensystems bzw. den Winkel zwischen Z Achse und der Strecke zwischen Zentrum und Koordinate $^{0}s$ $\overline{sp}$ \cite{SphericalCoordinates}. \autoref{fig:6} veranschaulicht die genannten Winkel der sphärischen Polarkoordinaten im kartesischen Koordinatensystem als Referenz farblich. Bezogen auf die Orientierung eines Körpers, können die Polarwinkel als Gier und Nick Winkel eines Objektes interpretiert werden, welche die \autoref{fig:1} illustriert. Der zusätzlichen Roll Winkel $Roll \in [ \, 0,2 \pi \,)$ komplettiert die Rotation des Endeffektors und somit die Menge $OR_{total}$. Die Kardinalität der Menge $^{0}OR_{total}$ ist durch die Menge aller sphärischen Vektoren $^{0}s \in S^{2}$ und zusätzlicher Betrachtung der $Roll$ Winkel bestimmt, was der \autoref{eq:14} zu entnehmen ist. +\label{sec:ORS} +Die Menge $^{0}OR_{total}$ beschreibt alle möglichen Orientierungen, die für jeden Vektor generiert werden. Ein grundlegendes Verfahren hierzu ist die sphärische Diskretisierung, welche die Approximation einer Sphäre durch Vektoren $^{0}s \in \R^{3}$ um ein Zentrum $^{0}p \in \R^{3}$ mit festem Abstand $r \in \R_{+}$, beschreibt und in \autoref{fig:OR} durch den Ursprung eines Grünen Pfeils angedeutet ist. Ziel dieses Prozesses ist die Entwicklung aller Orientierungen für einen Vektor $^{0}p$ aus den sphärischen Polarkoordinaten , die $^{0}s \in S^{2}$ auf der erzeugten Sphäre definieren. Die Menge $S^{2}$ aller Vektoren auf einer Sphäre ist in \autoref{eq:13} formuliert. Hierzu beschreiben Azimut $\theta \in [ \, 0,2 \pi \,)$ und Colatitude $\phi \in [ \, 0,\pi \,]$ den Winkel auf der XY Ebene des kartesischen Koordinatensystems bzw. den Winkel zwischen Z Achse und der Strecke zwischen Zentrum und Koordinate $\overline{sp}$ \footnote{\url{https://mathworld.wolfram.com/SphericalCoordinates.html}}. \autoref{fig:6} veranschaulicht die genannten Winkel der sphärischen Polarkoordinaten im kartesischen Koordinatensystem als Referenz farblich. Bezogen auf die Orientierung eines Körpers, können die Polarwinkel als Gier und Nick Winkel eines Objektes interpretiert werden, welche die \autoref{fig:1} illustriert. Der zusätzlichen Roll Winkel $Roll \in [ \, 0,2 \pi \,)$ komplettiert die Rotation des Endeffektors und somit die Menge $OR_{total}$. Die Kardinalität der Menge $^{0}OR_{total}$ ist durch die Menge aller sphärischen Vektoren $^{0}s \in S^{2}$ und zusätzlicher Betrachtung der $Roll$ Winkel bestimmt, was der \autoref{eq:14} zu entnehmen ist. \begin{figure}[h!] \centering @@ -110,15 +129,11 @@ S^{2} := \{s \in \R^{3} | \vert \overline{sp} \vert = r\} \end{equation} \subsubsection{Orientierungen durch Zufallsverfahren} -Sphärische Diskretisierungen, deren Verteilung bezüglich dessen Vektoren $s$ keinem Schema folgen oder an räumliche Parameter gebunden sind, verteilen diese nach Pseudozufallsverfahren. Die Anzahl der zu verteilenden Vektoren ist durch eine Zahl $n \in \N$ gekennzeichnet und stellt die Initialisierung des Verfahrens da. Die Kardinalität von $S^{2}$ lässt sich daher eindeutig bestimmen. +Sphärische Diskretisierungen, deren Verteilung bezüglich dessen Vektoren $s$ keinem Schema folgen oder an räumliche Parameter gebunden sind, verteilen diese nach Pseudozufallsverfahren. Die Anzahl der zu verteilenden Vektoren ist durch eine Zahl $n \in \N$ gekennzeichnet und stellt die Initialisierung des Verfahrens da. Die Kardinalität von $S^{2}$ lässt sich daher analog zur \autoref{eq:kadi} eindeutig bestimmen. -\begin{equation} -\label{eq:15} -\vert S^{2} \vert = n -\end{equation} \subsubsection{ungleichmäßige sphärische Diskretisierung} -Das Abtasten der Winkelintervalle von $\theta$ und $\phi$ hinsichtlich fester Auflösungen $p, q \in \R_{+}$ generiert ein sphärisches Gitter, indem die sphärischen Vektoren $^{0}s$ nicht äquidistant und demzufolge ungleichmäßig Verteilt sind. Die Kardinalität von $ S^{2}$ dieses Verfahrens ist anhand der gewählten Auflösungen $p, q$ und der Winkelintervalle nach \autoref{eq:16} berechenbar. +Das Abtasten der Winkelintervalle von $\theta$ und $\phi$ hinsichtlich fester Auflösungen $p, q \in \R_{+}$ generiert ein sphärisches Gitter, indem die sphärischen Vektoren $^{0}s$ nicht äquidistant und demzufolge ungleichmäßig verteilt sind. Die Kardinalität von $ S^{2}$ dieses Verfahrens ist anhand der gewählten Auflösungen $p, q$ und der Winkelintervalle nach \autoref{eq:16} berechenbar. \begin{equation} \label{eq:16} @@ -126,10 +141,10 @@ Das Abtasten der Winkelintervalle von $\theta$ und $\phi$ hinsichtlich fester Au \end{equation} \subsubsection{gleichmäßige sphärische Diskretisierung} -Die äquidistante Verteilung einer festen Zahl $n \in \N$ Vektoren auf einer Sphäre ist ein mathematisches Problem, dessen Lösung eine umfassende Bedeutung für Forschungsfelder der Biologie, Chemie und Physik aufweist. Beispielsweise basieren das Tammes Problem oder hard-spheres Problem auf dieser Problemstellung, bezogen auf der Modellierung fluider Partikel \cite{Saff1997}. Algorithmen zu diesem Kontext approximieren dieses Ziel, wobei die Distanzen auf der Sphäre minimalen Abweichungen unterliegen und in Spezialfällen die Anforderung einer äquidistanten Verteilung erfüllen. Die Kardinalität $\vert S^{2} \vert$ ist analog zu \autoref{eq:15} durch $n$ gegeben. +Die äquidistante Verteilung einer festen Zahl $n \in \N$ Vektoren auf einer Sphäre ist ein mathematisches Problem, dessen Lösung eine umfassende Bedeutung für Forschungsfelder der Biologie, Chemie und Physik aufweist. Beispielsweise basieren das Tammes Problem oder hard-spheres Problem auf dieser Problemstellung, bezogen auf der Modellierung fluider Partikel \cite{Saff1997}. Algorithmen zu diesem Kontext approximieren dieses Ziel, wobei die Distanzen auf der Sphäre minimalen Abweichungen unterliegen und in Spezialfällen die Anforderung einer äquidistanten Verteilung erfüllen. Die Kardinalität $\vert S^{2} \vert$ ist analog zu \autoref{eq:kadi} durch $n$ gegeben. \subsection{Arbeitsraumcharakterisierung} -\autoref{eq:18} formuliert die Metrik $D_{Reach}$ zur Klassifizierung der Erreichbarkeit eines Vektors $^{0}v_{Endeff}$. Hierzu erfolgt eine Gegenüberstellung der Menge $OR_{v}$ aller Orientierungen $R$ zu einem Vektor $v$, die der Endeffektor für den Vektor $^{0}v_{Endeff}$ während der Berechnungen erfolgreich einnehmen kann, gegenüber der Menge $OR_{total}$ aller generierten und operierten Orientierungen für $^{0}v_{Endeff}$ \cite{Porges2015}. +\autoref{eq:18} formuliert die Erreichbarkeitsmetrik $D_{Reach}$ zur Bewertung der Erreichbarkeit eines Vektors $^{0}v_{Endeff}$ durch den Endeffektor. Hierzu erfolgt eine Gegenüberstellung der Menge $OR_{v}$ aller Orientierungen $R$, die der Endeffektor für den Vektor $^{0}v_{Endeff}$ während der Berechnungen erfolgreich einnehmen kann, gegenüber der Menge $OR_{total}$ aller generierten und bearbeiteten Orientierungen für $^{0}v_{Endeff}$ \cite{Porges2015}. Demnach impliziert ein geringer Wert $D_{Reach}$ für einen Vektor, dass dieser vom Koordinatenursprung des Frames $Frame_{0}$ weniger erreichbar ist als ein Vektor mit höherer Bewertung. \begin{equation} \label{eq:17} @@ -139,13 +154,13 @@ OR_{v} := \{R | ((v, R),f _{kin}) \in RM \land f_{kin}((v, R)) = 1 \} \subset \begin{equation} \label{eq:18} \begin{split} -D_{Reach}:v & \to [0,1] \\ +D_{Reach}:P_{total} & \to [0,1] \\ D_{Reach}(v) &= \frac{\vert OR_{v} \vert}{\vert OR_{total} \vert} \end{split} \end{equation} \section{Arbeitsrauminversion} -Der invertierte Arbeitsraum beschreibt nach Vahrenkamp et al. alle Posen relativ zum Endeffektor, sphärisch um den Ursprung des $Frame_{0}$ verteilt, von denen aus das robotische System den Ursprung Frames erreicht \cite{Vahrenkamp2013}. Dazu beschreibt dieser Prozess die Invertierung aller persistierten Transformationen $^{0}T_{Endeff}$ der $RM$ und Portierung der zugehörigen Metrik $D_{Reach}(^{0}v_{Endeff})$. Die resultierende Menge enthält nach \autoref{eq:5} Transformationen der Form $^{Endeff}T_{0}$, welche dessen Basispose beschreiben. Diese Informationen werden als \emph{inverse reachability map (IRM)} persistiert, welche durch die \autoref{eq:19} definiert ist. +Der invertierte Arbeitsraum beschreibt nach Vahrenkamp et al. alle Posen relativ zum Endeffektor, sphärisch um den Ursprung des $Frame_{0}$ verteilt, von denen aus das robotische System den Ursprung Frames erreicht \cite{Vahrenkamp2013}. Dazu beschreibt dieser Prozess die Invertierung aller persistierten Transformationen $^{0}T_{Endeff}$ der $reachability map$ und Portierung der zugehörigen Metrik $D_{Reach}(^{0}v_{Endeff})$. Die resultierende Menge enthält nach \autoref{eq:5} Transformationen der Form $^{Endeff}T_{0}$, welche dessen Basispose beschreiben. Diese Informationen werden als \emph{inverse reachability map (IRM)} persistiert, welche durch die \autoref{eq:19} definiert ist. \begin{equation} \label{eq:19} @@ -153,36 +168,37 @@ IRM := \{((v,R)^{-1}, D_{Reach}(v)) | ((v,R),d) \in RM \} \end{equation} \section{Positionsanalyse} -Inhalt dieses Abschnittes ist die Analyse aller potentiellen Basispositionen $P_{Base}$ eines robotischen Systems, welche sich nach \autoref{eq:20} aus der Matrix Multiplikation mit dem invertierten Arbeitsraum IRM und Posen spezifischer Aufgaben $^{0}a \in A$ ergeben. Die Metrik wird dabei nach der Kalkulation auf $P_{Base}$ abgebildet und beschreibt die Zuverlässigkeit dieser Posen \cite{Makhal2018}. Komplexe Aufgaben mit der Anzahl $n \in \N : n > 2$ Posen $^{0}a$, welche der Roboter operieren muss, skalieren die Menge aller Basen $P_{Base}$ um $n$. Dabei ist die Verteilung der Vektoren $^{0}p_{Base} \in P_{Base}$ unregelmäßig, beispielsweise gruppieren sie aufeinander oder sind vernachlässigbar voneinander distanziert. Eine Methode, innerhalb dieser enormen Datenmenge eine optimale Position zur Operation der Aufgaben zu entnehmen, ist die Voxelisierung. Dieses Verfahren, ähnlich der kubischen Diskretisierung, erzeugt einen Hyperwürfel innerhalb eines festen Intervalls $[-q, q] \in \R$, welcher aus $k \in \N$ disjunkten Voxeln besteht, dessen Zentren jeweils um eine Auflösung $u \in \R_{+}$ distanziert sind. Jeder Voxel enthält eine disjunkte Teilmenge $TP_{0} ...TP_{k-1}\subset P_{Base}$ der Basispositionen, die abstrahiert voneinander betrachtet werden. Diese Abbildung der Vektoren $^{0}p_{Base}$ auf Voxel implementiert eine Abtastung, wobei die Wertigkeit eines Voxels durch sein Zentrum und dessen Inhalt repräsentiert wird. Gleichermaßen dezimiert diese Darstellung das Lösungsspektrum der Aufgabenabhängigen Kardinalität von $P_{Base}$ auf die Anzahl aller Voxel $k$. +\label{sec:positioning} +Dieser Abschnitt dokumentiert die Analyse aller potentiellen Basispositionen $P_{Base}$ eines robotischen Systems, welche sich nach \autoref{eq:20} aus der Matrix Multiplikation mit dem invertierten Arbeitsraum IRM und Posen spezifischer Aufgaben $^{0}t \in A$ ergeben. Die Metrik wird dabei nach der Kalkulation auf $P_{Base}$ abgebildet und beschreibt die Zuverlässigkeit dieser Posen \cite{Makhal2018}. Komplexe Aufgaben mit der Anzahl $n \in \N : n > 2$ Posen $^{0}t$, welche der Roboter operieren muss, skalieren die Menge aller Basen $P_{Base}$ um $n$. Dem \autoref{sec:partitionierung} ist das Verfahren der Voxelisierung zu entnehmen, welches diese enormen Datenmenge räumlich in separate Volumen teilt. Das Resultat dieser Abtastung ist eine Menge aus $k \in \N$ disjunkten Voxeln $V$, dessen Zentren jeweils um eine Auflösung $u \in \R_{+}$ distanziert sind und den Zeigern auf alle Basispositionen, die sich räumlich innerhalb eines Voxels befinden. Demnach erfolgt eine Spaltung der Basispositionen $P_{Base}$ in disjunkte Teilmenge $TP_{0} ...TP_{k-1}\subset P_{Base}$, die abstrahiert voneinander betrachtet werden können. Da jeder Vektor $^{0}p_{Base} \in P_{Base}$ auf seine Bewertung zeigt, ist der Wert eines Voxels implizit durch die disjunkte Teilmenge gegeben, in die Menge der Basispositionen $P_{Base}$ teilt. Gleichermaßen reduziert diese Darstellung das Lösungsspektrum von der aufgabenabhängigen Kardinalität von $P_{Base}$ auf die Anzahl aller Voxel $k$. \begin{equation} \label{eq:20} \begin{split} -P_{Base} &:= \{(a \times t, d) | (t, d) \in IRM , a \subset SE(3) \} \\ -\vert P_{Base} \vert &= \vert a \vert \times \vert IRM \vert | a \subset SE(3) +P_{Base} &:= \{(t \times T, d) | (T, d) \in IRM , t \subset A \} \\ +\vert P_{Base} \vert &= \vert A \vert \times \vert IRM \vert \end{split} \end{equation} \subsection{Charakterisierung eines Voxels} -Nun erfolgt die Eruierung des Algorithmus von Makhal et al. \cite{Makhal2018}, welcher alle $k$ Voxel charakterisiert und anhand der enthaltenen Vektoren $^{0}p_{Base}$ bestimmt, welches Voxel sich bevorzugt für die Platzierung des robotischen Systems eignet. Dies erfolgt durch eine weitere Metrik $D_{voxel}$ in \autoref{eq:22}, die sich aus dem arithmetischen Mittel $D_{Reach}^{average}$ eines Voxels und dem Verhältnis von dessen Mächtigkeit zum maximalen Voxel $TP_{max}$ ergibt. +Makhal et al. \cite{Makhal2018} beschreibt einen Algorithmus, welcher alle $k$ Voxel charakterisiert und anhand deren Vektoren $^{0}p_{Base} \in TP$ bestimmt, welches Voxel sich bevorzugt für die Platzierung des robotischen Systems eignet. Dies erfolgt durch eine weitere Metrik $D_{voxel}$ in \autoref{eq:22}, die sich aus dem arithmetischen Mittel $D_{Reach}^{mittel}$ eines Voxels und dem Verhältnis von dessen Kardinalität zum maximalen Voxel $TP_{max}$ ergibt. $TP_{max}$ ist das Resultat einer Iteration über alle $k$ Voxel, welche im Anschluss an die Voxelisierung erfolgt. Demnach lässt ein Voxel mit hoher Kardinalität darauf schließen, dass Posen verschiedener Aufgaben von ihm erreichbar sind. \begin{equation} \label{eq:22} \begin{split} -D_{voxel}:TP & \to [0,1] \\ -D_{voxel}(TP) &= D_{voxel}^{average}(TP) \times \frac{\vert TP \vert}{\vert TP_{max} \vert} +D_{voxel}:V & \to [0,1] \\ +D_{voxel}(TP) &= D_{voxel}^{mittel}(TP) \times \frac{\vert TP \vert}{\vert TP_{max} \vert} \end{split} \end{equation} -$TP_{max}$ ist das Resultat einer Iteration Über alle $k$ Voxel, welche im Anschluss an die Voxelisierung erfolgt. Demnach lässt ein Voxel mit hoher Kardinalität darauf schließen, dass Posen verschiedener Aufgaben von ihm erreichbar sind. Die Metrik $D_{voxel}$ bewertet demnach ein Voxel anhand dessen Inhalts und Mächtigkeit, wobei die Voxel mit der maximalen Wertigkeit eine valide Position des robotischen Systems für alle $n = \vert a \vert$ Posen der Aufgabe darstellt. +Die Metrik $D_{voxel}$ bewertet demnach ein Voxel anhand dessen Inhalts und Kardinalität, wobei ein Voxel mir maximaler Bewertung eine valide Position des robotischen Systems für alle $n = \vert A \vert$ Posen der Aufgabe darstellt. \begin{equation} \begin{split} \label{eq:21} -D_{voxel}^{average}:TP & \to [0,1] \\ -D_{voxel}^{average}(TP)&= \frac{\displaystyle\sum_{i=0}^{\vert TP \vert } D_{Reach}(p_{i})}{\vert TP \vert} +D_{voxel}^{mittel}:V & \to [0,1] \\ +D_{voxel}^{mittel}(TP)&= \frac{\displaystyle\sum_{i=0}^{\vert TP \vert } D_{Reach}(p_{i})}{\vert TP \vert} \end{split} \end{equation} diff --git a/thesis.tex b/thesis.tex index 0cc2a65..a5a3f7c 100644 --- a/thesis.tex +++ b/thesis.tex @@ -17,7 +17,7 @@ url=false, doi=true, isbn=false, - hyperref, + hyperref, ]{biblatex} \addbibresource{bibliography.bib} \AtEveryBibitem{% @@ -98,6 +98,8 @@ \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \usepackage{hyperref} +\usepackage{mathtools} + \usepackage{bera}% optional: just to have a nice mono-spaced font \usepackage{listings} -- GitLab