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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_87</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_86.html">< previous page</a></td> <td id="ebook_previous" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_87</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_88.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 87</font></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></td> <td colspan="3" height="12"></td> <td rowspan="5"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="2">which is the closed-form solution. When F</font><font face="Times New Roman, Times, Serif" size="1"><sub>33</sub></font><font face="Times New Roman, Times, Serif" size="2"> can be approximated as zero, the following reduction results:</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"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></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="ae15a1f65833afbec93ff81e627b31f6.gif" border="0" alt="0087-01.GIF" width="336" height="65" /></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"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></td> <td colspan="3" height="12"></td> <td rowspan="5"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="2">If the state-transition matrix is required for a time interval [0, <i>T</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>m</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></font><font face="Times New Roman, Times, Serif" size="2">] of duration long enough that the partitions of the F matrix cannot be considered constant, then it is possible to proceed by subdividing the interval. The interval can be decomposed into subintervals 0 < <i>t</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="2"> < <i>t</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>2</sub></font><font face="Times New Roman, Times, Serif" size="2"> <聽.聽.聽. < <i>T</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>m</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></font><font face="Times New Roman, Times, Serif" size="2">, where </font><font face="Symbol" size="2"><i>t</i></font><font face="Times New Roman, Times, Serif" size="2"> = max(<i>t</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>n</sub></font><font face="Times New Roman, Times, Serif" size="2"> - t</font><font face="Times New Roman, Times, Serif" size="1"><sub>n</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub>-1</sub></font><font face="Times New Roman, Times, Serif" size="2">) and the partitions of the F matrix can be considered constant over duration intervals of less than </font><font face="Symbol" size="2"><i>t</i></font><font face="Times New Roman, Times, Serif" size="2">. By the properties of state-transition 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"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></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="12086e3b8026c14b4098dbb34b62e011.gif" border="0" alt="0087-02.GIF" width="492" height="24" /></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"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></td> <td colspan="3" height="12"></td> <td rowspan="5"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="2">where </font><font face="Symbol" size="2">F</font><font face="Times New Roman, Times, Serif" size="2">(<i>t</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>n</sub></font><font face="Times New Roman, Times, Serif" size="2">, t</font><font face="Times New Roman, Times, Serif" size="1"><sub>n</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub>-1</sub></font><font face="Times New Roman, Times, Serif" size="2">) is defined by either Eq. (3.153) or (3.154) and F</font><font face="Times New Roman, Times, Serif" size="1"><sub>12</sub></font><font face="Times New Roman, Times, Serif" size="2">, F</font><font face="Times New Roman, Times, Serif" size="1"><sub>23</sub></font><font face="Times New Roman, Times, Serif" size="2">, and F</font><font face="Times New Roman, Times, Serif" size="1"><sub>33</sub></font><font face="Times New Roman, Times, Serif" size="2"> are defined by the data for the [<i>t</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>n</sub></font><font face="Times New Roman, Times, Serif" size="2">, t</font><font face="Times New Roman, Times, Serif" size="1"><sub>n</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub>-1</sub></font><font face="Times New Roman, Times, Serif" size="2">) interval. The transition matrix </font><font face="Symbol" size="2">F</font><font face="Times New Roman, Times, Serif" size="2">(<i>t</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>n</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub>-1</sub></font><font face="Times New Roman, Times, Serif" size="2">, <i>t</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="2">) is defined from previous iterations of Eq. (3.155). If the partitioned form of Eq. (3.155) is used advantageously, then the computation can be quite efficient. The iteration of Eq. (3.155) is initialized with </font><font face="Symbol" size="2">F</font><font face="Times New Roman, Times, Serif" size="2">(<i>t</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="2">, <i>t</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="2">)=I and continues for the interval of time propagation to yield </font><font face="Symbol" size="2">F</font><font face="Times New Roman, Times, Serif" size="2">(<i>T</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>m</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></font><font face="Times New Roman, Times, Serif" size="2">, 0).</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"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></td> <td colspan="3" height="12"></td> <td rowspan="5"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="2">Equation (3.152) corresponds to the F matrix for certain INS error models after simplification. In this case, F</font><font face="Times New Roman, Times, Serif" size="1"><sub>33</sub></font><font face="Times New Roman, Times, Serif" size="2"> is a skew-symmetric matrix. Therefore the exponential of F</font><font face="Times New Roman, Times, Serif" size="1"><sub>33</sub></font><font face="Times New Roman, Times, Serif" size="2"> can be calculated in closed form as was discussed relative to direction-cosine matrix computations in Chap. 2.</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>3.4.4<br />Mean and Covariance Propagation</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">From Eqs. (3.91), if the mean of the state vector is known at some time <i>k</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>0</sub></font><font face="Times New Roman, Times, Serif" size="3">, then the mean can be propagated forward according to</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="723de71c40c93e56dc57a10f707d5d6d.gif" border="0" alt="0087-03.GIF" width="367" height="67" /></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 assumption that <i>E<</i>w(<i>k</i>)> = 0 has been invoked. Intuitively, this formula states that since nothing is known <i>a priori</i> about the instantiation of the process noise, the mean of the state is propagated according to the state equations (3.42).</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 state covariance is defined as P(<i>k</i>) = <i>E</i><[x(<i>k</i>) - </font><font face="Symbol" size="3"><i>m</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>x</sub></font><font face="Times New Roman, Times, Serif" size="3">][x(<i>k</i>) - </font><font face="Symbol" size="3"><i>m</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>x</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">>, where </font><font face="Symbol" size="3"><i>m</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>x</sub></font><font face="Times New Roman, Times, Serif" size="3"> = <i>E</i>(x(<i>k</i>)>. Therefore the state covariance is propagated according to</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="0029f2da05bbfbe321f7ce80ed6d3734.gif" border="0" alt="0087-04.GIF" width="404" height="67" /></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_86.html">< previous page</a></td> <td id="ebook_next" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_87</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_88.html">next page ></a></td> </tr> </table> </body> </html>⌨️ 快捷键说明
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