**Part 1**

**Periodic Tasks and Locomotion Control** 

**0**

**1**

*Slovenia*

**Performing Periodic Tasks: On-Line**

Andrej Gams, Tadej Petriˇc, Aleš Ude and Leon Žlajpah

**with External Signals**

*Jožef Stefan Institute, Ljubljana*

**Learning, Adaptation and Synchronization**

One of the central issues in robotics and animal motor control is the problem of trajectory generation and modulation. Since in many cases trajectories have to be modified on-line when goals are changed, obstacles are encountered, or when external perturbations occur,

This chapter addresses some of the issues related to trajectory generation and modulation, including the supervised learning of periodic trajectories, and with an emphasis on the learning of the frequency and achieving and maintaining synchronization to external signals. Other addressed issues include robust movement execution despite external perturbations, modulation of the trajectory to reuse it under modified conditions and adaptation of the learned trajectory based on measured force information. Different experimental scenarios on

For the learning of a periodic trajectory without specifying the period and without using traditional off-line signal processing methods, our approach suggests splitting the task into two sub-tasks: (1) frequency extraction, and (2) the supervised learning of the waveform. This is done using two ingredients: nonlinear oscillators, also combined with an adaptive Fourier waveform for the frequency adaptation, and nonparametric regression <sup>1</sup> techniques for shaping the attractor landscapes according to the demonstrated trajectories. The systems are designed such that after having learned the trajectory, simple changes of parameters allow modulations in terms of, for instance, frequency, amplitude and oscillation offset, while keeping the general features of the original trajectory, or maintaining synchronization with an

The system we propose in this paper is based on the motion imitation approach described in (Ijspeert et al., 2002; Schaal et al., 2007). That approach uses two dynamical systems like the system presented here, but with a simple nonlinear oscillator to generate the phase and the amplitude of the periodic movements. A major drawback of that approach is that it requires the frequency of the demonstration signal to be explicitly specified. This means that the frequency has to be either known or extracted from the recorded signal by signal

<sup>1</sup> The term "nonparametric" is to indicate that the data to be modeled stem from very large families of distributions which cannot be indexed by a finite dimensional parameter vector in a natural way. It

the notions of trajectory generation and trajectory modulation are tightly coupled.

**1. Introduction**

external signal.

various robotic platforms are described.

does not mean that there are no parameters.
