Perception-Action Loop Latency: Phase-Plane & Stability Margin

Every real robot's perception-action loop has a delay in it — camera capture, detection and localization all take time before the controller learns the true state. This 2D companion to the 3D chase-robot simulator strips the problem down to its control-theory core: a 1-DOF PD-controlled mass tracks a sinusoidal setpoint through an adjustable round-trip perception latency, implemented as a genuine discrete delay buffer, not a cosmetic slow-down. Rather than re-drawing a top-down chase, it plots the tracking-error phase portrait directly — the classic view control engineers use to diagnose stability — alongside an independently derived analytic prediction: a first-order Padé approximation of the delay turns the closed loop into a cubic characteristic polynomial whose Routh-Hurwitz test predicts the latency at which the spiral should flip from converging to diverging. Enable the Smith predictor to watch the controller forward-replay its own known dynamics through the delay window and collapse the effective lag back toward zero, restoring the tight inward spiral even at latencies that would otherwise blow the loop up.