## 4 Conclusions  
A mathematical model was developed to investigate the transportation of objects upward on an inclined plane that harmonically oscillates in its longitudinal direction by applying an additional asymmetry achieved through dry friction dynamic manipulations. The modeling results have demonstrated that the asymmetry of frictional conditions that is achieved through dry friction dynamic manipulations results in a net frictional force that outweighs the gravitational force. Therefore, the object can be transported upwards. The object's velocity and direction can be changed by dynamically manipulating the dry friction coefficient to achieve the required frictional asymmetry. Also, the findings have identified the control parameters that define the moving object's directional motion. The phase shift \(\phi\) has proven its effectiveness in controlling and changing the direction of the moving object at different inclination angles, where the average velocity was significantly increased at \(\alpha=0.035\) rad between \(\phi=0\) and \(\phi=\pi/2\) and reached its maximum at \(\lambda=\pi\). However, \(\lambda\), which corresponds to the part of the period in which the friction coefficient is dynamically decreased, is a crucial parameter in changing the velocity of the object, as the average velocity was gradually increased and reached a maximum between \(\lambda=\pi\) and \(\lambda=2\pi/3\) at \(\phi=\pi/4\). Moreover, the results have demonstrated that the ratio of the modified effective dry friction coefficient \(\langle\mu_{m}\rangle\) to the nominal dry friction coefficient \(\mu_{0}\) has a significant effect on the transportation velocity of the object. At inclination angle \(\alpha=0.035\) rad, the average velocity was significantly decreased as this ratio increased, where the average velocity of the moving object reached the maximum at \(\mu_{0}=0.15\). Experimental investigations have been carried out to prove the technical feasibility of the proposed method. The experiment has verified that periodic dynamic dry friction manipulations create an additional asymmetry that results in a net frictional force that outweighs the gravitational force acting on an object placed on an inclined plane. Therefore, the object can be transported upwards to the inclined plane. The experimental findings have shown that the control parameters \(\phi\) and \(\lambda\) have a significant influence in controlling and changing the directional motion of the moving object. In agreement with the theoretical findings, the experiments demonstrated that the phase shift \(\phi\) is very effective in controllingand changing the direction of the moving object at different inclination angles, and \(\lambda\) has revealed a significant influence in changing the velocity of the object being transported at different inclination angles. The experimental findings qualitatively verified the modeling by demonstrating a similar response of motion parameters to the control parameters as it was obtained in the modeling results. The proposed transportation method is beneficial for various modern industries engaged in transportation and manipulation tasks with objects spanning a broad range of sizes, including those operating at small scales for applications in lab-on-a-chip technology, micro-assembly lines, micro-feeder systems and other delicate component manipulation systems. The method's straightforward technical implementation makes it a cost-effective alternative to more intricate devices like microgrippers. The presented research advances the classical theories of vibrational transportation on inclined surfaces. Future research will focus on further developing and optimizing the proposed transportation method to handle large quantities of small-scale materials, microelectronics components and similar micro-scale particles. Conceptualization, R.E.B., A.F. and S.K. ; methodology, R.E.B., A.F. and S.K. ; software, R.E.B., K.L. and S.K. ; validation, R.E.B. ; formal analysis, R.E.B., K.L., V.L. and S.K. ; investigation, R.E.B.