Fourier Series Representation of Periodic Functions

2000 ◽  
pp. 9-12
Author(s):  
Burkhard Buttkus
Keyword(s):  
Fourier Series ◽  
Series Representation ◽  
Periodic Functions

Optimizing Fourier Series Based Gaits for Modular Robots Using an Evolutionary Algorithm

2013 ◽  
Author(s):  
David Ko ◽  
Harry H. Cheng
Keyword(s):  
Genetic Algorithm ◽  
Fourier Series ◽  
Evolutionary Algorithm ◽  
Periodic Motion ◽  
Degrees Of Freedom ◽  
Series Representation ◽  
Robotic System ◽  
Experimental Results ◽  
New Method ◽  
Modular Robots

A new method of controlling and optimizing robotic gaits for a modular robotic system is presented in this paper. A robotic gait is implemented on a robotic system consisting of three Mobot modules for a total of twelve degrees of freedom using a Fourier series representation for the periodic motion of each joint. The gait implementation allows robotic modules to perform synchronized gaits with little or no communication with each other making it scalable to increasing numbers of modules. The coefficients of the Fourier series are optimized by a genetic algorithm to find gaits which move the robot cluster quickly and efficiently across flat terrain. Simulated and experimental results show that the optimized gaits can have over twice as much speed as randomly generated gaits.

24.—Mean-square Convergence of Non-harmonic Trigonometrical Series

1976 ◽  
Vol 75 (4) ◽  
pp. 297-323
Author(s):  
J. Cossar
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Mean Square ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Real Numbers ◽  
Trigonometrical Series ◽  
Fixed Positive Number ◽  
Mean Square Convergence ◽  
The Difference

SynopsisThe series considered are of the form , where Σ | cn |2 is convergent and the real numbers λn (the exponents) are distinct. It is known that if the exponents are integers, the series is the Fourier series of a periodic function of locally integrable square (the Riesz-Fischer theorem); and more generally that if the exponents are not necessarily integers but are such that the difference between any pair exceeds a fixed positive number, the series is the Fourier series of a function of the Stepanov class, S2, of almost periodic functions.We consider in this paper cases where the exponents are subject to less stringent conditions (depending on the coefficients cn). Some of the theorems included here are known but had been proved by other methods. A fuller account of the contents of the paper is given in Sections 1-5.

A study of the Fourier‐series representation for internal rotation

1988 ◽  
Vol 88 (3) ◽  
pp. 1764-1774 ◽  
Author(s):  
Alice Chung‐Phillips
Keyword(s):  
Fourier Series ◽  
Internal Rotation ◽  
Series Representation
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