  We investigate quantum phase transitions in the frustrated orthogonal-dimer
chain with an arbitrary spin $S \geq 1/2$. When the ratio of the competing
exchange couplings is varied, first-order phase transitions occur 2S times
among distinct spin-gap phases. The introduction of single-ion anisotropy
further enriches the phase diagram. The phase transitions described by the
present model possess most of the essential properties inherent in frustrated
quantum spin systems.
