Several lines of experimental evidence suggest that the conventional kinesin 1 walks by an asymmetric hand-over-hand mechanism, although it is a homodimer. In the previous study, we examined several important force-dependent features of the hand-over-hand mechanism of kinesin. In this study, we focus on the asymmetry in the hand-over-hand mechanism. We show that the experimentally observed kinesin limping can be explained in our model by the variation of the neck linker lengths in the kinesin stepping (which has also been suggested earlier by others). We also study the experimentally observed processive motion of a mutant heterodimer of kinesin, in which only one of the two heads has the capability of ATP hydrolysis, as well as the walking of wild-type kinesin in the presence of both ATP and its analogue AMPPNP. We show that the possible processive walking of the heterodimeric kinesin can be explained by introducing a force-generating intermediate, the kinesin-ATP complex, which is different from the posthydrolytic species, kinesin-ADP/Pi.