The three motions of the attachment point are related to 

 the motions of the platform CG by the inverse of equation (8), 

 If the two coordinate systems coincide initially, the displace- 

 ments of the point are 



{x } 

 c 



{x^} 



([A] 



[i]){x,} 



(13) 



Here {x, } = translations of the CG = (x, , x„ , x^) . 



[A] = coordinate transformation defined by (8). 



{X } = coordinates of anchor line end, (X , Y , Z ).. 

 c c c c 



[I] = unit matrix. 

 For a linear motions analysis, we retain only first order terms 



in (13) and replace the trigonometric functions by the small angle 



approximation, sin6 ^ 9, cos6 si 1. Noting that (8) constitutes 



an orthogonal transformation, therefore the inverse of [A] equals 



the transpose, and after multiplying and rearrancring , we obtain 



{x } = {x.} + 

 c 1 



Z -Y 

 c 



- Z X 

 c 



Y -X 

 c c 



c 



K 



c 



K 





1^3. 



(14) 



Equation (14) may be rewritten as 

 {x„} = [D] • {x} 



(15) 



where 



{x} = vector of six platform motions = (x , 



1' "2' 



, e^} 



[D] = 



1 















Z 

 c 



-^c 







1 







-z 



c 







X 

 c 











1 



Y 



-X 







138 



