<b>Example</b>. $k=5$, $d=3$, $100$ (substantive) iterations; center of mass pink, time growing downward:
<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="http://cs.smith.edu/~orourke/MathOverflow/InchWorm100.jpg" alt="InchWorm100" />
<br />
It appears the center of mass performs a scaled random walk, scaled as a function of $k$ and $d$.