As a composite optical field, Airy vortex beam possessing transverse and longitudinal superimposed modes has abundant intensity and phase distributions. However, the composite optical field is limited to the specific propagation direction and complex modulation of the propagation trajectory. To address these issues, based on the multicoordinate transformations, one-dimensional Airy beam, two-dimensional Airy beam and Airy vortex beam have been transformed in different coordinate systems, respectively. We have realized the orientation modulation of the two lobes of Airy beam and the embedded optical vortex in the range of 0-2π, independently. We have studied the propagation properties in the propagation distances of 0, 2, 6, and 10 cm in the case of the obtuse and acute angles between two lobes of the Airy beam and the Airy vortex beam with orthogonal lobes, respectively. The results of experiments and numerical simulations show that the generated optical field can propagate along arbitrarily appointed parabolic trajectories and directions in free space. In addition, a method for measuring topological charge is proposed by adding the shift factor d when the directions of the lobes are opposite. This proposed method enables in situ determination of the embedded optical vortexes' topological charge without the need for additional optical elements, and it is immune to environmental vibration and parasitic interference. Our results promote the applications of Airy vortex beam in particle manipulation and optical cleaning and provide an alternative scheme for the modulation of composite optical field.Measurement of the topological charge and the intensity profiles at different propagation distance via the experimental setup.