Stereo 3d Simulation Of Rigid Body Inertia... x;(Ix-4)+yilxy+l=0 (12) +w-+=0 xlx+l+-)=0 x+听+子=1 System(12)enables yi and i to be expressed through xi,using the first three equations,and substituting in the last equation: (a-A--1g1E y=w-1 (13) x-)业g-Ile =e.-1w-1= From the last equation of system(13)X;is found as follows: 1 (14) x=士 (Ix-4y-Igle w-x-1= =-1w-1x1e We find the values of y and by substituting x;in the first two equations of(13).Note that the components x,,of eigenvector n,do not have defined sign.There are two possibilities yielding an eigenvector n, along a defined line,but with two possible opposite directions.The correct direction will be found later.After finding all three eigenvectors n,n2 and n3 their components are substituted in(11)to generate the sought rotation matrix R (see Figure 1): 0.910.380.157 R=0.40-0.91-0.14 0.080.19-0.98 However,if the incorrect directions(signs in(14))were chosen,matrix R would yield an inverse(improper) rotation.This fact can be verified by calculating the determinant of R,which should be +1: R=1 This means that matrix R is a proper rotation.In the case when the determinant is-1,matrix R should be corrected to a proper rotation by multiplying it with a negative identity (changing direction of all eigenvectors) or by changing the sign of one of its rows (changing direction of one of the eigenvectors).Visit http://ialms.net/sim/to simulate other examples. III.CONCLUSION The described in this article simulation of rigid body properties in a stereo 3D virtual environment enables researchers and engineers to observe setups that are impossible to be created in laboratory conditions. Such an approach reveals important insights for the scientists working on unmanned helicopter autopilot analysis,synthesis and tuning. Authors of the presented material express their gratitude to Assoc.Professor Vesela Decheva. www.ijesi.org 34|PageStereo 3d Simulation Of Rigid Body Inertia... www.ijesi.org 34 | Page (12)       1 0 0 0 2 2 2                i i i i xz i yz i zz i i xy i yy i i yz i xx i i xy i xz x y z x I y I z I x I y I z I x I y I z I    System (12) enables i y and i z to be expressed through i x , using the first three equations, and substituting in the last equation: (13)                 1 1 2 2 2                                              z z i xy xz yz xx i yz xy xz yy i xz xy yz xx i yz xy xz i z z i xy xz yz xx i yz xy xz i i yy i xz xy yz xx i yz xy xz i i I I I I I I I I I I I I I I I I x I I I I I I I I z x I I I I I I I I y x         From the last equation of system (13) i x is found as follows: (14)         2 2 1 1                             z z i xy xz yz xx i yz xy xz yy i xz xy yz xx i yz xy xz i I I I I I I I I I I I I I I I I x     We find the values of i y and i z by substituting i x in the first two equations of (13). Note that the components i i i x , y ,z of eigenvector ni do not have defined sign. There are two possibilities yielding an eigenvector ni along a defined line, but with two possible opposite directions. The correct direction will be found later. After finding all three eigenvectors n1 , n2 and n3 their components are substituted in (11) to generate the sought rotation matrix R (see Figure 1):               0.08 0.19 0.98 0.40 0.91 0.14 0.91 0.38 0.15 R However, if the incorrect directions (signs in (14)) were chosen, matrix R would yield an inverse (improper) rotation. This fact can be verified by calculating the determinant of R , which should be 1 : R 1 This means that matrix R is a proper rotation. In the case when the determinant is 1, matrix R should be corrected to a proper rotation by multiplying it with a negative identity (changing direction of all eigenvectors) or by changing the sign of one of its rows (changing direction of one of the eigenvectors). Visit http://ialms.net/sim/ to simulate other examples. III. CONCLUSION The described in this article simulation of rigid body properties in a stereo 3D virtual environment enables researchers and engineers to observe setups that are impossible to be created in laboratory conditions. Such an approach reveals important insights for the scientists working on unmanned helicopter autopilot analysis, synthesis and tuning. Authors of the presented material express their gratitude to Assoc. Professor Vesela Decheva