This paper aims at presenting a general and compact matrix expression of the exact kinetic energy operator in polyspherical coordinates adapted to the study of semirigid molecules. The internal coordinates of an N atom system are expressed by a set of N - 1 relative position vectors. The operator can be applied to whatever the set of vectors (Jacobi, Radau, valence, satellite, etc., or a combination of these vectors), and whatever the number of atoms. It includes the rotation and the Coriolis coupling. Such a formulation gives the opportunity to develop a general code for calculating the ro-vibrational spectra in a curvilinear description including all the vibrational, rotational, and Coriolis couplings.