Prelab 1

RHex’s gait is produced by alternating phases, a slow stance phase followed by a fast recirculation phase, which together comprise the Buehler Clock. When appropriately phased amongst the robot’s six legs, these motions create an “alternating tripod” gait that produces stable locomotion.

Figure 1: The Buehler clock, plotted on the torus. The horizontal axis corresponds to phase (like time, but cyclic) while the vertical axis is leg motor angle.

1. Figure 1 shows a plot of the Buehler clock for a single leg, noting several gait parameters, $\phi_{0}$, $\phi_{s}$, and $\delta$. What do these various parameters control?
2. Write down the mathematical function $\phi = f(\theta)$ which produces a Buehler clock. $\phi$ is the motor angle output, while $\theta$ is the input phase, similar to time.
3. Let $\theta = s\,t\ (mod\ 2\pi )$ where $t$ is time. What does the parameter $s$ control?
4. What is $\frac{d}{dt}\,f(\theta)$? Assume that $\theta$ is defined based upon time, $t$, in the above question.
5. Let $f_1 (\theta)$ be the Buehler clock for one set of the legs. How would one create $f_2 (\theta)$, the clock function for the other set of legs where the two sets of legs are perfectly out of phase? To which legs do you apply functions $f_1$ and $f_2$ ?