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TeXDict begin

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1568 y Fr(Ac)m(kno)m(wledgemen)m(t)0 1781 y Fq(The)g(author)g
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(Branc)m(h)h(at)g(Marshall)e(Space)h(Fligh)m(t)0 1894
y(Cen)m(ter)h(for)f(their)g(useful)f(discussions)f(that)j(led)f(to)h
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b(ailable)28 b(to)h(me.)40 b(I)28 b(w)m(elcome)h(an)m(y)g(suggestions)f
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2120 y(placed)h(in)f(the)h(bibliograph)m(y)-8 b(.)0 2406
y Fp(Con)l(ten)l(ts)0 2610 y Fr(1)84 b(Quaternions)3154
b(2)136 2723 y Fq(1.1)94 b(De\014nition)61 b Fo(:)46
b(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h
(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)
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b(Multiplication)28 b(of)i(quaternions)72 b Fo(:)46 b(:)g(:)f(:)h(:)g
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b(Quaternions)29 b(as)i(rotations)39 b Fo(:)46 b(:)g(:)f(:)h(:)g(:)f(:)
h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h
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3061 y(1.4)94 b(Application)29 b(of)h(quaternions)g(in)f(co)s(ordinate)
h(frame)g(transformations)53 b Fo(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)
h(:)g(:)f(:)131 b Fq(6)345 3174 y(1.4.1)106 b(Euler)29
b(angles)i Fo(:)46 b(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:)
g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h
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b Fo(:)g(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)131
b Fq(6)136 3400 y(1.5)94 b(Time)30 b(deriv)-5 b(ativ)m(e)30
b(of)g(a)h(quaternion)42 b Fo(:)k(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h
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g(:)f(:)85 b Fq(12)1927 5656 y(1)p eop
2 1 bop 0 100 a Fn(A.)31 b(S.)f(Ho)s(del:)40 b(Quaternions)29
b(\(In)m(tro)s(duction\))g Fo(R)q(ev)s(ision)d Fq(:)f(1)p
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y Fp(1)135 b(Quaternions)0 601 y Fq(A)32 b(con)m(v)m(enien)m(t)g(w)m(a)
m(y)h(to)f(represen)m(t)f(the)h(relativ)m(e)g(orien)m(tation)f(of)h(t)m
(w)m(o)g(frames)g(is)e(through)h(the)h(use)f(of)g(quater-)0
714 y(nions.)0 958 y Fm(1.1)112 b(De\014nition)0 1129
y Fr(De\014nition)35 b(1.1)46 b Fq(A)31 b(quaternion)e(is)g(a)i(4-v)m
(ector)i(de\014ned)c(b)m(y)1268 1334 y Fo(q)f Fq(=)d
Fo(a)1474 1313 y Fq(^)1481 1334 y Fo(i)c Fq(+)f Fo(b)1662
1313 y Fq(^)1663 1334 y Fo(j)26 b Fq(+)19 b Fo(c)1857
1310 y Fq(^)1855 1334 y Fo(k)24 b Fq(+)c Fo(d;)46 b(a;)15
b(b;)g(c;)g(d)27 b Fl(2)e Fq(I)-9 b(R)0 1538 y(where)256
1517 y(^)263 1538 y Fo(i)p Fq(,)348 1517 y(^)350 1538
y Fo(j)5 b Fq(,)450 1514 y(^)448 1538 y Fo(k)33 b Fq(are)e(quan)m
(tities)f(satisfying)1498 1517 y(^)1505 1538 y Fo(i)1536
1505 y Fk(2)1601 1538 y Fq(=)1696 1517 y(^)1697 1538
y Fo(j)1739 1505 y Fk(2)1804 1538 y Fq(=)1903 1514 y(^)1900
1538 y Fo(k)1950 1505 y Fk(2)2015 1538 y Fq(=)25 b Fl(\000)p
Fq(1)31 b(and)1476 1721 y(^)1483 1742 y Fo(i)1513 1721
y Fq(^)1515 1742 y Fo(j)f Fq(=)1681 1718 y(^)1678 1742
y Fo(k)1846 1721 y Fq(^)1847 1742 y Fo(j)1892 1718 y
Fq(^)1890 1742 y Fo(k)e Fq(=)2054 1721 y(^)2061 1742
y Fo(i)2213 1718 y Fq(^)2211 1742 y Fo(k)2254 1721 y
Fq(^)2261 1742 y Fo(i)e Fq(=)2412 1721 y(^)2413 1742
y Fo(j)1372 1859 y Fq(^)1373 1880 y Fo(j)6 b Fq(\0201)25
b(=)g Fl(\000)1681 1856 y Fq(^)1679 1880 y Fo(k)1814
1856 y Fq(^)1812 1880 y Fo(k)1860 1859 y Fq(^)1862 1880
y Fo(j)31 b Fq(=)25 b Fl(\000)2090 1859 y Fq(^)2097 1880
y Fo(i)2204 1859 y Fq(^)2211 1880 y Fo(i)2244 1856 y
Fq(^)2242 1880 y Fo(k)k Fq(=)24 b Fl(\000)2483 1859 y
Fq(^)2484 1880 y Fo(j)0 2092 y Fr(Remark)34 b(1.1)46
b Fq(Notice)35 b(that)1058 2072 y(^)1065 2092 y Fo(i)p
Fq(,)1154 2072 y(^)1156 2092 y Fo(j)5 b Fq(,)35 b(and)1440
2068 y(^)1438 2092 y Fo(k)h Fq(are)e(scalar)g(quan)m(tities)e(equal)h
(to)2718 2021 y Fl(p)p 2793 2021 117 4 v 2793 2092 a(\000)p
Fq(1)q(;)i(ho)m(w)m(ev)m(er,)3341 2072 y(^)3348 2092
y Fo(i)p Fq(,)3437 2072 y(^)3438 2092 y Fo(j)6 b Fq(,)35
b(and)3723 2068 y(^)3720 2092 y Fo(k)i Fq(do)0 2205 y(not)g(comm)m(ute)
h(under)d(m)m(ultiplication.)56 b(Therefore)36 b(quaternions)g(do)g
(not)h(comm)m(ute)h(under)d(m)m(ultiplication)0 2318
y(\()p Fo(q)76 2332 y Fk(1)115 2318 y Fo(q)156 2332 y
Fk(2)221 2318 y Fl(6)p Fq(=)25 b Fo(q)358 2332 y Fk(2)397
2318 y Fo(q)438 2332 y Fk(1)497 2318 y Fq(in)20 b(general\).)38
b(This)19 b(is)h(consisten)m(t)h(with)f(the)h(in)m(terpretation)f(of)h
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b(b)s(e)f(discussed)f(in)g Fl(x)p Fq(1.4.2.)0 2644 y
Fr(M-\014le)46 b Fj(quaternion)57 b Fq(Create/extract)34
b(quaternion)29 b(information:)227 2869 y Fj(quaternion:)45
b(construct)g(or)i(extract)f(a)i(quaternion)227 2982
y(w)g(=)f(a*i)g(+)h(b*j)e(+)i(c*k)f(+)g(d)h(from)e(given)h(data.)227
3207 y(calling)f(formats:)227 3320 y([a,b,c,d])141 b(=)47
b(quaternion\(w\))570 b(-or-)227 3433 y([vv,theta])45
b(=)j(quaternion\(w\))227 3546 y(w)525 b(=)47 b(quaternion\(a,b,c,d\))
227 3659 y(w)525 b(=)47 b(quaternion\(vv,theta\))227
3884 y Fq(\()p Fj(vv,theta)29 b Fq(format)h(to)i(b)s(e)d(explained)g
(later\))0 4128 y Fm(1.2)112 b(Multiplication)34 b(of)k(quaternions)0
4299 y Fq(F)-8 b(rom)31 b(De\014nition)e(1.1,)j(w)m(e)e(ha)m(v)m(e)533
4503 y Fo(q)574 4517 y Fk(1)613 4503 y Fo(q)654 4517
y Fk(2)776 4503 y Fq(=)83 b(\()p Fo(a)1013 4517 y Fk(1)1046
4483 y Fq(^)1053 4503 y Fo(i)20 b Fq(+)g Fo(b)1234 4517
y Fk(1)1272 4483 y Fq(^)1274 4503 y Fo(j)26 b Fq(+)20
b Fo(c)1467 4517 y Fk(1)1509 4479 y Fq(^)1506 4503 y
Fo(k)k Fq(+)c Fo(d)1715 4517 y Fk(1)1754 4503 y Fq(\)\()p
Fo(a)1872 4517 y Fk(2)1905 4483 y Fq(^)1913 4503 y Fo(i)g
Fq(+)g Fo(b)2094 4517 y Fk(2)2132 4483 y Fq(^)2133 4503
y Fo(j)26 b Fq(+)20 b Fo(c)2326 4517 y Fk(2)2368 4479
y Fq(^)2366 4503 y Fo(k)j Fq(+)d Fo(d)2574 4517 y Fk(2)2614
4503 y Fq(\))776 4641 y(=)83 b(\()p Fo(a)1013 4655 y
Fk(1)1053 4641 y Fo(d)1100 4655 y Fk(2)1160 4641 y Fq(+)20
b Fo(b)1290 4655 y Fk(1)1329 4641 y Fo(c)1368 4655 y
Fk(2)1428 4641 y Fl(\000)g Fo(c)1558 4655 y Fk(1)1598
4641 y Fo(b)1637 4655 y Fk(2)1697 4641 y Fq(+)f Fo(d)1834
4655 y Fk(1)1874 4641 y Fo(a)1922 4655 y Fk(2)1962 4641
y Fq(\))p Fo(i)i Fq(+)e(\()p Fo(c)2213 4655 y Fk(1)2254
4641 y Fo(a)2302 4655 y Fk(2)2361 4641 y Fl(\000)h Fo(a)2500
4655 y Fk(1)2540 4641 y Fo(c)2579 4655 y Fk(2)2639 4641
y Fq(+)g Fo(b)2769 4655 y Fk(1)2808 4641 y Fo(d)2855
4655 y Fk(2)2915 4641 y Fq(+)g Fo(d)3053 4655 y Fk(1)3093
4641 y Fo(b)3132 4655 y Fk(2)3171 4641 y Fq(\))3205 4620
y(^)3206 4641 y Fo(j)930 4779 y Fq(+\()p Fo(a)1084 4793
y Fk(1)1124 4779 y Fo(b)1163 4793 y Fk(2)1222 4779 y
Fl(\000)g Fo(b)1352 4793 y Fk(1)1392 4779 y Fo(a)1440
4793 y Fk(2)1499 4779 y Fq(+)g Fo(c)1629 4793 y Fk(1)1669
4779 y Fo(d)1716 4793 y Fk(2)1776 4779 y Fq(+)g Fo(d)1914
4793 y Fk(1)1954 4779 y Fo(c)1993 4793 y Fk(2)2032 4779
y Fq(\))2069 4755 y(^)2067 4779 y Fo(k)k Fq(+)c(\()p
Fl(\000)p Fo(a)2383 4793 y Fk(1)2423 4779 y Fo(a)2471
4793 y Fk(2)2530 4779 y Fl(\000)g Fo(b)2660 4793 y Fk(1)2700
4779 y Fo(b)2739 4793 y Fk(2)2798 4779 y Fl(\000)g Fo(c)2928
4793 y Fk(1)2968 4779 y Fo(c)3007 4793 y Fk(2)3067 4779
y Fq(+)g Fo(d)3205 4793 y Fk(1)3245 4779 y Fo(d)3292
4793 y Fk(2)3331 4779 y Fq(\))347 b(\(1.1\))21 5018 y(De\014ne)21
b Fo(v)337 5032 y Fk(1)401 5018 y Fq(=)497 4924 y Fi(h)578
5018 y Fo(a)626 5032 y Fk(1)749 5018 y Fo(b)788 5032
y Fk(1)910 5018 y Fo(c)949 5032 y Fk(1)1030 4924 y Fi(i)1070
4947 y Fh(T)1145 5018 y Fq(and)g Fo(v)1357 5032 y Fk(2)1421
5018 y Fq(=)1517 4924 y Fi(h)1598 5018 y Fo(a)1646 5032
y Fk(2)1769 5018 y Fo(b)1808 5032 y Fk(2)1930 5018 y
Fo(c)1969 5032 y Fk(2)2050 4924 y Fi(i)2089 4947 y Fh(T)2145
5018 y Fq(.)37 b(Then)20 b([Mul)o(])h(equation)g(\(1.1\))h(ma)m(y)g(b)s
(e)e(rewritten)0 5131 y(as)1017 5244 y Fo(q)1058 5258
y Fk(1)1097 5244 y Fo(q)1138 5258 y Fk(2)1202 5244 y
Fq(=)25 b Fo(d)1345 5258 y Fk(1)1385 5244 y Fo(d)1432
5258 y Fk(2)1492 5244 y Fq(+)20 b Fo(d)1630 5258 y Fk(1)1669
5244 y Fo(v)1713 5258 y Fk(2)1773 5244 y Fq(+)g Fo(d)1911
5258 y Fk(2)1951 5244 y Fo(v)1995 5258 y Fk(1)2054 5244
y Fl(\000)g(h)q Fo(v)2225 5258 y Fk(1)2264 5244 y Fo(;)15
b(v)2348 5258 y Fk(2)2388 5244 y Fl(i)21 b Fq(+)f(\()p
Fo(v)2614 5258 y Fk(1)2674 5244 y Fl(\002)f Fo(v)2808
5258 y Fk(2)2848 5244 y Fq(\))830 b(\(1.2\))p eop
3 2 bop 0 100 a Fn(A.)31 b(S.)f(Ho)s(del:)40 b(Quaternions)29
b(\(In)m(tro)s(duction\))g Fo(R)q(ev)s(ision)d Fq(:)f(1)p
Fo(:)p Fq(1)436 b Fn(\(c\))31 b(1998)902 b Fq(3)0 399
y(where)30 b(the)g(cross)h(pro)s(duct)1385 622 y(\()p
Fo(v)1464 636 y Fk(1)1524 622 y Fl(\002)20 b Fo(v)1659
636 y Fk(2)1699 622 y Fq(\))1765 572 y Fk(\001)1759 622
y Fq(=)1855 424 y Fi(\014)1855 474 y(\014)1855 524 y(\014)1855
574 y(\014)1855 624 y(\014)1855 674 y(\014)1855 723 y(\014)1883
428 y(2)1883 574 y(6)1883 627 y(4)2001 492 y Fq(^)2008
513 y Fo(i)2167 492 y Fq(^)2168 513 y Fo(j)2328 489 y
Fq(^)2326 513 y Fo(k)1980 625 y(a)2028 639 y Fk(1)2150
625 y Fo(b)2189 639 y Fk(1)2312 625 y Fo(c)2351 639 y
Fk(1)1980 738 y Fo(a)2028 752 y Fk(2)2150 738 y Fo(b)2189
752 y Fk(2)2312 738 y Fo(c)2351 752 y Fk(2)2432 428 y
Fi(3)2432 574 y(7)2432 627 y(5)2487 424 y(\014)2487 474
y(\014)2487 524 y(\014)2487 574 y(\014)2487 624 y(\014)2487
674 y(\014)2487 723 y(\014)0 881 y Fq(This)29 b(v)m(ector-based)j(in)m
(terpretation)d(will)f(b)s(ecome)j(more)f(imp)s(ortan)m(t)g(in)f
Fl(x)p Fq(1.4.2.)0 1040 y Fr(M-\014le)46 b Fj(qmult)227
1231 y(function)g(c)h(=)h(qmult\(a,b\))227 1344 y(multiply)e(two)h
(quaternions)0 1534 y Fr(De\014nition)35 b(1.2)46 b Fq(The)26
b Fg(c)-5 b(onjugate)26 b Fq(of)g(a)h(quaternion)e Fo(q)j
Fq(=)d Fo(a)2077 1514 y Fq(^)2084 1534 y Fo(i)12 b Fq(+)f
Fo(b)2247 1514 y Fq(^)2248 1534 y Fo(j)17 b Fq(+)11 b
Fo(c)2425 1510 y Fq(^)2423 1534 y Fo(k)k Fq(+)c Fo(d)26
b Fq(is)f Fo(q)2771 1501 y Ff(\003)2836 1534 y Fq(=)g
Fl(\000)p Fo(a)3044 1514 y Fq(^)3051 1534 y Fo(i)11 b
Fl(\000)g Fo(b)3213 1514 y Fq(^)3214 1534 y Fo(j)17 b
Fl(\000)11 b Fo(c)3391 1510 y Fq(^)3389 1534 y Fo(k)k
Fq(+)c Fo(d)p Fq(.)40 b(Notice)0 1664 y(that)35 b Fo(q)s(q)289
1631 y Ff(\003)361 1664 y Fq(=)d Fo(q)508 1631 y Ff(\003)547
1664 y Fo(q)k Fq(=)c Fo(a)775 1631 y Fk(2)837 1664 y
Fq(+)23 b Fo(b)970 1631 y Fk(2)1033 1664 y Fq(+)f Fo(c)1165
1631 y Fk(2)1228 1664 y Fq(+)h Fo(d)1369 1631 y Fk(2)1447
1614 y(\001)1441 1664 y Fq(=)32 b Fl(j)p Fo(q)s Fl(j)1639
1622 y Fk(2)1678 1664 y Fq(.)54 b(F)-8 b(or)35 b(quaternions)f
Fo(q)j Fq(with)c Fo(q)s(q)2799 1631 y Ff(\003)2871 1664
y Fl(6)p Fq(=)f(0)j(w)m(e)g(de\014ne)f(the)h Fg(inverse)0
1821 y(quaternion)c Fo(q)498 1783 y Ff(\000)p Fk(1)618
1821 y Fq(=)746 1759 y Fo(q)790 1726 y Ff(\003)p 724
1799 128 4 v 724 1883 a Fo(q)s(q)812 1857 y Ff(\003)861
1821 y Fq(.)40 b(Notice)32 b(that)f Fo(q)s(q)1497 1788
y Ff(\000)p Fk(1)1616 1821 y Fq(=)25 b Fo(q)1756 1788
y Ff(\000)p Fk(1)1850 1821 y Fo(q)j Fq(=)d(1.)0 2037
y Fr(M-\014le)46 b Fj(qinv)59 b Fq(Compute)30 b(the)h(in)m(v)m(erse)f
(of)g(a)h(quaternion)227 2228 y Fj(function)46 b(b)h(=)h(qinv\(a\))227
2341 y(return)e(the)h(inverse)f(of)h(a)h(quaternion)227
2454 y(a)g(=)95 b([w,x,y,z])45 b(=)j(w*i)f(+)g(x*j)g(+)g(y*k)g(+)h(z)
227 2567 y(qmult\(a,qinv\(a\)\))c(=)j(1)h(=)f([0)g(0)h(0)f(1])0
2757 y Fr(Example)34 b(1.1)46 b Fq(Compute)30 b(and)f(m)m(ultiply)f
(the)j(in)m(v)m(erse)f(quaternion:)0 2916 y Fj(octave:1>)45
b(q)j(=)f(quaternion\(1,2,3,4\);)42 b(qi)48 b(=)f(qinv\(q\))0
3029 y(qi)g(=)h(-0.033333)93 b(-0.066667)g(-0.100000)140
b(0.133333)0 3142 y(octave:2>)45 b(qmult\(qi,q\))0 3255
y(ans)i(=)g(0)96 b(0)f(0)g(1)0 3368 y(octave:3>)45 b(qmult\(q,qi\))0
3481 y(ans)i(=)g(0)96 b(0)f(0)g(1)0 3640 y Fq(In)30 b(b)s(oth)f(cases)j
(the)e(pro)s(duct)f(is)h(the)g(iden)m(tiy)f(quaternion)h(0)2101
3619 y(^)2108 3640 y Fo(i)21 b Fq(+)f(0)2295 3619 y(^)2296
3640 y Fo(j)26 b Fq(+)20 b(0)2497 3616 y(^)2495 3640
y Fo(k)k Fq(+)19 b(1)26 b(=)f(1.)0 3878 y Fm(1.3)112
b(Quaternions)38 b(as)g(rotations)0 4062 y Fr(De\014nition)d(1.3)46
b Fq(A)31 b Fg(unit)h(quaternion)g Fo(q)c Fq(=)d Fo(a)1617
4041 y Fq(^)1624 4062 y Fo(i)20 b Fq(+)g Fo(b)1804 4041
y Fq(^)1805 4062 y Fo(j)26 b Fq(+)20 b Fo(c)2000 4038
y Fq(^)1998 4062 y Fo(k)k Fq(+)19 b Fo(d)31 b Fq(satis\014es)2572
3964 y Fi(\015)2572 4014 y(\015)2572 4064 y(\015)2618
3967 y(h)2699 4062 y Fo(a)83 b(b)g(c)g(d)3163 3967 y
Fi(i)3202 3964 y(\015)3202 4014 y(\015)3202 4064 y(\015)3248
4118 y Fk(2)3313 4062 y Fq(=)25 b(1.)0 4254 y Fr(Remark)34
b(1.2)46 b Fq(Unit)40 b(quaternions)f(ma)m(y)h(b)s(e)g(considered)f(as)
h(represen)m(tations)g(of)h(a)f(rotation)h(of)f Fo(\022)i
Fq(degrees)0 4367 y(ab)s(out)f(a)g(giv)m(en)g(unit)f(v)m(ector)j
Fo(v)h Fq(\(see)e(routine)e Fj(quaternion)p Fq(\).)70
b(The)41 b(transformation)f(is)g(accomplished)g(b)m(y)0
4496 y(m)m(ultiplying)27 b Fo(v)529 4510 y Fh(r)592 4496
y Fq(=)e Fo(q)s(v)s(q)823 4463 y Ff(\000)p Fk(1)917 4496
y Fq(,)31 b(where)f Fo(v)e Fq(=)1404 4402 y Fi(h)1485
4496 y Fo(x)83 b(y)j(z)h Fq(0)1967 4402 y Fi(i)2006 4496
y Fq(.)0 4684 y(With)30 b(this)f(in)m(terpretation,)h(unit)f
(quaternions)g(ma)m(y)i(b)s(e)f(written)f(as)890 4964
y Fo(q)f Fq(=)1055 4870 y Fi(\020)1105 4964 y Fo(ai)21
b Fq(+)e Fo(b)1333 4944 y Fq(^)1334 4964 y Fo(j)26 b
Fq(+)20 b Fo(c)1529 4940 y Fq(^)1527 4964 y Fo(k)1578
4870 y Fi(\021)1643 4964 y Fq(sin)n(\()p Fo(\022)s(=)p
Fq(2\))h(+)f(cos)q(\()p Fo(\022)s(=)p Fq(2\))2432 4914
y Fk(\001)2426 4964 y Fq(=)2522 4771 y Fi(0)2522 4917
y(B)2522 4970 y(@)2594 4771 y(2)2594 4917 y(6)2594 4970
y(4)2691 4851 y Fo(a)2696 4964 y(b)2696 5077 y(c)2781
4771 y Fi(3)2781 4917 y(7)2781 4970 y(5)2851 4964 y Fo(;)15
b(\022)2937 4771 y Fi(1)2937 4917 y(C)2937 4970 y(A)3713
4964 y Fq(\(1.3\))0 5279 y(Notice)26 b(that)f(this)f(implies)e(that)
1137 5185 y Fi(h)1218 5279 y Fo(a)83 b(b)g(c)1552 5185
y Fi(i)1615 5279 y Fq(is)24 b(also)h(a)g(unit)e(v)m(ector.)41
b(This)23 b(is)g(consisten)m(t)j(with)d(m)m(ultiplication)0
5407 y(b)m(y)30 b(the)h(iden)m(tit)m(y)f(quaterion)g
Fo(q)e Fq(=)d(1,)31 b(since)e(cos)q(\(0\))d(=)f(1)31
b(and)f(sin)o(\(0\))c(=)f(0.)p eop
4 3 bop 0 100 a Fn(A.)31 b(S.)f(Ho)s(del:)g(Quaternions)f(\(In)m(tro)s
(duction\))g Fo(R)q(ev)s(ision)d Fq(:)f(1)p Fo(:)p Fq(1)439
b Fn(\(c\))32 b(1998)908 b Fq(4)0 399 y Fr(Example)34
b(1.2)46 b Fq(Unit)33 b(quaternions)g(ma)m(y)h(b)s(e)g(m)m(ultiplied)c
(to)35 b(obtain)e(a)i(single)e(equiv)-5 b(alen)m(t)33
b(co)s(ordinate)h(rota-)0 511 y(tion.)47 b(Let)33 b Fo(q)435
525 y Fk(3)506 511 y Fq(b)s(e)f(a)h(90)800 478 y Ff(\016)872
511 y Fq(rotation)g(ab)s(out)f(the)h Fo(x)p Fl(\000)p
Fq(axis.)46 b(F)-8 b(ollo)m(w)33 b(this)e(rotation)i(b)m(y)g
Fo(q)2982 525 y Fk(2)3021 511 y Fq(,)g(a)g(90)3247 478
y Ff(\016)3319 511 y Fq(rotation)g(ab)s(out)0 624 y(the)k(\(new\))g
Fo(y)i Fq(axis.)60 b(Finally)-8 b(,)37 b(follo)m(w)f(this)g(rotation)h
(b)m(y)f Fo(q)2067 638 y Fk(1)2106 624 y Fq(,)j(a)e(90)2342
591 y Ff(\016)2419 624 y Fq(rotation)g(ab)s(out)g(the)g(\(newly)f
(revised\))g Fo(z)0 737 y Fq(axis.)k(Then)30 b Fo(q)e
Fq(=)d Fo(q)664 751 y Fk(1)703 737 y Fo(q)744 751 y Fk(2)783
737 y Fo(q)824 751 y Fk(3)893 737 y Fq(is)k(a)i(90)g(degree)g(rotation)
g(ab)s(out)f(the)h(original)d Fo(y)s Fq(-axis.)0 925
y Fj(octave:1>)45 b(degrees)h(=)i(pi/180;)0 1038 y(octave:2>)d(q1)i(=)h
(quaternion\([0,0,1],90*de)o(gree)o(s\);)0 1151 y(octave:3>)d(q2)i(=)h
(quaternion\([0,1,0],90*de)o(gree)o(s\);)0 1264 y(octave:4>)d(q3)i(=)h
(quaternion\([1,0,0],90*de)o(gree)o(s\);)0 1377 y(octave:5>)d(q)j(=)f
(qmult\(q1,qmult\(q2,q3\)\);)0 1489 y(octave:6>)e([vv,th])h(=)i
(quaternion\(q\))0 1602 y(vv)f(=)h(0)f(1)h(0)0 1715 y(th)f(=)h(1.5708)0
1828 y(octave:7>)d(th/degrees)0 1941 y(ans)i(=)g(90.000)118
2154 y Fq(The)26 b(rotation)g(ab)s(out)g(a)h(v)m(ector)h(axis)e(in)m
(terpretation)f(of)i(quaternions)e(requires)g(care:)39
b(the)26 b(axis)g(of)h(rotation)0 2267 y(is)40 b(tak)m(en)j(based)e(on)
h(the)f(co)s(ordinate)g(frame)h(for)f(whic)m(h)f(the)i(quaternion)e(w)m
(as)i(written.)73 b(F)-8 b(or)42 b(example,)j(a)0 2379
y(rotation)30 b(around)e(the)i(inertial)d Fo(y)s Fq(-axis)i(follo)m(w)m
(ed)g(b)m(y)g(a)h(rotation)g(around)e(the)i(inertial)d
Fo(x)j Fq(axis)e(can)i(b)s(e)f(written)0 2492 y(as)949
2623 y Fo(q)f Fq(=)d Fo(q)1155 2637 y Fh(x)1199 2623
y Fo(q)1240 2637 y Fh(y)1306 2623 y Fq(=)1402 2504 y
Fi(\022)1463 2529 y(h)1544 2623 y Fq(1)83 b(0)h(0)1888
2529 y Fi(i)1927 2552 y Fh(T)1982 2623 y Fo(;)15 b(\022)2065
2637 y Fh(x)2109 2504 y Fi(\023)g(\022)2246 2529 y(h)2327
2623 y Fq(0)83 b(1)h(0)2671 2529 y Fi(i)2710 2552 y Fh(T)2765
2623 y Fo(;)15 b(\022)2848 2637 y Fh(y)2890 2504 y Fi(\023)0
2825 y Fq(This)31 b(is)i(inconsisten)m(t)f(with)g(the)h(de\014nition)e
(of)j(Euler)d(angles,)k(in)c(whic)m(h)h(eac)m(h)j(angle)e(is)f
(de\014ned)g(b)m(y)h(a)h(frame)0 2938 y(based)41 b(up)s(on)f(the)i
(previous)e(rotation.)75 b(This)40 b(problem)g(is)h(addressed)f(b)m(y)i
(co)s(ordinate)f(transformation)g(b)m(y)0 3051 y(quaternions:)0
3264 y Fr(M-\014le)46 b Fj(qtrans)59 b Fq(T)-8 b(ransform)29
b(a)i(quaternion)e(in)g(one)i(frame)f(to)h(the)g(co)s(ordinate)f(basis)
f(of)i(another)f(frame.)275 3489 y Fj(function)46 b(v)h(=)g
(qtrans\(v,q\))275 3602 y(transform)e(the)i(vector)f(v)i(\(in)f
(quaternion)e(form\))h(by)h(the)g(quaternion)e(q;)275
3714 y(v)i(=)h([x)f(y)h(z)f(0],)g(q)g(=)h(transformation)c(quaternion)
275 3827 y(returns)i(v)h(=)h(q*v/q)0 4052 y Fr(Remark)34
b(1.3)46 b Fq(Giv)m(en)34 b(a)h(quaternion)e Fo(q)1425
4067 y Fh(f)1504 4052 y Fq(describing)f(the)j(rotation)f(from)g(an)g
(inertial)e(frame)i(\()p Fo(x)3463 4066 y Fk(1)3503 4052
y Fo(;)15 b(y)3588 4066 y Fk(1)3628 4052 y Fo(;)g(z)3710
4066 y Fk(1)3750 4052 y Fq(\))34 b(to)0 4165 y(second)24
b(frame)f(\()p Fo(x)622 4179 y Fk(2)662 4165 y Fo(;)15
b(y)747 4179 y Fk(2)787 4165 y Fo(;)g(z)869 4179 y Fk(2)909
4165 y Fq(\),)25 b(a)f(quaternion)f Fo(q)1555 4179 y
Fh(r)1616 4165 y Fq(de\014ned)f(in)h(the)h(second)f(frame)h(ma)m(y)g(b)
s(e)f(applied)f(b)m(y)h(m)m(ultiplying)0 4278 y Fo(q)41
4293 y Fh(f)86 4278 y Fo(q)127 4292 y Fh(r)164 4278 y
Fo(q)205 4293 y Fh(f)250 4245 y Ff(\000)p Fk(1)345 4278
y Fq(:)111 4466 y(1.)46 b(T)-8 b(ransform)30 b(\(rotate\))i(frame)f(2)f
(\()p Fo(x)1424 4480 y Fk(2)1464 4466 y Fo(;)15 b(y)1549
4480 y Fk(2)1589 4466 y Fo(;)g(z)1671 4480 y Fk(2)1711
4466 y Fq(\))30 b(bac)m(k)h(to)h(the)e(inertial)e(frame.)111
4653 y(2.)46 b(P)m(erform)31 b(the)f(rotation)h Fo(q)1124
4667 y Fh(r)1161 4653 y Fq(.)111 4841 y(3.)46 b(Rotate)33
b(bac)m(k)e(to)g(the)f(second)h(frame)f(via)g Fo(q)1745
4856 y Fh(f)1790 4841 y Fq(.)0 5074 y(That)40 b(is,)h(if)e
Fo(q)499 5088 y Fh(r)577 5074 y Fq(=)689 4955 y Fi(\022)750
4980 y(h)831 5074 y Fq(0)83 b(1)h(0)1175 4980 y Fi(i)1214
5003 y Fh(T)1269 5074 y Fo(;)15 b(\022)1352 5088 y Fh(r)1390
4955 y Fi(\023)1491 5074 y Fq(is)38 b(a)i(rotation)h(ab)s(out)e(the)h
Fo(y)j Fq(axis)c(in)f(the)i(frame)g(\()p Fo(x)3385 5088
y Fk(2)3425 5074 y Fo(;)15 b(y)3510 5088 y Fk(2)3549
5074 y Fo(;)g(z)3631 5088 y Fk(2)3671 5074 y Fq(\),)43
b(the)0 5235 y(quaternion)29 b Fo(q)498 5250 y Fh(f)543
5235 y Fo(q)584 5249 y Fh(r)622 5235 y Fo(q)663 5250
y Fh(f)708 5202 y Ff(\000)p Fk(1)832 5235 y Fq(is)h(the)g(equiv)-5
b(alen)m(t)30 b(rotation)h(in)e(the)i(frame)f(\()p Fo(x)2466
5249 y Fk(1)2506 5235 y Fo(;)15 b(y)2591 5249 y Fk(1)2630
5235 y Fo(;)g(z)2712 5249 y Fk(1)2752 5235 y Fq(\).)p
eop
5 4 bop 0 100 a Fn(A.)31 b(S.)f(Ho)s(del:)40 b(Quaternions)29
b(\(In)m(tro)s(duction\))g Fo(R)q(ev)s(ision)d Fq(:)f(1)p
Fo(:)p Fq(1)436 b Fn(\(c\))31 b(1998)902 b Fq(5)0 399
y Fr(Example)34 b(1.3)46 b(Quaternion)27 b(algebra)d
Fl(\000)p Fq(90)1672 366 y Ff(\016)1736 399 y Fq(rotation)h(ab)s(out)e
Fo(x)h Fq(follo)m(w)m(ed)g(b)m(y)g(90)2963 366 y Ff(\016)3028
399 y Fq(rotation)g(ab)s(out)g(revised)0 522 y Fo(y)33
b Fq(and)d Fo(z)k Fq(axes)d(is)f(equiv)-5 b(alen)m(t)30
b(to)h(a)g(rotation)f(ab)s(out)g(the)2007 428 y Fi(h)2088
522 y Fq(1)83 b(1)h(1)2432 428 y Fi(i)2501 522 y Fq(v)m(ector.)0
711 y Fj(qx)95 b(=)47 b(quaternion\([1,0,0],-pi/2\);)184
b(#)48 b(elementary)d(rotations)0 824 y(qy1)i(=)g
(quaternion\([0,1,0],pi/2\);)0 936 y(qz2)g(=)g
(quaternion\([0,0,1],pi/2\);)0 1162 y(qyi)g(=)g(qtrans\(qy1,qx\);)712
b(#)48 b(transform)d(back)i(to)g(original)e(coordinates)0
1275 y(qzi)i(=)g(qtrans\(qtrans\(qz2,qy1\),qx\))o(;)0
1388 y(qzi2=)f(qtrans\(qz2,qyi\);)664 b(#)48 b(NOT)f(THE)f(SAME)h(as)g
(qzi!)0 1501 y(qzi3=)f(qtrans\(qz2,qmult\(qyi,qx\)\);)184
b(#)48 b(This)e(matches)94 b(qzi)0 1727 y([qyiv,thyi])45
b(=)i(quaternion\(qyi\))330 b(#)48 b(check)e(vectors,)g(angles)0
1840 y([qziv,thzi])f(=)i(quaternion\(qzi\))0 1953 y([qzi2v,thz2i])d(=)k
(quaternion\(qzi2\))0 2066 y([qzi3v,thz3i])c(=)k(quaternion\(qzi3\))0
2291 y(qtot)f(=)g(qmult\(qzi,qmult\(qyi,qx\)\))0 2404
y(qtotchk)f(=)h(qmult\(qx,)f(qmult\(qy1,qz2\)\))118 2582
y Fq(Output:)0 2741 y Fj(qyiv)h(=)g(0.00000)141 b(0.00000)94
b(-1.00000)0 2854 y(thyi)47 b(=)g(1.5708)0 3013 y Fo(q)41
3027 y Fh(y)76 3036 y Fe(1)145 3013 y Fq(in)29 b(the)h
Fo(q)448 3027 y Fh(x)522 3013 y Fq(frame)g(is)g(equiv)-5
b(alen)m(t)29 b(to)j(a)e(rotation)h(ab)s(out)f(the)h(original)d
Fl(\000)p Fo(z)35 b Fq(axis.)0 3172 y Fj(qziv)47 b(=)g(1.0000e+00)141
b(1.5701e-16)f(3.4863e-32)0 3285 y(thzi)47 b(=)g(1.5708)0
3398 y(qzi2v)f(=)i(1.5701e-16)140 b(-1.5701e-16)188 b(1.0000e+00)0
3511 y(thz2i)46 b(=)i(1.5708)0 3624 y(qzi3v)e(=)i(1.0000e+00)140
b(1.9626e-16)h(3.9252e-17)0 3737 y(thz3i)46 b(=)i(1.5708)0
3896 y Fq(One)29 b(ma)m(y)h(b)s(e)f(tempted)g(to)h(view)f
Fo(q)1227 3910 y Fh(y)1262 3920 y Fd(i)1321 3896 y Fq(as)h(the)g
(rotation)f(to)i(the)e(basis)f(frame)i(of)f Fo(q)2821
3910 y Fh(z)2854 3919 y Fe(2)2892 3896 y Fq(,)h(but)f(this)f(is)h
(incorrect.)40 b Fo(q)3835 3910 y Fh(y)3870 3920 y Fd(i)0
4009 y Fq(merely)33 b(represen)m(ts)g(the)g(rotation)h
Fo(q)1282 4023 y Fh(y)1317 4032 y Fe(1)1388 4009 y Fq(in)e(the)i
(reference)f(frame;)i(b)s(oth)e(rotations)g Fo(q)2972
4023 y Fh(x)3049 4009 y Fq(and)g Fo(q)3270 4023 y Fh(y)3305
4033 y Fd(i)3368 4009 y Fq(m)m(ust)g(b)s(e)g(used)0 4122
y(to)e(bac)m(k-transform)g Fo(q)782 4136 y Fh(z)815 4145
y Fe(2)883 4122 y Fq(in)m(to)g(the)f(reference)h(frame.)41
b(This)28 b(is)i(sho)m(wn)f(as)i(follo)m(ws:)611 4292
y Fo(q)652 4306 y Fh(y)687 4316 y Fd(i)800 4292 y Fq(=)83
b Fo(q)995 4306 y Fh(x)1038 4292 y Fo(q)1079 4306 y Fh(y)1114
4315 y Fe(1)1153 4292 y Fo(q)1194 4306 y Fh(x)1237 4254
y Ff(\000)p Fk(1)613 4449 y Fo(q)654 4463 y Fh(z)687
4473 y Fd(i)800 4449 y Fq(=)g Fo(q)995 4463 y Fh(x)1053
4355 y Fi(\020)1103 4449 y Fo(q)1144 4463 y Fh(y)1179
4472 y Fe(1)1217 4449 y Fo(q)1258 4463 y Fh(z)1291 4472
y Fe(2)1329 4449 y Fo(q)1370 4463 y Fh(y)1405 4472 y
Fe(1)1444 4411 y Ff(\000)p Fk(1)1538 4355 y Fi(\021)1603
4449 y Fo(q)1644 4463 y Fh(x)1687 4411 y Ff(\000)p Fk(1)578
4614 y Fo(q)619 4628 y Fh(z)652 4638 y Fd(i)678 4628
y Fk(2)800 4614 y Fq(=)g Fo(q)995 4628 y Fh(y)1030 4638
y Fd(i)1060 4614 y Fo(q)1101 4628 y Fh(z)1134 4637 y
Fe(2)1172 4614 y Fo(q)1213 4628 y Fh(y)1248 4638 y Fd(i)1278
4577 y Ff(\000)p Fk(1)800 4788 y Fq(=)g Fo(q)995 4802
y Fh(x)1038 4788 y Fo(q)1079 4802 y Fh(y)1114 4811 y
Fe(1)1153 4788 y Fo(q)1194 4802 y Fh(x)1237 4751 y Ff(\000)p
Fk(1)1331 4788 y Fo(q)1372 4802 y Fh(z)1405 4811 y Fe(2)1443
4694 y Fi(\020)1493 4788 y Fo(q)1534 4802 y Fh(x)1577
4788 y Fo(q)1618 4802 y Fh(y)1653 4811 y Fe(1)1692 4788
y Fo(q)1733 4802 y Fh(x)1776 4751 y Ff(\000)p Fk(1)1870
4694 y Fi(\021)1920 4717 y Ff(\000)p Fk(1)2040 4788 y
Fq(=)25 b Fo(q)2177 4802 y Fh(x)2220 4788 y Fo(q)2261
4802 y Fh(y)2296 4811 y Fe(1)2334 4788 y Fo(q)2375 4802
y Fh(x)2419 4751 y Ff(\000)p Fk(1)2513 4788 y Fo(q)2554
4802 y Fh(z)2587 4811 y Fe(2)2625 4788 y Fo(q)2666 4802
y Fh(x)2710 4788 y Fo(q)2751 4802 y Fh(y)2786 4811 y
Fe(1)2824 4751 y Ff(\000)p Fk(1)2918 4788 y Fo(q)2959
4802 y Fh(x)3003 4751 y Ff(\000)p Fk(1)3122 4788 y Fl(6)p
Fq(=)g Fo(q)3259 4802 y Fh(z)3292 4812 y Fd(i)578 4958
y Fo(q)619 4972 y Fh(z)652 4982 y Fd(i)678 4972 y Fk(3)800
4958 y Fq(=)83 b(\()p Fo(q)1030 4972 y Fh(y)1065 4982
y Fd(i)1095 4958 y Fo(q)1136 4972 y Fh(x)1180 4958 y
Fq(\))15 b Fo(q)1271 4972 y Fh(z)1304 4981 y Fe(2)1342
4958 y Fq(\()q Fo(q)1419 4972 y Fh(y)1454 4982 y Fd(i)1484
4958 y Fo(q)1525 4972 y Fh(x)1568 4958 y Fq(\))1603 4916
y Ff(\000)p Fk(1)1723 4958 y Fq(=)25 b Fo(q)1860 4972
y Fh(y)1895 4981 y Fe(1)1933 4958 y Fo(q)1974 4972 y
Fh(x)2018 4958 y Fo(q)2059 4972 y Fh(z)2092 4981 y Fe(2)2130
4958 y Fo(q)2171 4972 y Fh(x)2214 4921 y Ff(\000)p Fk(1)2309
4958 y Fo(q)2350 4972 y Fh(y)2385 4981 y Fe(1)2423 4921
y Ff(\000)p Fk(1)2542 4958 y Fq(=)g Fo(q)2679 4972 y
Fh(z)2712 4982 y Fd(i)0 5274 y Fj(qtot)47 b(=)g(-5.5511e-17)188
b(7.0711e-01)g(1.2326e-32)g(7.0711e-01)0 5387 y(qtotchk)46
b(=)h(-5.5511e-17)188 b(7.0711e-01)g(5.5511e-17)h(7.0711e-01)p
eop
6 5 bop 0 100 a Fn(A.)31 b(S.)f(Ho)s(del:)40 b(Quaternions)29
b(\(In)m(tro)s(duction\))g Fo(R)q(ev)s(ision)d Fq(:)f(1)p
Fo(:)p Fq(1)436 b Fn(\(c\))31 b(1998)902 b Fq(6)0 399
y Fm(1.4)112 b(Application)35 b(of)j(quaternions)g(in)e(co)s(ordinate)i
(frame)f(transformations)0 570 y Fq(In)m(terpretations)30
b(of)h(quaternions.)0 810 y Fr(1.4.1)105 b(Euler)35 b(angles)0
982 y Fq(\(y)m(a)m(w-pitc)m(h-roll\))71 b(ma)m(y)36 b(b)s(e)f(used)g
(to)h(construct)g(an)f(inertial-to-b)s(o)s(dy)e(quaterion)i
Fo(q)2995 997 y Fh(bi)3088 982 y Fq(as)h(follo)m(ws.)55
b(The)35 b(y)m(a)m(w)0 1095 y(rotation)c(is)e(ab)s(out)h(the)h
Fo(z)j Fq(axis)c(in)f(the)i(inertial)d(frame)1508 1299
y Fo(q)1549 1313 y Fh(y)1615 1299 y Fq(=)1711 1205 y
Fi(\020h)1841 1299 y Fq(0)84 b(0)f(1)2185 1205 y Fi(i)2240
1299 y Fo(;)15 b( )2342 1205 y Fi(\021)0 1519 y Fq(The)30
b(pitc)m(h)g(rotation)g(is)g(ab)s(out)g(the)g Fo(y)k
Fq(axis)29 b(in)h(frame)g(1)h(\()p Fo(y)s Fq(-axis)f(after)h(the)f(y)m
(a)m(w)i(rotation)e(is)g(made\))1500 1739 y Fo(q)1541
1753 y Fh(p)1577 1762 y Fe(1)1640 1739 y Fq(=)1736 1644
y Fi(\020h)1866 1739 y Fq(0)84 b(1)f(0)2210 1644 y Fi(i)2265
1739 y Fo(;)15 b(\022)2351 1644 y Fi(\021)0 1953 y Fq(The)30
b(roll)f(rotation)h(is)g(made)g(ab)s(out)g(the)h Fo(x)f
Fq(axis)g(in)f(frame)h(2)1495 2158 y Fo(q)1536 2172 y
Fh(r)1568 2181 y Fe(2)1631 2158 y Fq(=)1727 2063 y Fi(\020h)1857
2158 y Fq(1)84 b(0)f(0)2201 2063 y Fi(i)2256 2158 y Fo(;)15
b(')2355 2063 y Fi(\021)0 2372 y Fq(In)28 b(order)g(to)h(construct)g(a)
g(single)e(quaterion)h Fo(q)1626 2387 y Fh(bi)1712 2372
y Fq(relating)g(the)g(inertial)f(to)i(b)s(o)s(dy)e(axis)h(rotation,)h
(rotations)g Fo(q)3826 2386 y Fh(p)3862 2395 y Fe(1)0
2485 y Fq(and)h Fo(q)218 2499 y Fh(r)250 2508 y Fe(2)318
2485 y Fq(m)m(ust)g(b)s(e)g(expressed)g(in)f(the)i(inertial)d(frame:)
1199 2689 y Fo(q)1240 2703 y Fh(p)1363 2689 y Fq(=)82
b Fo(q)1557 2703 y Fh(y)1598 2689 y Fo(q)1639 2703 y
Fh(p)1675 2712 y Fe(1)1713 2689 y Fo(q)1754 2703 y Fh(y)1795
2652 y Ff(\000)p Fk(1)1201 2827 y Fo(q)1242 2841 y Fh(r)1363
2827 y Fq(=)g Fo(q)1557 2841 y Fh(y)1598 2827 y Fo(q)1639
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1.000 UP
1.000 UL
LT1
4151 4417 M
(y \(body\)) Rshow
4235 4417 M
399 0 V
2737 2625 M
693 514 V
2737 2625 Pls
3430 3139 Pls
4434 4417 Pls
1.000 UP
1.000 UL
LT2
4151 4277 M
(z \(body\)) Rshow
4235 4277 M
399 0 V
2737 2625 M
365 -894 V
2737 2625 Crs
3102 1731 Crs
4434 4277 Crs
1.000 UP
1.000 UL
LT0
4151 4137 M
(x \(inertial\)) Rshow
4235 4137 M
399 0 V
3467 2625 M
-730 0 V
3467 2625 Star
2737 2625 Star
4434 4137 Star
1.000 UP
1.000 UL
LT1
4151 3997 M
(y \(inertial\)) Rshow
4235 3997 M
399 0 V
2737 2625 M
632 516 V
2737 2625 Star
3369 3141 Star
4434 3997 Star
1.000 UP
1.000 UL
LT2
4151 3857 M
(z \(inertial\)) Rshow
4235 3857 M
399 0 V
2737 2625 M
365 -894 V
2737 2625 Box
3102 1731 Box
4434 3857 Box
1.000 UL
LT5
4151 3717 M
(eigenaxis) Rshow
4235 3717 M
399 0 V
2737 2625 M
-365 894 V
78 5 V
10 10 V
6 10 V
3 9 V
-3 7 V
-6 6 V
-10 3 V
-14 2 V
-16 -1 V
-18 -3 V
-20 -6 V
1.000 UL
LT3
3200 1215 M
-1460 0 V
1010 3003 L
2274 4035 L
1460 0 V
4464 2247 L
3200 1215 L
4464 2247 L
-1460 0 V
2274 4035 L
3004 2247 L
1740 1215 L
1.000 UL
LT4
3734 4035 M
2470 3003 L
-1460 0 V
1460 0 V
3200 1215 L
stroke
grestore
end
showpage
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(-2) Cshow
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1705 560 M
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2221 560 M
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2737 560 M
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3253 560 M
0 63 V
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0 -4207 R
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0 63 V
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LTb
672 560 M
4130 0 V
0 4130 V
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672 560 L
2737 210 M
(rotate about eigenaxis 10.00 deg) Cshow
2737 4900 M
(yaw=10.00 deg, pitch= 0.00 deg) Cshow
1.000 UP
1.000 UL
LT0
4151 4557 M
(x \(body\)) Rshow
4235 4557 M
399 0 V
3346 2535 M
-609 90 V
3346 2535 Pls
2737 2625 Pls
4434 4557 Pls
1.000 UP
1.000 UL
LT1
4151 4417 M
(y \(body\)) Rshow
4235 4417 M
399 0 V
2737 2625 M
749 508 V
2737 2625 Pls
3486 3133 Pls
4434 4417 Pls
1.000 UP
1.000 UL
LT2
4151 4277 M
(z \(body\)) Rshow
4235 4277 M
399 0 V
2737 2625 M
365 -894 V
2737 2625 Crs
3102 1731 Crs
4434 4277 Crs
1.000 UP
1.000 UL
LT0
4151 4137 M
(x \(inertial\)) Rshow
4235 4137 M
399 0 V
3467 2625 M
-730 0 V
3467 2625 Star
2737 2625 Star
4434 4137 Star
1.000 UP
1.000 UL
LT1
4151 3997 M
(y \(inertial\)) Rshow
4235 3997 M
399 0 V
2737 2625 M
632 516 V
2737 2625 Star
3369 3141 Star
4434 3997 Star
1.000 UP
1.000 UL
LT2
4151 3857 M
(z \(inertial\)) Rshow
4235 3857 M
399 0 V
2737 2625 M
365 -894 V
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3102 1731 Box
4434 3857 Box
1.000 UL
LT5
4151 3717 M
(eigenaxis) Rshow
4235 3717 M
399 0 V
2737 2625 M
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1.000 UL
LT3
3200 1215 M
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-1460 0 V
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3004 2247 L
1740 1215 L
1.000 UL
LT4
3734 4035 M
2470 3003 L
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1460 0 V
3200 1215 L
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end
showpage
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showpage
 @endspecial eop
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b(\(In)m(tro)s(duction\))g Fo(R)q(ev)s(ision)d Fq(:)f(1)p
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showpage
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1188 560 M
0 63 V
0 4067 R
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0 -4207 R
(-1.5) Cshow
1705 560 M
0 63 V
0 4067 R
0 -63 V
0 -4207 R
(-1) Cshow
2221 560 M
0 63 V
0 4067 R
0 -63 V
0 -4207 R
(-0.5) Cshow
2737 560 M
0 63 V
0 4067 R
0 -63 V
0 -4207 R
(0) Cshow
3253 560 M
0 63 V
0 4067 R
0 -63 V
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(0.5) Cshow
3770 560 M
0 63 V
0 4067 R
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0 -4207 R
(1) Cshow
4286 560 M
0 63 V
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0 -4207 R
(1.5) Cshow
4802 560 M
0 63 V
0 4067 R
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0 -4207 R
(2) Cshow
LTb
672 560 M
4130 0 V
0 4130 V
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672 560 L
2737 210 M
(rotate about eigenaxis 20.61 deg) Cshow
2737 4900 M
(yaw=20.00 deg, pitch= 5.00 deg) Cshow
1.000 UP
1.000 UL
LT0
4151 4557 M
(x \(body\)) Rshow
4235 4557 M
399 0 V
3173 2527 M
-436 98 V
3173 2527 Pls
2737 2625 Pls
4434 4557 Pls
1.000 UP
1.000 UL
LT1
4151 4417 M
(y \(body\)) Rshow
4235 4417 M
399 0 V
2737 2625 M
844 485 V
2737 2625 Pls
3581 3110 Pls
4434 4417 Pls
1.000 UP
1.000 UL
LT2
4151 4277 M
(z \(body\)) Rshow
4235 4277 M
399 0 V
2737 2625 M
405 -906 V
2737 2625 Crs
3142 1719 Crs
4434 4277 Crs
1.000 UP
1.000 UL
LT0
4151 4137 M
(x \(inertial\)) Rshow
4235 4137 M
399 0 V
3467 2625 M
-730 0 V
3467 2625 Star
2737 2625 Star
4434 4137 Star
1.000 UP
1.000 UL
LT1
4151 3997 M
(y \(inertial\)) Rshow
4235 3997 M
399 0 V
2737 2625 M
632 516 V
2737 2625 Star
3369 3141 Star
4434 3997 Star
1.000 UP
1.000 UL
LT2
4151 3857 M
(z \(inertial\)) Rshow
4235 3857 M
399 0 V
2737 2625 M
365 -894 V
2737 2625 Box
3102 1731 Box
4434 3857 Box
1.000 UL
LT5
4151 3717 M
(eigenaxis) Rshow
4235 3717 M
399 0 V
2737 2625 M
-375 886 V
90 18 V
5 10 V
1 9 V
-3 7 V
-7 5 V
-11 3 V
-15 1 V
-17 -1 V
-18 -4 V
-20 -6 V
-20 -8 V
1.000 UL
LT3
3200 1215 M
-1460 0 V
1010 3003 L
2274 4035 L
1460 0 V
4464 2247 L
3200 1215 L
4464 2247 L
-1460 0 V
2274 4035 L
3004 2247 L
1740 1215 L
1.000 UL
LT4
3734 4035 M
2470 3003 L
-1460 0 V
1460 0 V
3200 1215 L
stroke
grestore
end
showpage
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(-1.5) Cshow
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0 63 V
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2737 560 M
0 63 V
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3253 560 M
0 63 V
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(0.5) Cshow
3770 560 M
0 63 V
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0 63 V
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(1.5) Cshow
4802 560 M
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LTb
672 560 M
4130 0 V
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672 560 L
2737 210 M
(rotate about eigenaxis 22.34 deg) Cshow
2737 4900 M
(yaw=20.00 deg, pitch=10.00 deg) Cshow
1.000 UP
1.000 UL
LT0
4151 4557 M
(x \(body\)) Rshow
4235 4557 M
399 0 V
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4434 4557 Pls
1.000 UP
1.000 UL
LT1
4151 4417 M
(y \(body\)) Rshow
4235 4417 M
399 0 V
2737 2625 M
844 485 V
2737 2625 Pls
3581 3110 Pls
4434 4417 Pls
1.000 UP
1.000 UL
LT2
4151 4277 M
(z \(body\)) Rshow
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399 0 V
2737 2625 M
441 -911 V
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1.000 UP
1.000 UL
LT0
4151 4137 M
(x \(inertial\)) Rshow
4235 4137 M
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3467 2625 M
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1.000 UP
1.000 UL
LT1
4151 3997 M
(y \(inertial\)) Rshow
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2737 2625 M
632 516 V
2737 2625 Star
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LT2
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(z \(inertial\)) Rshow
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LT5
4151 3717 M
(eigenaxis) Rshow
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LT3
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LT4
3734 4035 M
2470 3003 L
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3200 1215 L
stroke
grestore
end
showpage
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showpage
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showpage
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showpage
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Fq(\001)p Fo(t=)p Fq(2\).)41 b(F)-8 b(or)0 1661 y(\001)p
Fo(t)30 b Fq(su\016cien)m(tly)f(small,)g(cos)q(\()p Fo(!)s
Fq(\001)p Fo(t=)p Fq(2\))d Fl(\031)f Fq(1)31 b(and)f(sin)n(\()p
Fo(!)s Fq(\001)p Fo(t=)p Fq(2\))c Fl(\031)f Fo(!)s Fq(\001)p
Fo(t=)p Fq(2,)31 b(so)g Fo(q)2713 1675 y Fh(r)2781 1661
y Fq(ma)m(y)g(b)s(e)f(appro)m(ximated)g(as)1576 1892
y Fo(q)1617 1906 y Fh(r)1680 1892 y Fl(\031)25 b Fq(1)c(+)1932
1773 y Fi(\022)2004 1830 y Fo(!)s Fq(\001)p Fo(t)p 2004
1871 169 4 v 2065 1954 a Fq(2)2182 1773 y Fi(\023)2268
1869 y Fq(\026)2258 1892 y(\012)0 2117 y(Th)m(us)1239
2325 y Fo(q)1280 2340 y Fh(b)1314 2325 y Fq(\()p Fo(t)g
Fq(+)e(\001)p Fo(t)p Fq(\))83 b(=)g Fo(q)1915 2340 y
Fh(b)1949 2325 y Fq(\()p Fo(t)p Fq(\))2067 2206 y Fi(\022)2129
2325 y Fq(1)21 b(+)2285 2206 y Fi(\022)2357 2264 y Fo(!)s
Fq(\001)p Fo(t)p 2357 2304 V 2418 2388 a Fq(2)2535 2206
y Fi(\023)2621 2302 y Fq(\026)2611 2325 y(\012)f(+)g
Fl(\001)15 b(\001)g(\001)2894 2206 y Fi(\023)950 2559
y Fo(q)991 2574 y Fh(b)1025 2559 y Fq(\()p Fo(t)20 b
Fq(+)g(\001)p Fo(t)p Fq(\))g Fl(\000)g Fo(q)1500 2574
y Fh(b)1534 2559 y Fq(\()p Fo(t)p Fq(\))83 b(=)g Fo(q)1915
2574 y Fh(b)1949 2559 y Fq(\()p Fo(t)p Fq(\))2067 2440
y Fi(\022)2139 2497 y Fo(!)s Fq(\001)p Fo(t)p 2139 2538
V 2200 2621 a Fq(2)2317 2440 y Fi(\023)2404 2536 y Fq(\026)2393
2559 y(\012)20 b(+)g Fl(\001)15 b(\001)g(\001)940 2737
y Fo(q)981 2752 y Fh(b)1015 2737 y Fq(\()p Fo(t)20 b
Fq(+)g(\001)p Fo(t)p Fq(\))g Fl(\000)g Fo(q)1490 2752
y Fh(b)1524 2737 y Fq(\()p Fo(t)p Fq(\))p 940 2778 688
4 v 1229 2861 a(\001)p Fo(t)1720 2799 y Fq(=)1874 2680
y Fi(\022)1945 2737 y Fo(!)p 1945 2778 60 4 v 1952 2861
a Fq(2)2015 2680 y Fi(\023)2091 2799 y Fo(q)2132 2814
y Fh(b)2166 2799 y Fq(\()p Fo(t)p Fq(\))2279 2776 y(\026)2269
2799 y(\012)h(+)f Fl(\001)15 b(\001)g(\001)0 3024 y Fq(W)-8
b(e)32 b(tak)m(e)g(the)e(limit)e(at)j(\001)p Fo(t)25
b Fl(!)g Fq(0)31 b(to)g(obtain)978 3171 y Fo(dq)1066
3186 y Fh(b)p 978 3212 123 4 v 999 3295 a Fo(dt)1136
3233 y Fq(=)1232 3114 y Fi(\022)1303 3171 y Fo(!)p 1303
3212 60 4 v 1310 3295 a Fq(2)1373 3114 y Fi(\023)1449
3233 y Fo(q)1490 3248 y Fh(b)1524 3233 y Fq(\()p Fo(t)p
Fq(\))1637 3210 y(\026)1627 3233 y(\012)25 b(=)1824 3171
y Fo(!)p 1824 3212 V 1831 3295 a Fq(2)1909 3164 y Fi(\000)1947
3233 y Fo(q)1988 3248 y Fh(b;)p Fk(4)2087 3210 y Fq(\026)2077
3233 y(\012)20 b Fl(\000)g Fq(\()7 b(\026)-52 b Fo(q)2330
3248 y Fh(b)2384 3233 y Fl(\001)2440 3210 y Fq(\026)2430
3233 y(\012)o(\))21 b Fl(\000)2652 3210 y Fq(\026)2642
3233 y(\012)f Fl(\002)26 b Fq(\026)-51 b Fo(q)2860 3248
y Fh(b)2894 3164 y Fi(\001)0 3592 y Fq(where)30 b Fo(x)21
b Fl(\001)f Fo(y)29 b Fq(=)552 3528 y Fi(P)655 3592 y
Fo(x)707 3607 y Fh(`)740 3592 y Fo(y)785 3607 y Fh(`)848
3592 y Fq(is)h(the)h(v)m(ector)h(inner)d(\(dot\))j(pro)s(duct)d(and)h
Fo(x)21 b Fl(\002)f Fo(y)29 b Fq(=)2689 3395 y Fi(\014)2689
3445 y(\014)2689 3494 y(\014)2689 3544 y(\014)2689 3594
y(\014)2689 3644 y(\014)2689 3694 y(\014)2716 3398 y(2)2716
3544 y(6)2716 3597 y(4)2836 3462 y Fq(^)2843 3483 y Fo(i)3010
3462 y Fq(^)3012 3483 y Fo(j)3185 3459 y Fq(^)3182 3483
y Fo(k)2813 3596 y(x)2865 3610 y Fk(1)2987 3596 y Fo(x)3039
3610 y Fk(2)3162 3596 y Fo(x)3214 3610 y Fk(3)2817 3709
y Fo(y)2862 3723 y Fk(1)2991 3709 y Fo(y)3036 3723 y
Fk(2)3166 3709 y Fo(y)3211 3723 y Fk(3)3295 3398 y Fi(3)3295
3544 y(7)3295 3597 y(5)3350 3395 y(\014)3350 3445 y(\014)3350
3494 y(\014)3350 3544 y(\014)3350 3594 y(\014)3350 3644
y(\014)3350 3694 y(\014)3409 3592 y Fq(is)g(the)i(v)m(ector)0
3815 y(cross)f(pro)s(duct.)40 b(In)30 b(matrix)f(form)h(this)g(is)1011
3922 y Fi(2)1011 4068 y(6)1011 4118 y(6)1011 4168 y(6)1011
4221 y(4)1125 3996 y Fq(_)-42 b Fo(q)1149 4011 y Fh(b)p
Fk(1)1125 4109 y Fq(_)g Fo(q)1149 4124 y Fh(b)p Fk(2)1125
4222 y Fq(_)g Fo(q)1149 4237 y Fh(b)p Fk(3)1125 4335
y Fq(_)g Fo(q)1149 4350 y Fh(b)p Fk(4)1260 3922 y Fi(3)1260
4068 y(7)1260 4118 y(7)1260 4168 y(7)1260 4221 y(5)1340
4166 y Fq(=)1446 4104 y Fo(!)p 1446 4145 V 1453 4228
a Fq(2)1531 3922 y Fi(2)1531 4068 y(6)1531 4118 y(6)1531
4168 y(6)1531 4221 y(4)1688 3996 y Fq(0)180 b Fo(!)1970
4010 y Fk(3)2127 3996 y Fl(\000)p Fo(!)2255 4010 y Fk(2)2377
3996 y Fo(!)2434 4010 y Fk(1)1628 4109 y Fl(\000)p Fo(!)1756
4123 y Fk(3)1938 4109 y Fq(0)g Fo(!)2220 4123 y Fk(1)2377
4109 y Fo(!)2434 4123 y Fk(2)1663 4222 y Fo(!)1720 4236
y Fk(2)1878 4222 y Fl(\000)p Fo(!)2006 4236 y Fk(1)2188
4222 y Fq(0)144 b Fo(!)2434 4236 y Fk(3)1628 4335 y Fl(\000)p
Fo(!)1756 4349 y Fk(1)1878 4335 y Fl(\000)p Fo(!)2006
4349 y Fk(2)2127 4335 y Fl(\000)p Fo(!)2255 4349 y Fk(3)2402
4335 y Fq(0)2515 3922 y Fi(3)2515 4068 y(7)2515 4118
y(7)2515 4168 y(7)2515 4221 y(5)2585 3922 y(2)2585 4068
y(6)2585 4118 y(6)2585 4168 y(6)2585 4221 y(4)2682 3996
y Fo(q)2723 4011 y Fh(b)p Fk(1)2682 4109 y Fo(q)2723
4124 y Fh(b)p Fk(2)2682 4222 y Fo(q)2723 4237 y Fh(b)p
Fk(3)2682 4335 y Fo(q)2723 4350 y Fh(b)p Fk(4)2834 3922
y Fi(3)2834 4068 y(7)2834 4118 y(7)2834 4168 y(7)2834
4221 y(5)0 4522 y Fq(Notice)31 b(that)g(the)g(4)21 b
Fl(\002)e Fq(4)31 b(matrix)f(is)f(sk)m(ew)i(symmetric)f(\(hence)h(its)e
(matrix)h(exp)s(onen)m(tial)f(is)h(orthogonal\).)0 4806
y Fp(References)0 5009 y Fq([Mul])127 b(Larry)30 b(Mullins.)37
b(Course)30 b(4160:)43 b(Quaternions.)c(Course)30 b(notes.)0
5192 y([R)-10 b(W91])47 b(M.)41 b(H.)g(Rheinfurth)d(and)h(H.)i(B.)g
(Wilson.)70 b(Metho)s(ds)40 b(of)h(applied)d(dynamics.)69
b(T)-8 b(ec)m(hnical)40 b(Rep)s(ort)337 5305 y(NASA)30
b(RP-1262,)j(NASA,)e(George)h(C.)e(Marshall)f(Space)h(Fligh)m(t)g(Cen)m
(ter,)h(1991.)p eop
14 13 bop 0 527 a Fp(Index)0 709 y Fq(conjugate)166 821
y(quaternion,)30 b(3)0 1017 y(Euler)f(angles)166 1130
y(quaternions,)g(6)0 1326 y(Octa)m(v)m(e)166 1439 y Fj(qinv)p
Fq(,)h(3)166 1552 y Fj(qmult)p Fq(,)f(3)166 1665 y Fj(qtrans)p
Fq(,)g(4)166 1778 y(qtransv,)h(8)166 1891 y(qtransvmat,)h(8)166
2004 y Fj(quaternion)p Fq(,)d(2)0 2200 y(quaternions)166
2313 y(as)j(co)s(ordinate)f(rotations,)h(3)166 2425 y(co)s(ordinate)f
(transformation,)g(4)166 2538 y(co)s(ordinate)g(transformation)g
(matrices,)h(6)166 2651 y(cross)g(pro)s(ducts,)e(2)166
2764 y(de\014nition,)f(2)166 2877 y(deriv)-5 b(ativ)m(es,)30
b(12)166 2990 y(Euler)f(angles,)h(6)166 3103 y(m)m(ultiplication,)e(2)
166 3216 y(unit)h(quaternions,)g(3)1905 5656 y(14)p eop
end
userdict /end-hook known{end-hook}if