pbio/doc/control: Copy-edit math.
Fix a few typos.
@@ -506,7 +506,7 @@ gray regions of Figure \ref{fig:positions}, which holds if:
%
\begin{align}
\label{eq:a:decreaselimit}
- - \dfrac{1}{2 \a_2}\lr{\w_0^2 - \w_3^2} < \th_3 - \th_0
+ \dfrac{1}{2 \a_2}\lr{\w_3^2 - \w_0^2} < \th_3 - \th_0
\end{align}
%
If it holds, then there is a nonzero constant speed phase ($\th_1 \neq \th_2$)
@@ -534,15 +534,15 @@ can feasibly slow down without overshooting. To handle this, we cut the
initial speed value to within the allowed bound causing an abrupt deceleration:
%
\begin{align}
- \w_0 := \sqrt{\w_3^2 - \dfrac{1}{2\a_2}\lr{\th_3-\th_0}}
+ \w_0 := \sqrt{\w_3^2 - 2\a_2\lr{\th_3-\th_0}}
\end{align}
%
By definition, this means that there is only a deceleration phase, so
-$\th_1 = \th_2 = \th_3$, $\w_1=\w_2=\w_3$, $t_1=t_2=t_3$ with:
+$\th_1 = \th_2 = \th_0$, $\w_1=\w_2=\w_0$, $t_1=t_2=t_0$ with:
%
\begin{align}
\label{eq:a:decreaselimittime}
- t_1 = \dfrac{\w_0-\w_3}{\a_2}
+ t_3 - t_2 = \dfrac{\w_0-\w_3}{\a_2}
\end{align}
\subsubsection{Selecting the standard, cut short, or decreasing case}
@@ -558,8 +558,6 @@ The procedure is then as follows:
\ref{sec:a:cutshortw3is0} or \ref{sec:a:cutshortw3isw1}).
\end{enumerate}
%
-For efficiency, the decreasing and standard case can share the same code if the
-sign of $\a_0$ is set appropriately.
\subsubsection{Reversing and unreversing the final and target speed}
\label{sec:a:reversing}
@@ -570,7 +568,7 @@ calculate the trajectory, and map the final result back to obtain the
originally requested command.
\begin{itemize}
- \item Let the boolean $a := \wt < 0$.
+ \item Let the boolean $a := \th_3 < \th_0$.
\item If $a$, then invert targets as:
$\th_3 := 2 \th_0 - \th_3$, $\wt := -\wt$, $\w_0 := -\w_0$,
$\w_3 := -\w_3$.
