  We construct models of the differential $KO$-theory and the twisted
differential $KO$-theory, by refining Karoubi's $KO$-theory [Kar78] in terms of
gradations on Clifford modules. In order for this, we set up the generalized
Clifford superconnection formalism which generalizes the Quillen's
superconnection formalism [Qui85]. One of our models can be regarded as
classifying "fermionic mass terms" in physics.
