Organizations: School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China. · Department of Advanced Energy, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, 277-8561, Japan. · Faculty of Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, 999077, China.
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 Lap3 to 25.50 s on Lap12, with drift first emerging on Lap11. Lap12 reaches 16.5∘ 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.
Figures & tables
Figure 1: Planar three-degree-of-freedom single-track model with vehicle sideslip angle β .
Figure 2: Continuous grip-to-drift evolution. (a) Normalized Fiala lateral force (top) and incremental cornering stiffness (bottom) for χx=0 , 0.5, and 0.8; open circles mark sliding onset. (b) (β,r) trajectories through the critical corner for Laps 3, 5, 7, and 9 through 12 at μ=0.6 . Open circles linked by the dashed line identify the fixed location s/L=0.467 , marked by a star on the inset track. Gray dotted lines denote the ∣β∣ and ∣r∣ thresholds. The inset track is rotated and reflected for display.
Figure 3: BE-LMPC framework for repetitive lap-time improvement under a fixed vehicle model and global speed bound. Recoverable laps update the empirical safe set, and the recovery rule expands the sideslip and yaw-rate bounds. The demand-to-capacity ratio characterizes the resulting grip or drift response.
Vehicle parameters
Controller parameters
Parameter
Symbol
Value
Parameter
Symbol
Value
Vehicle mass
m
3.2kg
Power regularization
vϵ
0.50m/s
Yaw moment of inertia
Iz
0.04kgm2
Prediction horizon
N
25
CG-to-axle distances
lf/lr
0.16/0.17 m
Control period
Ts
0.05s
CG height
h
0.08m
Global speed bound
vmax
10.0m/s
Cornering stiffness/load
cαf/cαr
2.3/2.3 rad -1
Maximum steering angle
δmax
0.41rad
Table 1: Principal vehicle and controller parameters
Test
Condition
Purpose
Main case
BE-LMPC, μ=0.6
Evaluate lap-wise evolution from grip to drift
Friction sweep
μ=0.6 , 0.8, 1.0, 1.2
Identify friction conditions associated with emergence
Method comparison
BE-LMPC, grip constraint, fixed envelope
Isolate the effects of boundary exploration and drift freedom
Table 2: Simulation test matrix
Figure 4: Lap-wise drift emergence at μ=0.6 . (a) Lap times for Laps 3 through 12; blue and red bars denote grip and drift, respectively, and labels give the envelope level. (b) Track trajectories for grip Lap 10 and first-drift Lap 11; red segments satisfy the joint criterion. The map is rotated and reflected for display. (c) Threshold-normalized sideslip and rear-axle utilization with the envelope level; R and X denote accepted and rejected laps.
Figure 5: State and control responses for grip Lap 10 and first-drift Lap 11 at μ=0.6 : (a) speed, (b) sideslip angle, (c) yaw rate, (d) steering angle, and (e) longitudinal acceleration. Gray dotted lines mark the steering constraint. In (e), dotted and solid curves denote acmd and realized ax , respectively. Light-red shading marks intervals satisfying the joint drift criterion.
Varied parameter
Values
First drift lap
βth / ( ∘ )
6, 8, 10, 12
11, 11, 11, 11
ρth
0.75, 0.82, 0.90
9, 11, –
Td / s
0.20, 0.30, 0.50
10, 11, 12
Table 3: Effect of drift-criterion parameters on the first detected drift lap
Figure 6: Effect of tire-road friction on drift emergence. (a) Lap-wise outcomes: recoverable grip (open circles), recoverable drift (filled squares), and rejection (crosses). (b) Final-lap maximum sideslip and rear-axle utilization normalized by their thresholds; the hatched region satisfies both criteria. (c) Final recoverable lap time (solid circles, left axis) and peak speed (dashed triangles, right axis).
μ
First drift lap
Final recoverable lap
Lap time (s)
Max. ∣β∣ / ( ∘ )
Max. ρr
Final state
0.6
11
12
25.50
16.49
0.894
Drift
0.8
–
12
25.25
12.66
0.699
Limit grip
1.0
–
12
25.20
11.50
0.572
Limit grip
1.2
–
12
25.10
10.82
0.483
Limit grip
Table 4: Quantitative drift-emergence results under different friction conditions
Figure 7: Comparison at μ=0.6 of BE-LMPC, grip-constrained sideslip, and a fixed envelope. (a) Lap-time convergence; the dash-dotted vertical line marks the first BE-LMPC drift lap. (b) Final-lap sideslip and rear-utilization criterion plane; the hatched region satisfies both thresholds. (c) Lap outcomes: recoverable grip (open circles), recoverable drift (filled squares), and rejection (crosses). (d) Final trajectories with an enlargement at the location of maximum path separation.
School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China · School of Mechanical and Aerospace Engineering, Nanyang Technological University, 639798, Singapore
Professorship of Autonomous Vehicle Systems, TUM School of Engineering and Design, Technical University of Munich, 85748 Garching, Germany; Munich Institute of Robotics and Machine Intelligence (MIRMI)