2450:
reference vectors are usually known directions (e.g. stars, Earth magnetic field, gravity vector, etc.). Body fixed vectors are the measured directions as observed by an on-board sensor (e.g. star tracker, magnetometer, etc.). With advances in micro-electronics, attitude determination algorithms such as TRIAD have found their place in a variety of devices (e.g. smart phones, cars, tablets, UAVs, etc.) with a broad impact on modern society.
23:
is the earliest published algorithm for determining spacecraft attitude, which was first introduced by Harold Black in 1964. Given the knowledge of two vectors in the reference and body coordinates of a satellite, the TRIAD algorithm obtains the direction cosine matrix relating to both frames. Harold
2449:
TRIAD was used as an attitude determination technique to process the telemetry data from the
Transit satellite system (used by the U.S. Navy for navigation). The principles of the Transit system gave rise to the global positioning system satellite constellation. In an application problem, the
24:
Black played a key role in the development of the guidance, navigation, and control of the U.S. Navy's
Transit satellite system at Johns Hopkins Applied Physics Laboratories. TRIAD represented the state of practice in spacecraft attitude determination before the advent of
1476:
1314:
2259:
are also left-handed because of the one-one correspondence between the vectors. This is because of the simple fact that, in
Euclidean geometry, the angle between any two vectors remains invariant to coordinate transformations. Therefore, the determinant
932:
are noisy and the orthogonality condition of the attitude matrix (or the direction cosine matrix) is not preserved by the above procedure. TRIAD incorporates the following elegant procedure to redress this problem. To this end, one defines unit vectors,
1959:
It is of consequence to note that the TRIAD method always produces a proper orthogonal matrix irrespective of the handedness of the reference and body vectors employed in the estimation process. This can be shown as follows: In a matrix form given
532:
1946:
Note that computational efficiency has been achieved in this procedure by replacing the matrix inverse with a transpose. This is possible because the matrices involved in computing attitude are each composed of a TRIAD of
392:
transforms vectors in the body fixed frame into the frame of the reference vectors. Among other properties, rotational matrices preserve the length of the vector they operate on. Note that the direction cosine matrix
1149:
1033:
2441:
This is quite useful in practical applications since the analyst is always guaranteed a proper orthogonal matrix irrespective of the nature of the reference and measured vector quantities.
231:
930:
164:
1927:
1339:
1177:
2295:
2422:
370:
1993:
166:
be the corresponding measured directions of the reference unit vectors as resolved in a body fixed frame of reference. Following that, they are then related by the equations,
1501:). Their cross product is used as the third column in the linear system of equations obtaining a proper orthogonal matrix for the spacecraft attitude given by the following:
2217:
2112:
106:
70:
2367:
1686:
832:
422:
2257:
2237:
869:
280:
2338:
2315:
569:
411:
390:
300:
2650:
1058:
942:
2645:
2560:
1717:) are not necessary, they have been carried out to achieve a computational advantage in solving the linear system of equations in (
2459:
28:. and its several optimal solutions. Covariance analysis for Black's solution was subsequently provided by Markley.
175:
877:
111:
1471:{\displaystyle {\hat {m}}={\frac {{\vec {r}}_{1}\times {\vec {r}}_{2}}{||{\vec {r}}_{1}\times {\vec {r}}_{2}||}}}
1309:{\displaystyle {\hat {M}}={\frac {{\vec {R}}_{1}\times {\vec {R}}_{2}}{||{\vec {R}}_{1}\times {\vec {R}}_{2}||}}}
1732:
2263:
2464:
2384:
2524:
Black, Harold (July–August 1990). "Early
Developments of Transit, the Navy Navigation Satellite System".
2606:
309:
1969:
2533:
2498:
527:{\displaystyle {\vec {R}}_{1}\times {\vec {R}}_{2}=A\left({\vec {r}}_{1}\times {\vec {r}}_{2}\right)}
2117:
2015:
75:
39:
2343:
1510:
580:
2469:
25:
303:
2242:
2222:
854:
2587:
2541:
2506:
1723:). Thus an estimate of the spacecraft attitude is given by the proper orthogonal matrix as
2489:
Black, Harold (July 1964). "A Passive System for
Determining the Attitude of a Satellite".
253:
2578:
Wahba, Grace (July 1966). "A Least
Squares Estimate of Satellite Attitude, Problem 65.1".
2340:
depending on whether its columns are right-handed or left-handed respectively (similarly,
2537:
2502:
2320:
2300:
1948:
554:
396:
375:
285:
874:
The solution presented above works well in the noise-free case. However, in practice,
2639:
2607:"Attitude Determination Using Vector Observations: A Fast Optimal Matrix Algorithm"
1144:{\displaystyle {\hat {s}}={\frac {{\vec {r}}_{1}}{||{\vec {r}}_{1}||}}}
1028:{\displaystyle {\hat {S}}={\frac {{\vec {R}}_{1}}{||{\vec {R}}_{1}||}}}
2591:
2545:
2510:
36:
Firstly, one considers the linearly independent reference vectors
1951:
basis vectors. "TRIAD" derives its name from this observation.
2369:). Taking determinant on both sides of the relation in Eq. (
1495:
to be used in place of the first two columns of equation (
551:
TRIAD proposes an estimate of the direction cosine matrix
16:
Solution to the spacecraft attitude determination problem
302:
is a rotation matrix (sometimes also known as a proper
2561:"Attitude Determination Using Two Vector Measurements"
571:
as a solution to the linear system equations given by
413:
also transforms the cross product vector, written as,
2387:
2346:
2323:
2303:
2266:
2245:
2225:
2120:
2018:
1972:
1735:
1513:
1342:
1180:
1061:
945:
880:
871:
have been used to separate different column vectors.
857:
583:
557:
425:
399:
378:
312:
288:
256:
178:
114:
78:
42:
1955:
TRIAD Attitude Matrix and
Handedness of Measurements
2416:
2361:
2332:
2309:
2289:
2251:
2231:
2211:
2106:
1987:
1921:
1680:
1470:
1308:
1143:
1027:
924:
863:
826:
563:
526:
405:
384:
364:
294:
274:
225:
158:
100:
64:
2239:form a left-handed TRIAD, then the columns of
226:{\displaystyle {\vec {R}}_{i}=A{\vec {r}}_{i}}
925:{\displaystyle {\vec {r}}_{1},{\vec {r}}_{2}}
159:{\displaystyle {\vec {r}}_{1},{\vec {r}}_{2}}
8:
1922:{\displaystyle {\hat {A}}=\left.\left^{T}}
2526:Journal of Guidance, Control and Dynamics
2386:
2345:
2322:
2302:
2265:
2244:
2224:
2190:
2189:
2175:
2174:
2154:
2153:
2133:
2132:
2119:
2088:
2087:
2073:
2072:
2052:
2051:
2031:
2030:
2017:
1971:
1913:
1897:
1896:
1882:
1881:
1861:
1860:
1840:
1839:
1814:
1813:
1799:
1798:
1778:
1777:
1757:
1756:
1737:
1736:
1734:
1662:
1661:
1647:
1646:
1626:
1625:
1605:
1604:
1577:
1576:
1562:
1561:
1541:
1540:
1520:
1519:
1512:
1460:
1455:
1449:
1438:
1437:
1427:
1416:
1415:
1409:
1404:
1396:
1385:
1384:
1374:
1363:
1362:
1358:
1344:
1343:
1341:
1298:
1293:
1287:
1276:
1275:
1265:
1254:
1253:
1247:
1242:
1234:
1223:
1222:
1212:
1201:
1200:
1196:
1182:
1181:
1179:
1133:
1128:
1122:
1111:
1110:
1104:
1099:
1092:
1081:
1080:
1077:
1063:
1062:
1060:
1017:
1012:
1006:
995:
994:
988:
983:
976:
965:
964:
961:
947:
946:
944:
916:
905:
904:
894:
883:
882:
879:
856:
808:
797:
796:
786:
775:
774:
753:
742:
741:
725:
714:
713:
685:
674:
673:
663:
652:
651:
630:
619:
618:
602:
591:
590:
582:
556:
513:
502:
501:
491:
480:
479:
461:
450:
449:
439:
428:
427:
424:
398:
377:
317:
311:
287:
255:
217:
206:
205:
192:
181:
180:
177:
150:
139:
138:
128:
117:
116:
113:
92:
81:
80:
77:
56:
45:
44:
41:
2481:
2290:{\displaystyle det\left(\Gamma \right)}
1705:While the normalizations of equations (
2614:The Journal of Astronautical Sciences
7:
2417:{\displaystyle det\left(A\right)=1.}
2377:
1962:
1725:
1503:
1332:
1170:
1051:
935:
573:
415:
168:
2605:Markley, Landis (April–June 1993).
2347:
2280:
2246:
2226:
2121:
2019:
1982:
1973:
365:{\displaystyle A^{T}A=I,det(A)=+1}
14:
1988:{\displaystyle \Gamma =A\Delta }
2565:1999 Flight Mechanics Symposium
2212:{\displaystyle \Delta =\left.}
2195:
2180:
2159:
2138:
2107:{\displaystyle \Gamma :=\left}
2093:
2078:
2057:
2036:
1902:
1887:
1866:
1845:
1819:
1804:
1783:
1762:
1742:
1667:
1652:
1631:
1610:
1582:
1567:
1546:
1525:
1461:
1456:
1443:
1421:
1410:
1405:
1390:
1368:
1349:
1299:
1294:
1281:
1259:
1248:
1243:
1228:
1206:
1187:
1134:
1129:
1116:
1105:
1100:
1086:
1068:
1018:
1013:
1000:
989:
984:
970:
952:
910:
888:
802:
780:
747:
719:
679:
657:
624:
596:
507:
485:
455:
433:
350:
344:
211:
186:
144:
122:
101:{\displaystyle {\vec {R}}_{2}}
86:
65:{\displaystyle {\vec {R}}_{1}}
50:
1:
2567:: 2 – via ResearchGate.
2460:Attitude Dynamics and Control
2362:{\displaystyle \Delta =\pm 1}
2651:Rotation in three dimensions
2371:
2219:Note that if the columns of
1681:{\displaystyle \left=A\left}
827:{\displaystyle \left=A\left}
2646:Spacecraft attitude control
2559:Markley, F. Landis (1999).
1719:
1713:
1707:
1497:
2667:
2375:), one concludes that
2252:{\displaystyle \Delta }
2232:{\displaystyle \Gamma }
864:{\displaystyle \vdots }
2465:Orientation (Geometry)
2418:
2363:
2334:
2311:
2291:
2253:
2233:
2213:
2108:
1989:
1923:
1682:
1472:
1310:
1145:
1029:
926:
865:
828:
565:
528:
407:
386:
366:
296:
276:
227:
160:
102:
66:
2419:
2364:
2335:
2312:
2292:
2254:
2234:
2214:
2109:
1990:
1924:
1683:
1473:
1311:
1146:
1030:
927:
866:
829:
566:
529:
408:
387:
367:
297:
277:
275:{\displaystyle i=1,2}
228:
161:
103:
67:
2385:
2344:
2321:
2301:
2264:
2243:
2223:
2118:
2016:
1970:
1733:
1511:
1340:
1178:
1059:
943:
878:
855:
581:
555:
423:
397:
376:
310:
286:
254:
176:
112:
76:
40:
2538:1990JGCD...13..577B
2503:1964AIAAJ...2.1350.
2414:
2359:
2333:{\displaystyle -1}
2330:
2307:
2287:
2249:
2229:
2209:
2104:
1985:
1919:
1678:
1468:
1306:
1141:
1025:
922:
861:
824:
561:
524:
403:
382:
362:
292:
272:
223:
156:
98:
62:
2439:
2438:
2310:{\displaystyle 1}
2198:
2183:
2173:
2167:
2162:
2152:
2146:
2141:
2096:
2081:
2071:
2065:
2060:
2050:
2044:
2039:
2010:
2009:
1944:
1943:
1905:
1890:
1880:
1874:
1869:
1859:
1853:
1848:
1822:
1807:
1797:
1791:
1786:
1776:
1770:
1765:
1745:
1703:
1702:
1670:
1655:
1645:
1639:
1634:
1624:
1618:
1613:
1585:
1570:
1560:
1554:
1549:
1539:
1533:
1528:
1493:
1492:
1466:
1446:
1424:
1393:
1371:
1352:
1331:
1330:
1304:
1284:
1262:
1231:
1209:
1190:
1166:
1165:
1139:
1119:
1089:
1071:
1050:
1049:
1023:
1003:
973:
955:
913:
891:
849:
848:
805:
783:
767:
761:
750:
739:
733:
722:
682:
660:
644:
638:
627:
616:
610:
599:
564:{\displaystyle A}
549:
548:
510:
488:
458:
436:
406:{\displaystyle A}
385:{\displaystyle A}
304:orthogonal matrix
295:{\displaystyle A}
248:
247:
214:
189:
147:
125:
89:
53:
2658:
2630:
2629:
2627:
2625:
2611:
2602:
2596:
2595:
2575:
2569:
2568:
2556:
2550:
2549:
2521:
2515:
2514:
2497:(7): 1350–1351.
2486:
2433:
2423:
2421:
2420:
2415:
2407:
2378:
2368:
2366:
2365:
2360:
2339:
2337:
2336:
2331:
2316:
2314:
2313:
2308:
2296:
2294:
2293:
2288:
2286:
2258:
2256:
2255:
2250:
2238:
2236:
2235:
2230:
2218:
2216:
2215:
2210:
2205:
2201:
2200:
2199:
2191:
2185:
2184:
2176:
2171:
2165:
2164:
2163:
2155:
2150:
2144:
2143:
2142:
2134:
2113:
2111:
2110:
2105:
2103:
2099:
2098:
2097:
2089:
2083:
2082:
2074:
2069:
2063:
2062:
2061:
2053:
2048:
2042:
2041:
2040:
2032:
2004:
1994:
1992:
1991:
1986:
1963:
1938:
1928:
1926:
1925:
1920:
1918:
1917:
1912:
1908:
1907:
1906:
1898:
1892:
1891:
1883:
1878:
1872:
1871:
1870:
1862:
1857:
1851:
1850:
1849:
1841:
1829:
1825:
1824:
1823:
1815:
1809:
1808:
1800:
1795:
1789:
1788:
1787:
1779:
1774:
1768:
1767:
1766:
1758:
1747:
1746:
1738:
1726:
1697:
1687:
1685:
1684:
1679:
1677:
1673:
1672:
1671:
1663:
1657:
1656:
1648:
1643:
1637:
1636:
1635:
1627:
1622:
1616:
1615:
1614:
1606:
1592:
1588:
1587:
1586:
1578:
1572:
1571:
1563:
1558:
1552:
1551:
1550:
1542:
1537:
1531:
1530:
1529:
1521:
1504:
1487:
1477:
1475:
1474:
1469:
1467:
1465:
1464:
1459:
1454:
1453:
1448:
1447:
1439:
1432:
1431:
1426:
1425:
1417:
1413:
1408:
1402:
1401:
1400:
1395:
1394:
1386:
1379:
1378:
1373:
1372:
1364:
1359:
1354:
1353:
1345:
1333:
1325:
1315:
1313:
1312:
1307:
1305:
1303:
1302:
1297:
1292:
1291:
1286:
1285:
1277:
1270:
1269:
1264:
1263:
1255:
1251:
1246:
1240:
1239:
1238:
1233:
1232:
1224:
1217:
1216:
1211:
1210:
1202:
1197:
1192:
1191:
1183:
1171:
1160:
1150:
1148:
1147:
1142:
1140:
1138:
1137:
1132:
1127:
1126:
1121:
1120:
1112:
1108:
1103:
1097:
1096:
1091:
1090:
1082:
1078:
1073:
1072:
1064:
1052:
1044:
1034:
1032:
1031:
1026:
1024:
1022:
1021:
1016:
1011:
1010:
1005:
1004:
996:
992:
987:
981:
980:
975:
974:
966:
962:
957:
956:
948:
936:
931:
929:
928:
923:
921:
920:
915:
914:
906:
899:
898:
893:
892:
884:
870:
868:
867:
862:
843:
833:
831:
830:
825:
823:
819:
818:
814:
813:
812:
807:
806:
798:
791:
790:
785:
784:
776:
765:
759:
758:
757:
752:
751:
743:
737:
731:
730:
729:
724:
723:
715:
700:
696:
695:
691:
690:
689:
684:
683:
675:
668:
667:
662:
661:
653:
642:
636:
635:
634:
629:
628:
620:
614:
608:
607:
606:
601:
600:
592:
574:
570:
568:
567:
562:
543:
533:
531:
530:
525:
523:
519:
518:
517:
512:
511:
503:
496:
495:
490:
489:
481:
466:
465:
460:
459:
451:
444:
443:
438:
437:
429:
416:
412:
410:
409:
404:
391:
389:
388:
383:
371:
369:
368:
363:
322:
321:
301:
299:
298:
293:
281:
279:
278:
273:
242:
232:
230:
229:
224:
222:
221:
216:
215:
207:
197:
196:
191:
190:
182:
169:
165:
163:
162:
157:
155:
154:
149:
148:
140:
133:
132:
127:
126:
118:
107:
105:
104:
99:
97:
96:
91:
90:
82:
71:
69:
68:
63:
61:
60:
55:
54:
46:
2666:
2665:
2661:
2660:
2659:
2657:
2656:
2655:
2636:
2635:
2634:
2633:
2623:
2621:
2609:
2604:
2603:
2599:
2592:10.1137/1008080
2577:
2576:
2572:
2558:
2557:
2553:
2546:10.2514/3.25373
2523:
2522:
2518:
2488:
2487:
2483:
2478:
2470:Wahba's problem
2456:
2447:
2431:
2397:
2383:
2382:
2342:
2341:
2319:
2318:
2299:
2298:
2276:
2262:
2261:
2241:
2240:
2221:
2220:
2131:
2127:
2116:
2115:
2029:
2025:
2014:
2013:
2002:
1968:
1967:
1957:
1936:
1838:
1834:
1833:
1755:
1751:
1731:
1730:
1695:
1603:
1599:
1518:
1514:
1509:
1508:
1485:
1436:
1414:
1403:
1383:
1361:
1360:
1338:
1337:
1323:
1274:
1252:
1241:
1221:
1199:
1198:
1176:
1175:
1158:
1109:
1098:
1079:
1057:
1056:
1042:
993:
982:
963:
941:
940:
903:
881:
876:
875:
853:
852:
841:
795:
773:
772:
768:
740:
712:
711:
707:
672:
650:
649:
645:
617:
589:
588:
584:
579:
578:
553:
552:
541:
500:
478:
477:
473:
448:
426:
421:
420:
395:
394:
374:
373:
313:
308:
307:
284:
283:
252:
251:
240:
204:
179:
174:
173:
137:
115:
110:
109:
79:
74:
73:
43:
38:
37:
34:
26:Wahba's problem
17:
12:
11:
5:
2664:
2662:
2654:
2653:
2648:
2638:
2637:
2632:
2631:
2597:
2570:
2551:
2532:(4): 577–585.
2516:
2511:10.2514/3.2555
2480:
2479:
2477:
2474:
2473:
2472:
2467:
2462:
2455:
2452:
2446:
2443:
2437:
2436:
2427:
2425:
2413:
2410:
2406:
2403:
2400:
2396:
2393:
2390:
2358:
2355:
2352:
2349:
2329:
2326:
2306:
2285:
2282:
2279:
2275:
2272:
2269:
2248:
2228:
2208:
2204:
2197:
2194:
2188:
2182:
2179:
2170:
2161:
2158:
2149:
2140:
2137:
2130:
2126:
2123:
2102:
2095:
2092:
2086:
2080:
2077:
2068:
2059:
2056:
2047:
2038:
2035:
2028:
2024:
2021:
2008:
2007:
1998:
1996:
1984:
1981:
1978:
1975:
1956:
1953:
1942:
1941:
1932:
1930:
1916:
1911:
1904:
1901:
1895:
1889:
1886:
1877:
1868:
1865:
1856:
1847:
1844:
1837:
1832:
1828:
1821:
1818:
1812:
1806:
1803:
1794:
1785:
1782:
1773:
1764:
1761:
1754:
1750:
1744:
1741:
1701:
1700:
1691:
1689:
1676:
1669:
1666:
1660:
1654:
1651:
1642:
1633:
1630:
1621:
1612:
1609:
1602:
1598:
1595:
1591:
1584:
1581:
1575:
1569:
1566:
1557:
1548:
1545:
1536:
1527:
1524:
1517:
1491:
1490:
1481:
1479:
1463:
1458:
1452:
1445:
1442:
1435:
1430:
1423:
1420:
1412:
1407:
1399:
1392:
1389:
1382:
1377:
1370:
1367:
1357:
1351:
1348:
1329:
1328:
1319:
1317:
1301:
1296:
1290:
1283:
1280:
1273:
1268:
1261:
1258:
1250:
1245:
1237:
1230:
1227:
1220:
1215:
1208:
1205:
1195:
1189:
1186:
1164:
1163:
1154:
1152:
1136:
1131:
1125:
1118:
1115:
1107:
1102:
1095:
1088:
1085:
1076:
1070:
1067:
1048:
1047:
1038:
1036:
1020:
1015:
1009:
1002:
999:
991:
986:
979:
972:
969:
960:
954:
951:
919:
912:
909:
902:
897:
890:
887:
860:
847:
846:
837:
835:
822:
817:
811:
804:
801:
794:
789:
782:
779:
771:
764:
756:
749:
746:
736:
728:
721:
718:
710:
706:
703:
699:
694:
688:
681:
678:
671:
666:
659:
656:
648:
641:
633:
626:
623:
613:
605:
598:
595:
587:
560:
547:
546:
537:
535:
522:
516:
509:
506:
499:
494:
487:
484:
476:
472:
469:
464:
457:
454:
447:
442:
435:
432:
402:
381:
361:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
320:
316:
291:
271:
268:
265:
262:
259:
246:
245:
236:
234:
220:
213:
210:
203:
200:
195:
188:
185:
153:
146:
143:
136:
131:
124:
121:
95:
88:
85:
59:
52:
49:
33:
30:
15:
13:
10:
9:
6:
4:
3:
2:
2663:
2652:
2649:
2647:
2644:
2643:
2641:
2619:
2615:
2608:
2601:
2598:
2593:
2589:
2585:
2581:
2574:
2571:
2566:
2562:
2555:
2552:
2547:
2543:
2539:
2535:
2531:
2527:
2520:
2517:
2512:
2508:
2504:
2500:
2496:
2492:
2485:
2482:
2475:
2471:
2468:
2466:
2463:
2461:
2458:
2457:
2453:
2451:
2444:
2442:
2435:
2428:
2426:
2424:
2411:
2408:
2404:
2401:
2398:
2394:
2391:
2388:
2380:
2379:
2376:
2374:
2373:
2356:
2353:
2350:
2327:
2324:
2304:
2283:
2277:
2273:
2270:
2267:
2206:
2202:
2192:
2186:
2177:
2168:
2156:
2147:
2135:
2128:
2124:
2100:
2090:
2084:
2075:
2066:
2054:
2045:
2033:
2026:
2022:
2006:
1999:
1997:
1995:
1979:
1976:
1965:
1964:
1961:
1954:
1952:
1950:
1940:
1933:
1931:
1929:
1914:
1909:
1899:
1893:
1884:
1875:
1863:
1854:
1842:
1835:
1830:
1826:
1816:
1810:
1801:
1792:
1780:
1771:
1759:
1752:
1748:
1739:
1728:
1727:
1724:
1722:
1721:
1716:
1715:
1710:
1709:
1699:
1692:
1690:
1688:
1674:
1664:
1658:
1649:
1640:
1628:
1619:
1607:
1600:
1596:
1593:
1589:
1579:
1573:
1564:
1555:
1543:
1534:
1522:
1515:
1506:
1505:
1502:
1500:
1499:
1489:
1482:
1480:
1478:
1450:
1440:
1433:
1428:
1418:
1397:
1387:
1380:
1375:
1365:
1355:
1346:
1335:
1334:
1327:
1320:
1318:
1316:
1288:
1278:
1271:
1266:
1256:
1235:
1225:
1218:
1213:
1203:
1193:
1184:
1173:
1172:
1169:
1162:
1155:
1153:
1151:
1123:
1113:
1093:
1083:
1074:
1065:
1054:
1053:
1046:
1039:
1037:
1035:
1007:
997:
977:
967:
958:
949:
938:
937:
934:
917:
907:
900:
895:
885:
872:
858:
845:
838:
836:
834:
820:
815:
809:
799:
792:
787:
777:
769:
762:
754:
744:
734:
726:
716:
708:
704:
701:
697:
692:
686:
676:
669:
664:
654:
646:
639:
631:
621:
611:
603:
593:
585:
576:
575:
572:
558:
545:
538:
536:
534:
520:
514:
504:
497:
492:
482:
474:
470:
467:
462:
452:
445:
440:
430:
418:
417:
414:
400:
379:
359:
356:
353:
347:
341:
338:
335:
332:
329:
326:
323:
318:
314:
305:
289:
269:
266:
263:
260:
257:
244:
237:
235:
233:
218:
208:
201:
198:
193:
183:
171:
170:
167:
151:
141:
134:
129:
119:
93:
83:
57:
47:
31:
29:
27:
22:
2622:. Retrieved
2620:(2): 261–280
2617:
2613:
2600:
2583:
2579:
2573:
2564:
2554:
2529:
2525:
2519:
2494:
2491:AIAA Journal
2490:
2484:
2448:
2445:Applications
2440:
2429:
2381:
2370:
2011:
2000:
1966:
1958:
1945:
1934:
1729:
1718:
1712:
1706:
1704:
1693:
1507:
1496:
1494:
1483:
1336:
1321:
1174:
1167:
1156:
1055:
1040:
939:
873:
850:
839:
577:
550:
539:
419:
249:
238:
172:
35:
21:TRIAD method
20:
18:
2586:: 385–386.
2580:SIAM Review
1949:orthonormal
2640:Categories
2476:References
2624:April 18,
2354:±
2348:Δ
2325:−
2281:Γ
2247:Δ
2227:Γ
2196:^
2187:×
2181:^
2169:⋮
2160:^
2148:⋮
2139:^
2122:Δ
2094:^
2085:×
2079:^
2067:⋮
2058:^
2046:⋮
2037:^
2020:Γ
1983:Δ
1974:Γ
1903:^
1894:×
1888:^
1876:⋮
1867:^
1855:⋮
1846:^
1820:^
1811:×
1805:^
1793:⋮
1784:^
1772:⋮
1763:^
1743:^
1668:^
1659:×
1653:^
1641:⋮
1632:^
1620:⋮
1611:^
1583:^
1574:×
1568:^
1556:⋮
1547:^
1535:⋮
1526:^
1444:→
1434:×
1422:→
1391:→
1381:×
1369:→
1350:^
1282:→
1272:×
1260:→
1229:→
1219:×
1207:→
1188:^
1117:→
1087:→
1069:^
1001:→
971:→
953:^
911:→
889:→
859:⋮
803:→
793:×
781:→
763:⋮
748:→
735:⋮
720:→
680:→
670:×
658:→
640:⋮
625:→
612:⋮
597:→
508:→
498:×
486:→
456:→
446:×
434:→
212:→
187:→
145:→
123:→
87:→
51:→
2454:See also
306:, i.e.,
282:, where
2534:Bibcode
2499:Bibcode
2012:where
32:Summary
2172:
2166:
2151:
2145:
2070:
2064:
2049:
2043:
1879:
1873:
1858:
1852:
1796:
1790:
1775:
1769:
1644:
1638:
1623:
1617:
1559:
1553:
1538:
1532:
851:where
766:
760:
738:
732:
643:
637:
615:
609:
108:. Let
2610:(PDF)
2114:and
1711:) - (
2626:2012
1168:and
250:for
72:and
19:The
2588:doi
2542:doi
2507:doi
2317:or
2297:is
372:).
2642::
2618:41
2616:.
2612:.
2582:.
2563:.
2540:.
2530:13
2528:.
2505:.
2493:.
2432:11
2412:1.
2372:10
2023::=
2003:10
2628:.
2594:.
2590::
2584:8
2548:.
2544::
2536::
2513:.
2509::
2501::
2495:2
2434:)
2430:(
2409:=
2405:)
2402:A
2399:(
2395:t
2392:e
2389:d
2357:1
2351:=
2328:1
2305:1
2284:)
2278:(
2274:t
2271:e
2268:d
2207:.
2203:]
2193:m
2178:s
2157:m
2136:s
2129:[
2125:=
2101:]
2091:M
2076:S
2055:M
2034:S
2027:[
2005:)
2001:(
1980:A
1977:=
1939:)
1937:9
1935:(
1915:T
1910:]
1900:m
1885:s
1864:m
1843:s
1836:[
1831:.
1827:]
1817:M
1802:S
1781:M
1760:S
1753:[
1749:=
1740:A
1720:8
1714:7
1708:4
1698:)
1696:8
1694:(
1675:]
1665:m
1650:s
1629:m
1608:s
1601:[
1597:A
1594:=
1590:]
1580:M
1565:S
1544:M
1523:S
1516:[
1498:3
1488:)
1486:7
1484:(
1462:|
1457:|
1451:2
1441:r
1429:1
1419:r
1411:|
1406:|
1398:2
1388:r
1376:1
1366:r
1356:=
1347:m
1326:)
1324:6
1322:(
1300:|
1295:|
1289:2
1279:R
1267:1
1257:R
1249:|
1244:|
1236:2
1226:R
1214:1
1204:R
1194:=
1185:M
1161:)
1159:5
1157:(
1135:|
1130:|
1124:1
1114:r
1106:|
1101:|
1094:1
1084:r
1075:=
1066:s
1045:)
1043:4
1041:(
1019:|
1014:|
1008:1
998:R
990:|
985:|
978:1
968:R
959:=
950:S
918:2
908:r
901:,
896:1
886:r
844:)
842:3
840:(
821:]
816:)
810:2
800:r
788:1
778:r
770:(
755:2
745:r
727:1
717:r
709:[
705:A
702:=
698:]
693:)
687:2
677:R
665:1
655:R
647:(
632:2
622:R
604:1
594:R
586:[
559:A
544:)
542:2
540:(
521:)
515:2
505:r
493:1
483:r
475:(
471:A
468:=
463:2
453:R
441:1
431:R
401:A
380:A
360:1
357:+
354:=
351:)
348:A
345:(
342:t
339:e
336:d
333:,
330:I
327:=
324:A
319:T
315:A
290:A
270:2
267:,
264:1
261:=
258:i
243:)
241:1
239:(
219:i
209:r
202:A
199:=
194:i
184:R
152:2
142:r
135:,
130:1
120:r
94:2
84:R
58:1
48:R
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.