Post-Collision Trajectory Restoration for a Single-track Ackermann Vehicle using Heuristic Steering and Tractive Force Functions

Post-collision trajectory restoration is a safety-critical capability for autonomous vehicles, as impact-induced lateral motion and yaw transients can rapidly drive the vehicle away from the intended path. This paper proposes a structured heuristic recovery control law that jointly commands steering and tractive force for a generalized single-track Ackermann vehicle model. The formulation explicitly accounts for time-varying longitudinal velocity in the lateral-yaw dynamics and retains nonlinear steering-coupled interaction terms that are commonly simplified in the literature. Unlike approaches that assume constant longitudinal speed, the proposed design targets the transient post-impact regime where speed variations and nonlinear coupling significantly influence recovery. The method is evaluated in simulation on the proposed generalized single-track model and a standard 3DOF single-track reference model in MATLAB, demonstrating consistent post-collision restoration behaviour across representative initial post-impact conditions.

💡 Research Summary

The paper addresses a critical gap in autonomous vehicle safety: the ability to restore a vehicle’s intended trajectory after an impact that imparts lateral velocity and yaw rate. While most existing literature focuses on collision avoidance or assumes post‑impact dynamics are negligible, this work explicitly tackles the transient recovery phase where longitudinal speed varies and nonlinear coupling between steering angle and vehicle dynamics is significant.

A generalized single‑track Ackermann model, previously introduced by Ghosh et al. (2023), forms the basis of the analysis. Unlike simplified models, the authors retain the full set of nonlinear terms: time‑varying longitudinal velocity (v_x(t)), steering‑dependent drag, speed‑dependent tire friction ((\mu_0+\mu_1 v_x^2)), and the interaction of steering angle (\delta_s) with lateral and yaw dynamics. The resulting equations of motion (Eqs. 7‑9) capture the coupling that becomes dominant immediately after a collision.

To counteract the induced lateral drift and yaw, the authors propose an open‑loop, heuristic control law consisting of two sinusoidal pulses for steering and one sinusoidal pulse for longitudinal force. The steering command (\delta_s(t)) is the sum of (\delta_1(t)) (a “recovery” pulse that drives the vehicle back toward the desired line) and (\delta_2(t)) (a “stabilization” pulse that maintains heading after the vehicle has re‑aligned). Each pulse is shaped by a unit‑step window, allowing the designer to specify start and end times ((\tau_0,\tau_1,\tau_2,\tau_3)), amplitude ((A_1, A_2)), period ((T_1, T_2)), and phase delay. The longitudinal force command (F_{xt}(t)) is similarly split into an initial impact force (F_i(t)) and a corrective force (F_c(t)) that follows a sinusoidal windowed profile. The control parameters are tuned manually for each simulation case; no feedback from vehicle states is used.

Simulation experiments are carried out in MATLAB on two vehicle representations: (1) the full generalized Ackermann model and (2) a conventional 3‑DOF single‑track model used as a reference. Vehicle parameters (mass 1750 kg, cornering stiffness (C_{\alpha f}=C_{\alpha r}=1.2\times10^5) N/rad, aerodynamic drag coefficient 0.98, etc.) are listed in Table 1. Two collision scenarios are examined:

• Case 1 – Lateral impact at the vehicle’s center of gravity (CG). The vehicle initially travels at (v_{x0}=30) m/s, receives a lateral velocity of 20 m/s, and zero yaw rate.
• Case 2 – Lateral impact off the CG. The vehicle receives a lateral velocity of 10 m/s and a yaw rate of 0.35 rad/s.

In both cases, the control law is applied with parameter sets given in Table 2 (different values for the generalized model and the reference model because of the stronger nonlinear coupling in the former). Results (Figures 3‑6) show that without control the lateral velocity and yaw rate persist, causing the vehicle to drift away from the intended straight line. With the heuristic steering and force functions, the lateral velocity and yaw rate decay rapidly, the heading angle (\psi) returns to zero, and the vehicle’s lateral position converges to the desired path. The longitudinal speed recovers while the vehicle is steered back, demonstrating simultaneous trajectory correction and forward motion.

The discussion highlights three main contributions: (i) a simple, analytically tractable control structure that can be implemented without complex state estimation; (ii) explicit consideration of longitudinal speed variations, which are shown to affect lateral‑yaw dynamics and thus control authority; (iii) validation across two distinct vehicle models, indicating the method’s versatility. Limitations are also acknowledged: the open‑loop nature makes the approach sensitive to modeling errors, actuator saturation, sensor latency, and unmodeled roll/pitch dynamics. Real collisions may also damage vehicle structure, violating the assumption of unchanged parameters.

The authors conclude that the proposed heuristic steering‑force shaping strategy successfully restores post‑collision trajectory for a high‑speed vehicle, even when nonlinear dynamics are present. Future work is suggested in the direction of feedback‑based adaptive control, online parameter identification, and experimental validation on a physical test vehicle. Such extensions would address robustness concerns and bring the concept closer to deployment in safety‑critical autonomous driving systems.