Kinematic analysis of flexible bipedal robotic systems

In spite of its intrinsic complexities, the passive gait of bipedal robots on a sloping ramp is a subject of interest for numerous researchers. What distinguishes the present research from similar works is the consideration of flexibility in the constituent links of this type of robotic systems. This is not a far-fetched assumption because in the transient(impact) phase, due to the impulsive forces which are applied to the system, the likelihood of exciting the vibration modes increases considerably. Moreover,the human leg bones that are involved in walking are supported by viscoelastic muscles and ligaments. Therefore, for achieving more exact results, it is essential to model the robot links with viscoelastic properties. To this end, the Gibbs-Appell formulation and Newton’s kinematic impact law are used to derive the most general form of the system’s dynamic equations in the swing and transient phases of motion. The most important issue in the passive walking motion of bipedal robots is the determination of the initial robot configuration with which the system could accomplish a periodic and stable gait solely under the effect of gravitational force. The extremely unstable nature of the system studied in this paper and the vibrations caused by the impulsive forces induced by the impact of robot feet with the inclined surface are some of the very serious challenges encountered for achieving the above-mentioned goal. To overcome such challenges, an innovative method that uses a combination of the linearized equations of motion in the swing phase and the algebraic motion equations in the transition phase is presented in this paper to obtain an eigenvalue problem. By solving this problem, the suitable initial conditions that are necessary for the passive gait of this bipedal robot on a sloping surface are determined. The effects of the characteristic parameters of elastic links including the modulus of elasticity and the Kelvin-Voigt coefficient on the walking stability of this type of robotic systems are also studied. The findings of this parametric study reveal that the increase in the Kelvin-Voigt coefficient enhances the stability of the robotic system, while the increase in the modulus of elasticity has an opposite effect.