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Martin Dieblich
73 lines
2.0 KiB
TeX
73 lines
2.0 KiB
TeX
\section{Scalar}
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For scalar values are a few functions available
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\subsection{Multiplication and Rightshift}
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Represents $ a \multiplication b $ with a right shift about \textit{r}. This becomes close to
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\begin{equation}
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2^{-r} a \multiplication b
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\end{equation}
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but it is not the same.
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\inHfile{INT\_MULT\_RSHIFT(a, b, r)}{pprz\_algebra\_int}
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\subsection{$\sqrt x$ Squareroot}
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Calculates the squareroot $y = \sqrt x$. The function uses the Babylonian method.
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\begin{equation}
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y_{n+1} = \frac 1 2 \left( y_n + \frac{x}{y_n} \right)
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\end{equation}
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\inHfile{INT32\_SQRT(out,in)}{pprz\_algebra\_int}
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\subsection{atan2() 4-quadrant arctangent}
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Calculates the 4-quadrant arctangent of two values, x and y:
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\begin{equation}
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a = atan2(y,x)
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\end{equation}
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The function uses a trick, which is desribed in detail at
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\begin{itemize}
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\item http://www.dspguru.com/comp.dsp/tricks/alg/fxdatan2.htm
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\end{itemize}
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In short:
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\begin{figure}[h!]
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\centering
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\begin{tabular}{ccc}
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\begin{minipage}{4cm}
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\centering
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\includegraphics[width=4cm]{xyvalues}
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\end{minipage}
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&
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\begin{minipage}{4cm}
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\centering
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\includegraphics[width=4cm]{ratiofunction}
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\end{minipage}
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&
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\begin{minipage}{5cm}
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\centering
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\includegraphics[width=4cm]{atan2_alternate}
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\end{minipage}
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\\
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(a) x/y-values &
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(b) ratio function &
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(c) comparison of the result (blue)
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\\
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& & and the real value (red)
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\end{tabular}
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\caption{alternate atan2 function} \label{alternate atan2 function}
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\end{figure}
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If you have a set of x/y values (figure \ref{alternate atan2 function}a), you can compute the ratio (figure \ref{alternate atan2 function}b) of them:
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\begin{equation}
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r = \frac{x+y}{x-y}
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\end{equation}
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and transform this ratio very close to the real values (figure \ref{alternate atan2 function}c) using
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\begin{equation}
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\alpha = \tfrac \pi 4 (1-r)
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\end{equation}
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or (more accurate) using
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\begin{equation}
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\alpha_2 = 0.1963 \multiplication r^3 -0.9817 \multiplication r + \tfrac \pi 4
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\end{equation}
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\inHfile{INT32\_ATAN2(a, y, x)}{pprz\_algebra\_int}
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\inHfile{INT32\_ATAN2\_2(a, y, x)}{pprz\_algebra\_int} |