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.

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.

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.

Answer the following questions:

  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 $f_2$ ?