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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.