Timed Rule-Based Supervision of an End-to-End Autonomous Parking Policy

I. INTRODUCTION

• The study asks whether a manually calibrated supervisor can correct repeatable failures of an existing policy in a known simulated lot.Observed failures included boundary exits, alternating brake and throttle after stopping, and delayed braking near obstacles.Source: §I · p. 1
• The work specifies a timed, hand-calibrated output supervisor for the previously published E2E Parking policy and documents its guards, overrides, and parameter values.The implementation is released to support replication and evaluation in further settings.Source: §I · p. 1
• 48 additional target successes in 384 repeated attempts are reported for the supervised policy compared with the same retrained policy without supervision, within one CARLA lot.The retrained policy’s target success was 85.16%; with the PSS it was 97.66%.Source: §I · p. 1

A. Transformer Model

• Transformer parking approaches differ in decision layer and evaluation environment, so their published success rates are not directly comparable with this control-output intervention.E2E Parking supplies the nominal architecture and CARLA pipeline, while ParkingE2E and HOPE report different approaches and settings.Source: A. Transformer Model · p. 1

B. BEV Representation

• The study retains the published E2E Parking network’s BEV path and changes only its executed control commands.Source: B. BEV Representation · p. 1

C. Rule-based Methods

• The PSS applies fixed, manually calibrated guards to outputs of an unchanged learned policy, and its runtime correction retains limitations of manual guards and calibrated thresholds.The architecture figure distinguishes inherited components from the hand-designed PSS, which also uses CARLA world coordinates and vehicle velocity.Source: C. Rule-based Methods · p. 2, Figure 1 · p. 2

III. METHOD

• The method retains Yang et al.’s network architecture as the nominal controller and adds the post-inference PSS as the new component.Figure 1 separates the inherited architecture from the added supervisor.Source: §III · p. 2

A. Problem Definition

• The policy is trained with expert trajectories pairing sequential sensor data with control signals, and its policy maps observations to control outputs under a loss function.Source: A. Problem Definition · p. 2

B. Inputs and Outputs

• Inputs include four-camera images, vehicle velocity and acceleration, and target-slot coordinates that are initially user-set and then tracked through segmentation.Source: B. Inputs and Outputs · p. 2
• The model forecasts control signals for the next four steps at 10 Hz, including normalized acceleration, steering angle, and gear selection.Acceleration and steering are discretized into 201 values; the sequence also includes gear state.Source: B. Inputs and Outputs · p. 2

C. End-to-End Neural Network Architecture

• The described network stages are reproduced for implementation context; the new PSS is separate from the inherited E2E Parking policy.Figure 2 depicts the inherited network alone, while Figure 1 shows the complete method.Source: C. End-to-End Neural Network… · p. 2, Figure 2 · p. 3

1) BEV Feature Generation:

• Camera features and discrete depth distributions are lifted into 3D frustums, back-projected using camera calibration, and height-sum-pooled into a BEV feature map.Source: 1) BEV Feature Generation · p. 2

2) Multi-Modal Feature Fusion:

• The BEV map is concatenated with a binary target-slot grid, compressed into visual tokens, and fused with projected vehicle kinematics using a self-attention Transformer encoder.Source: 2) Multi-Modal Feature Fusion · p. 2

3) Auto-Regressive Control Prediction:

• The inherited Transformer decoder uses cross-attention to generate control tokens autoregressively, then decodes them into throttle, brake, steer, and gear commands before the PSS.Source: 3) Auto-Regressive Control Prediction · p. 3

D. Multi-Task Learning Loss Functions

• The network’s multi-task losses include discrete control cross-entropy, BEV semantic segmentation, and auxiliary depth supervision.The segmentation centroid supplies the target for the next inference step; depth is supervised using projected LiDAR points.Source: D. Multi-Task Learning Loss Functions · p. 3

E. Timed Rule-Based Output Supervision via PSS

• The PSS is a manually designed timed state machine whose guards were selected from observed failures and whose overrides replace part or all of the nominal command.Its numerical values were calibrated for the study lot and vehicle, and it provides no invariant proof, reachability analysis, or formal safety guarantee.Source: E. Timed Rule-Based Output Supervision… · p. 3
• The supervisor runs at 10 Hz, and a discrete counter tracks time spent in selected modes; the specified conditions do not define a proved safe set.Source: E. Timed Rule-Based Output Supervision… · p. 3

1) Formal Formulation of the Safety Shield:

• The finite-state supervisor includes nominal pass-through, kinematic regulation, stall prevention, target alignment, boundary guarding, and forced recovery modes.The forced reverse recovery follows boundary guarding; the listed control mappings include pass-through and stopping, edge, and recovery overrides.Source: 1) Formal Formulation of the Safety… · p. 3

• The study uses 16 hand-chosen values; parameterization makes them visible but does not make the transition logic environment independent.Source: 2) System Calibration and Generalization · p. 4
• Yboundary is an absolute CARLA-map coordinate, while the terminal guard uses the goal’s x coordinate and overrides use fixed actuator values and durations.Source: 2) System Calibration and Generalization · p. 4
• A different layout, vehicle, localization system, or sensor suite may require changes to parameters and guards; required retuning was not measured.Source: 2) System Calibration and Generalization · p. 4

A. Experimental Setup

• The network retains the original architecture and is trained with batch size 16, Adam, learning rate 7.5 × 10−5, and weight decay 1 × 10−4.Training used four NVIDIA RTX 4090 GPUs.Source: A. Experimental Setup · p. 4

B. Dataset Details

• Gen 2B contains 248 training and 104 validation trajectories, generated across 16 even-indexed slots; odd-indexed slots are reserved for closed-loop evaluation.Fig. 4 shows the CARLA parking lot.Source: B. Dataset Details · p. 4, Figure 4 · p. 4
• Across seeds 0–47, nominal trajectories use fixed BEV initial poses and additional distance-randomized lateral poses.Seeds 0–31 provide 192 training and 64 validation trajectories; seeds 32–47 add 32 of each.Source: B. Dataset Details · p. 4
• Four shadowed scenarios use seeds 11, 27, 12, and 28 for target slots 3-8 and 3-10, contributing 24 training and 8 validation trajectories.Fig. 5 depicts the four scenes.Source: B. Dataset Details · p. 4

C. Metrics

• TSR counts parking within the target slot’s position and orientation tolerances; TFR counts arrivals outside those acceptable thresholds.Source: C. Metrics · p. 4
• NTR combines non-target success and failure rates, while OR and TR measure lot exits and failure to park within 30 seconds.The original work aggregates OR and TR as TR.Source: C. Metrics · pp. 4–5
• Fig. 5 depicts shadowed scenes for slots 3-8 and 3-10 under seeds 11, 27, 12, and 28.Source: Figure 5 · p. 5
• APE and AOE measure final position and heading errors among successful attempts, while APT is their mean parking duration.Source: C. Metrics · p. 5
• AIT is average inference time per step, and CR is the frequency of collisions during parking.Source: C. Metrics · pp. 4–5

D. Closed-Loop Evaluation Results

• Each configuration comprises 384 attempts across 16 held-out slots, six poses, and four rounds, all using the same lot, vehicle, sensor arrangement, and success rule.The 30 s time limit applies; aggregated attempts lack per-round variability and are treated as descriptive results.Source: D. Closed-Loop Evaluation Results · p. 5
• Observed baseline failures include boundary exits, delayed braking near streetlight-adjacent slots, and brake–throttle chattering after stopping that can lead to timeout.Source: D. Closed-Loop Evaluation Results · p. 5
• At 7.5 km/h, Fig. 6 reports 97.66% TSR and 21.34 s APT; 5.0 km/h has the same TSR with longer APT, while 12.5 km/h has lower TSR and shorter APT.Source: Figure 6 · p. 5, D. Closed-Loop Evaluation Results · p. 5

2) Test with Rule-based Correction:

• TR decreases from 7.552% to 0.781%, while TFR rises from 0 to 0.260%; among successful attempts, APE changes from 0.237 to 0.211 m and AOE from 0.335° to 0.325°.The aggregate table does not establish statistical significance, and accuracy/time means condition on successful attempts whose composition differs between configurations.Source: 2) Test with Rule-based Correction · p. 5
• TSR improves or stays unchanged for 14 of 16 slots, but decreases for slots 2-7 and 2-15.Some collisions and timeouts remain.Source: 2) Test with Rule-based Correction · p. 5
• 97.656% (375/384) TSR with PSS versus 85.156% (327/384) without it, a descriptive increase of 12.5 percentage points or 48 target successes.The results cover 384 repeated attempts in one CARLA lot.Source: 2) Test with Rule-based Correction · p. 5, Table II · p. 6

3) Visualization of Comparisons:

• For target slot 2-1, the original policy exhibits boundary failures including stopping in a non-target slot, collision, or leaving the lot.The observed cases include an attempted stop on the road, described as an extremely dangerous operation.Source: 3) Visualization of Comparisons · p. 6, Figure 7 · p. 6
• In a timeout case, target-slot segmentation disappears or shifts away from slot 2-1, and problematic vehicle segmentation also leads to incorrect control outputs.The vehicle eventually stops between slot 3-1 and the sidewalk, with the predicted target segmentation displaced from the ground-truth target.Source: 3) Visualization of Comparisons · p. 6, Figure 9 · p. 7
• In the outbound case, target segmentation moves outside the lot at T = 8.6 s; forced reverse can gradually restore target-slot segmentation and lead to successful parking.The corrected trajectory also uses earlier braking and a subsequent 2.5-second timed automaton signal to complete the maneuver without further intervention.Source: 3) Visualization of Comparisons · pp. 6–7, Figure 8 · p. 6, Figure 12 · p. 7
• After stopping at target slot 3-9, alternating throttle and brake can cause a timeout; rule-based correction holds brake at 0.9 to keep the vehicle static.A zero-throttle output for 2 seconds is required to determine successful parking.Source: 3) Visualization of Comparisons · p. 7, Figure 11 · p. 7

4) Comparison Scope and Recent Parking Methods:

• Table IV's Yang et al. and expert results provide context, not PSS-isolating baselines, because their data/model availability or evaluation conditions differ.Expert rows are demonstration-quality references, not deployable policy baselines.Source: 4) Comparison Scope and Recent Parking… · p. 8
• Published success rates from ParkingE2E and HOPE cannot be defensibly ranked against this method because sensors, outputs, vehicles, success definitions, and evaluation environments differ.A direct head-to-head comparison would require reimplementing both methods under a shared protocol; those experiments are not reported.Source: 4) Comparison Scope and Recent Parking… · p. 8

5) Failure Case Analysis and Limitations:

• Figure 15 identifies abnormal BEV features during forced braking at target slot 3-9.Figure 16 depicts a closed-loop failure after target-slot overshoot.Source: Figure 15 · p. 8, Figure 16 · p. 8
• After severe slot overshoot, the PSS prevents out-of-boundary run-off but may leave the vehicle parked in an incorrect non-target slot.The limitation is illustrated as a closed-loop failure after target-slot overshoot.Source: 5) Failure Case Analysis and Limitations · p. 8, Figure 16 · p. 8
• Conflicting neural outputs and conservative PSS regulations can trigger alternating throttle and braking, ending in terminal timeout failures.The vehicle may still be guided safely into the target slot despite the efficiency loss.Source: 5) Failure Case Analysis and Limitations · p. 8

E. Ablation Study

• The peer-work table uses different published evaluation protocols, and only the two Gen 2B policy rows are the within-study PSS comparison.These tables do not establish a head-to-head performance comparison with other methods.Source: Table III · p. 9, Table IV · p. 9
• Table V reports target success rates of 93.49% without target alignment, 91.93% without speed limiting, and 89.84% without geofencing, versus 97.66% with the complete PSS.The corresponding APTs are 21.66, 18.07, 21.54, and 21.34 s, respectively.Source: E. Ablation Study · p. 8, Table V · p. 9

V. CONCLUSION

• 327 to 375 out of 384 repeated attempts are the retrained policy’s target successes with and without the PSS, respectively, in the single-lot CARLA evaluation.The conclusion states that neither the ablations nor the state-machine specification establish formal safety or generalization beyond this configuration.Source: §V · p. 9
• The evaluation uses one CARLA lot, vehicle, and sensor setup, with hand-selected rules and map-specific coordinates; no different environment or real-world test was performed.Simulator world coordinates would need to be estimated with uncertainty for real-vehicle use, and no formal safety guarantee is established.Source: §V · p. 9
• A direct same-protocol experiment against ParkingE2E or HOPE remains outstanding.Source: §V · p. 9
• Future evaluation should test more independently sampled lots, obstacle layouts, vehicles, and sensors, and report round- and scenario-level outcomes.Real-vehicle use also requires a localization source, uncertainty handling, and safety analysis.Source: §V · p. 9
• The authors release the framework’s full implementation to support reproducibility and further research.Source: §V · p. 9