<p>Bio-inspired structural adhesives composed of micropillar arrays show superior adhesion performances and have been extensively applied in various fields, including medical and health care, grasping manipulators, climbing robots, etc. Accurate predictions of the adhesion properties of the micropillar arrays are crucial for their optimal design and practical applications. The nano-asperities are widespread in most natural and artificial surfaces and significantly affect the adhesion properties of the micropillar arrays. However, most existing theoretical models for investigating the adhesion behavior of a micropillar array treat the pillars as linear springs, leading to a lack of consideration for the surface roughness. In this study, a novel theoretical model incorporating the contact adhesion between a single pillar and external surfaces decorated with nano-asperities is established to quantify the adhesion properties of micropillar array on rough surfaces with multi-scale roughnesses. It is shown that the adhesion force and efficiency (defined as ratio of the adhesion force to the preload) of a micropillar array on rough surfaces are governed by the stiffness parameter (ratio of the height to the elastic modulus of the micropillar) and the surface elastic modulus, where the former determines the number of the contact micropillars and the latter dictates the detachment force of individual micropillars. The homogeneous micropillars composed of either stiff or soft materials face an inherent conflict between the adhesion force and efficiency because the increasing elastic modulus benefits the adhesion strength but hinders the axial deformation and surface contact area. Accordingly, we propose bio-inspired graded micropillars with modulus gradient along the axial direction to address such a conflict. The stiffness parameter and surface modulus of the graded micropillars can be independently modulated to improve the adhesion performance on rough surfaces. Moreover, a theoretical model incorporating the stability against lateral collapse and the modified Greenwood-Williamson (G-W) model is further established to quantitatively evaluate the adhesion performance of the graded micropillar array on rough surfaces. Based on the developed model, the quantitative optimization strategies for the design of graded micropillar array with arbitrary modulus transition are achieved, for the first time, to simultaneously improve the adhesion force and efficiency. The obtained results combined not only deepen our understanding for the underlying mechanisms governing the adhesion behavior of the micropillar array on rough surfaces, but also extend the design space for optimizing the bio-inspired micropillar-structure adhesives to better accommodate the complex external environments.</p>