Using bilateral symmetry of the biped robot mechanism for efficient and faster optimal gait learning on uneven terrain

Abstract

This paper, describes the relation between dynamically similar right gait (RG) and left gait (LG) of a 14 degrees of freedom (DOF) biped robot on uneven terrain by using the dynamic similarity principle of gaits (DSPG). The existence of DSPG is ingrained in bipedal systems’ bilateral body plan and the mirror image transformation of polar and axial vectors. The concept of fundamental basis vectors associated with the support foot plane act as a reference frame to compare the dynamic similarity among gaits for different walking conditions on uneven terrain. The DSPG discerns the generation of all possible optimal gaits on uneven terrain from a small primitive set of gaits. The identification of primitive walk dataset results in saving of both walk dataset generation time and learning time. The proposed DSPG is validated by generating the optimal RG of a 14-DOF biped robot using a genetic algorithm (GA) and utilizing DSPG to obtain corresponding dynamically similar LG. Simulation experiments of the bipedal walk on uneven terrain is validated by generating a primitive walk dataset for a 14-DOF biped robot and learning it by a feed forward neural-network (NN). Using the trained NN and DSPG, biped robot can successfully walk on 3D uneven ground by generating real-time optimal gaits.

Appendices

Appendix-A

Detailed inertial and geometric parameters of the links of a new model of the biped robot is given in Tables 12, 13, 14, 15 and 16. The new model of the biped robot has complicated distribution of mass within each link and is more massive compared to the biped robot model described in Sect. 2. The generalized link model of the biped robot is shown in Fig. 10. The new model of the biped robot link have masses $m_{C_{i}}$, $m_{O_{i}}$, and $m_{A_{i}}$ located at points ${C_{i}}$, ${O_{i}}$, and ${A_{i}}$ respectively, the total mass of the ith link is: $m_i = m_{C_{i}}+ m_{O_{i}} +m_{A_{i}}$ whereas the ith link of the biped robot model of the Sect. 2 has only one mass $m_{C_{i}} = m_i$ located at its geometric center point ${C_{i}}$.

Table 12 Inertial and geometric parameters of link-0, link-1 and link-2

Table 13 Inertial and geometric parameters of link-3, link-4, and link-5

Table 14 Inertial and geometric parameters of link-6,link-7, and link-8

Table 15 Inertial and geometric parameters of link-9,link-10, and link-11

Table 16 Inertial and geometric parameters of link-12, link-13 , link-14

Appendix-B

In this section DSPG has been verified for a new 14-DOF biped robot model discussed in Appendix-A. Optimized one step RG with walking conditions: ${\mathcal {G}}^{R} = \{ {\mathcal {S}} = 0$, $\theta = 15^{\circ }$, $\phi = 150^{\circ }$, $\psi = 120 ^{\circ }$, ${\mathcal {X}}_{\scriptscriptstyle {DSP1}}= -120 \text {mm}$, ${\mathcal {Y}}_{\scriptscriptstyle {DSP1}}= 85 \text {mm}$, ${\mathcal {Z}}_{\scriptscriptstyle {DSP1}}= 15 \text {mm}$, ${\mathcal {X}}_{\scriptscriptstyle {DSP2}} = 125 \text {mm}$, ${\mathcal {Y}}_{\scriptscriptstyle {DSP2}}= 85 \text {mm}$, ${\mathcal {Z}}_{\scriptscriptstyle {DSP2}}=-25\text {mm} \}$ is obtained by using a genetic algorithm. Stick diagram is shown in Fig. 45a. To traverse gait ${\mathcal {G}}^{R}$ biped robot consumes $W_{{\mathcal {G}}^{R}} = 3.75$ Joules of energy. Its dynamically similar gait ${\mathcal {G}}^{L}$, obtained by applying the principle of DSPG is shown in Fig. 45b along with ZMP, as this gait ${\mathcal {G}}^{L} \sim {\mathcal {G}}^{R}$, the computed work done $W_{{\mathcal {G}}^{L}} = 3.75$ Joules is exactly equal to $W_{{\mathcal {G}}^{R}}$ and zero moment points are mirror image of each other. Joints torque profile of new biped robot model for gait ${{\mathcal {G}}^{R}}$ are shown in Fig. 46a, b. Joint torques for its dynamically similar gait ${{\mathcal {G}}^{L}}$ it is shown in Fig. 47a, b. The ZMP state during gaits ${{\mathcal {G}}^{R}}$ and ${{\mathcal {G}}^{L}}$ are shown in Fig. 48a, b respectively.

Fig. 45

Stick diagram and ZMP plot of right and left gait of the new 14-DOF biped robot model

Fig. 46

Joint torques of right gait ${\mathcal {G}}^{R}$ obtained by optimization (new biped robot model of Appendix-A)

Fig. 47

Joint torques of left gait ${\mathcal {G}}^{L}$, where ${\mathcal {G}}^{L} \sim {\mathcal {G}}^{R}$ determined by DSPG (new biped robot model of Appendix-A)

Fig. 48

ZMP of right gait and its dynamically similar left gait of the new 14-DOF biped robot model of Appendix-A