A Stopping-Distance Bound for Phone-Grade Obstacle Warning for White-Cane Users: Which Hazards a Cheap Sensor Can Honestly Cover

A white cane finds ground-level obstacles but misses head-level ones and gives only contact warning of drop-offs. This paper computes the warning range a walking person needs and what phone-grade sensors deliver. With a human reaction time of 0.55 s and stopping deceleration of 2.0 m/s squared, time-of-flight sensing (latency 0.08 s) needs 0.77 m of warning range at 0.6 m/s, 1.18 m at 1.0 m/s and 1.67 m at 1.4 m/s; monocular depth estimation (latency 0.30 s) needs 1.40 m at 1.0 m/s and 1.98 m at 1.4 m/s. Broad targets are covered by all three classes: a 0.40 m sign board is detected at 3.40 m ultrasonically, 4.00 m by time-of-flight and 2.01 m monocularly. Thin targets are covered by none: a 0.05 m pole is detected at 0.93 m, 0.95 m and 0.57 m respectively, and a 0.03 m branch end at 0.56 m, 0.57 m and 0.34 m, all below the range needed at 1.0 m/s. Dark targets are worse, with time-of-flight range on a broad dark target falling to 1.41 m and on a dark pole to 0.34 m. For drop-offs, a chest sensor 1.20 m high at 45 degrees intercepts the ground 1.20 m ahead and is in time only up to 1.02 m/s, while a 30-degree tilt intercepts at 2.08 m and up to 1.69 m/s; a cane mount at 0.75 m and 45 degrees intercepts at 0.75 m and only up to 0.58 m/s. SIMULATED, NOT MEASURED: all numbers are reproducible from the deposited compute.py.