Neuromuscular model achieving speed control and steering with a 3D bipedal walker

Abstract

Nowadays, very few humanoid robots manage to travel in our daily environments. This is mainly due to their limited locomotion capabilities, far from the human ones. Recently, we developed a bio-inspired torque-based controller recruiting virtual muscles driven by reflexes and a central pattern generator. Straight walking experiments were obtained in a 3D simulation environment, resulting in the emergence of human-like and robust gait patterns, with speed modulation capabilities. In this paper, we extend this model, in order to control the steering direction and curvature. Based on human turning strategies, new control pathways are introduced and optimized to reach the sharpest possible turns. In sum, tele-operated motions can be achieved through the control of two scalar inputs (i.e. forward speed and heading). This is particularly relevant for steering the robot on-line, and navigating in cluttered environments. Finally, the biped demonstrated significant robustness during blind walking experiments.

Appendices

Appendices

1.1 Appendix 1: Stimulations for curved motion

The stimulations computation rules from Van der Noot et al. (2018) affected by the curved motion updates (see Sect. 3) are summarized here. More details (e.g. time delays) are provided in that former contribution. Variables presented in Sect. 3 are not detailed here.

The hip lateral joints are controlled by the HAB and HAD muscles. First, the HAB muscles receive stimulations coming from the CPG. These stimulations are mainly proportional to $[x_E]^{+}$ (excited by $u_E$) for the right leg and to $[x_F]^{+}$ (excited by $u_F$) for the left leg.

During the leg supporting phase, the following proportional-derivative (PD) control is applied: $\varDelta _{\varPsi ,\{R,L\}} = (k_{p,\varPsi }\,(\delta \,{ }\varPsi _{ref,\{R,L\}}^{*} - \varPsi _t) - k_{d,\varPsi }\,\dot{\varPsi }_t)\,\tilde{F}_{gd,\{R,L\}}$, where $k_{p,\varPsi }$ and $k_{d,\varPsi }$ are parameters to optimize, $\varPsi _t$ is the torso lateral lean angle and $\dot{\varPsi }_t$ is its derivative. $\delta$ equals 1 for the right leg and $-\,1$ for the left one. Finally, $\tilde{F}_{gd,\{R,L\}}$ is the vertical force below the corresponding foot, normalized to the walker weight. Then, HAB and HAD muscles are mainly commanded by a stimulation equal to $[\varDelta _{\varPsi ,\{R,L\}}]^{+}$ or $[\varDelta _{\varPsi ,\{R,L\}}]^{-}$.

During the contralateral leg supporting phase, a hip lateral reference angle $\varphi _{h,l,ref,\{R,L\}}$ is computed as $- k_{p,\varLambda ,h} (-\delta \,\varLambda _{ref,h,\{R,L\}}^{*} - \varDelta _{com,\{L,R\}}) + k_{d,\varLambda ,h}\,\dot{\varDelta }_{com,\{L,R\}}$, where $k_{p,\varLambda ,h}$ and $k_{d,\varLambda ,h}$ are control parameters to optimize, $\varDelta _{com,L}$ is the COM lateral position, relative to the left foot and $\dot{\varDelta }_{com,L}$ its derivative (similar for $\varDelta _{com,R}$ and $\dot{\varDelta }_{com,R}$ relative to the right foot). The resulting local angle reference $\varphi _{h,l,ref,\{R,L\}}$ is later maintained by using a similar PD control rule as described above (i.e. for the supporting phase), with similar stimulations sent to HAB and HAD.

The hip transverse joints are controlled with the following PD computation: $\varDelta _{trans,\{R,L\}} = 500 \, (\varphi _{h,t,ref,\{R,L\}} - \varphi _{h,t,\{R,L\}}) - 20 \, \dot{\varphi }_{h,t,\{R,L\}}$, where $\varphi _{h,t,\{R,L\}}$ is the hip joint transverse position and $\dot{\varphi }_{h,t,\{R,L\}}$ is its derivative. Stimulation equal to $[\varDelta _{trans,\{R,L\}}]^{+}$ or $[\varDelta _{trans,\{R,L\}}]^{-}$ are then sent to the HER and HIR muscles.

1.2 Appendix 2: Optimization parameters

The parameters to be optimized in the controller, and their ranges are reported Table 2: the transverse (t) and lateral (l) leg parameters, as well as the CPG-related parameters. The speed dependent parameters are computed as follows: $k_{y,in} = K_{y,in} + L_{y,in}\,v_{*}$; $k_{y,out} = K_{y,out} + L_{y,out}\,v_{*} + M_{y,out}\,v_{*}^2$; $\varDelta _\varLambda = K_{\varDelta ,\varLambda } + L_{\varDelta ,\varLambda }\,v_{*}$; $\varDelta _\varPsi = K_{\varDelta ,\varPsi } + L_{\varDelta ,\varPsi }\,v_{*} + M_{\varDelta ,\varPsi }\,v_{*}^2$; $\eta _{o} = K_{\eta ,o} + L_{\eta ,o}\,v_{*}$; $\nu _{l} = K_{\nu ,l} + L_{\nu ,l}\,v_{*} + M_{\nu ,l}\,v_{*}^2$, where $v_{*} = v_{{ ref}} - 0.65$ and $v_{{ ref}}$ is the target forward speed. The parameters optimized for the reference controller are provided in the simulation code extension (Online Resource 1).

Table 2 Optimization parameters and their bounds