1. Car-Following Theory Glossary Car-following theories
Ca-following theory applications
Traffic stability

2. Glossary Distance headway 车头间距 Density 密度 Car-following theory 跟车理论
Reaction time 反应时间
Stimuli 刺激
Sensitivity 敏感度
Traffic stability 交通稳定性
Local stability 局部稳定性
Asymptotic stability 渐进稳定性

3. Car-following theories
Notations and definitions
Acceleration
Speeds
Following vehicle
Lead vehicle
Positions
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4. Pipes’ theory
A good rule for following another vehicle at a safe distance is to allow yourself at least the length of a car between your vehicle and the vehicle ahead for every ten miles per hour of speed at which you are traveling.
Let (feet) Or
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5. Forbes’ theory
The minimum time headway is equal to the reaction time and the time required for the lead vehicle to traverse a distance equivalent to its length. Or
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6. What is the sensitivity term?
General Motors’ theories
A law of psychology:
response = func(sensitivity, stimuli)
Response acceleration
Stimuli relative velocity
What is the sensitivity term?
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7. If sensitivity = constant, then
It is called the linear car-following model.
If sensitivity
Then
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8. If sensitivity
Then
Generalized model
l, m = parameters

9. Car-following theory applications
An example
Lead vehicle: accelerate at a constant rate of 3.3 ft/sec2 until a speed of 44 ft/sec is reached; maintain the constant speed 44 ft/sec for 10 sec; decelerate at a constant rate of 4.6 ft/sec2 until it is stopped.
Following vehicle: follow the lead vehicle with a safe minimum distance headway.
Initial distance headway: 25 ft
Reaction time: 1.0 sec
Sensitivity parameter: 0.5
How about the trajectory of the following vehicle?
11/21/05
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10. T = time slice Equations Speed Distance Lead vehicle Following vehicle
Acceleration

11. Traffic stability “Unstable” behavior “Stable” behavior
When a driver of a following vehicle responds slowly, he might finally responds, over responds by exerting significantly large acceleration or deceleration rates.
It is characterized by higher reaction times and higher sensitivity responses.
“Stable” behavior
A driver of a following vehicle is very attentive to the car-following process and does not resort to sudden accelerations or decelerations except in extreme emergencies.
It is characterized by lower reaction times and lower sensitivity responses.
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12. C = a(Dt) is used to classify stable and unstable behaviors
Local stability
The stability of a single following car.
Lead vehicle: decelerates for 2 seconds then accelerates for 2 seconds, return back to original speed.
Following vehicle: react…decelerate…accelerate…decelerate…
Distance headway: lager…smaller…larger…smaller
Local stability: nonoscillatory, damped oscillatory, increased oscillatory
Asymptotic stability
The stability of a line of following vehicles.
Asymptotic stability: damped oscillatory and increased oscillatory.
In case of increased oscillatory, the variance of distance between successive two vehicles becomes larger and larger, collision would occur.