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In this paper, functional analysis is employed to develop an ideal path of a vehicle undergoing a limit lane-change maneuver. Inputs to the problem are the lane width, tire-road coefficient of friction and either vehicle velocity or total longitudinal lane-change distance. Vehicle velocity is assumed to be constant. The problem is formulated using the calculus of variations. The solution technique relies on elliptic functions to achieve a closed-form solution. The synthesis of an ideal lane-change trajectory is treated as a minimal-energy-curve optimization problem with prescribed continuity and boundary conditions. The concept of critical speed is employed to limit the maximum curvature of any specified lane-change, thereby ensuring that the synthesized trajectory function describes a path that can be traversed under realistic road conditions. The analytical solution is confirmed by comparison to a numerical solution and a validated 8 degree-of-freedom vehicle model simulation.

This paper provides an exposition of the basic and some refined inertial critical speed estimation formulas. A literature review of existing inertial formulas for estimating critical cornering speed were identified for the ultimate purpose of developing a useful, compact, and more accurate speed estimation formula. Background information is presented covering the general definitions and utility of critical speed formulas. First, as a point of reference, the basic critical speed formulas are derived. Included is a list of the key assumptions on which the basic formulas are based. It is shown that the basic formulas are founded on the fundamental principles of physics and engineering mechanics; namely, Newton's Second Law and centrifugal force.

This paper covers briefly the theory of tire-road friction, coefficient of friction measurement techniques, and the vagaries of tire-road friction as they relate to critical speed estimation. A literature review of tire-road friction studies was conducted to identify the primary factors effecting the tire-road coefficient of friction. Background information is presented covering general definitions and the connection between the basic critical speed formulas and the coefficient of friction. The primary components of tire-road friction, adhesion and hysteresis, are discussed along with minor effects such as tearing, wear, waves, and roll formation. Common coefficient of friction field measuring techniques are described, including the skid-to-stop test and drag sled. Influential factors such as tire characteristics, tire inflation pressure, road conditions, and dynamic factors are reviewed.

This paper seeks to compare various models of desired, or ideal, vehicle lane-change trajectories or paths and to determine which is best based on selected criteria. Background information is presented covering the utility of lane-change maneuvers, lane-change terminology, and known desired open-loop trajectories. The performance of a vehicle may be assessed by measuring its deviation from an ideal path during simulated or actual limit lane-change maneuvers. Therefore, several techniques for assessing vehicle performance against an ideal trajectory are demonstrated. Similarly, in the present effort, performance indices such as integral penalty (cost) functions are used for assessing candidate lane-change trajectories. Therefore, ideal trajectory comparison is essentially treated as an optimization problem with prescribed continuity and boundary conditions, where a path's length, curvature, and rate of change of curvature are taken as its costs.