34.4. General motor’s car following model

The basic philosophy of car following model is from Newtonian mechanics, where the acceleration may be regarded as the response of a matter to the stimulus it receives in the form of the force it receives from the interaction with other particles in the system. Hence, the basic philosophy of car-following theories can be summarised by the following equation:

[Response] α [Stimulus] (34.2)

The car following models are rooted in the principles of Newtonian physics. At its core, these models suggest that a driver's acceleration (how fast they speed up) depends on the stimuli they experience; this could be the position of the car in front, its speed, or the distance between the two vehicles. This relationship can be expressed mathematically, indicating that how a driver reacts (the response) is proportionate to the forces they perceive (the stimulus). For instance, if a driver sees the car in front slow down, they will likely respond by slowing down too, applying the principle of cause and effect.

Imagine a line of people walking. If the person at the front suddenly stops, everyone else behind them will also slow down or stop, showing a direct reaction to the stimulus (the person stopping). This concept mirrors how drivers must respond to the actions of the vehicle ahead of them.

The car following model proposed by General motors is based on follow-the leader concept. This is based on two assumptions; (a) higher the speed of the vehicle, higher will be the spacing between the vehicles and (b) to avoid collision, driver must maintain a safe distance with the vehicle ahead.

Let ∆xt isthe gapavailablefor(n+1)th vehicle, and let ∆x is the safe distance, vt and vt are the n+1 safe n+1 n velocities, the gap required is given by,

∆xt = ∆x + τvt (34.4)

where τ is a sensitivity coefficient.

This part of the model identifies that as vehicles move faster, they need to maintain an increased distance from the car in front to ensure safety. The equation provided identifies the gap needed for a following vehicle based on a 'safe distance' and factors in how fast the leading vehicle is going (captured by the term τvt). This indicates that each driver must evaluate their speed and the distance to the vehicle in front to avoid collisions and ensure sufficient reaction time.

Think about how drivers adjust their distance when driving on a highway compared to city driving. On highways, where speeds are higher, cars are spaced farther apart than in city traffic where speeds are lower. This spacing allows more time to react to sudden stops or slowdowns. The model captures this essential behavior of drivers adapting to their environment for safety.

The above equation can be written as

x_n = x_(n+1) + τv_(n+1) (34.5)

Differentiating the above equation with respect to time, we get

v_n - v_(n+1) = τ.a_t (34.6)

This part provides a mathematical representation of how the distance between vehicles and their speeds change over time. The first equation explains how even as a vehicle (n) reacts to the speed of the car in front (n+1), it still needs to account for acceleration (how quickly it is speeding up or slowing down). The differentiation indicates how velocities adjust as time passes, linking speeds and gaps directly to a sensitivity coefficient.

Imagine a car following a scooter. As the scooter speeds up, the car must adjust its speed; if it doesn't respond quickly enough, the gap between them decreases, increasing the chance of a collision. Just as a driver would instinctively adjust their speed based on the vehicle ahead, these mathematical expressions mirror that instinctive behavior.

General Motors has proposed various forms of sensitivity coefficient term resulting in five generations of models. The most general model has the form:

at = [ α * (vt_n - vt_n+1) ^ m ] / [ (xt_n - xt_n+1) ^ l ] (34.8)

where l is a distance headway exponent and can take values from +4 to -1, m is a speed exponent and can take values from -2 to +2, and α is a sensitivity coefficient. These parameters are to be calibrated using field data.

This model describes how changes in vehicle dynamics are quantitatively expressed. It recognizes that a vehicle's acceleration (at) can depend on the relative speed (between n and n+1 vehicles) and the distance separating them. The terms l and m represent how responsive these factors are relative to changes in distance and speed. Through calibration, these models can be fine-tuned to reflect real-world driving behavior, ensuring they provide accurate simulations.

Think about how a rubber band stretches when you pull it - the more you stretch it, the more effort it takes to hold it in that state. In the same way, as the distance between vehicles decreases, or as speed differences increase, the driver must exert more control (change acceleration) to maintain a safe following distance and avoid collisions.

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