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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd"> <html> <head> <title>page_50</title> <link rel="stylesheet" href="reset.css" type="text/css" media="all"> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> </head> <body> <table summary="top nav" border="0" width="100%"> <tr> <td align="left" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_49.html">< previous page</a></td> <td id="ebook_previous" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_50</strong></td> <td align="right" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_51.html">next page ></a></td> </tr> <tr> <td id="ebook_page" align="left" colspan="3" style="background: #ffffff; padding: 20px;"> <table border="0" width="100%" cellpadding="0"><tr><td align="center"> <table border="0" cellpadding="2" cellspacing="0" width="100%"><tr><td align="left"></td> <td align="right"></td> </tr></table></td></tr><tr><td align="left"><p></p><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="right"><font face="Times New Roman, Times, Serif" size="2" color="#FF0000">Page 50</font></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">Define</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="21864bc8e214ccff84c8c85acb3d733b.gif" border="0" alt="0050-01.GIF" width="366" height="40" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">which can be considered as the vector transformation from frame <i>a</i> at time <i>t</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>k</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub>-1</sub></font><font face="Times New Roman, Times, Serif" size="3"> to frame <i>a</i> at time <i>t</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>k</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></font><font face="Times New Roman, Times, Serif" size="3">. Consider the error incurred by approximating the sine and the cosine functions by their truncated Taylor series expansions. In this case,</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="4f7c27a8c5a4e6d73cce82290cb5deb9.gif" border="0" alt="0050-02.GIF" width="344" height="31" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">where </font><font face="Symbol" size="3"><i>s</i></font><font face="Times New Roman, Times, Serif" size="3"> = ||</font><font face="Symbol" size="3"><i>u</i></font><font face="Times New Roman, Times, Serif" size="3">||, </font><font face="Symbol" size="3">娄</font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="3">(</font><font face="Symbol" size="3">s</font><font face="Times New Roman, Times, Serif" size="3">) </font><font face="Symbol" size="3">禄</font><font face="Times New Roman, Times, Serif" size="3"> [sin(</font><font face="Symbol" size="3"><i>s</i></font><font face="Times New Roman, Times, Serif" size="3">)/</font><font face="Symbol" size="3"><i>s</i></font><font face="Times New Roman, Times, Serif" size="3">] and </font><font face="Symbol" size="3">娄</font><font face="Times New Roman, Times, Serif" size="1"><sub>2</sub></font><font face="Times New Roman, Times, Serif" size="3">(</font><font face="Symbol" size="3"><i>s</i></font><font face="Times New Roman, Times, Serif" size="3">) </font><font face="Symbol" size="3">禄</font><font face="Times New Roman, Times, Serif" size="3"> {[1 - cos(</font><font face="Symbol" size="3"><i>s</i></font><font face="Times New Roman, Times, Serif" size="3">)]/</font><font face="Symbol" size="3"><i>s</i></font><font face="Times New Roman, Times, Serif" size="2"><sup>2</sup></font><font face="Times New Roman, Times, Serif" size="3">}. Let R represent the <i>a</i> to <i>b</i> transformation that would result from using </font><font face="Symbol" size="3">L</font><font face="Times New Roman, Times, Serif" size="3"> and let <img src="c7964d70a8c372381587469ddb438545.gif" border="0" alt="RCIRC.GIF" width="16" height="18" /> represent the <i>a</i> to <i>b</i> transformation that would result from using <img src="43894091edf0135841295aa09951d21b.gif" border="0" alt="C0050-01.GIF" width="15" height="17" /> Assume that R and <img src="c7964d70a8c372381587469ddb438545.gif" border="0" alt="RCIRC.GIF" width="16" height="18" /> are related by</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="ad0d9e3be21256aaf0e6e0197b02e727.gif" border="0" alt="0050-03.GIF" width="285" height="20" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">where P is an arbitrarily structured perturbation matrix. The matrix P is assumed small since it represents the effects of computation errors. Higher powers of P are neglected in the subsequent analysis.</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">Since R is an orthonormal matrix, R</font><font face="Times New Roman, Times, Serif" size="2"><sup><i>T</i></sup></font><font face="Times New Roman, Times, Serif" size="3">R</font><font face="Times New Roman, Times, Serif" size="3"> = I. Forming the same product for <img src="c7964d70a8c372381587469ddb438545.gif" border="0" alt="RCIRC.GIF" width="16" height="18" /> yields</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="e9e062bca00ec891e547768b8c2d1427.gif" border="0" alt="0050-04.GIF" width="328" height="20" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="25ed1dae4ca6e669dd986f2839b3b010.gif" border="0" alt="0050-05.GIF" width="296" height="20" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">where the second-order P</font><font face="Times New Roman, Times, Serif" size="2"><sup><i>T</i></sup></font><font face="Times New Roman, Times, Serif" size="3">P</font><font face="Times New Roman, Times, Serif" size="3"> term has been neglected. If the calculation of <img src="c7964d70a8c372381587469ddb438545.gif" border="0" alt="RCIRC.GIF" width="16" height="18" /> includes an orthonormalization step, then <img src="5bc2ec79565a97c62e6d5d28c28a014f.gif" border="0" alt="C0050-02.GIF" width="58" height="18" />, which implies that P must be skew symmetric. In this case, P represents the error of <img src="c7964d70a8c372381587469ddb438545.gif" border="0" alt="RCIRC.GIF" width="16" height="18" /> from the inverse of R</font><font face="Times New Roman, Times, Serif" size="2"><sup>T</sup></font><font face="Times New Roman, Times, Serif" size="3">, which is referred to as the drift error [116]. Note that the direction-cosine approach does not automatically produce an orthonormal transformation matrix. Ensuring orthonormality requires extra calculation. When orthonormality is not enforced, P will not be skew symmetric and may contain skew and scale-factor errors [116]. The product of R and <img src="c7964d70a8c372381587469ddb438545.gif" border="0" alt="RCIRC.GIF" width="16" height="18" /> yields a means of calculating P:</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="1c9998656575e9eeb175446e186a1efd.gif" border="0" alt="0050-06.GIF" width="296" height="21" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="1199ed9ed23179c787a4a157a46f8121.gif" border="0" alt="0050-07.GIF" width="277" height="21" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">Substituting the expressions for R and <img src="c7964d70a8c372381587469ddb438545.gif" border="0" alt="RCIRC.GIF" width="16" height="18" /> into Eq. (2.93) and reducing yields</font><font face="Times New Roman, Times, Serif" size="3" color="#FFFF00"></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="8780e604d63922c9c73b18454218ce6c.gif" border="0" alt="0050-08.GIF" width="484" height="80" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr></table><p><font size="0"></font></p>聽 </td> </tr> <tr> <td align="left" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_49.html">< previous page</a></td> <td id="ebook_next" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_50</strong></td> <td align="right" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_51.html">next page ></a></td> </tr> </table> </body> </html>⌨️ 快捷键说明
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