# moment of inertial taken at different position

what is difference between Principal axes of inertia and principal moments of inertia Taken at the center of mass,Moments of inertia Taken at the center of mass and aligned with the output coordinate system and Moments of inertia Taken at the output coordinate system ?

And why is the momet of inertia taken at center of mass and aligned with output coordinate system is used in urdf in ros?

Principal axes of inertia and principal moments of inertia: ( grams *  square millimeters )
Taken at the center of mass.
Ix = ( 0.89,  0.45,  0.00)     Px = 177217.10
Iy = (-0.45,  0.89, -0.01)     Py = 632385.17
Iz = ( 0.00,  0.00,  1.00)     Pz = 780108.86

Moments of inertia: ( grams *  square millimeters )
Taken at the center of mass and aligned with the output coordinate system.
Lxx = 268506.70 Lxy = 182259.67 Lxz = -4.63
Lyx = 182259.67 Lyy = 541099.71 Lyz = -851.31
Lzx = -4.63 Lzy = -851.31   Lzz = 780104.72

Moments of inertia: ( grams *  square millimeters )
Taken at the output coordinate system.
Ixx = 364622.20 Ixy = -112574.17    Ixz = -2115.49
Iyx = -112574.17    Iyy = 1445552.39    Iyz = -163.21
Izx = -2115.49  Izy = -163.21   Izz = 1780663.05

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An inertial tensor matrix can be expressed in any coordinate system, and expresses the torque required to rotate the object about the origin of that coordinate system. All inertia tensors are diagonalizable, i.e. a rotation exists that will bring all their off diagonal values to zero, it is then expressed in its principal axes. The first section in your post above is this rotation matrix and a vector of the principal moments, multiplying these together produces the inertia tensor matrix.

I'd assume the different forms of the inertia tensor shown above are used at different times to simply the equations of motion. They can all be calculated from each other if the relevant transformations are known, but it is simpler to have the values to hand.

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