  A quantum kicked rotor model is one of the promising systems to realize
various Floquet topological phases. We consider a double-kicked rotor model for
a one-dimensional quasi-spin-1/2 Bose-Einstein condensate with spin-dependent
and spin-independent kicks which are implementable for cold atomic experiments.
We theoretically show that the model can realize all the Altland-Zirnbauer
classes with nontrivial topology in one dimension. In the case of class CII, we
show that a pair of winding numbers $(w_0,w_\pi)\in 2\mathbb{Z}\times
2\mathbb{Z}$ featuring the edge states at zero and $\pi$ quasienergy,
respectively, takes various values depending on the strengths of the kicks. We
also find that the winding numbers change to $\mathbb{Z}$ when we break the
time-reversal and particle-hole symmetries by changing the phase of a kicking
lattice. We numerically confirm that the winding numbers can be obtained by
measuring the mean chiral displacement in the long-time limit in the present
case with four internal degrees of freedom. We further propose two feasible
methods to experimentally realize the spin-dependent and spin-independent kicks
required for various topological phases.
