Guaranteed Escape for a Bouncing Robot in Pipe Chains

We study the symmetric bouncing of a point robot within orthogonally-joined rectangles with equal width, which we refer to as pipes. We provide an exhaustive case analysis of every trajectory pattern inside a single rectangular pipe segment, identifying the conditions under which the robot exits. We then extend the analysis to L-shaped pipes and, more generally, to linear chains of $k$ orthogonally connected pipe segments. We prove exit guarantees for the special angle $\alpha = \pi/4$. Furthermore, these results extend to pipes with curved joints.