The self-driven movement of biped robots along an inclined surface is a topic that, despite its inherent complexities, has attracted the attention of many researchers. However, what distinguishes this research from similar works is the consideration of flexibility in the links that constitute such robotic systems. This assumption is not far-fetched, as in the transition phase (impact), the impulse force applied to the system significantly increases the likelihood of exciting vibrational modes. On the other hand, the bones involved in walking are enveloped by muscles, which have viscoelastic properties. Therefore, to achieve more accurate results, modeling the links with viscoelastic properties becomes inevitable. In the discussion of the autonomous movement of bipedal robots, the most important issue is determining the initial configuration of the robot so that the system can perform a periodic and stable motion solely under the influence of gravitational attraction. The highly unstable nature of the system under study, along with the vibrations created by the impact force resulting from the foot striking the inclined surface, constitutes one of the very serious challenges in the progress of this research. Nevertheless, this paper overcomes these challenges by presenting a completely systematic method with very strong mathematical frameworks. Finally, the effect of the intrinsic parameters of elastic links, including the modulus of elasticity and the Kelvin-Voigt coefficient, on the stability of the movement of such robotic systems is examined.