The Foucault pendulum model in textbooks is generalized to a small vibration system with spherical constraint, where the vibration center undergoes slow motion along any closed orbit on the sphere. Starting from the Lagrangian function of the generalized Foucault pendulum, the adiabatic approximate solution and precession angle are derived. The results show that under the adiabatic approximation, the precession angle is independent of motion details, being only a functional of the closed orbit, which geometrically originates from the solid angle of the spherical area enclosed by the closed orbit. This paper avoids the application of theories of differential geometry such as surface vector translation.