1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
|
@c DO NOT EDIT! Generated automatically by munge-texi.
@c Copyright (C) 1996, 1997 John W. Eaton
@c This is part of the Octave manual.
@c For copying conditions, see the file gpl.texi.
@node Quaternions
@chapter Quaternions
Quaternions are hypercomplex numbers used to represent spatial
rotations in three dimensions. This set of routines provides a useful
basis for working with quaternions in Octave. A tutorial is in the
Octave source, scripts/quaternion/quaternion.ps.
These functions were written by A. S. Hodel, Associate Professor,
Auburn University.
@anchor{doc-quaternion}
@deftypefn {Function File} {[@var{a}, @var{b}, @var{c}, @var{d}] =} quaternion (w)
@deftypefnx {Function File} {[@var{vv}, @var{theta}] =} quaternion (w)
@deftypefnx {Function File} {@var{w} =} quaternion (@var{a}, @var{b}, @var{c}, @var{d})
@deftypefnx {Function File} {@var{w} =} quaternion (@var{vv}, @var{theta})
Construct or extract a quaternion
@example
w = a*i + b*j + c*k + d
@end example
@noindent
from given data.
@end deftypefn
@anchor{doc-qconj}
@deftypefn {Function File} {} qconj (@var{q})
Conjugate of a quaternion.
@example
q = [w, x, y, z] = w*i + x*j + y*k + z
qconj (q) = -w*i -x*j -y*k + z
@end example
@end deftypefn
@anchor{doc-qderiv}
@deftypefn {Function File} {} qderiv (omega)
Derivative of a quaternion.
Let Q be a quaternion to transform a vector from a fixed frame to
a rotating frame. If the rotating frame is rotating about the
[x, y, z] axes at angular rates [wx, wy, wz], then the derivative
of Q is given by
@example
Q' = qderivmat (omega) * Q
@end example
If the passive convention is used (rotate the frame, not the vector),
then
@example
Q' = -qderivmat (omega) * Q
@end example
@end deftypefn
@anchor{doc-qderivmat}
@deftypefn {Function File} {} qderivmat (@var{omega})
Derivative of a quaternion.
Let Q be a quaternion to transform a vector from a fixed frame to
a rotating frame. If the rotating frame is rotating about the
[x, y, z] axes at angular rates [wx, wy, wz], then the derivative
of Q is given by
@example
Q' = qderivmat (omega) * Q
@end example
If the passive convention is used (rotate the frame, not the vector),
then
@example
Q' = -qderivmat (omega) * Q.
@end example
@end deftypefn
@anchor{doc-qinv}
@deftypefn {Function File} {} qinv (@var{q})
Return the inverse of a quaternion.
@example
q = [w, x, y, z] = w*i + x*j + y*k + z
qmult (q, qinv (q)) = 1 = [0 0 0 1]
@end example
@end deftypefn
@anchor{doc-qmult}
@deftypefn {Function File} {} qmult (@var{a}, @var{b})
Multiply two quaternions.
@example
[w, x, y, z] = w*i + x*j + y*k + z
@end example
@noindent
identities:
@example
i^2 = j^2 = k^2 = -1
ij = k jk = i
ki = j kj = -i
ji = -k ik = -j
@end example
@end deftypefn
@anchor{doc-qtrans}
@deftypefn {Function File} {} qtrans (@var{v}, @var{q})
Transform the unit quaternion @var{v} by the unit quaternion @var{q}.
Returns @code{@var{v} = @var{q}*@var{v}/@var{q}}.
@end deftypefn
@anchor{doc-qtransv}
@deftypefn {Function File} {} qtransv (@var{v}, @var{q})
Transform the 3-D vector @var{v} by the unit quaternion @var{q}.
Return a column vector.
@example
vi = (2*real(q)^2 - 1)*vb + 2*imag(q)*(imag(q)'*vb)
+ 2*real(q)*cross(imag(q),vb)
@end example
@noindent
Where imag(q) is a column vector of length 3.
@end deftypefn
@anchor{doc-qtransvmat}
@deftypefn {Function File} {} qtransvmat (@var{qib})
Construct a 3x3 transformation matrix from quaternion @var{qib} that
is equivalent to rotation of th radians about axis @var{vv}, where
@code{[@var{vv}, @var{th}] = quaternion (@var{qib})}.
@end deftypefn
@anchor{doc-qcoordinate_plot}
@deftypefn {Function File} {} qcoordinate_plot (@var{qf}, @var{qb}, @var{qv})
Plot in the current figure a set of coordinate axes as viewed from
the orientation specified by quaternion @var{qv}. Inertial axes are
also plotted:
@table @var
@item qf
Quaternion from reference (x,y,z) to inertial.
@item qb
Quaternion from reference to body.
@item qv
Quaternion from reference to view angle.
@end table
@end deftypefn
|
