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

Perception Loop

Round-trip latency L 120 ms

Proportional gain K_p 30

Derivative gain K_d 10

Live Readouts

Tracking stable

RMS tracking error

0.00 m

Peak error (5 s)

0.00 m

Delay seen by controller

120 ms

Analytic stability margin (Padé + Routh-Hurwitz)

– ms

How it works

This is the phase-plane counterpart to the 3D chase-robot simulator: instead of watching a robot follow a target across a floor, it plots the tracking-error dynamics themselves — the quantity control theory actually cares about — and adds an independent analytic stability prediction the 3D version never computed.

plant (1 DOF): a = (F − c·v)/m
error:  e = r(t) − x,   ė = ṙ(t) − v
controller: F = Kp·e_meas − Kd·v_meas

The perception pipeline reports the plant's state from L seconds ago, not now — a pure time-delay e^−sL in the loop's transfer function. Approximating that delay with a first-order Padé model (e^−sL ≈ (2−sL)/(2+sL)) turns the closed loop into a cubic characteristic polynomial a₃s³+a₂s²+a₁s+a₀ with

a3 = m·L
a2 = 2m + L·(c − Kd)
a1 = 2·(c + Kd) − Kp·L
a0 = 2·Kp

The Routh-Hurwitz test (a2·a1 > a3·a0, all coefficients positive) predicts the exact latency at which the loop should start to ring — shown as the "analytic stability margin" readout. It is only an approximation of the true delay, so it lands close to but not exactly on the latency where the phase portrait actually starts to spiral outward instead of inward; that gap is itself a genuine property of the Padé approximation, not a bug.

Smith predictor. The controller keeps a log of every force it already issued. Starting from the stale measurement, it forward-integrates the known plant model through the commands sent during the delay window, reconstructing the true current state before the PD law ever sees it — collapsing the "delay seen by controller" toward zero even though the true round-trip latency L never changed.

Phase portrait (top) — error e on the horizontal axis, error rate ė on the vertical axis. A trajectory spiralling into the center is a stable loop; one spiralling outward or orbiting a fixed loop is unstable/ringing.
Time strip (bottom) — setpoint r(t) and actual position x(t) over the last 8 seconds, the same instability made visible as a growing tracking lag/overshoot.

Error trajectory (phase) Setpoint r(t) Actual x(t)