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Using the moving axes theorem about which we willleam Iaterin this chapter, we
have
Then,
dd  =C70lb X Zb
               :  .
(4.144)
ddj, = 03~b X jb                          (4.145)
dSc = col,z, x kb
dd  = C117b + C12]b + c13rCb
(4.146)
+ ClltOlb X lb + C12702b X /b + C13COlb x kb         (4.147)
                                     -. Clllb + C12Jb + C13kb + COlb X 71                                 (4.148)
       t-
    ecause ii is a vector ofmagnitude unity and fixed direction in the inertial system,
                              dd   = Cll Zb + C12 Jb + C13kb + 7.ol.b X z: = 0               (4.149)
S:imilarly,
                                  d/t = C212b + C22 Jb + C23~b + COlb X -lr  = 0                 (4.150)
                                  d/c = C3171  + C32Jb + C33kb + 03lb X kr  = O                 (4.151)

 

     O = SiD-I(-C13)
,= si.-, (~r,.,:,)
1/, = s,.-,  (~r,,)
342              PERFORMANCE, STABIUTY, DYNAMICS, AND CONTROL
Now,
C70l.b X Lr' =
Then,
or
Similarly,
  >
zb
 p
Cll
Jb
 q
C12
kb
 r
C13
(4.152)
= lb(C13q - C12r) - jb(C13p - Ciir) + kb(C12P - Ciiq)    (4.153)
ddt .= (CII + C13q - C12r)rt, + (C12 + Ciir - C13P) jb
+ (C13 + C12P - Ciiq)k,
-0

where
Cll = C12r - Ct3q
C12 = C13 p - Ciir
C13 = Ciiq - C12P
C21 = C22r - C23q
C22 - C23 p - C21r
C23 = C21q - C22P
C31 - C32r - C33q
C32 - C33p - C31r
C33 = C3iq - C32P
(4.154)
(4.155)
(4.15 6)
(4.157)
(4.158)
(4.15 9)
(4.160)
(4.161)
(4.162)
(4.163)
(4.164)
  C11 C12 <       -r  q
- C21 C22 'g'j,l[_q -. -,P] ,4.165,
  C31 C32 (       p  0
C;, =  Cb S2lb
gZtbt = [_q
(4.166)
(4.167)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 343
Here, S2lb iS called the skew-symmetric form of angular velocity vector COb,.b. Equa-
tion (4.166) is the required relation for propagating the direction cosine matrix C~
forward in time given its initial val.ue. Generally,. the initial value of C~ is not
directly given. Instead, the irutial values of the Euler angles are given. With this
information, the elements of C~ can be obtained using Eqs. (4.128-4.136).
      The skew-symmetric form of a given vector has the property that premultiplying
it to another vector gives the vector cross product of the first 'vector with the second
vector. For example,
                                      S2{.bA = ojb X A                            (4,168)
   Another property of the skew-symmetric form of angular velocity matrix can
be illustrated as foLlows. Suppose we are given 02{. b and are asked to find qZ:,b, we
can do this as follows:
                                    cLcib = I                           (4.169)
　
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