Commit d995fa05 authored by s_mara's avatar s_mara
Browse files

new shit

parent f4739052
......@@ -19,6 +19,7 @@
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\OT1/cmr/bx/n/12 Punkt-zu-Punkt-Steuerung (PTP) \OT1/cmr/m/n/12 bes-timmt an-ha
nd vorgegebene Robotere-in-schr^^?ankungen
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......@@ -14,7 +14,10 @@
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......@@ -23,11 +23,14 @@
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......@@ -30,16 +30,19 @@
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......@@ -260,19 +260,19 @@ x(t) = f(\theta(t)) \Rightarrow x(t) \in \mathbb{R}^6
\end{align*}
Somit ergibt sich aus der Differentation von Position $x$ unter Zeit, die Endeffektorgeschwindigkeit zur Zeitpunkt $t$:
\begin{align*}
\frac{\delta x}{\delta t} = \frac{\delta f(\theta(t))}{\delta t} \stackrel{\text{Kettenregel}}{=} J_f(\theta(t)) \cdot \dot{\theta(t)}
\frac{\partial x}{\partial t} = \frac{\partial f(\theta(t))}{\partial t} \stackrel{\text{Kettenregel}}{=} J_f(\theta(t)) \cdot \dot{\theta(t)}
\end{align*}
Die Berechnung von $J_f(\theta)$ ist:
\begin{align*}
J_f(\theta) = (\frac{\delta f}{\delta \theta_1} ... \frac{\delta f}{\delta \theta_n}) \in \mathbb{R}^{6 \times n}
J_f(\theta) = (\frac{\partial f}{\partial \theta_1} ... \frac{\partial f}{\partial \theta_n}) \in \mathbb{R}^{6 \times n}
\end{align*}
wobei man von der einzelne Ableitungen, zwei Berechnungen unterscheiden kann:
\begin{enumerate}
\item
Wenn das $j$-te Gelenk eine Translation in Richtung $v_j \in \mathbb{R}^3$ macht
\begin{align*}
\frac{\delta f}{\delta \theta_j} = \begin{pmatrix}
\frac{\partial f}{\partial \theta_j} = \begin{pmatrix}
v_j \\
0
\end{pmatrix} \in \mathbb{R}^6
......@@ -281,7 +281,7 @@ v_j \\
\item
Wenn das $j$-te Gelenk eine Rotation in um die Achse $v_j \in \mathbb{R}^3$ an der Position $p_j \in \mathbb{R}^3 $ macht
\begin{align*}
\frac{\delta f}{\delta \theta_j} = \begin{pmatrix}
\frac{\partial f}{\partial \theta_j} = \begin{pmatrix}
v_j \times (f(\theta) - p_j)\\
0
\end{pmatrix} \in \mathbb{R}^6
......@@ -301,7 +301,43 @@ muss $\Theta$ gefunden werden. \par
Hier gibt es verschiedene Lösungsverfahren, und jeder hat ihre eigene Vorteile im Vergleich zu der andere. Z.B, Numerische Verfahren für die Allgemeine Lösung von Gleichungssysteme sind langsamer und haben höhere Aufwand im Vergleich zur Spezielle Lösungen wie trigonometrisch basierte Lösungen, aber die wirken dagegen als kräftiger denn man kann die numerische Verfahren überall Anwenden und die spezielle Lösungen stehen speziell für eine begrenzte Anzahl von Roboter.
\section{Dynamik}
%TODO
\subsection{Allgemeine Bewegungsgleichung}
\begin{align*}
\tau = M(q) \cdot \ddot{q} + c(\dot{q},q) + g(q)
\end{align*}
\begin{multicols}{2}
\begin{itemize}
\item $\tau: n \times 1$ Vektor der generalisierten Kräfte
\item $M(q): n \times n$ Massenträgheitsmatrix
\item $c(\dot{q},q): n \times 1$ Zentripetalkomponenten
\item $g(q): n \times 1$ Vektor der Gravitationskomponenten
\end{itemize}
\end{multicols}
\subsection{Methode nach Lagrange}
Ziel hier ist: Ermittle für jedes Gelenk $i$ eines Roboters die Bewegungsgleichung mittels:
\begin{align*}
\tau_{i} = \frac{d}{dt}(\frac{\partial L}{\partial \dot{q_i}}) - \frac{\partial L}{\partial q_i}
\end{align*}
Danach $E_{kin}$ und $E_{pot}$ und die beide in der Form ausdrucken:
\begin{align*}
L(q,\dot{q}) = E_{kin}(q, \dot{q}) - E_{pot}(q)
\end{align*}
Und schließlich berechne die Ableitungen.
\begin{prereq}
Die Kinematische Energie eine Rotierende Körper ist:
\begin{align*}
E_{kin} = \frac{1}{2}mv^2 + \frac{1}{2}J_{t}\omega^2
\end{align*}
wobei die $J_t$ die spezifische Trägeheitsmoment eine Körper sei. Für eine Stab ist es
\begin{align*}
J_t=\frac{1}{12}ml^2
\end{align*}
\end{prereq}
\subsection{Newton}
Im Vergleich zu Lagrange ist Newton deutlich schneller mit einer Aufwand von $O(n)$ im Gegensatz zur $O(n^3)$. Hier wird mit Impulsen/Momenten berechnet, statt Energien, und Newton hat eine festgelegte Verfahren/Algorithmus während Lagrange ist rein Physische Berechnungen.
\section{Regelung}
Unter einer Regelung versteht man eine Anordnung, durch welche bei unvollständig bekannter Strecke, insbesondere unvollständiger Kenntnis der Störgröße, die Regelgröße, d.h. die Ausgangsgröße der Strecke, laufend erfasst und mit der Führungsgröße verglichen wird, um mittels der so gebildeten Differenz die Regelgröße an den Sollverlauf anzugleichen. Dieser ganze Regelung wird üblicherweise in kleiner Regler gepackt die ganz low level Berechnungen durchführen müssen (wie z.B ein Motorregler der die Geschwindigkeit einer Motor konstant behalten soll, unter der Störgröße von Reibung, usw). Für High Level Rechnungen/Regelungen, werden komplette Rechner benutzt statt kleine Regler. \par
......
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