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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_39</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_38.html">< previous page</a></td> <td id="ebook_previous" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_39</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_40.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 39</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">second axis of the resultant frame of the first rotation, and </font><font face="Symbol" size="3"><i>dq</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="3"> about the first axis of the resultant frame of the second rotation. Denote this infinitesimal rotation by <img src="54f08efb0e86399efd82bb1f5efe7d16.gif" border="0" alt="C0039-01.GIF" width="135" height="20" />. The vector transformation from frame <i>a</i> to frame <i>b</i> is defined by the series of three rotations: [</font><font face="Symbol" size="3"><i>dq</i></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="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"><i>dq</i></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="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>dq</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>3</sub></font><font face="Times New Roman, Times, Serif" size="3">]</font><font face="Times New Roman, Times, Serif" size="1"><sub>3</sub></font><font face="Times New Roman, Times, Serif" size="3">. Because each angle is infinitesimal [which implies that cos(</font><font face="Symbol" size="3"><i>dq</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>i</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></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, sin(</font><font face="Symbol" size="3"><i>dq</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>i</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></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"></font><font face="Symbol" size="3"><i>dq</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>i</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></font><font face="Times New Roman, Times, Serif" size="3">, and </font><font face="Symbol" size="3"><i>dq</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>i</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></font><font face="Times New Roman, Times, Serif" size="3"></font><font face="Symbol" size="3"><i>dq</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>j</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></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"> 0 for <i>i, j</i> = 1, 2, 3], the order in which the rotations occur will not be important. The matrix representation of the vector transformation is</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="36258e1b469650135639a247300da02f.gif" border="0" alt="0039-01.GIF" width="430" height="113" /></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="94a4ef5ac06488d9393d01b206db703c.gif" border="0" alt="0039-02.GIF" width="433" height="23" /></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 <img src="ab6cbb00d652811a830681ddb7730505.gif" border="0" alt="C0039-02.GIF" width="93" height="19" /> is the skew-symmetric representation of <img src="7759925512954a6035ee85abd0b2d59e.gif" border="0" alt="C0039-03.GIF" width="27" height="19" />, as defined in App. A.</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"><i>2.4.2<br />Quaternions</i></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">A theorem due to Euler states that any given sequence of rotations can be represented as a single rotation about a single fixed axis [60]. For a vehicle experiencing arbitrary angular motion, the effective axis and rotation angle will evolve over time. In this section the quaternion method for parameterizing the effective axis and rotation angle is introduced. From the properties of quaternions [41, 60, 116, 138, 142], it is possible to derive another parameterization of the direction-cosine matrix. Quaternion parameterizations may appear less intuitively appealing, but they are singularity free and more computationally efficient than the Euler angle or direction-cosine approaches.</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">The quaternion is a four vector b = [<i>b</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="3">, <i>b</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>2</sub></font><font face="Times New Roman, Times, Serif" size="3">, <i>b</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>3</sub></font><font face="Times New Roman, Times, Serif" size="3">, <i>b</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>4</sub></font><font face="Times New Roman, Times, Serif" size="3">]</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"> that can be partitioned as</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="8fe2ad912ac6bb7577cdd22b2f8a2fd9.gif" border="0" alt="0039-03.GIF" width="113" 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">where E is a unit vector and </font><font face="Symbol" size="3">z</font><font face="Times New Roman, Times, Serif" size="3"> is a positive rotation about E. If the quaternion b represents the rotational transformation from reference frame <i>a</i> to reference frame <i>b</i>, then frame <i>a</i> is aligned with frame <i>b</i> when frame <i>a</i> is rotated by </font><font face="Symbol" size="3">z</font><font face="Times New Roman, Times, Serif" size="3"> radians about E. Note that b has the normality property that ||b|| = 1. Therefore the quaternion also has only 3 degrees of freedom.</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">A quaternion b = (<i>b</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="3">, <i>b</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>2</sub></font><font face="Times New Roman, Times, Serif" size="3">, <i>b</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>3</sub></font><font face="Times New Roman, Times, Serif" size="3">, <i>b</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>4</sub></font><font face="Times New Roman, Times, Serif" size="3">) can be represented by a generalized (four-component) complex number:</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="3797fd08c65d6d2566f339162cfdd298.gif" border="0" alt="0039-04.GIF" width="151" height="21" /></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_38.html">< previous page</a></td> <td id="ebook_next" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_39</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_40.html">next page ></a></td> </tr> </table> </body> </html>⌨️ 快捷键说明
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