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The method originates from non-linear dynamical systems for frequency extraction, which are based on adaptive frequency oscillators in a feedback loop. In previous work, we had developed a method that could extract separate frequency components by using several adaptive frequency oscillators in a loop, but that method required a logical algorithm to identify the basic frequency. The novel method presented here uses a Fourier series representation in the feedback loop combined with a single oscillator. In this way it can extract the frequency and the phase of an unknown periodic signal in real time and without any additional signal processing or preprocessing. The method determines the Fourier series coefficients and can be used for dynamic Fourier series implementation. The proposed method can be used for the control of rhythmic robotic tasks, where only the extraction of the basic frequency is crucial. For demonstration several highly non-linear and dynamic periodic robotic tasks are shown, including also a task where an electromyography (EMG) signal is used in a feedback loop.<\/jats:p>","DOI":"10.1177\/0278364911421511","type":"journal-article","created":{"date-parts":[[2011,9,22]],"date-time":"2011-09-22T20:21:02Z","timestamp":1316722862000},"page":"1775-1788","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":78,"title":["On-line frequency adaptation and movement imitation for rhythmic robotic tasks"],"prefix":"10.1177","volume":"30","author":[{"given":"Tadej","family":"Petri\u010d","sequence":"first","affiliation":[{"name":"Department for Automation, Biocybernetics and Robotics, Jo\u017eef Stefan Institute, Ljubljana, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrej","family":"Gams","sequence":"additional","affiliation":[{"name":"Department for Automation, Biocybernetics and Robotics, Jo\u017eef Stefan Institute, Ljubljana, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Auke Jan","family":"Ijspeert","sequence":"additional","affiliation":[{"name":"School of Engineering, Institute of Bioengineering, EPFL \u2013 \u00c9cole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Leon","family":"\u017dlajpah","sequence":"additional","affiliation":[{"name":"Department for Automation, Biocybernetics and Robotics, Jo\u017eef Stefan Institute, Ljubljana, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2011,9,22]]},"reference":[{"key":"bibr1-0278364911421511","volume-title":"Model-based Control of A Robot Manipulator","author":"An C","year":"1988"},{"key":"bibr2-0278364911421511","unstructured":"Bailey A (2004) Biomimetic Control with a Feedback Coupled Nonlinear Oscillator: Insect Experiments, Design Tools, and Hexapedal Robot Adaptation Results. 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