Delays in perception and action for improving walk–run transition stability in bipedal gait

Abstract

This paper focuses on a transition motion between bipedal walking and running, whose characteristics have been revealed through numerous biological experiments. Although hysteresis in walk-to-run and run-to-walk transitions, the amount of which is proportional to the magnitude of acceleration/deceleration, is observed, it has not been elucidated yet. This is due to difficulty to conduct any biological experiments without the hysteresis in locomotion. From the viewpoint of nonlinear dynamics, the hysteresis is caused by dependency of the state history. Delays in perception and action for the walk–run transition can be assumed as the cause of the hysteresis. This paper therefore investigates i) whether the delays really cause the hysteresis in the walk–run transition, and ii) how they contribute to locomotion performance. To this end, in numerical simulations, we employ a controller for bipedal gait, named unified bipedal gait (UBG), which has capabilities of walking, running, and transitions between them like bipeds do. UBG is first optimized in pursue of biologically plausible controller with the same characteristics about the transition motion for reliable investigation. This optimized UBG reveals that the delays in perception and action actually cause the hysteresis with linearity of its amount and the magnitude of acceleration/deceleration. In addition, the delays play the key role of a bridge between latent representations of walking and running, thereby improving the success rate of the transition by nearly 40% from the case without the delays.

Appendices

Appendix

A Details of unified bipedal gait

Let us define the detailed equations of UBG (their perfect derivations are shown in the original papers [27, 29]). First, $\ell (\theta )$ is given by solving Eq. (1) with regard to $\theta$ as follows:

$$\begin{aligned} \ell (\psi (\theta )) =\,&\mathrm{e}^{-{{\hat{\zeta }}} {{\hat{\omega }}}_0 \psi } \left( A_1 \cos {{{\hat{\omega }}}_d \psi } + A_2 \sin {{{\hat{\omega }}}_d \psi } \right) \nonumber \\&+\, \frac{k}{{{\hat{k}}}} \ell _r - \frac{mg}{\sqrt{{{\hat{c}}}^2 + \left( {{\hat{k}}} - {{\hat{m}}} \right) ^2}} \cos (\theta (\psi ) \pm \alpha _g)\nonumber \\ \end{aligned}$$

(12)

where all the variables with hat, i.e., ${{\hat{m}}}$, ${{\hat{k}}}$, ${{\hat{c}}}$, ${{\hat{\zeta }}}$, ${{\hat{\omega }}}_0$, ${{\hat{\omega }}}_d$, are approximated as constants to solve Eq. (1) with regard to $\theta$, although their physical meanings are similar to the original variables without hat. $A_1$ and $A_2$ are set according to the initial states of each gait step. $\alpha _g$ is the phase difference equal to $\tan ^{-1}\{{{\hat{c}}} / ({{\hat{k}}} - {{\hat{m}}})\}$. $\psi (\theta )$ is an alternate variable of $\theta$ for monotonically increasing property.

When giving $\ell (\theta )$, ${{\dot{\theta }}} = F_1(\theta )$ and $\dot{\phi }= F_2(\theta )$ are derived from their natural dynamics as the following equations, respectively.

$$\begin{aligned} F_1(\theta ) \!\!&\,=\,\!\!\, {\mp } \frac{ 1 }{ m \ell ^2(\theta ) } \sqrt{ 2 \left( \!\! -\frac{ C_\phi }{ \sin ^2{\theta } } \! + \! m^2 g \!\! \int \!\! \ell ^3(\theta ) \sin {\theta } \mathrm{d}\theta \! + \! C_\theta \!\! \right) } \nonumber \\ \!\!&\,=\,\!\! {\mp }\, \frac{ \sqrt{ 2 \left( - C(\theta ) + L(\theta ) + C_\theta \right) } }{ M(\theta ) } \end{aligned}$$

(13)

$$\begin{aligned} F_2(\theta ) \!\!&\,=\,\!\! \pm \, \frac{ \sqrt{ 2 C_\phi } }{ m \ell ^2(\theta ) \sin ^2{\theta } } = \pm \frac{ \sqrt{ 2 C(\theta ) } }{ M(\theta ) \sin {\theta } } \end{aligned}$$

(14)

where $C_\theta$ and $C_\phi$ are conserved quantities of integrals, which are fixed in each gait step. $C_\phi /\sin ^2{\theta }$ is substituted for $C(\theta )$, and the inertia term $m \ell ^2(\theta )$ is also substituted for $M(\theta )$. $L(\theta ) = m^2 g \int \ell ^3(\theta ) \sin {\theta } \mathrm{d}\theta$ is solved approximately using Eq. (12). All the signs, ± and ${\mp }$, are given by the actual pendulum behavior: whether the pendulum is about to rise or going to fall. The autonomous systems are updated at the time of starting new gait step and generates the COM trajectory autonomously.

B Gait speed controllers

Gait speed control is required to investigate PTS. To this end, two types of controllers are implemented in our simulation model. We briefly introduce them in this section. To simplify the description, we omitted control methods for sideward and turning gait speeds from original papers, and several variable notations were changed according to this paper.

1.1 B.1 Adaptive speed controller

Let us briefly introduce the principle of the adaptive speed controller. To control the forward speed, the momentum of the swing leg (with mass $m_{L}$) is injected into D-SLIP model. If the swing leg swings over a touchdown position, its momentum would accelerate the gait speed forward; otherwise, the gait speed would be decelerated. The forward swing-up and touchdown positions, $p_{Lx}^{su}$ and $p_{Lx}^{td}$, are simply designed based on the capture point defined in Pratt et al.

$$\begin{aligned} p_{Lx}^{su} = ( \alpha _x + \iota _x ) \frac{F_r^{\mathrm {ref}}}{\omega } , p_{Lx}^{td} = \alpha _{x} \frac{F_r^{\mathrm {prd}}}{\omega } \end{aligned}$$

(15)

where $F_r^{\mathrm {prd}}$ is the predictive gait speed at the end of a step. $\alpha _x = 0.8$ is the constant to keep locomotion, $\omega$ is the natural frequency of the inverted pendulum $\omega =\sqrt{g/(\ell \cos \theta )}$, and $\iota _x$ is the integral term accumulating the deviation of the reference and actual gait speeds to adapt the swing-up amount for the optimal momentum that yields $F_r^{\mathrm {ref}}$.

1.2 B.2 Quasi-passive dynamic autonomous control

To enhance the performance of the gait speed control, we additionally employed quasi-passive autonomous control. This controller injects the additional angular momenta into $C_\theta$ and $C_\phi$ in Eqs. (13) and (14). The magnitude of $C_\phi$ increases the forward speed. The magnitude of $C_\theta$ is correlated to the time $T_\mathrm {sup}$. Hence, two angular momenta, $L_\phi$ and $L_\theta$, are designed for the gait speed control and the gait step time stabilization, respectively.

$$\begin{aligned} L_\phi&= \left\{ \frac{1}{2} \left( \frac{2 G_\phi }{ 1 + \exp (- 2 G_\phi \varDelta F_r ) } - G_\phi \right) \right\} C_\phi \end{aligned}$$

(16)

$$\begin{aligned} L_\theta&= \left\{ \frac{2 G_\theta }{ 1 + \exp (- 2 G_\theta \varDelta T ) } - G_\theta \right\} C_\theta \end{aligned}$$

(17)

where $\varDelta F_r$ and $\varDelta T$ are deviations of the reference and actual gait speeds and the reference and actual gait step times, respectively. $G_\phi = 0.2$ and $G_\theta = 0.002$ are the maximal magnifications. By injecting $L_\phi$ and $L_\theta$ into the autonomous systems in Eqs. (13) and (14) as follows, locomotion will be changed toward that $\varDelta F_r$ and $\varDelta T$ are equal to 0.

$$\begin{aligned} C_\phi \Leftarrow C_\phi + L_\phi , \ C_\theta \Leftarrow C_\theta + L_\theta \end{aligned}$$

(18)