directional cosine matrix for coordinate transformation

8 views (last 30 days)
Good Afternoon All,
I know this is highly engineering based but the photo shown below is a reference paper to calculate the Torque Roll Axis. It sounds all crazy fancy but to my understanding its a coordinate transformation to make the new x axis (x') parallel to the x moment of inertia. I have applied his formulas to create this new DCM for transformation. I have called his directional cosine vector for x' Tx, the directional cosine vector for y' Ty and the directional cosine vector for z' Tz. I am not sure if I am producing his equations correctly in matlab. i.e maybe using the wrong functions. For example when applying an arbitrary value it shouldn't matter what value I choose correct? However when changing the arbitrary values it does change my DCM. I have also checked to see if the TRA_Transform_Matrix is orthogonal and it does meet that requirement.
If anyone could take a look at the code and tell me if I am on the right track I would greatly appreciate it. I know its a lot to ask but I am getting a crazy DCM.
Thanks so much.
Here is just some sample test data:
CG=[1.429925 .01714458 .7631665];
Mount=[2.32062 -.10742 .51072];
I=[17.24533 66.20312 62.17456 -.5598926 13.70156 1.363275]
%I is the only variable that serves an actual importance.
function[TRA_Transform]= TRA_Transformation_Matrix(CG,Mount,I)
%This program calculates the Transformation Matrix for the TRA
%
% User Input
%This is a subfunction of the main program of PT_Wizard_NF_Calculator.
%Please see the main routine document for further information of inputs.
%
% CG: Center of Gravity Position Matrix expressed as [xc,yc,zc]
% Mount: Any Mount Position Matrix expressed as [x,y,z]
% I: Moment of Inertia Matrix expressed as [Ixx,Iyy,Izz,Ixy,Ixz,Iyz]
%
%Arbitrary Values in TRA_Transform (distances)
ry=CG(2)-Mount(2);
ryz=CG(2)-Mount(3);
rz=CG(3)-Mount(3);
%Mass Inertia Matrix
M_Int=zeros(3,3);
M_Int(1,1)=I(1);
M_Int(1,2)=-I(4);
M_Int(1,3)=-I(5);
M_Int(2,1)=-I(4);
M_Int(2,2)=I(2);
M_Int(2,3)=-I(6);
M_Int(3,1)=-I(5);
M_Int(3,2)=-I(6);
M_Int(3,3)=I(3);
%Inverse of Mass-Inertia Matrix to find TRA Direction
Inv_M_Int=inv(M_Int);
%Formulating the norm vector used for finding the normalization constant
Norm_Vec=[Inv_M_Int(1,1) Inv_M_Int(2,1) Inv_M_Int(3,1)];
%Calculating the normalization constant (a)
Norm_Constant_a=norm(Norm_Vec);
%Applying normalization to M_Rot_Inv
Norm_Inv_M_Int=Inv_M_Int/Norm_Constant_a;
%TRA direction (assuming torque applied to x-axis)
TRA_Direction=[0 0 0 Norm_Inv_M_Int(1,1) Norm_Inv_M_Int(2,1) Norm_Inv_M_Int(3,1)];
%TRA DCM for x
TRA_Xx=TRA_Direction(4);
TRA_Xy=TRA_Direction(5);
TRA_Xz=TRA_Direction(6);
%TRA DCM for y
TRA_Yxi=(-(TRA_Xy*ry+TRA_Xz*ryz)/TRA_Xx);
TRA_Yyi=ry;
TRA_Yzi=ryz;
%Creating TRA Transform Y vector for norm
TRA_Y_Vec=[TRA_Yxi TRA_Yyi TRA_Yzi];
%TRA Transform for y
TRA_Yx=TRA_Yxi/norm(TRA_Y_Vec);
TRA_Yy=TRA_Yyi/norm(TRA_Y_Vec);
TRA_Yz=TRA_Yzi/norm(TRA_Y_Vec);
%TRA DCM for z
%Initial Matrix of TRA Transform z
Int_TRA_z=[TRA_Xx TRA_Xy; TRA_Yx TRA_Yy];
%Solving for the z elements for TRA Transform
TRA_z=-inv(Int_TRA_z)*[TRA_Xz; TRA_Yz]*(rz);
TRA_Zxi=TRA_z(1);
TRA_Zyi=TRA_z(2);
TRA_Zzi=rz;
%Creating TRA Transform Y vector for norm
TRA_Z_Vec=[TRA_Zxi TRA_Zyi TRA_Zzi];
%TRA Transform for z
TRA_Zx=TRA_Zxi/norm(TRA_Z_Vec);
TRA_Zy=TRA_Zyi/norm(TRA_Z_Vec);
TRA_Zz=TRA_Zzi/norm(TRA_Z_Vec);
%TRA DCM Formulation
TRA=zeros(3,3);
TRA(1,1)=TRA_Xx;
TRA(1,2)=TRA_Xy;
TRA(1,3)=TRA_Xz;
TRA(2,1)=TRA_Yx;
TRA(2,2)=TRA_Yy;
TRA(2,3)=TRA_Yz;
TRA(3,1)=TRA_Zx;
TRA(3,2)=TRA_Zy;
TRA(3,3)=TRA_Zz;
%Completed TRA Transformation Matrix
zeros_temp=zeros(3,3);
TRA_Transform=[TRA zeros_temp; zeros_temp TRA];

Answers (0)

Categories

Find more on Cartesian Coordinate System Conversion in Help Center and File Exchange

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!