Free mechanical set: defining the position

Position of the free mechanical set

The free motion is defined by a list of positions of the mobile mechanical set.

To define these positions, the mobile mechanical set will be dependent on the reference coordinate system.


Let us consider:

  • RREF the defining coordinate system of the mobile mechanical set (reference coordinate system)
  • RA the defining coordinate system of a particular position A of the mobile mechanical set

The RA coordinate system is defined in relation with the RREF coordinate system by means of the quantities presented in the table below.

Information Description
Position Coordinates of the center of the RA coordinate system in the RREF coordinate system: (Xc, Yc, Zc)
Orientation Components * of RA coordinate system in the RREF reference coordinate system: (Ux, Uy, Uz), (Vx, Vy, Vz) (Wx, Wy, Wz)
Note: * see a more accurate definition in the block below.


Let us consider:

  • three unitary orthogonal vectors Ex, Ey, Ez and an origin O defining the initial coordinate system (of reference) of the mechanical set: RREF
  • three unitary orthogonal vectors E1, E2, E3 and an origin C defining the position A of the mechanical set: RA

The second coordinate system is known in the first one, which means that the triplets (Ux, Uy, Uz), (Vx, Vy, Vz) et (Wx, Wy, Wz) are known as:

E1 = Ux*Ex + Uy*Ey + Uz*Ez

E2 = Vx*Ex + Vy*Ey + Vz*Ez

E3 = Wx*Ex + Wy*Ey + Wz*Ez

Note: The third triplet (Wx, Wy, Wz) is not necessary, as, when the first and the second triplets are known, it is easy then to know this last one, given that the coordinate system is orthonormaled. Therefore: E3 = E1⊥E2.