Frontal-Sagittal Dynamic Coupling in the Optimal Design of a Passive Bipedal Walker

Abstract

The parametric design of passive bipedal walkers (PBWs) has been addressed by considering its dynamics in sagittal plane only. Nevertheless, with the aim of obtaining a complete parametric description, frontal plane dynamics must also be involved. Hence, in order to obtain a synergetic design of a PBW that simultaneously couples operating requirements of both planes, a concurrent design problem is proposed. The concurrent design approach is established as a nonlinear discontinuous dynamic optimization problem with mixed design variables and is solved by using four variants of differential evolution (DE) algorithm. The reduction in differences among Poincaré mapping values related to consecutive gait cycles is proposed as the performance function, where its minimization must induce the convergence toward a limit cycle in PBW frontal and sagittal planes. A comparative analysis between the proposed concurrent design and a sequential design process is carried out for a particular case study: the PBW with single joint and curved feet. The simulation results show the advantages of the obtained concurrent design solutions to provide stable gait cycles in both PBW planes. In addition, the statistical analysis of the applied DE variants indicates that DE/best/1/bin promotes an efficient tradeoff between exploration and exploitation of solutions in the objective space. For the particular problem, this results in a high design reconfigurability, mainly associated with the material assignment diversity provided by the optimizer.

Integrated Structure-Control Design of a Bipedal Robot Based on Passive Dynamic Walking

“…For this second condition σ = f c is assigned in (1). For more details about single support phase model, see Appendix B of [14].…”

“…This equation is evaluated in the sagittal and frontal plane collisions and is only computed once per transition event. For the detailed description of the double support phase equations, see Appendix B of [14].…”

“…This coupling is established to promote a periodic movement that permits the gait cycle's development. According to [14], the movement periodicity can be promoted by minimizing the differences among the system state values of consecutive transition events. Hence, the double support phases of the frontal and sagittal plane are established as Poincaré Sections to evaluate a cyclic behavior of the system along with the PWS and SPWS.…”

“…With the aim of evaluating the general S-C design objective (12), the frontal and sagittal plane dynamic response of the SPBR must be numerically simulated. Based on [14] the dynamics simulation is carried out to evaluate the convergence of the SPBR movements and to measure the energetic requirements provided by the semi-passive control strategy. Furthermore, the existence of dynamic coupling between the frontal and sagittal plane movements is checked by calculating the oscillation periods T f ro , T p sag and T sp sag related to the frontal plane dynamics, the PWS and SPWS of the sagittal plane, respectively.…”

“…The second research trend studies the optimization-based structural design of bipedal systems, where the problem of finding the optimal structure parameters for different types of walking systems was proposed. For instance, in the design of the passive bipedal walker leg that performs limit cycles in both the frontal and sagittal planes [14], in the design of an eight-bar mechanism to fulfill the desired locomotion task with a minimum force transmission during the stance phase [15,16], in the design of Stephenson III six-bar mechanism for tracking of a gait trajectory [17] and in the optimal mass distribution for passive dynamic biped robot [18]. Although both research trends have shown their own advantages, the trade-off between the natural dynamics of the structure and the control signal features related to the walking performance has not been addressed.…”