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Current-driven domain wall motion from pinning sites in nanostrips with perpendicular magnetic anisotropy is studied by using micromagnetic simulations, supported by a one-dimensional model of wall dynamics. The threshold current density of perpendicular anisotropy strips is much smaller than that of in-plane anisotropy strips, and is almost independent of the pinning potential strength. This results from the narrower domain wall width, smaller hard-axis anisotropy, and the larger ratio of the depinning field and hard-axis anisotropy. In the one-dimensional model with a zero damping constant, the threshold current density is found to be about 0.72 of the intrinsic threshold current density for a perfect strip in a strong pinning regime that corresponds to strips with perpendicular magnetic anisotropy. The fact that the threshold current density from the pinning sites is smaller than the intrinsic current density is because the effective field, equivalent to the pinning potential, enhances a breakdown in the pinning site. Moreover, in the strong pinning regime, an opposite-direction depinning hardly ever occurs after current pulse is turned off. These features of strips with perpendicular magnetic anisotropy are attractive for magnetic random access memories where the domain wall should be moved stably between the pinning sites with the small current pulse.