Engineering entrainment and adaptation in limit cycle systems - Buchli, Richetti Ijspeert, 2006

This paper present some tools towards the beginning of designing a control system composed of networks of coupled oscillators. This is the problem of making the desired behavior of a system a limit cycle of its dynamics.

There approach is to discuss the phase space of the problem, the space of the oscillators and provide tools for determining the transform between these 'coordinate systems'

They also do a lot fo the work at reviewing and comparing different models of oscillators, to give a sense of their utility for different kinds of problems. oscillators they look at: Phase oscillator, isochronous oscillator, amplitude-controlled phase oscillator, harmonic oscillator, piecewise linear oscillator, linear system with reset, relaxation oscillator. This later involves the point to the over excited researcher that they may not need a strongly nonlinear oscillator; "is a "neural" oscillator really the right on eto achieve my goal?"

The math gets over my head, but my project may well beinside this paper, so give it another go.

"Oscillators are of interest for engineers for several reasons. They can be exploited for timing and swquencing. They can synchronize to external signals and show coordinated behabior with perturbations and other oscillators. Connecting them into networks or lattices, they can form coordicated yet flexible sptiotemporal patterns. These networks can act as pattern generators that can reduce the dimensionality of a given control problem, in the sense that a small number of simple (scalar) parameters cn control multidimenstional output patterns. Of course they also exhibit all the common features of structurally stable synamical systems such as smooth changes under parameter variation. The structural stability makes it possible to fuse in input without destroying the autonomous dynamics of the system; i.i., the resulting dynamics is a combination of internal and external dynamics."

"...Because of these interesting roperties, CPGs are being increasingly used in robotics (see for instance Kimura et al. 1999; Wilbur et al. 2002; Wndo et al. 2005; Buchli et al. 2006; Righetti and Ijspeert 2006b)."

To read:

• Big Dog (learn the still existing problems. if it doesn't hit the military; why?)
• "Finally, another difficulty with oscillators is that while there are many tools for analyzing the behavior of an oscillator (see Kuramoto 1984; Winfree 2001; Glass and Mackey 1988; Pikovsky et al. 2001 for some outstanding textbooks), there is a lack of methodologies for designing them to exhibit a particular behavior"
• structurally unstable dynamics in robots