# Biorobotics Laboratory BioRob

## Ludovic Righetti - Master Thesis

## Control and Synchronization with Nonlinear Dynamical Systems for an application to Humanoid Robotics

### (Winter 2003-2004)

## Abstract

The movement generation in animals is done by Central Pattern Generators
(CPG), which are distributed units of neurons that are able to produce, in a
quasi automatic manner, specific patterns of motor neuron impulses. However,
recent investigations have shown that CPGs also adapt to the conditions of the
environment. This project, inspired by these recent investigations, enhance a
previous model of an artificial CPG (used in the humanoid robotics project), based on nonlinear systems, to add these adaptation abilities. The interest of the model is that, instead of encoding impulse patterns, it already encodes rhythmic movement trajectories. This trajectories are encoded as limit cycles, thus providing strong robustness to perturbations. Due to the properties of the nonlinear model, new movements can be generated by system's parameters change. First, we developed feedback loops so that the CPG can react to external perturbations, in a quite efficient
way. Then we analytically demonstrated the stability of the approach. In a second
time, we developed, in addition to the CPG, a discrete nonlinear dynamical
system that is able to synchronize with an external signal. The originality of
the approach is that the system can synchronize with any signal, without any
frequency information or any signal processing. Moreover the system does not
only synchronize but also learns the frequency of the external source, so that it
can change its own intrinsic frequency.

Report of the project (pdf)

## Results

We test synchronization capabilities with a simple phase oscillator coupled to an arbitrary periodic input F

Then we studied plotted the Arnold tongues of the system for a simple sinusoidal input, in order to understand of the oscillator was entrained. The next plot is a plot of the measured frequency of the oscillator, when coupled to a sinusoidal input of frequency 30. On the y-axis it is the coupling strength and on the x-axis is the intrinsic frequency of the oscillator.

By studying the entrainment phenomenon and the Arnold Tongues of the system, we discovered that the measured frequency of the oscillator is always bit closer to the pertubing frequency than the intrinsic frequency. The next graph a slice of the previous Arnlod tongue plot, for a fixed coupling strength. We clearly see the entrainment basin and a synchronization algorithm follows.

Then we derived the following discrete dynamical system to track frequency changes and to learn input frequencies.

Note that the intrinsic frequency of the oscillator is made a dynamical system, then we have a dynamic learning of the nearest frequency component of an arbitrary input signal.
We test synchronization and learning capabilities of the coupled oscillator with the following noisy input signal. It is composed of 2 sinus of non-proportionnal frequencies and white noise.

The following figure show the evolution of the intrinsic frequency of the oscillator, learning the frequency of the input. The main frequency of the input is 10 at the beginning, the becomes 60 at iteration 100 and fall to 45 at iteration 250. It is represented in red. In blue we can see the evolution of the intrinsic frequency of the oscillator, tracking and learning input frequency changes.

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