2 Automatic generation of gait control tables

Control tables are generated using a wave propagation model. The algorithm is as follows (figure ). Having a waveform in its initial state, $f(x,t_{0})$ (in the figure, sinusoidal waves are drawn, but other waveforms could be used) and a worm with all its articulations over the $x$ axis (figure -1). Let $(x_{i},y_{i})$ be the coordinates of the articulation $i$, at some instant $t$. The angular position vector for the initial time, $\overrightarrow{\varphi(t_{o})}$, is calculated fitting the articulations to the wave, so that $y_{i}=f(x_{i},t_{0})$ for all i. The distance L between articulations is maintained. It could be said that ``the worm fits the wave'' (figure -2). Next, the wave is shifted (instant $t_{1}$. Figure -3) and the worm fits the wave again, obtaining $\overrightarrow{\varphi(t_{1})}$ (figure -4). Points 3 and 4 are repeated until the wave reach its initial phase. After $m$ instant of time, all the vector that comprises the table are generated.

Figure: The algorithm used to generate the control tables

By means of this algorithm, control tables are obtained, regardless of the waveform used, $f(x,t)$. In the locomotion test, sinusoidal and semi-sinusoidal waves (just the positive part of the sinusoidal wave) have been used.