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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. Validation is necessary because the optimal trajectory function is based on a point-mass model that does not account explicitly for the yaw torque induced by tire forces during cornering. Optimal lane-change trajectories are compared at two speeds to paths produced by a validated vehicle simulation model. The proven simulation model is comprised of an eight-degree-of-freedom (8-DOF) vehicle model with lagged tire forces, controlled by a nonlinear continuous-gain-optimized controller, and subject to a step input signal for lateral displacement. Sensitivity analysis is conducted to determine how sensitive the optimal emergency lane-change trajectory is to errors in the coefficient of friction.
Sledge et al. (Mon,) studied this question.