Unpredictable as dice: analyzing riddled basin structures in a passive dynamic walker

Two situations are considered in the controlling of a non-linear system. One is that the the final outcome of the system is predictable or controllable by fine tuning its initial state or parameter; another is that its prediction or control is never successful by fine tuning the initial state or parameter. Such a property, derived by non-linearity, is called final state sensitivity and is distinguished from initial state sensitivity of chaos, which is also unpredictability derived by non-linearity. The final state sensitivity is characterized by a fractal basin boundary. In this paper, we show that a locomotion dynamics of a simple biped robot has the final state sensitivity. Our model is a passive dynamic walker, which can naturally walk without input signals. We numerically calculate the model and show the existence of initial state regions that have final state sensitivity. A quantitative method using uncertainty exponents calculated from fractal dimension of a basin of attraction is employed. We confirm fractal basin boundary and final state sensitivity in both periodic-gait parameters and chaotic-gait parameters. Our result suggests that, in some sets of parameter region, the behavior of a robot becomes unpredictable as a dice, which is qualitatively different from other parameter set in its nature, requiring infinite precisions to set its initial condition to control its fate.