Reversible Dynamics, Asymmetric Actions: Parking as a Trajectory-Level Affordance

Abstract

Drivers are often told to back into a tight parking space, not drive in nose first. The advice looks paradoxical, because a slow car is dynamically reversible: reverse its speed and replay the steering backward, and it retraces a feasible path. Played backward, though, reverse-in shows a car driving forward out of the space. Reverse-in and forward-in are different tasks: they share a start position but end facing opposite ways. We separate three clearances between car and obstacles. Static fit $F$ is the clearance the parked car keeps at its best acceptable destination pose. Eventual access $A_\infty$ is the clearance it can keep all the way there with unlimited corrections. Resource-bounded access $A_K$ is the clearance that survives a budget of $K$ reversals, switches between forward and reverse. They satisfy $F\ge A_\infty\ge A_K$. In one fully specified scene (one car, bay, aisle and start), a certified argument proves $A_\infty=F=0.5000$ m for both tasks. In this scene, all certified forward/reverse asymmetry therefore arises at finite $K$: it lies in how many corrections each task needs, not in the clearance either can eventually keep. With one reversal, an explicit path gives reverse-in at least $0.4410$ m clearance. A certified reachability argument caps forward-in's one-reversal clearance at $d_0<0.1720002$ m. With two reversals, forward-in gets at least $0.2833$ m. To keep a clearance strictly above any $d$ with $d_0\le d<0.2833$ m, reverse-in thus needs exactly one reversal and forward-in exactly two when both tasks move forward first. If both move in reverse first, the counts are two and three. The contrast depends on the start position (Section 4.15). Proposed experiments test whether observers distinguish fit, eventual access and budgeted access. They compare these with simpler cues, such as how the turning body sweeps, and learned manoeuvre labels.