A Dynamical Systems Approach to Learning: A Frequency-Adaptive Hopper Robot

Abstract

Abstract. We present an example of the dynamical systems approach to learning and adaptation. Our goal is to explore how both control and learning can be embedded into a single dynamical system, rather than having a separation between controller and learning algorithm. First, we present our adaptive frequency Hopf oscillator, and illustrate how it can learn the frequencies of complex rhythmic input signals. Then, we present a controller based on these adaptive oscillators applied to the control of a simulated 4-degrees-of-freedom spring-mass hopper. By the appropriate design of the couplings between the adaptive oscillators and the mechanical system, the controller adapts to the mechanical properties of the hopper, in particular its resonant frequency. As a result, hopping is initiated and locomotion similar to the bound emerges. Interestingly, efficient locomotion is achieved without explicit inter-limb coupling, i.e. the only effective inter-limb coupling is established via the mechanical system and the environment. Furthermore, the self-organization process leads to forward locomotion which is optimal with respect to the velocity/power ratio.

Finding Resonance: Adaptive Frequency Oscillators for Dynamic Legged Locomotion

“…Our research aims at finding controllers which are autonomously capable of finding and exploiting such intrinsic locomotion capabilities. We have shown in simulation how very simple dynamical systems can find intrinsic locomotion modalities and elicit them [15]. This robot allows to test the presented approach on a real robot.…”

“…As input to the oscillator one of the sensors of the robot is used. The closest match to the setup as in [15] is using one of the knee sensors (sensor 2 or 4) as input. In Fig.…”

“…In previous work, we have demonstrated in simulation that the adaptive frequency oscillators can be used to adapt to the resonant frequency of the body and initiate locomotion in crawling and hopping robots [10], [15]. In particular, we illustrated how it could adapt to changes in body weight.…”

Engineering entrainment and adaptation in limit cycle systems

“…It would of course be interesting to exploit such substrates for engineering applications. algorithmic dynamic algorithmic dynamic (Ermentrout & Kopell, 1994) R 1a (Williamson, 1998) R 1a R 1a (Matsuoka, 1985(Matsuoka, , 1987 R 1a (Schöner, Jiang, & Kelso, 1990) R 1a (Schöner & Kelso, 1988) R 1a (Endo et al, 2005) R 1a (Morimoto et al, 2006) R 1a (Righetti & Ijspeert, 2006a) R 1b (Taga, 1994(Taga, , 1995b(Taga, , 1995a R 1a (Buchli & Ijspeert, 2004a) R 1a (22, 2003 R 1a (Schöner & Santos, 2001;Santos, 2003Santos, , 2004 R 1a (Collins & Richmond, 1994) R 1a (Ijspeert, 2001) R 1a/2 (Zegers & Sundareshan, 2003) R 2b (Okada, Tatani, & Nakamura, 2002;Okada, Nakamura, & Nakamura, 2003) R 2a (Ijspeert, Nakanishi, & Schaal, 2002) R 2b (Ruiz, Owens, & Townley, 1998) R 2 (Galicki, Leistritz, & Witte, 1999;Leistritz, Galicki, Witte, & Kochs, 2002) R 2 (Righetti & Ijspeert, 2006b) R 1a/2b (Marbach & Ijspeert, 2005) A 1a (Nishii, 1999) A 1c/1a (Ermentrout, 1991) A 1c (Large, 1994(Large, , 1996 A 1a 1c (Buchli & Ijspeert, 2004b;Righetti, Buchli, & Ijspeert, 2006;Buchli, Righetti, & Ijspeert, 2005;Buchli et al, 2006) A 1c Table 3 Table classifying …”

Performing Periodic Tasks: On-Line Learning, Adaptation and Synchronization with External Signals

“…Combining several adaptive frequency oscillators in a feedback loop allows extraction of several frequency components (Buchli et al, 2008;Gams et al, 2009). Applications vary from bipedal walking to frequency tuning of a hopping robot (Buchli et al, 2005). Such feedback structures can be used as a whole imitation system that both extracts the frequency and learns the waveform of the input signal.…”