Vehiole dynamics Associated with Submarine Rescue 



maneuvers. The positive direction of this axis is downward com-^ 

 mensurate with standard submarine dynamics analysis. The y-axis 

 is through the x-axis z-axis intersection with positive direction to 

 the starboard to provide a right handed orthogonal system. This 

 X, y, z body axis frame is related to an inertial axis system, 

 X, Y, Z, through the ordered rotations ^, 0, and $ about the 

 z, y and x axis. The origin of inertial axis system is located at 

 the origin of the vehicle axis system at the start of a computation 

 and the X Y plane is parallel to the water surface with the vehicle 

 axis in the X Z plane. 



The Euler angles are formed in the following mianner. With 

 the two systems initially coincident, a first rotation, (Z^), is 

 performed giving the system (xj, Y\> Z). Next a rotation, (y, 9) , 

 is performed about the y, axis resulting in the system (x, y,, Zj). 

 A third rotation, (x <^) , about the x axis brings the body axis 

 system, (x, y, z) to the final position. The transformation matrix 

 relating the (X, Y, Z) system to the (x, y, z) system through the 

 above ordered rotations is then 



cosecosi|j cosesini|j sin4> 



sine sin«^cosijj-cos0 cosii; sin0 sin<i>sini|j+cos<^cos4i sin<J>cose 

 _sin0cos<f»cos4j+alni|jsin«^ sin0 cos<j)sini|j-sin4>cos^|j cos(j> cos0 



It remains to relate the Euler angle rates to the roll, pitch and ya^ 

 rates, (p,q,r) respectively. The relation is 



1115 



