Example. $k=5$, $d=3$, $100$ (substantive) iterations; center of mass pink, time growing downward:
InchWorm100 http://cs.smith.edu/~orourke/MathOverflow/InchWorm100.jpg
It appears the center of mass performs a scaled random walk, scaled as a function of $k$ and $d$.
Joseph O'Rourke
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