Cooperative Adaptive Cruise Control: Platoon String Stability

Autopilot Mode

Platoon Control

Time gap h 1.0 s

Sensor / actuator lag τ 0.60 s

Live Telemetry

String stability STABLE

Amplification (rear/lead)

1.00×

Min gap in platoon

30.0 m

Lead speed

90.0 km/h

Rear speed

90.0 km/h

How it works

Each follower i tracks the vehicle ahead with a constant time-gap spacing policy: the desired bumper-to-bumper gap grows with speed, s_des = s₀ + h·v_i. A linear feedback law converts the spacing error and closing rate into an acceleration command:

e_i  = (x_{i-1} - x_i - L) - (s₀ + h·v_i)      spacing error
ė_i  = v_{i-1} - v_i                            closing-rate error
a_i  = k_p·e_i(t-τ) + k_d·ė_i(t-τ)  [ACC]
a_i  = k_p·e_i(t-τ) + k_d·ė_i(t-τ) + a_{i-1}(t-τ_ff)  [CACC]

Under ACC, vehicle i only has its own radar/camera, so every signal is delayed by the sensor+actuator lag τ set on the slider — the same lag every human-following driver has, just automated. That delay means each vehicle over-reacts slightly to what the one ahead already did, and — unless the time gap h is generously large — the correction overshoots and the next vehicle overshoots more, so a single brake pulse from the leader amplifies down the string.

Under CACC, every vehicle also receives the leader's actual acceleration command over V2V radio (near-zero delay, τ_ff ≈ 0.05 s) and feeds it forward directly — it starts braking with the leader instead of reacting to the gap closing afterward. This feed-forward term is what breaks the delay-driven amplification chain: real CACC platoons (Ploeg et al. 2011; the 2011 Grand Cooperative Driving Challenge) hold string-stable at time gaps under 0.6 s, roughly a third of what unconnected ACC needs to stay safe.

Amplification ratio — the rear vehicle's rolling speed-variation divided by the leader's. Above 1.0× a disturbance is growing as it propagates backward; a ratio that keeps climbing toward the back of a long enough platoon means a rear-end collision under ACC even though every individual controller is locally well-tuned.
Min gap — the tightest bumper-to-bumper spacing anywhere in the platoon right now; watch it dip hard during a brake pulse in ACC mode with a short time gap.

ACC · radar only

Lead vehicle On target gap Gap error (too close/far)

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