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We investigate the effect of an additional modulation parameter $\delta$ onthe mobility properties of quasiperiodic lattices described by a generalizedGaneshan-Pixley-Das Sarma model with two on site modulation parameters. For thecase with bounded quasiperiodic potential, we unveil the existence ofself-duality relation, independent of $\delta$. By applying Avila's globaltheory, we analytically derive Lyapunov exponents in the whole parameter space,which enables us to determine mobility edges or anomalous mobility edgesexactly. Our analytical results indicate that the mobility edge equation isdescribed by two curves and their intersection with the spectrum gives the truemobility edge. Tuning the strength parameter $\delta$ can change the spectrumof the quasiperiodic lattice, and thus engineers the mobility of quasi-periodicsystems, giving rise to completely extended, partially localized, andcompletely localized regions. For the case with unbounded quasiperiodicpotential, we also obtain the analytical expression of the anomalous mobilityedge, which separates localized states from critical states. By increasing thestrength parameter $\delta$, we find that the critical states can be destroyedgradually and finally vanishes.

Engineering mobility in quasiperiodic lattices with exact mobility edges