Gipps' model


Gipps's model is a car-following model named after Peter G. Gipps who developed it in the late-1970s under S.R.C. grants at the Transport Operations Research Group at the University of Newcastle-Upon-Tyne and the Transport Studies Group at the University College London. It is based directly on driver behavior and expectancy for vehicles in a stream of traffic. Limitations on driver and vehicle parameters for safety purposes mimic the traits of vehicles following vehicles in the front of the traffic stream. Gipps' model is differentiated by other models in that Gipps uses a timestep within the function equal to to reduce the computation required for numerical analysis.

Introduction

The method of modeling individual cars along a continuous space originates with Chandler et al., Gazis et al., Lee and Bender and Fenton, though many other papers proceeded and have since followed. In turn, these papers have bases in several works from the mid-1950s. Of special importance are a few that have analogies to fluid dynamics and movement of gases and Richards postulated the density of traffic to be a function of position; Newell. First mention of simulating traffic with “high speed computers” is given by Gerlough and Mathewson and Goode.

Definition

The impetus for modeling vehicles in a stream of traffic and their subsequent actions and reactions comes from the need to analyze changes to roadway parameters. Indeed, many factors affect how traffic behaves. Gipps describes models current to that time to be in the general form of:
which is defined primarily by one vehicle following another ; reaction time of the following vehicle; the location, speed and acceleration of the following vehicle; and finally, model constants to adjust the model to real-life conditions. Gipps’ new and improved model should reflect the following properties:
Gipps sets limitations on the model through safety considerations and assuming a driver would estimate his or her speed based on the vehicle in front to be able to come to a full and safe stop if needed. Pipes and many others have defined following characteristics placed into models based on various driver department codes defining safe following speeds, known informally as a “2 second rule,” though is formally defined through code.
;Model notation
;Constraints leading to development
Gipps defines the model by a set of limitations. The following vehicle is limited by two constraints: that it will not exceed its driver's desired speed and its free acceleration should first increase with speed as engine torque increases then decrease to zero as the desired speed is reached.
The third constraint, braking, is given by
for vehicle at point, where must ensure that the difference between point where vehicle n-1 stops and the effective size of vehicle n-1 is greater than the point where vehicle n stops. However, Gipps finds the driver of vehicle n allows for an additional buffer and introduces a safety margin, of delay when driver n is traveling at speed. Thus the braking limitation is given by
Because a driver in traffic cannot estimate, it is replaced by an estimated value. Therefore, the above after replacement yields,
If the introduced delay,, is equal to half of the reaction time,, and the driver is willing to brake hard, a model system can continue without disruption to flow. Thus, the previous equation can be rewritten with this in mind to yield
If the final assumption is true, that is, the driver travels as fast and safely as possible, the new speed of the driver's vehicle is given by the final equation being Gipps' model:
where the first argument of the minimization regimes describes an uncongested roadway and headways are large, and the second argument describes congested conditions where headways are small and speeds are limited by followed vehicles.
These two equations used to determine the velocity of a vehicle in the next timestep represent free-flow and congested conditions, respectively. If the vehicle is in free-flow, the free-flow branch of the equation indicates that the speed of the vehicle will increase as a function of its current speed, the speed at which the driver intends to travel, and the acceleration of the vehicle. Analyzing the variables in these two equations, it becomes apparent that as the gap between two vehicles decreases the velocity given by the congested branch of the equation will decrease and is more likely to prevail.

Using numerical methods to generate time-space diagrams

After determining the velocity of the vehicle at the next timestep, its position at the next timestep should be calculated. There are several numerical methods that can be used to do this, depending on the accuracy to which the user would prefer. Using higher order methods to calculate a vehicle's position in the next timestep will yield a result with higher accuracy. Numerical methods can also be used to find positions of vehicles in other car following models, such as the intelligent driver model.
Eulers Method can be used to obtain accurate results, but the timestep would have to be very small, resulting in a greater amount of computation. Also, as a vehicle comes to a stop and the following vehicle approaches it, the term underneath the square root in the congested part of the velocity equation could potentially fall below zero if Euler's method is being used and the timestep is too large. The position of the vehicle in the next timestep is given by the equation:
x= x +vτ
Higher order methods not only use the velocity in the current timestep, but velocities from the previous timestep to generate a more accurate result. For instance, Huen’s Method averages the velocity from the current and previous timestep to determine the next position of a vehicle:
Butchers Method uses an even more elegant solution to solve the same problem:
x = x + τ
k1 = v
k3 = v + - v)
k4 = v + - v)
k5 = v + - v)
k6 = v
Using higher-order methods reduces the probability that the term under the square root in the congested branch of the velocity equation will fall below zero.
For the purpose of simulation, it is important to make sure the velocity and position of every vehicle has been calculated for a timestep before determining the moving along to the next timestep.
In 2000, Wilson used Gipp's model for simulating driver behavior on a ring road. In this case, every vehicle in the system is following another vehicle – the leader follows the last vehicle. The results of the experiment showed that the cars followed a free-flow time-space trajectory when the density on the ring road was low. However, as the number of vehicles on the road increases, kinematic waves begin to form as the congested part of the Gipps’ Model velocity equation prevails.