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Vehicle Drift Emergence: Continuous Evolution from Grip Driving to the Handling Limit via Boundary Exploration Learning Model Predictive Control
Sheng Zhao, Binh-Minh Nguyen, Hangyu Lu, Xiaodong Wu
TL;DR
Existing autonomous drift control typically prescribes a drift target, while the continuous emergence of drift from grip driving remains unresolved. This paper uses boundary-exploration LMPC in a repetitive lap-time task without drift references, finding that drift emerges conditionally as performance demand approaches tire capacity.
Problem
Existing controllers establish how to execute prescribed drift but leave its emergence from ordinary driving under a drift-independent objective insufficiently characterized.
Method
BE-LMPC learns from completed laps using an empirical safe set, locally shifted terminal cost, progress optimization, and recoverability-gated expansion of the handling envelope.
Results
At µ = 0.6, lap time decreases from 49.95 s on Lap 3 to 25.50 s on Lap 12, with drift first emerging on Lap 11; no drift is detected for µ = 0.8 to 1.2.
Takeaways & Limitations
Drift is characterized as a conditional continuation of limit handling that emerges when performance demand approaches available tire capacity, rather than as a separately prescribed motion mode.
Takeaways & Limitations
The conclusions are limited to a deterministic matched-model study with simplified vehicle dynamics, fixed force allocation, one track, and no rigorous control-invariance guarantee for the empirical safe set.
Abstract
from arXiv · showhide
Automated drift controllers commonly track a prescribed drift equilibrium, sideslip reference, or trajectory. These formulations establish how to execute drift, whereas the continuous transition from grip driving to drift near the handling limit remains unresolved. This paper defines drift emergence in a repetitive lap time minimization task, where neither the controller objective nor the reward contains an explicit drift reference. A boundary exploration learning model predictive controller (BE-LMPC) constructs an empirical safe set and a locally shifted terminal cost from completed laps. By iteratively improving spatial speed allocation under a fixed global speed bound, the controller progressively explores larger sideslip and yaw rate envelopes while preserving recoverability. As lap performance improves, sustained sideslip and pronounced yaw motion emerge while the rear axle approaches saturation. Analysis shows that, when external conditions vary smoothly, the transition from tire adhesion to sliding does not itself cause abrupt changes in tire force or vehicle state. The combined-slip Fiala model satisfies this continuity condition at the transition. At a tire road friction coefficient of 0.6, lap time decreases from 49.95 s on Lap~3 to 25.50 s on Lap~12, with drift first emerging on Lap~11. Lap~12 reaches 16.5$^\circ$ sideslip and 0.894 rear axle utilization. In contrast, no drift is detected for friction coefficients from 0.8 to 1.2; at 1.2, a similar peak speed is achieved with only 0.483 rear axle utilization. These results characterize drift as a conditional continuation of limit handling that emerges when increasing performance demand approaches the available tire capacity, rather than as a separately prescribed motion mode.
1 Introduction
Prior autonomous drift controllers generally prescribe and track drift references, leaving the emergence of high-sideslip motion from ordinary driving under performance-seeking control insufficiently characterized. This paper studies whether repeated lap-time optimization can produce drift autonomously under unchanged control objectives and specific operating conditions.
- Prescribed drift equilibria, sideslip references, or trajectories let autonomous controllers execute drift but do not explain its emergence from ordinary driving.
- Stability-envelope studies focus on admissible regions or avoiding instability, rather than observing high-dynamic responses selected near the boundary.
- Autonomous racing with repeated laps provides a controlled setting for learning-based controllers to improve performance from completed trajectories without specifying the full optimal trajectory.
- The paper studies whether unchanged closed-loop control can evolve from grip cornering to sustained high-sideslip motion with high rear-axle utilization on a low-friction surface.
- The proposed contribution combines continuous-dynamics analysis of drift emergence with boundary-exploration LMPC that expands sideslip and yaw-rate envelopes while preserving recoverability.
2 Vehicle Dynamics Model and Assumptions
The vehicle is modeled as a planar three-degree-of-freedom single-track system with front steering, longitudinal actuation, combined-slip tire forces, and a fixed 50:50 front-rear force split. The matched plant and controller omit several physical dynamics, limiting the results to the stated configuration.
- Vehicle configuration: The model uses planar three-degree-of-freedom single-track dynamics with front-axle steering and longitudinal actuation at both axles.
- State and dynamics: x = [X, Y, ψ, v, r, β]T and u = [acmd, δ]T define the vehicle state and control input.
- State and dynamics: The reduced speed dynamics use ˙v ≈ ax, applied unchanged across laps and friction cases independently of grip-to-drift classification.
- Model assumptions: Below 0.9 m/s, dynamics blend smoothly with a regularized kinematic model, while reported laps start at or above 1.2 m/s and use the nonlinear model.
- Longitudinal actuation: λx = 0.5 fixes a 50:50 front-rear longitudinal-force split, abstracting powertrain and braking allocation at the axle level.
- Combined-slip tire model: Combined loading reduces lateral tire capacity, with Fiala forces computed using one constant friction coefficient for both axles in each simulation.
- Model assumptions: The matched plant and controller omit body roll, pitch, suspension motion, left-to-right load transfer, tire temperature, wheel or driveline states, and explicit actuator dynamics.
3 Problem Formulation and Preliminary Analysis
Drift emergence is defined as a sustained high-sideslip, high-yaw response under an unchanged controller, and analyzed as a continuous transition through the handling boundary. The transition is conditional: it becomes reachable when performance demand approaches tire capacity.
- 3.1 Operational Definition of Drift Emergence: Drift emergence requires sustained high sideslip, pronounced yaw motion, and near-saturation rear-axle operation under an unchanged closed-loop formulation.Large |β| alone is insufficient, and the drift label is applied only after persistence for Td = 0.30 s.
- 3.2 Continuity Between Grip and Drift Responses: The continuity analysis assumes locally Lipschitz tire-force maps, bounded piecewise-continuous inputs, and continuously varying external parameters.Under these conditions, crossing an adhesion-to-sliding branch boundary does not itself introduce a discontinuity in the vehicle vector field.
- 3.2 Continuity Between Grip and Drift Responses: The combined-slip Fiala model remains continuous at sliding onset, while incremental cornering stiffness approaches zero smoothly.Increasing χx lowers the attainable lateral-force plateau and advances sliding onset to smaller tire slip.
- 3.2 Continuity Between Grip and Drift Responses: At µ = 0.6, phase-plane trajectories progress from low sideslip on Lap 3 into the high-sideslip, high-yaw-rate region on Lap 11.The same plant model and controller feedback channel are retained throughout the progression.
- 3.3 Conditionality of Drift Emergence: BE-LMPC exposes this conditional response by updating an empirical safe set while expanding sideslip and yaw-rate bounds after recoverable laps.The framework retains only recoverable experience and characterizes the realized response through demand-to-capacity ratio.
- 3.3 Conditionality of Drift Emergence: When R approaches unity, tire utilization increases and declining incremental cornering stiffness makes the drift-emergence region reachable.R ≪ 1 leaves the trajectory far from the tire-capacity boundary; lower µ reduces vgrip, while iterative improvement increases vatt.
4 Boundary-Exploration Learning Model Predictive Control
BE-LMPC improves repetitive lap performance using completed-lap experience, local terminal guidance, and recoverability-gated boundary expansion. Drift references and drift rewards are absent from the optimization, so the response is selected by progress demand and vehicle dynamics.
- 4 Boundary-Exploration Learning Model Predictive Control: BE-LMPC uses recoverable completed laps to construct an empirical safe set and terminal cost for subsequent repetitive-racing iterations.The safe set represents successful closed-loop experience rather than a formal control-invariance certificate.
- 4 Boundary-Exploration Learning Model Predictive Control: The implementation selects neighboring samples from the two most recent accepted laps and forms their convex combinations as a local empirical safe set.Local construction keeps online optimization compact while guiding terminal states toward previously supported regions.
- 4 Boundary-Exploration Learning Model Predictive Control: The global speed bound remains fixed across laps, while nonlinear dynamics, track constraints, and learned terminal costs determine spatial speed allocation.The boundary level ηj is updated after each complete lap according to recoverability.
- 4 Boundary-Exploration Learning Model Predictive Control: A lap enters the experience set only when track margin, solver status, combined utilization, and state-finiteness acceptance conditions are satisfied.These tolerances exclude failed exploratory trajectories from subsequent terminal experience data.
- 4 Boundary-Exploration Learning Model Predictive Control: The online problem is a linear time-varying quadratic program over predicted states, inputs, track slacks, convex weights, and terminal slack.The plant remains nonlinear, while one local affine model is used per control update and OSQP solves the resulting QP.
- 4 Boundary-Exploration Learning Model Predictive Control: No β or r reference, countersteering reward, saturation reward, or drift label appears in the optimization.The offline drift detector reports labels only; it does not affect control inputs or boundary updates.
5 Simulation Tests and Evaluation
The simulations evaluate repetitive racing with matched nonlinear plant and controller models under a fixed global speed bound. Performance and response metrics compare learning behavior, constrained grip operation, and drift-related handling conditions.
- 5 Simulation Tests and Evaluation: The closed track is 80.48 m long, uses 0.75 m half-width, and evaluates Laps 3 through 12 after two initial 1.2 m/s experience laps.Control updates occur at 0.05 s, while the nonlinear plant is integrated with a 0.005 s fourth-order Runge–Kutta step.
- 5 Simulation Tests and Evaluation: The controller and plant use matched vehicle and Fiala tire models, so the tests assess mechanism consistency rather than robustness to structural model mismatch.Every learning lap uses the same 10 m/s global speed bound, above observed peak speeds.
- 5 Simulation Tests and Evaluation: The grip-constrained comparison fixes the predicted |β| bound at 7°, below the 8° drift-detector threshold, while the nonlinear plant reaches 8.62° between samples.The measured exceedance follows from discrete enforcement on a locally affine prediction model.
- 5 Simulation Tests and Evaluation: Lap time is the primary performance metric, complemented by sideslip, rear-axle utilization, sustained-drift fraction and duration, track margin, and recoverability.A drift label requires every condition in the operational criterion to persist.
6 Results and Discussion
Under low friction, BE-LMPC progresses from grip driving to sustained drift without changing the global speed bound or handling-envelope level. Drift appears when sideslip, yaw motion, and rear-axle utilization jointly reach the criterion, while higher-friction cases remain below the rear-utilization boundary.
- Drift Emerges Continuously During Low-Friction Learning: 49.95 s to 25.50 s: Lap time decreases from Lap 3 to Lap 12 at µ = 0.6, while drift first appears on Lap 11.The envelope fully expands on Lap 8, but performance continues improving before emergence under the unchanged global speed bound.
- State and Control Responses Before and After Emergence: 26.15 s: Lap 11 is the first detected drift lap, reaching 14.6° sideslip and 0.859 rear-axle utilization while remaining near Lap 10’s speed level.Its sideslip spans −12.8° to 14.6°, indicating a distinct nonlinear attitude response rather than a large speed change.
- Drift Emerges Continuously During Low-Friction Learning: 16.5° sideslip and 0.894 rear-axle utilization are reached on Lap 12, with every displayed lap remaining recoverable.The joint drift criterion persists for 1.00 s on Lap 12, following 0.70 s on Lap 11.
- State and Control Responses Before and After Emergence: The first drift event occurs with the same global speed bound and fully expanded handling envelope used for the preceding grip lap.Lap 10 remains grip while adjacent Lap 11 becomes the first detected drift lap, and the controller uses no separate drift channel.
- Sensitivity of the Drift Criterion: Increasing the sideslip threshold from 6° to 12° leaves first detection on Lap 11, whereas persistence and rear-utilization thresholds shift the detected lap.Detection moves to Lap 9 with a 0.75 rear-utilization threshold, disappears within ten laps at 0.90, and shifts from Lap 10 to Lap 12 as Td increases from 0.20 s to 0.50 s.
- Tire-Road Friction Determines Whether the Drift Branch Is Reached: 25.10 s to 25.50 s: Final lap times are similar across friction cases, but drift occurs only at µ = 0.6 because higher-friction cases remain below the rear-utilization threshold.Rear-axle utilization decreases from 0.894 at µ = 0.6 to 0.483 at µ = 1.2, while peak speeds remain approximately 2.96 m/s.
- Comparison With Grip-Constrained Learning: 4.5% and 4.7%: BE-LMPC reaches 25.50 s by Lap 12, compared with 26.70 s and 26.75 s for grip-constrained and fixed-envelope methods.BE-LMPC also reaches 16.49° sideslip and 0.894 rear-axle utilization, while both baselines remain below the rear-saturation condition.
- Comparison With Grip-Constrained Learning: The low-friction lap-time advantage accompanies greater sideslip and rear utilization within the same track corridor rather than an infeasible shortcut.All laps are accepted, and the three final trajectories remain within the same track corridor.
7 Conclusion
The paper characterizes drift emergence as a conditional, continuous response that develops from conventional driving as demand approaches the handling limit. BE-LMPC produces this behavior without drift references, while the tested conclusions remain bounded by the matched-model simulation setting.
- At µ = 0.6, the vehicle evolves from grip driving on Lap 3 to first-detected drift on Lap 11, then completes recoverable Lap 12 in 25.50 s.
- Across µ = 0.8 to 1.2, every lap remains outside the complete joint drift criterion.
- Future work should separate prediction and plant models and include wheel-speed, driveline, actuator, and tire-parameter uncertainty.
- Hardware-in-the-loop and scaled-vehicle experiments are required to test whether the simulated emergence mechanism persists under sensing errors and real-vehicle effects.