# sln2 - EAS 4510 HW#2 Solutions Spring 2005 1 The direction...

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EAS 4510 HW#2 Solutions Spring 2005 1) The direction cosines matrix for a 3-1-3 rotation through Euler angles , i , ω is: () 313 3, 1, cos( ) sin( ) 0 1 0 0 cos( ) sin( ) 0 sin( ) cos( ) 0 0 cos( ) sin( ) sin( ) cos( ) 0 0 0 1 0 sin( ) cos( ) 0 0 1 cos( )cos( ) sin( )sin( )cos( ) cos( )sin( ) sin CCC i C Ci i ii i C ω ωω −− =Ω ΩΩ ⎡⎤ ⎢⎥ =− ⎣⎦ Ω− Ω+ = ( )cos( )cos( ) sin( )sin( ) sin( )cos( ) cos( )sin( )cos( ) sin( )sin( ) cos( )cos( )cos( ) cos( )sin( ) sin( )sin( ) cos( )sin( ) cos( ) i i −Ω + 2) Given the results of problem 1, the following DCM’s given correspond to the Euler angles , i , ω found below. Note that each possible solution is a complete set of all 3 angles, except for the case where two of the angles cannot be found independently, in which case we can still find some information about those angles. a) .5810 .7772 .2418 .8118 .5316 .2418 .0594 .3368 .9397 C 1 cos( ) .9397 cos (.9397) 20 20 i io r = == DD Case 1) for 20 i = D , 1 1 (1, 3) tan( ) 1 (2,3) tan (1) 45 225 cos( )sin( ) (2,3) .2418 .2418 cos 45 45 sin( 20 ) C C or iC or ⎛⎞ ⎜⎟ ⎝⎠ D To satisfy both conditions above, only 45 = D is possible. Now for : (3,1) .0594 tan( ) (3,2) .3368 10 190 cos( )sin( ) (3,2) .3368 C C or Ω= = = = 1 .3368 cos 10 10 sin(20 ) or = D To satisfy both conditions only 10 D is possible.

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Case 2) for 20 i =− D , 1 1 (1, 3) tan( ) 1 (2,3) tan (1) 45 225 cos( )sin( ) (2,3) .2418 .2418 cos 135 225 sin( 20 ) C C or iC or ω == ⎛⎞ ⎜⎟ ⎝⎠ DD D To satisfy both conditions above, only 225 = D is possible. Now for : (3,1) .0594 tan( ) (3,2) .3368 10 190 cos( )sin( ) (3,2) .3368 C C or Ω= = −Ω = = 1 .3368 cos 170 190 sin( 20 ) or = D To satisfy both conditions only 190 D is possible. b) .9659 .2588 0 .2588 .9659 0 00 1 C ⎡⎤ ⎢⎥ ⎣⎦ 1 1 cos( ) 1 180 cos( )cos( ) sin( )sin( ) cos( ) (1,1) .9659 cos (.9659) 15 15 cos( )sin( ) sin( )cos( ) sin( ) (1,2) .2588 sin (.2588) 15 165 i i C or C or ωωω ωω = Ω+ −Ω= = = Ω− = = = = D To satisfy both conditions above, the only possibility is 15 = D .
3) We will solve for the Euler angles ψ , θ , φ of the 3-2-1 rotation developed in class: () 321 1, 2, 3, 1 0 0 c o s () 0 s i n () c o s ( ) s i n ( ) 0 0 cos( ) sin( ) 0 1 0 sin( ) cos( ) 0 0 s i n o s () s i n () 0 c o s 0 0 1 cos( )cos( ) cos( )sin( ) sin( ) cos( )sin( ) si CC C C C C

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sln2 - EAS 4510 HW#2 Solutions Spring 2005 1 The direction...

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