Finding Resonance: Adaptive Frequency Oscillators for Dynamic Legged Locomotion

Abstract

Abstract-There is much to gain from providing walking machines with passive dynamics, e.g. by including compliant elements in the structure. These elements can offer interesting properties such as self-stabilization, energy efficiency and simplified control. However, there is still no general design strategy for such robots and their controllers. In particular, the calibration of control parameters is often complicated because of the highly nonlinear behavior of the interactions between passive components and the environment.In this article, we propose an approach in which the calibration of a key parameter of a walking controller, namely its intrinsic frequency, is done automatically. The approach uses adaptive frequency oscillators to automatically tune the intrinsic frequency of the oscillators to the resonant frequency of a compliant quadruped robot. The tuning goes beyond simple synchronization and the learned frequency stays in the controller when the robot is put to halt. The controller is model free, robust and simple. Results are presented illustrating how the controller can robustly tune itself to the robot, as well as readapt when the mass of the robot is changed. We also provide an analysis of the convergence of the frequency adaptation for a linearized plant, and show how that analysis is useful for determining which type of sensory feedback must be used for stable convergence. This approach is expected to explain some aspects of developmental processes in biological and artificial adaptive systems that "develop" through the embodied system-environment interactions.

Engineering entrainment and adaptation in limit cycle systems

“…This law allows the oscillator to adapt to the frequency of the perturbation p(t). Such an additional law for the parameter ω endows the system with many very nice properties and can be exploited for different tasks such as adaptation to body dynamics (Buchli & Ijspeert, 2004b;Buchli et al, , 2006 or programmable CPGs (Righetti & Ijspeert, 2006b). As can be seen in Table 3, the column with the adaption is only sparsely populated.…”

“…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 …”

“…Finally, CPGs can readily integrate sensory feedback signals in the differential equations, and show interesting properties such as entrainment by the mechanical body (Taga, 1998). Because of these interesting properties, CPGs are increasingly used in robotics (see for instance (Kimura, Akiyama, & Sakurama, 1999;Wilbur, Vorus, Cao, & Currie, 2002;Endo, Nakanishi, Morimoto, & Cheng, 2005;Buchli, Iida, & Ijspeert, 2006;Righetti & Ijspeert, 2006b)). …”

Biped Robot Walking using a Combination of Truncated Fourier Series and GALA (Genetic Algorithm parameters adaption using Learning Automata)

“…In CPG approach, special neural circuits take the role of the rhythmic walking controller using the non-linear equations to model the neural activities. Researchers usually focus on complex mathematical models like Hopf [9] or Matsuoka [10] to model these neural activities and generate rhythmic walk patterns (Gait).…”

Evolution of Biped Walking Using Truncated Fourier Series and Particle Swarm Optimization

“…In CPG approach, special neural circuits take the role of rhythmic walking controller using the non-linear equations to model the neural activities. Researchers usually focus on complex mathematical models like Hopf [6] or Matsuoka [7] to model these neural activities and generate rhythmic walk patterns (Gaits).…”