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Gipps' Model
Gipps' model is a mathematical model for describing car-following behaviour by motorists in the United Kingdom. The model is 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. Gipps' model 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 \tau to reduce the computation required for numerical analysis. Introduction The method of modeling individual cars along a continuous space originates with Chandler et al. (1958), Gazis et al. (1961), Lee (1966) and Bender and Fenton (1972),Gipps, P. G. 1981 A behavioural car ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microscopic Traffic Flow Model
Microscopic traffic flow models are a class of scientific models of vehicular traffic dynamics. In contrast, to macroscopic models, microscopic traffic flow models simulate single vehicle-driver units, so the dynamic variables of the models represent microscopic properties like the position and velocity of single vehicles. Car-following models Also known as ''time-continuous models'', all car-following models have in common that they are defined by ordinary differential equations describing the complete dynamics of the vehicles' positions x_\alpha and velocities v_\alpha. It is assumed that the input stimuli of the drivers are restricted to their own velocity v_\alpha, the net distance (bumper-to-bumper distance) s_\alpha = x_ - x_\alpha - \ell_ to the leading vehicle \alpha-1 (where \ell_ denotes the vehicle length), and the velocity v_ of the leading vehicle. The equation of motion of each vehicle is characterized by an acceleration function that depends on those input stimul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Method
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who treated it in his book ''Institutionum calculi integralis'' (published 1768–1870). The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size. The Euler method often serves as the basis to construct more complex methods, e.g., predictor–corrector method. Informal geometrical description Consider the problem of calculating the shape of an unknown curve which starts at a given point and satisfies a given differential equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simulation
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or process, whereas the simulation represents the evolution of the model over time. Often, computers are used to execute the computer simulation, simulation. Simulation is used in many contexts, such as simulation of technology for performance tuning or optimizing, safety engineering, testing, training, education, and video games. Simulation is also used with scientific modelling of natural systems or human systems to gain insight into their functioning, as in economics. Simulation can be used to show the eventual real effects of alternative conditions and courses of action. Simulation is also used when the real system cannot be engaged, because it may not be accessible, or it may be dangerous or unacceptable to engage, or it is being designed bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Runge–Kutta Methods
Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation :\frac = f(t, y). Explicit Runge–Kutta methods take the form :\begin y_ &= y_n + h \sum_^s b_i k_i \\ k_1 &= f(t_n, y_n), \\ k_2 &= f(t_n+c_2h, y_n+h(a_k_1)), \\ k_3 &= f(t_n+c_3h, y_n+h(a_k_1+a_k_2)), \\ &\;\;\vdots \\ k_i &= f\left(t_n + c_i h, y_n + h \sum_^ a_ k_j\right). \end Stages for implicit methods of s stages take the more general form, with the solution to be found over all s :k_i = f\left(t_n + c_i h, y_n + h \sum_^ a_ k_j\right). Each method listed on this page is defined by its Butcher tableau, which puts the coefficients of the method in a table as follows: : \begin c_1 & a_ & a_& \dots & a_\\ c_2 & a_ & a_& \dots & a_\\ \vdots & \vdots & \vdots& \ddots& \vdots\\ c_s & a_ & a_& \dots & a_ \\ \hline & b_1 & b_2 & \dots & b_s\\ \end For adaptive and implicit methods, the Butcher tableau is extended to give values of b^*_i, and the estim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intelligent Driver Model
In traffic flow modeling, the intelligent driver model (IDM) is a time-continuous car-following model for the simulation of freeway and urban traffic. It was developed by Treiber, Hennecke and Helbing in 2000 to improve upon results provided with other "intelligent" driver models such as Gipps' model, which loses realistic properties in the deterministic limit. Model definition As a car-following model, the IDM describes the dynamics of the positions and velocities of single vehicles. For vehicle \alpha, x_\alpha denotes its position at time t, and v_\alpha its velocity. Furthermore, l_\alpha gives the length of the vehicle. To simplify notation, we define the ''net distance'' s_\alpha := x_ - x_\alpha - l_, where \alpha - 1 refers to the vehicle directly in front of vehicle \alpha, and the velocity difference, or ''approaching rate'', \Delta v_\alpha := v_\alpha - v_. For a simplified version of the model, the dynamics of vehicle \alpha are then described by the following two o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newell's Car-following Model
In traffic flow theory, Newell’s car-following model is a method used to determine how vehicles follow one another on a roadway. The main idea of this model is that a vehicle will maintain a minimum space and time gap between it and the vehicle that precedes it. Thus, under congested conditions, if the leading car changes its speed, the following vehicle will also change speed at a point in time-space along the traffic wave speed, ''-w''.Newell G.F. (2002) A simplified car-following theory: a lower order model. Institute of Transportation Studies, University of California, Berkeley. Overview Assuming the fundamental diagram (flow-density) is a triangular function, a traffic state ''A'' with speed ''vA'' and density ''kA'' can be assumed in the congestion region. The density on the roadway can be determined using the spacing between vehicles and is computed simply the equation: ''kA = 1/sA'' Geometric relations from the fundamental diagram can be used to calculate the density ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics (physics), kinetics, not kinematics. For further details, see analytical dynamics. Kinematics is used in astrophysics to describe the motion of celestial bodies and collections of such bodies. In mechanical engin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heun's Method
In mathematics and computational science, Heun's method may refer to the improved or modified Euler's method (that is, the explicit trapezoidal rule), or a similar two-stage Runge–Kutta method. It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. Both variants can be seen as extensions of the Euler method into two-stage second-order Runge–Kutta methods. The procedure for calculating the numerical solution to the initial value problem: :y'(t) = f(t,y(t)), \qquad \qquad y(t_0)=y_0, by way of Heun's method, is to first calculate the intermediate value \tilde_ and then the final approximation y_ at the next integration point. :\tilde_ = y_i + h f(t_i,y_i) :y_ = y_i + \frac (t_i, y_i) + f(t_,\tilde_) : where h is the step size and t_=t_i+h. Description Euler's method is used as the foundation for Heun's method. Euler's method uses the line tangent to the function at the beginning of the interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity; both magnitude and direction are needed to define it. The scalar absolute value (magnitude) of velocity is called , being a coherent derived unit whose quantity is measured in the SI (metric system) as metres per second (m/s or m⋅s−1). For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector. If there is a change in speed, direction or both, then the object is said to be undergoing an ''acceleration''. Constant velocity vs acceleration To have a ''constant velocity'', an object must have a constant speed in a constant direction. Constant direction cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because . Every nonnegative real number has a unique nonnegative square root, called the ''principal square root'', which is denoted by \sqrt, where the symbol \sqrt is called the ''radical sign'' or ''radix''. For example, to express the fact that the principal square root of 9 is 3, we write \sqrt = 3. The term (or number) whose square root is being considered is known as the ''radicand''. The radicand is the number or expression underneath the radical sign, in this case 9. For nonnegative , the principal square root can also be written in exponent notation, as . Every positive number has two square roots: \sqrt, which is positive, and -\sqrt, which is negative. The two roots can be written more concisely using the ± sign as \plusmn\sqrt. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intelligent Driver Model
In traffic flow modeling, the intelligent driver model (IDM) is a time-continuous car-following model for the simulation of freeway and urban traffic. It was developed by Treiber, Hennecke and Helbing in 2000 to improve upon results provided with other "intelligent" driver models such as Gipps' model, which loses realistic properties in the deterministic limit. Model definition As a car-following model, the IDM describes the dynamics of the positions and velocities of single vehicles. For vehicle \alpha, x_\alpha denotes its position at time t, and v_\alpha its velocity. Furthermore, l_\alpha gives the length of the vehicle. To simplify notation, we define the ''net distance'' s_\alpha := x_ - x_\alpha - l_, where \alpha - 1 refers to the vehicle directly in front of vehicle \alpha, and the velocity difference, or ''approaching rate'', \Delta v_\alpha := v_\alpha - v_. For a simplified version of the model, the dynamics of vehicle \alpha are then described by the following two o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newcastle University
Newcastle University (legally the University of Newcastle upon Tyne) is a UK public university, public research university based in Newcastle upon Tyne, North East England. It has overseas campuses in Singapore and Malaysia. The university is a red brick university and a member of the Russell Group, an association of research-intensive UK universities. The university finds its roots in the School of Medicine and Surgery (later the College of Medicine), established in 1834, and the Edward Fenwick Boyd#College of Physical Science, College of Physical Science (later renamed Armstrong College), founded in 1871. These two colleges came to form the larger division of the federal University of Durham, with the Durham Colleges forming the other. The Newcastle colleges merged to form King's College in 1937. In 1963, following an Act of Parliament, King's College became the University of Newcastle upon Tyne. The university subdivides into three faculties: the Faculty of Humanities and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
