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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_53</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_52.html">< previous page</a></td> <td id="ebook_previous" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_53</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_54.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 53</font></td></tr></table><table cellpadding="0" cellspacing="0" border="0" width="100%"><tr><td height="12"></td></tr><tr><td><table cellspacing="0" width="642" cellpadding="4"><tr><td colspan="7" valign="top"><font face="Times New Roman, Times, Serif" size="2">TABLE 2.4 Coefficients for Approximate Quaternion Update for Algorithms of Orders One through Six</font></td></tr><tr><td valign="top"></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">First</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">Second</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">Third</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">Fourth</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">Fifth</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">Sixth</font></td></tr></table></td></tr><tr><td valign="top"><font face="Times New Roman, Times, Serif" size="2"><i>f</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>c</sub></font></i></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">1</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">1-<i>s</i><sup>2</sup>/2</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">1-<i>s</i><sup>2</sup>/2</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">1-<i>s</i><sup>2</sup>/2+<i>s</i><sup>4</sup>/24</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">1-<i>s</i><sup>2</sup>/2+<i>s</i><sup>4</sup>/24</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">1-<i>s</i><sup>2</sup>/2+<i>s</i><sup>4</sup>/24-<i>s</i><sup>6</sup>/720</font></td></tr></table></td></tr><tr><td valign="top"><font face="Times New Roman, Times, Serif" size="2"><i>f</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>s</sub></font></i></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">0.5</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">0.5</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">0.5(1-<i>s</i><sup>2</sup>/6)</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">0.5(1-<i>s</i><sup>2</sup>/6)</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">0.5(1-<i>s</i><sup>2</sup>/6+<i>s</i><sup>4</sup>/120)</font></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="center"><font face="Times New Roman, Times, Serif" size="2">0.5(1-<i>s</i><sup>2</sup>/6+<i>s</i><sup>4</sup>/120)</font></td></tr></table></td></tr></table></td></tr></table><br /><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">algorithm previously presented, then the only remaining error in the calculation is the drift error </font><font face="Symbol" size="3">D</font><font face="Times New Roman, Times, Serif" size="3">, which is a skew-symmetric matrix representation of the difference of <img src="f56451491274597467277b92c13bc8a0.gif" border="0" alt="C0053-01.GIF" width="60" height="20" /> from the inverse of <img src="2fa60f35c6635b966d5bc358a1c106cb.gif" border="0" alt="C0053-02.GIF" width="61" height="21" />. To quantify this error, consider the product of these two matrices:</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="f77b9948fafad2f4817c92306fc9a014.gif" border="0" alt="0053-01.GIF" width="368" height="22" /></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="3d0471a01801fbd906c8592e5a0ff53a.gif" border="0" alt="0053-02.GIF" width="248" 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"><img src="8988ecd022a52271b3069d595874ad9f.gif" border="0" alt="0053-03.GIF" width="267" height="22" /></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">Table 2.4 presents the power-series approximation for <i>f</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>c</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<i> f</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>s</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></font><font face="Times New Roman, Times, Serif" size="3"> for algorithms of orders one through six. In the table,<img src="6dd8c9aca9cda71069206a9b8059e208.gif" border="0" alt="C0053-03.GIF" width="131" height="19" />. The drift error incurred for the algorithms of each order are plotted in Fig. 2.14. The drift error is calculated as the norm of the three upper off-diagonal elements of </font><font face="Symbol" size="3">D</font><font face="Times New Roman, Times, Serif" size="3">. The plot shows the same general characteristics as Fig. 2.13 did for the direct direction-cosine approach; however, the quaternion error is always smaller than the direct approach for the same order algorithm and the same rotation magnitude. This is predominantly due to the quaternion expansions involving one half of the rotation angle.</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">2.5.3.5<br />Angle Update Integration</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 direction-cosine and quaternion update equations both involve the solution of differential equations involving the angular rate. In the previous sections solutions have been presented that are valid under the assumption that the direction of the angular rate vector was constant. It has been previously stated that switching the order in which rotations occur results in different resultant rotations. Therefore, when the angular rate vector does not have a fixed direction, the simple integral of the angular rate should not be expected to be accurate.</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">Under arbitrary motion, the angular rate vector cannot be assumed constant; however, the presented algorithms are usable if the angle vector is updated correctly. It can be shown [15, 130] that the angle update is correctly calculated 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="7b416395ea351f6119c54458ed619e42.gif" border="0" alt="0053-04.GIF" width="445" height="39" /></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">for ||</font><font face="Symbol" size="3"><i>a</i></font><font face="Times New Roman, Times, Serif" size="3">|| </font><font face="Symbol" size="3">鹿</font><font face="Times New Roman, Times, Serif" size="3"> <i>n</i></font><i><font face="Symbol" size="3">p</font><font face="Times New Roman, Times, Serif" size="3">, n</font></i><font face="Times New Roman, Times, Serif" size="3"> = </font><font face="Symbol" size="3">卤</font><font face="Times New Roman, Times, Serif" size="3">1, </font><font face="Symbol" size="3">卤</font><font face="Times New Roman, Times, Serif" size="3">2,聽.聽.聽. . In this expression, the first term coincides with the solution for a fixed angular rate direction vector. The second two terms account for the effects of noncommutativity (i.e., previous rotations represented</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></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_52.html">< previous page</a></td> <td id="ebook_next" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_53</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_54.html">next page ></a></td> </tr> </table> </body> </html>⌨️ 快捷键说明
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