Synchronous Transition in Complex Object Control

A complex object is a system with internal degrees of freedom, such as a cup of hot coffee hand-held by a human in walking. In spite of the natural ability of humans to handle complex objects, an understanding of how this is accomplished is lacking, yet the issue is fundamental to applied fields such as soft robotics. Recent virtual experiments on how humans handle a moving bowl with a mechanical ball inside have revealed that humans typically use two strategies to handle a complex object: a low-frequency strategy in which the motions of the bowl and ball synchronized in phase and a high-frequency strategy where antiphase synchronization occurs. Utilizing a nonlinear dynamical model of a pendulum attached to a moving cart, subject to external periodic forcing, we study the transition between in-phase and antiphase synchronization. We find that, in the weakly forcing regime, as the external driving frequency is varied, the transition is abrupt and occurs at the frequency of resonance, which can be fully understood using linear systems control theory. Beyond this regime, a transition region emerges in between in-phase and antiphase synchronization, where the motions of the cart and the pendulum are not synchronized. We also find that there is bistability in and near the transition region on the low-frequency side. Overall, our results indicate that humans are able to switch abruptly and efficiently from one synchronous attractor to another, a mechanism that can be exploited for designing smart robots to adaptively handle complex objects in a changing environment.

Figure 1

A schematic illustration of the cart-pendulum system to simulate humans’ handling of a complex object such as a cup of hot coffee: (a) a conceptual model of a ball rolling inside a circular cup and (b) a nonlinear mechanical model of a pendulum attached to a moving cart.

Figure 2

Representative time series of cart position $x(t)$ (blue) and pendulum angular position $\theta (t)$ (orange) for forcing amplitude $F=5$ N. (a)–(c) Driving frequencies $f=0.6$ Hz, $f=0.7$ Hz, and $f=0.8$ Hz, respectively. There is in-phase synchronization between the pendulum and the cart for $f=0.6$ Hz (a) and antiphase synchronization for $f=0.8$ Hz (c).

Figure 3

Representative phase-plane trajectories of cart position $x(t)$ and pendulum angular position $\theta (t)$. (a)–(c) Driving frequencies $f=0.6$ Hz, $f=0.7$ Hz, and $f=0.8$ Hz, respectively.

Figure 4

Absolute difference in the instantaneous phase between the pendulum and the cart displacement. The forcing amplitude is $F=5$ N. (a)–(c) Driving frequencies $f=0.6$ Hz, $f=0.7$ Hz, and $f=0.8$ Hz, respectively.

Figure 5

Transition scenarios between in-phase and antiphase synchronization in the parameter plane of forcing frequency and amplitude. Blue represents the in-phase synchronization region in which the phase difference between the pendulum and cart is zero; turquoise represents the antiphase synchronization region in which the phase difference is $\pi$; yellow represents the region in which the phase difference does not settle into a steady-state value. The initial condition is $x(0)=0$, $\theta (0)=0$, $\dot{x}(0)=0$, and $\dot{\theta }(0)=0$. There are two distinct transition scenarios: abrupt transition in the weakly forcing regime where the critical frequency value is the natural frequency of the pendulum, and gradual transition in which an intermediate frequency region of indeterminant phase difference arises between the in-phase and antiphase synchronization regions. The red stripes specify the region of bistability (to be discussed in Sec. 4b), calculated using initial conditions $x(0)$, $\theta (0)$, $\dot{x}(0)$, and $\dot{\theta }(0)$—all chosen uniformly from the interval $[-2,2]$.

Figure 6

Block diagram of the equivalent linear control system in the weak forcing regime. For summing junctions, the input is assumed to be additive unless otherwise indicated by a minus sign.

Figure 7

Frequency response of $T_{x\theta }$. Right: magnitude response. Left: phase response.

Figure 8

Illustration of bistability. Shown are two distinct dynamical behaviors in the bistability region, which are observed for the same set of parameter values ($f=0.62$ Hz and $F=5$ N) but with different initial conditions. (a1),(a2) Waveforms $x(t)$ and $\theta (t)$ and the corresponding phase space plot, respectively, from the initial conditions $x(0)=0$, $\theta (0)=0$, $\dot{x}(0)=0$, and $\dot{\theta }(0)=0$. This is an in-phase attractor. It is not exactly sinusoidal, but is very close to. (b1),(b2) Waveforms $x(t)$ and $\theta (t)$ and the corresponding phase space plot, respectively, from the initial conditions $x(0)=0.4$, $\theta (0)=0.3$, $\dot{x}(0)=-1.1$, and $\dot{\theta }(0)=1.7$. This is a “Lissajous” attractor. In the bistability regime, two distinct attractors as exemplified in (a2) and (b2) coexist.

Figure 9

Fractal basin boundaries. In the parameter region where there is bistability, the boundaries between the basins of attraction of the two coexisting attractors exhibit an apparently fractal structure. The parameter values are $f=0.62$ Hz, $F=7$ N, $\dot{x}(0)=1$, and $\dot{\theta }(0)=0.5$. The yellow regions are the basin of the in-phase attractor, and the blue regions represent the basin of the “Lissajous” attractor as exemplified in Fig. 8(b2).

Figure 10

Transition scenario from antiphase to in-phase attractors through bistability in the nonlinear regime. (a) Bifurcation diagram of the two attractors (in blue and green) for $F=6$ N in terms of the maximum value of the pendulum angle. (b) Bifurcation diagram in terms of the maximum value of the cart displacement. There are four distinct parameter regions, which are distinguished by the three vertical black dashed lines. The boundary between the Lissajous and antiphase attractors is determined by the emergence of new local maxima of $x$. For a fixed value of a relatively large forcing amplitude, as the driving frequency is reduced, the system undergoes the following transition sequence: antiphase attractor $\to$ Lissajous attractor $\to$ bistability characterized by the coexistence of a Lissajous attractor and an in-phase attractor $\to$ in-phase attractor.