As a rule, mathematical models use the following basic systems of coordinates: 1) one associated with the spacecraft with the active docking assembly (body 1), or system of coordinates 1; 2) one associated with the spacecraft with the passive docking assembly (body 2), or system of coordinates 2; 3) one associated with the buffer of the docking mechanism of the docking mechanism (rod, ring with guides) or system of coordinates 3; 4) inertial system of coordinates (for the most part necessary only in the construction of mathematical models of docking dynamics). Here it is assumed that are the main central axes of inertia, although this limitation is not necessary. Axes are directed so that they pass through the longitudinal axes of symmetry of the docking assemblies; after docking these axes lie on one line. To reduce the number of unknowns, the relative motion of the center of mass of body 2 is usually examined in system of coordinates 1. Rotation relative to the center of mass is described independently for each body so that the momenta created by the control system can be calculated. Transformation of vectors from the first to the second system of coordinates is done in the following sequence of rotations: yaw # (relative to axis j, pitch 0 (relative to the intermediate axis , roll (relative to axis ). The matrices of the transformation and the transposed matrix are as follows: The angular velocities 0, 0 are found from the component of relative velocity , in system of coordinates 2 with the matrix of transformation , and angles are found by integrating the appropriate angular velocities:
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