Optimal Control Analysis of Minimum-Time Maneuvers for Vehicles Recovering from Instability

Vehicles may enter highly unstable dynamic states due to lateral collisions, sudden loss of grip, or extreme steering disturbances. When such instability arises in congested road sections where obstacle avoidance is required, the safety risk to both the ego vehicle and surrounding traffic escalates significantly. In such scenarios, the vehicle must not only regain stability but also navigate the roadway in the shortest feasible time to prevent secondary collisions. This paper investigates the minimum-time maneuver of a vehicle starting from an unstable dynamic condition and constrained to travel within prescribed road boundaries. A single-track vehicle model with combined-slip nonlinear tire model is employed to capture the vehicle dynamics under high slip conditions. Phase-plane analysis is conducted to reveal how control inputs reshape the system’s vector field and influence the possibility and speed of stability recovery. An optimal control problem is formulated to compute the minimum-time control sequence subject to both dynamic and kinematic constraints, actuator limits and road boundary constraints. The optimal control problem accounts for both stabilization and rapid progression through the constrained road segment. Simulation results on straight and curved road sections show that the minimum-time maneuver consistently exhibits a two-stage structure. The vehicle initially undergoes a stabilization phase, characterized by spiral convergence in the (β,r) phase plane. After stability is restored, the optimal maneuver transitions into the second phase where the vehicle follows the minimum-time trajectory dominated by the road geometry. The findings suggest that, in emergency scenarios, stability recovery should be prioritized before attempting aggressive avoidance or cornering maneuvers.