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

Kejia Gao, Liguo Zhou, Lei Yu, Alois Knoll

Year: 2026Venue: arXiv preprintArea: cs.ROType: PreprintEmbeddings: 49

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%

Last extracted: 9/29/2026, 11:49:04 AM

Summary

This paper introduces a Timed Rule-Based Parametric Safety Shield (PSS) to correct recurring failures in an end-to-end autonomous parking policy based on a vision-based Transformer architecture. The PSS intervenes in control outputs using hand-calibrated thresholds for speed, position, and duration to address issues like boundary exits, delayed braking, and stalling. In closed-loop evaluations within the CARLA simulator, the PSS improved target success rates from 85.16% to 97.66% across 384 attempts, demonstrating significant error reduction in position and orientation, though results are specific to the simulated environment.

Entities (6)

CARLA · simulation-environment · 99%Parametric Safety Shield · systemmodule · 98%E2E Parking · modelpolicy · 95%Transformer · architecture · 95%E2E Parking Dataset · dataset · 92%Yang et al. · authorreference · 90%

Relation Signals (7)

E2E Parking → evaluatedin → CARLA

confidence 99% · We study whether a manually specified runtime supervisor can correct recurring failures of an existing end-to-end parking policy in a fixed CARLA parking lot.

Parametric Safety Shield → improves → Target Success Rate

confidence 98% · Target success increases from 327/384 (85.16%) for the retrained policy to 375/384 (97.66%) with the PSS

Parametric Safety Shield → corrects → E2E Parking

confidence 95% · We study whether a manually specified runtime supervisor can correct recurring failures of an existing end-to-end parking policy... our contribution is a timed, rule-based Parametric Safety Shield (PSS) applied to its control outputs.

E2E Parking → uses → Transformer

confidence 95% · The vision-based Transformer architecture is inherited from Yang et al.

Parametric Safety Shield → mitigates → boundary exits

confidence 90% · The PSS uses hand-calibrated speed, position, and duration thresholds to intervene in observed failure modes, including boundary exits, delayed braking, and stalled or oscillatory control.

Parametric Safety Shield → mitigates → delayed braking

Parametric Safety Shield → mitigates → stalled control

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Abstract

Abstract:We study whether a manually specified runtime supervisor can correct recurring failures of an existing end-to-end parking policy in a fixed CARLA parking lot. The vision-based Transformer architecture is inherited from Yang et al.; our contribution is a timed, rule-based Parametric Safety Shield (PSS) applied to its control outputs. The PSS uses hand-calibrated speed, position, and duration thresholds to intervene in observed failure modes, including boundary exits, delayed braking, and stalled or oscillatory control. In the reported closed-loop evaluation, 16 held-out target slots and six initial poses are each evaluated in four rounds (384 attempts per configuration). Target success increases from 327/384 (85.16%) for the retrained policy to 375/384 (97.66%) with the PSS; mean position and orientation errors among successful attempts are 0.21m and 0.33 degrees. These results show an improvement within this simulator setup. The repeated attempts share one map, vehicle, and sensor configuration, and the PSS uses simulator world coordinates; thus the results do not establish generalization to other lots or real vehicles, or a formal safety guarantee.

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Timed Rule-Based Supervision of an End-to-End Autonomous Parking Policy Kejia Gao Liguo Zhou* Lei Yu Alois Knoll Abstract We study whether a manually specified runtime supervisor can correct recurring failures of an existing end-to-end parking policy in a fixed CARLA parking lot. The vision-based Transformer architecture is inherited from Yang et al.; our contribution is a timed, rule-based Parametric Safety Shield (PSS) applied to its control outputs. The PSS uses hand-calibrated speed, position, and duration thresholds to intervene in observed failure modes, including boundary exits, delayed braking, and stalled or oscillatory control. In the reported closed-loop evaluation, 16 held-out target slots and six initial poses are each evaluated in four rounds (384 attempts per configuration). Target success increases from 327/384 (85.16%) for the retrained policy to 375/384 (97.66%) with the PSS; mean position and orientation errors among successful attempts are 0.21 m and 0.33∘. These results show an improvement within this simulator setup. The repeated attempts share one map, vehicle, and sensor configuration, and the PSS uses simulator world coordinates; thus the results do not establish generalization to other lots or real vehicles, or a formal safety guarantee. Index Terms: Autonomous Driving, Autonomous Parking, End-to-End Neural Network, Timed Automaton, Rule-based Correction. I Introduction End-to-end learning maps sensor inputs to control outputs and has been studied for autonomous parking. Parking is a structured, low-speed task, but collisions and incorrect stops remain possible. We examine a narrow question: whether a manually calibrated supervisor can correct repeatable failure modes of an existing policy in a known simulated lot. In closed loop, isolated control errors can accumulate into parking failures. The E2E Parking Dataset [1] supports reproduction of the E2E Parking model [2] in CARLA. The retrained model achieved 85.16% target success in the reported evaluation. Inspection of its failures in the study lot found boundary exits, alternating brake and throttle after stopping, and delayed braking near obstacles. These observations motivated the specific interventions studied here; they do not establish a generalization gap outside the tested setup. We attach a Parametric Safety Shield (PSS), a hand-designed timed supervisor, to the output of the existing Transformer policy. The supervisor overwrites selected commands when its speed, position, and timing guards fire. This is an engineering extension of the published policy rather than a new neural architecture or a proof of safety. In the evaluated CARLA configuration, target success rises from 85.16% to 97.66%, with mean position and orientation errors of 0.21 m and 0.33∘ among successful attempts. The main contributions of this paper are as follows: • We specify a timed, hand-calibrated output supervisor for the previously published E2E Parking policy and document its guards, overrides, and parameter values. • We report a closed-loop comparison with the same retrained policy without supervision, plus component ablations, within one CARLA lot. The main observed change is 48 additional target successes in 384 repeated attempts. • We release the implementation at https://github.com/KejiaGao/e2e-parking-carla-timed-automaton to support replication and evaluation in further settings. I Related Works We focus on parking policies and the perception and rule-based components relevant to the proposed supervisor. I-A Transformer Model Transformers support multi-modal feature fusion in driving systems, including TransFuser [3], InterFuser [4], and UniAD [5]. These works concern broader driving tasks and do not test the parking output supervisor studied here. Transformer models are also studied for parking. ParkPredict+ [6] forecasts waypoints. E2E Parking [2] provides the camera-to-control Transformer architecture and CARLA evaluation pipeline used as the nominal policy in this study. Our earlier E2E Parking Dataset work [1] released data and reported 85.16% target success after retraining that architecture. ParkingE2E [7] instead predicts future waypoints from camera images and reports real-vehicle evaluation in four garages. HOPE [8] combines reinforcement learning with Reeds–Shepp curves for parking path planning and evaluates diverse scenario difficulty, including real-world tests. These recent methods address different decision layers and evaluation environments; their published success rates are not directly comparable with our CARLA control-output intervention. I-B BEV Representation The inherited E2E Parking policy projects camera features into bird’s-eye view (BEV) using Lift-Splat-Shoot [9]. BEVDepth [10] demonstrates explicit depth supervision for multi-view BEV estimation. Our work retains the published parking network’s BEV path and changes only its executed control commands. I-C Rule-based Methods Rule-based planning has been combined with learned components in driving [11, 12]. HOPE [8] is especially relevant to parking because it combines a learned planner with geometric Reeds–Shepp structure. Our PSS is narrower: it applies fixed, manually calibrated guards to the outputs of an unchanged learned policy. Our scope is a runtime correction of the published E2E Parking controller’s outputs. This differs from changing the learned policy, but it retains the familiar limitations of manually designed guards and calibrated thresholds. I Method We retain the network architecture of Yang et al. [2] as the nominal controller. The new component in this paper is the post-inference PSS. Figure 1 shows the complete signal path and separates inherited and added components. Cameras, vehicle state, target slot Inherited E2E Parking Transformer Proposed timed PSS guards and overrides Vehicle in CARLA executed command next observation and simulator position/velocity for PSS Fig. 1: Complete closed-loop architecture. Blue boxes are inherited from E2E Parking [2]; the red box is the hand-designed component introduced here. The PSS uses CARLA world coordinates and vehicle velocity in addition to the nominal command. I-A Problem Definition The system is intended to output accurate control signal of parking by using input data of sensors. The model is trained in a supervised learning framework with expert demonstration dataset =τn∣n∈1,2,…,ND=\τ^n n∈\1,2,…,N\\, where each of the N trajectories τn _n consists of sequential sensor data and control signals: τn=(tn,tn)t=1Tnτ^n= \ (X_t^n,C_t^n ) \_t=1^T^n. tnX_t^n includes RGB images of four cameras, target slot coordinates and ego vehicle’s dynamic state in the nnth trajectory at time step t, while tnC_t^n consists of values of throttle, brake, steering and gear state in the nnth trajectory at time step t. The training objective is defined as follows: arg⁡minπ(,)∼[ℒ⁡(,π⁡())] _πE_(X,C) [L (C,π(X) ) ] (1) where π denotes the control policy that maps observations to control outputs and ℒL is the loss function. I-B Inputs and Outputs Inputs: The model’s input includes camera images, motion states (velocity and acceleration), and the target slot coordinates. Four surrounding cameras (front, left, right, rear) are deployed to capture the environmental images. The target slot is initially set by the user and then continuously tracked through segmentation task. Outputs: The model forecasts control signals for the next four steps at a frequency of 10 Hz10\,Hz. The outputs consist of normalized acceleration (acct∈[−1,1]acc_t∈[-1,1]), steering angle (steert∈[−1,1]steer_t∈[-1,1]), and gear selection (geart∈0,1gear_t∈\0,1\). Inspired by the approach of Pix2Seq [13], we regard control prediction as a language modeling task, where signals are tokenized and predicted sequentially. Acceleration and steering are discretized into 201 natural numbers [0,200][0,200], where 0 signifies full brake/left and 200 indicating full throttle/right, and 100 represents no brake, throttle or steering. For gear state, 0 means forward gear and 200 indicates reverse gear. The sequence format is: S=[BOS,c0,c1,c2,c3,EOS] S=[BOS,c_0,c_1,c_2,c_3,EOS] (2) cn=[accn,steern,gearn]. c_n=[acc_n,steer_n,gear_n]. I-C End-to-End Neural Network Architecture The following BEV generation, feature fusion, and control prediction stages describe the inherited E2E Parking network of Yang et al. [2]; they are reproduced for implementation context. Figure 2 depicts that network alone, not the complete method in Fig. 1. Four camera images EfficientNet-B4 and LSS BEV features Target slot and ego motion fusion Transformer encoder and decoder Control token sequence Fig. 2: Schematic of the inherited E2E Parking nominal policy, adapted from Yang et al. [2]. Auxiliary depth and segmentation losses are described in the text. The new PSS is shown separately in Fig. 1. We use this vision-based imitation learning network as the nominal policy. The description below summarizes its operations rather than claiming a new neural architecture. I-C1 BEV Feature Generation Given multi-view surrounding images ℐ=Inn=1NI=\I_n\_n=1^N (In∈ℝ3×H×WI_n ^3× H× W), an EfficientNet-B4 backbone extracts multi-view 2D feature maps Fimg,n=Backbone(In)∈ℝCi×Hi×WiF_img,n=Backbone(I_n) ^C_i× H_i× W_i. Following the LSS framework [9], a depth classification head predicts discrete distributions Dn∈ℝD×Hi×WiD_n ^D× H_i× W_i over D bins. A 3D context frustum Ffrus,nF_frus,n is generated via the outer product: Ffrus,n=Dn⊗Fimg,n∈ℝD×Ci×Hi×WiF_frus,n=D_n F_img,n ^D× C_i× H_i× W_i (3) Using camera intrinsics and extrinsics, these frustums are back-projected into a 3D voxel space and aggregated along the height axis via sum-pooling, yielding the BEV feature map Fbev∈ℝCb×Xb×YbF_bev ^C_b× X_b× Y_b. I-C2 Multi-Modal Feature Fusion The visual BEV map is unified with spatial intent and kinematics. The target parking slot is encoded into a binary grid Ftarget∈0,11×Xb×YbF_target∈\0,1\^1× X_b× Y_b and channel-wise concatenated with FbevF_bev to form Fbev′=[Fbev;Ftarget]∈ℝ(Cb+1)×Xb×YbF_bev =[F_bev\,;\,F_target] ^(C_b+1)× X_b× Y_b. To compress redundancy, Fbev′F_bev is passed through a ResNet18 network and flattened into a visual sequence Fbev_seq∈ℝCb_ds×LF_bev\_seq ^C_b\_ds× L, where Cb_dsC_b\_ds is the downsampled channel size and L is the token length. Concurrently, the vehicle’s kinematic vector Fego0∈ℝ1×NegoF_ego0 ^1× N_ego is projected by an MLP into motion features Fego∈ℝ2×LF_ego ^2× L. The final representation FfuseF_fuse is unified via a multi-head self-attention Transformer encoder to capture global spatial correlations: Ffuse=TransformerEncoder([Fbev_seqFego])∈ℝ(Cb_ds+2)×LF_fuse=TransformerEncoder ( bmatrixF_bev\_seq\\ F_ego bmatrix ) ^(C_b\_ds+2)× L (4) I-C3 Auto-Regressive Control Prediction A Transformer decoder maps FfuseF_fuse into actuator profiles autoregressively. The cross-attention mechanism utilizes FfuseF_fuse to populate the Key (K) and Value (V) matrices: K=WKFfuse,V=WVFfuseK=W_KF_fuse,V=W_VF_fuse, while an empty query sequence with positional embeddings serves as the Query (Q). The decoder iteratively generates control tokens, which a linear layer decodes into raw vehicle commands (throttle, brake, steer, gear) prior to entering the PSS layer. I-D Multi-Task Learning Loss Functions The network is optimized via a multi-task loss function: ℒtotal=λ1ℒctrl+λ2ℒseg+λ3ℒdepthL_total= _1L_ctrl+ _2L_seg+ _3L_depth (5) a. Control Loss (ℒctrlL_ctrl): A discrete cross-entropy loss enforces exact matching with expert demonstrations: ℒctrl=−∑t=1T∑kyt,klogy^t,kL_ctrl=- _t=1^T _ky_t,k y_t,k, where yt,ky_t,k and y^t,k y_t,k are the ground-truth and predicted probabilities for the k-th control bin at step t. b. Segmentation Loss (ℒsegL_seg): A semantic head categorizes the BEV grid into vehicle, target slot, and background using categorical cross-entropy. During inference, closed-loop tracking is maintained by extracting the spatial centroid (x¯,y¯)( x, y) of the predicted target slot to serve as FtargetF_target for the next step. c. Auxiliary Depth Loss (ℒdepthL_depth): Inspired by BEVDepth [10], discrete depth estimations DnD_n are supervised via binary cross-entropy using one-hot encoded LiDAR points projected onto camera planes, substantially enhancing spatial interpretability. I-E Timed Rule-Based Output Supervision via PSS The PSS is a manually designed state machine with duration counters. Its guards were selected from observed failure cases in the study lot, and its numerical values were calibrated for that lot and vehicle. When a guard fires, the PSS replaces part or all of the nominal command. Figure 3 shows the state topology. We use the term “shield” for this runtime supervisor; no invariant proof, reachability analysis, or formal safety guarantee is provided. Fig. 3: Topology of the PSS structured as a finite-state supervisor. The supervisor runs at 10Hz10\ Hz (Δt=0.1s t=0.1\ s). A discrete counter c1c_1 tracks time spent in selected modes. Its states (Z), guards (U), and control overrides (Y) are listed below to make the implemented intervention reproducible. These conditions do not define a proved safe set. I-E1 Formal Formulation of the Safety Shield a. Operational States (Z) • z1:Nominal Inferencez_1:Nominal Inference — The default state in which the Transformer command passes through; it does not imply that the state is safe. • z2:Kinematic Regulationz_2:Kinematic Regulation — Activated when the vehicle velocity violates the safe kinematic upper bound, overriding the throttle to enforce immediate deceleration. • z3:Stall Prevention (Boost)z_3:Stall Prevention (Boost) — Addresses the low-velocity dead-zone problem common in imitation learning by injecting a persistent micro-torque to overcome static friction. • z4:Target Alignment (Stop)z_4:Target Alignment (Stop) — Triggered upon reaching the precise terminal stopping threshold along the primary parking axis. • z5:Boundary Guardingz_5:Boundary Guarding — Initiated near the configured lot boundary to reduce observed run-off events. • z6:Forced Recoveryz_6:Forced Recovery — A structured, time-bounded reverse maneuver triggered sequentially after z5z_5 to guide the vehicle back into the safe operational region. TABLE I: Parameter Calibration of the PSS Parameter Description Calibrated Value vlimitv_limit Kinematic velocity upper bound 7.5km/h7.5\ km/h vstallv_stall Low-velocity stall threshold 2.0km/h2.0\ km/h δx _x Longitudinal alignment tolerance 0.075m0.075\ m YboundaryY_boundary Spatial boundary of the parking lot −239.4m-239.4\ m vy,thv_y,th Lateral velocity run-off threshold −0.01km/h-0.01\ km/h ϵf _f Numerical deadband floating zero 1×10−51× 10^-5 αrev_th _rev\_th Throttle lower bound in reverse mode 0.110.11 αboost _boost Stall prevention micro-throttle input 0.30.3 αrecover _recover Forced recovery throttle input 0.30.3 PstopP_stop Terminal alignment braking input 0.90.9 PedgeP_edge Boundary guarding braking input 0.30.3 TstallT_stall Maximum allowable stall duration 21steps21\ steps TboostT_boost Boost torque injection duration 10steps10\ steps TstopT_stop Hold duration for terminal stop 21steps21\ steps TedgeT_edge Reaction buffer at spatial edge 4steps4\ steps TrecoverT_recover Forced recovery phase duration 25steps25\ steps b. Parametric Guard Conditions (U) u~1: u_1\!: |vx|≥|vlimit| |v_x |≥ |v_limit | (6) u~2: u_2\!: (OPEN(¬R∧uth<ϵf)∨(R∧uth<αrev_th))∧ubr<ϵf∧|vx|<|vstall| aligned (&( R\; \;u_th< _f)\; \;(R\; \;u_th< _rev\_th) )\\ & \;u_br< _f\; \; |v_x |< |v_stall | aligned (7) u~3: u_3\!: ubr>ϵf u_br> _f (8) u~4: u_4\!: |x−xgoal|<δx |x-x_goal |< _x (9) u~5: u_5\!: ¬R∧|y|>|Yboundary|∧|vy|>|vy,th| R\; \; |y |> |Y_boundary |\; \; |v_y |> |v_y,th | (10) where vxv_x and vyv_y denote the vehicle longitudinal and lateral velocities; x and y represent the spatial coordinates; R∈True,FalseR∈\True,False\ is the reverse flag; uthu_th and ubru_br are the throttle and brake control commands, respectively. c. Control Output Mappings (Y) • y~1:No intervention (Pass-through neural policy outputs) y_1\!:~No intervention (Pass-through neural policy outputs) • y~2:throttle=0 y_2\!:~ throttle=0 • y~3:throttle=αboost y_3\!:~ throttle= _boost • y~4:throttle=0,brake=Pstop,steer=0,reverse=True y_4\!:~ throttle=0,\; brake=P_stop,\; steer=0,\; reverse=True • y~5:throttle=0,brake=Pedge,reverse=True y_5\!:~ throttle=0,\; brake=P_edge,\; reverse=True • y~6:throttle=αrecover,brake=0,steer=steer_set,reverse=True y_6\!:~ throttle= _recover,\; brake=0,\; steer= steer\_set,\; reverse=True where the recovery steering angle steer_set is adaptively computed based on the spatial error vector: steer_set=θmax,if x<xgoal−θmax,otherwise steer\_set= cases _max,&if x<x_goal\\ - _max,&otherwise cases (11) I-E2 System Calibration and Generalization Table I lists the 16 hand-chosen values used in this study. Parameterization makes these choices visible, but it does not make the state-transition logic environment independent. In particular, YboundaryY_boundary is an absolute coordinate in the CARLA map, the terminal guard uses the goal’s x coordinate, and the overrides use fixed actuator values and durations. A different layout, vehicle, localization system, or sensor suite may require changes to both parameters and guards. We have not measured the amount of retuning required. IV Experiments IV-A Experimental Setup The network architecture keeps the same as the original one and hyperparameters are slightly different. The network was trained on 4 NVIDIA RTX 4090 (total VRAM: 96 GB) with a batch size of 16. Adam optimizer is utilized with an initial learning rate of 7.5×10−57.5× 10^-5 and a weight decay of 1×10−41× 10^-4, while the beta values are 0.9 and 0.999 as default. IV-B Dataset Details The training dataset, Gen 2B of the E2E Parking Dataset [1], comprises 248 training and 104 validation trajectories. As illustrated in Fig. 4, training/validation slots (marked in red and brown) utilize even-indexed slots (2-2, …, 2-16; 3-2, …, 3-16) for expert data generation, while odd-indexed slots (2-1, …, 2-15; 3-1, …, 3-15) are reserved for closed-loop evaluation. The data generation process is consolidated as follows: • Nominal Trajectory Collection (Seeds 0–47): For each of the 16 slots, scenarios are iteratively instantiated across 48 random seeds. Seeds 0–31 yield 192 training and 64 validation trajectories by querying six fixed BEV initial poses (far, middle, and near across left/right sides) and two distance-randomized lateral poses. Seeds 32–47 contribute 32 training trajectories (far-left/right) and 32 validation trajectories (middle-left/right). • Illumination-Adverse Scenarios (Seeds 11, 12, 27, 28): To introduce visual corner cases, the solar altitude and azimuth angle are dynamically modulated to cast explicit streetlight shadows onto specific slots. Under seeds 11 and 27, shadows are cast on slot 3-8 (Fig. 5a, 5b); under seeds 12 and 28, shadows are projected onto slot 3-10 (Fig. 5c, 5d). Each of these four distinct adversarial scenarios contributes 6 training and 2 validation trajectories using the baseline spatial initializations, yielding 24 training and 8 validation trajectories in total. Fig. 4: Top view of the parking lot in CARLA. (a) (b) (c) (d) Fig. 5: (a) Scene with random seed = 11, when target slot is 3-8 with shadow. (b) Scene with random seed = 27, when target slot is 3-8 with shadow. (c) Scene with random seed = 12, when target slot is 3-10 with shadow. (d) Scene with random seed = 28, when target slot is 3-10 with shadow. IV-C Metrics The metrics used for model testing remain the same as the original ones [2]. Target Success Rate (TSR) indicates the probability of the ego vehicle successfully parking in the set target slot. A parking attempt is judged as successful if the vehicle’s center is within 0.6 meters horizontally and 1 meter longitudinally from the slot’s center, with an oriental error of not higher than 10 degrees. Target Failure Rate (TFR) indicates the likelihood that the ego vehicle arrives at the designated parking slot but errors beyond the acceptable threshold. Non-Target Rate (NTR) means the probability that the ego vehicle parks in a slot other than the intended target. NTR is the sum of the Non-Target Success Rate (NTSR) and Non-Target Failure Rate (NTFR), depending on the final position within the non-target slot meets acceptable error margins. Collision Rate (CR) measures the frequency of collisions that occur during the parking maneuver. Outbound Rate (OR) refers to the probability that the ego vehicle exits from the parking lot. Timeout Rate (TR) indicates the probability that the ego vehicle fails to park successfully within a predefined time constraint (30 s30\,s). In the original work, the authors aggregate OR and TR, referring to their sum as TR. Average Position Error (APE) is the mean Euclidean error between the ego vehicle’s final position and the center of the target parking slot in successful parking tasks. Average Orientation Error (AOE) reflects the average yaw angle deviation between the ego vehicle’s final heading and the desired orientation of the parking slot for successful attempts. Average Parking Time (APT) indicates the mean duration taken for successful parking maneuvers. Average Inference Time (AIT) represents the average computational time spent per step during model inference. IV-D Closed-Loop Evaluation Results We evaluate the retrained nominal policy and its PSS-supervised version in closed loop in CARLA: executed controls change the vehicle state and hence subsequent observations. Each reported configuration has 16×6×4=38416× 6× 4=384 attempts: 16 held-out target slots, six initial-pose trials per slot, and four evaluation rounds for each slot–pose combination. Thus 384 is the number of attempts, not the number of independent parking layouts or vehicle configurations. Obstacle placement can vary with the scenario seed, but all attempts use the same parking lot, vehicle, sensor arrangement, and success rule. The vehicle has a 30s30\ s time limit. The tables aggregate attempts across rounds; they do not report per-round variability, so we treat the differences as descriptive results for this test bed rather than evidence of a population-level reliability or cross-environment generalization claim. IV-D1 Test with Speed Limit and Pareto Trade-Off When the target slot is 2-1 or 3-1 with the ego vehicle facing outward, the baseline model frequently exits the operational zone and acts abnormally. Moreover, due to the slot distribution constraints set by the original authors, the model lacks exposure to streetlight-adjacent slots (e.g., 2-9, 3-9), resulting in delayed braking and subsequent collisions. In rare corner cases, an abnormal control chattering occurs where brake and throttle signals alternate repeatedly post-stopping, inducing a terminal timeout. We vary vlimitv_limit within the PSS in the same evaluation setting. Figure 6 reports a TSR of 97.66%97.66\% and APT of 21.34 s21.34\,s at 7.5km/h7.5\ km/h. The 5.0km/h5.0\ km/h setting has the same reported TSR but a longer APT (25.54 s25.54\,s); the 12.5km/h12.5\ km/h setting has a lower reported TSR (94.01%94.01\%) and shorter APT (18.37 s18.37\,s). We use 7.5km/h7.5\ km/h for the remaining results. This selection is specific to the tested lot and does not certify a universally safe speed. Fig. 6: Pareto frontier showcasing the trade-off between TSR and APT under different physical speed limits vlimitv_limit. IV-D2 Test with Rule-based Correction Table I reports a TSR of 97.656% (375/384), compared with 85.156% (327/384) for the same retrained policy without the PSS: a descriptive increase of 12.5 percentage points, or 48 target successes. TR decreases from 7.552% to 0.781%, while TFR rises from 0 to 0.260%; thus the supervisor does not improve every failure category. Among successful attempts, APE changes from 0.237 to 0.211 m and AOE from 0.335∘ to 0.325∘. APT changes from 22.080 to 21.341 s. The aggregate table alone does not establish statistical significance, and its accuracy/time means condition on successful attempts whose composition differs between configurations. At the slot level, reported TSR improves or stays unchanged for 14 of 16 slots and decreases for slots 2-7 and 2-15. The gains at slots 2-1, 2-9, 3-1, and 3-9 are consistent with the specific boundary-exit and delayed-braking failures used to design the guards. Some collisions and timeouts remain in Table I. TABLE I: Closed-loop results for the retrained policy with PSS: 384 repeated attempts (1616 slots × 66 initial-pose trials × 44 rounds) in one CARLA lot. TaskIdx TSR (%) TFR (%) NTSR (%) NTFR (%) CR (%) OR (%) TR (%) APE (m) AOE (deg) APT (s) AIT (s) 2-1 87.500 4.167 4.167 0.000 4.167 0.000 0.000 0.199 0.600 21.156 0.077 2-3 95.833 0.000 0.000 0.000 0.000 0.000 4.167 0.158 0.232 20.126 0.077 2-5 100.000 0.000 0.000 0.000 0.000 0.000 0.000 0.232 0.206 20.357 0.077 2-7 95.833 0.000 0.000 0.000 4.167 0.000 0.000 0.247 0.204 21.181 0.077 2-9 100.000 0.000 0.000 0.000 0.000 0.000 0.000 0.219 0.383 22.443 0.077 2-11 100.000 0.000 0.000 0.000 0.000 0.000 0.000 0.235 0.416 20.686 0.078 2-13 100.000 0.000 0.000 0.000 0.000 0.000 0.000 0.196 0.308 20.896 0.078 2-15 91.667 0.000 0.000 0.000 4.167 0.000 4.167 0.185 0.351 20.347 0.077 3-1 100.000 0.000 0.000 0.000 0.000 0.000 0.000 0.233 0.668 21.694 0.077 3-3 100.000 0.000 0.000 0.000 0.000 0.000 0.000 0.170 0.321 21.574 0.078 3-5 100.000 0.000 0.000 0.000 0.000 0.000 0.000 0.267 0.482 22.136 0.078 3-7 100.000 0.000 0.000 0.000 0.000 0.000 0.000 0.169 0.132 20.804 0.078 3-9 100.000 0.000 0.000 0.000 0.000 0.000 0.000 0.270 0.248 20.517 0.078 3-11 100.000 0.000 0.000 0.000 0.000 0.000 0.000 0.208 0.231 21.758 0.077 3-13 95.833 0.000 0.000 0.000 0.000 0.000 4.167 0.139 0.200 23.896 0.077 3-15 95.833 0.000 0.000 0.000 4.167 0.000 0.000 0.243 0.221 21.883 0.077 Avg 97.656 0.260 0.260 0.000 1.042 0.000 0.781 0.211 0.325 21.341 0.077 IV-D3 Visualization of Comparisons Parking Slot 2-1 Figure 7 illustrate multiple failure cases during the closed-loop test with the original model, when the target is parking slot 2-1. The model has various incorrect inferences when the vehicle is located at the edge of the parking lot, where a typical problem is that it cannot track the target slot. In Figure 7a, the vehicle fails to stop in a non-target slot 3-1 with activated throttle and collides with the static vehicle at the back. In Figure 7b and Figure 7c, the vehicle stops with activated brake outside of a non-target slot 3-1, as if there were an additional slot between slot 3-1 and the sidewalk. In Figure 7d, the vehicle exits from the parking lot and it attempts to be parked on the road, which is regarded as an extremely dangerous operation. Figure 8 depicts the corrected trajectory with the help of rule-based correction. With an earlier braking point, the vehicle stops at the correct location, and the subsequent 2.5-second timed automaton signal output is sufficient to correct its behavior, enabling it to complete the following parking maneuver without any further intervention. From the above cases, it can be judged that rule-based correction is necessary to get deployed to constrain the control signals from model inference. (a) Case a (b) Case b (c) Case c (d) Case d Fig. 7: Visualization of four failure cases. Fig. 8: Visualization of a corrected trajectory. (a) T=4.5 sT=$4.5\,s$ (b) T=4.6 sT=$4.6\,s$ (c) T=5.1 sT=$5.1\,s$ (d) T=8.8 sT=$8.8\,s$ (e) T=9.5 sT=$9.5\,s$ (f) T=16.8 sT=$16.8\,s$ (g) T=17.6 sT=$17.6\,s$ (h) T=21.8 sT=$21.8\,s$ (i) T=30 sT=$30\,s$ Fig. 9: BEV features of a timeout case. The failure cases can be explained by the generated BEV features. Figure 9 displays BEV features of a timeout case in different time steps, where drivable area is black, predicted target slot is write and obstacles are grey. The target in the test is slot 2-1. The BEV feature remains to be correct until T=4.5 sT=$4.5\,s$. After that, the segmentation output of the target parking slot disappears and moves to the area between slot 2-1 and sidewalk. At T=9.5 sT=$9.5\,s$, the vehicle stops at the edge of the parking lot and the segmentation of target is located at the vehicle’s current position. At T=16.8 sT=$16.8\,s$ the vehicle starts to reverse in the opposite direction of the target and the BEV feature is confusing. In the end the vehicle stops at the area between slot 3-1 and sidewalk, with the predicted target segmentation at the same position away from the ground-truth target. Meanwhile, the segmentation of other vehicles is problematic, which also leads to incorrect output signals. Figure 10 demonstrates BEV features of a outbound case in different time steps. The BEV features in the first 5 seconds remain the same as those in the timeout case. At T=8.6 sT=$8.6\,s$ the target segmentation moves outside of the parking lot and the vehicle continues moving out of the parking lot without slowing down. Afterwards, the vehicle approaches the green area and target segmentation is abnormal. It disappears unexpectedly and then appears at the vehicle’s current position, which is far away from the ground-truth target. (a) T=4.0 sT=$4.0\,s$ (b) T=4.1 sT=$4.1\,s$ (c) T=4.4 sT=$4.4\,s$ (d) T=6.8 sT=$6.8\,s$ (e) T=8.6 sT=$8.6\,s$ (f) T=9.5 sT=$9.5\,s$ Fig. 10: BEV features of an outbound case. Fig. 11: Comparison of abnormal and corrected control signal. Figure 12 demonstrates BEV features corrected by rules. The transformation of BEV feature at an early stage is similar to the above 2 cases shown in Figure 9a, 9b and 9c or Figure 10a, 10b and 10c. When the vehicle stops at the edge of the parking lot, the target segmentation disappears and the target becomes unknown to the model. In the process of forced reverse set by rule, the target slot area can be gradually segmented, which leads to a success parking in the end. The segmentation of target becomes explicit and correct after the activation of timed automaton. (a) T=8.6 sT=$8.6\,s$ (b) T=9.1 sT=$9.1\,s$ (c) T=9.4 sT=$9.4\,s$ (d) T=9.5 sT=$9.5\,s$ (e) T=9.8 sT=$9.8\,s$ (f) T=14.0 sT=$14.0\,s$ Fig. 12: BEV features corrected by rules. Parking Slot 3-9 Figure 11 compares the control signals with and without the intervention of rule-based correction after stopping of the vehicle. The y-axis is control value of throttle and brake, where positive values stand for throttle and negative values indicate brake. Although the vehicle is parked into the target slot with allowable positional and oriental errors, the vehicle may frequently get throttle signals after it stops, which is certainly considered as a dangerous behavior. This alternating output of throttle and brake is judged as a timeout, since maintaining a throttle output of zero for 2 seconds is a necessary condition for determining a successful parking. With the help of rule-based correction, the brake value remains to be 0.9 in order to force the vehicle to keep static state. The BEV feature may become abnormal in the end of the parking task. Since there is no segmentation label for streetlight, it is challenging to avoid collision with streetlight. In Figure 13, the segmentation result is distorted and the vehicle fails to stop at the right moment, colliding with the post of a streetlight. In Figure 14, the vehicle is forced to get braked at the designated position and the segmentation remains to be normal and explicit. In some cases where BEV features are abnormal (like Figure 15), the PSS overwrites the model’s wrong control signals influenced by distorted BEV features and forces the vehicle to stop at the right position. If there were no PSS, the shrinking of the target segmentation may cause unexpected problems. (a) T=16.8 sT=$16.8\,s$ (b) T=17.3 sT=$17.3\,s$ (c) T=18.2 sT=$18.2\,s$ Fig. 13: BEV features before collision at target slot 3-9. (a) T=15.8 sT=$15.8\,s$ (b) T=17.3 sT=$17.3\,s$ (c) T=18.7 sT=$18.7\,s$ Fig. 14: BEV features during forced brake at target slot 3-9. (a) T=10.7 sT=$10.7\,s$ (b) T=14.6 sT=$14.6\,s$ (c) T=18.3 sT=$18.3\,s$ Fig. 15: Abnormal BEV features during forced brake at target slot 3-9. IV-D4 Comparison Scope and Recent Parking Methods Table IV retains the reported Yang et al. model and expert results for context. Only the two Gen 2B policy rows isolate the PSS within the evaluation reported here. The Yang et al. models, expert demonstrations, and our policy use different data/model availability or evaluation conditions; numerical differences between those rows cannot be attributed to the PSS. The expert rows are demonstration quality references, not deployable policy baselines. Table I compares our approach with two recent peer-reviewed parking methods at the level of method, output, and evaluation domain. ParkingE2E [7] learns image-to-waypoint planning and reports 87.8% mean success across four real garages. HOPE [8] learns a hybrid path planner using reinforcement learning and Reeds–Shepp curves, tests difficulty-stratified scenarios, and includes a real-world demonstration. Our method instead modifies low-level controls from an inherited network and is evaluated in one CARLA lot. Different sensors, outputs, vehicles, success definitions, and environments prevent a defensible ranking from their published rates. Reimplementing both methods and evaluating them under a shared protocol would be required to satisfy a direct head-to-head comparison; such experiments are not part of the available results. TABLE I: Methodological comparison with recent peer-reviewed parking work. Published evaluations use different protocols; this table is not a head-to-head performance comparison. Method Core decision mechanism Output and inputs Reported evaluation scope ParkingE2E [7] Learned target-query Transformer Future waypoints from surround-view cameras Four real garages and real-vehicle tests; 87.8% mean parking success under its own protocol HOPE [8] Reinforcement learning plus Reeds–Shepp curves and action masking Parking path from its planner state representation Difficulty-stratified parking scenarios and real-world experiments This work Inherited E2E Parking Transformer plus hand-coded timed output supervisor Low-level controls from cameras, target, vehicle state, and CARLA world coordinates One CARLA lot, one vehicle/sensor setup; 16 slots × 6 initial poses × 4 rounds TABLE IV: Reported policy and expert results for context. Only the two Gen 2B policy rows are the within-study comparison of the PSS. TaskIdx TSR (%) TFR (%) CR (%) APE (m) AOE (deg) APT (s) E2E Parking [2] Unreleased Model 91.41 2.08 2.08 0.30 0.87 15.72 E2E Parking [2] Released Model 75.00 1.56 10.42 0.25 0.63 18.41 Model trained on Gen 2B [1] (Baseline) 85.16 0.00 2.87 0.24 0.34 22.08 Model trained on Gen 2B [1] + Rule-based Correction 97.66 0.26 1.04 0.21 0.33 21.34 Original Expert [2] 100.00 0.00 0.00 0.23 0.48 14.96 New Expert 100.00 0.00 0.00 0.13 0.05 16.95 IV-D5 Failure Case Analysis and Limitations Closed-loop evaluations reveal two primary limitations of the proposed framework. First, as qualitatively illustrated in Fig. 16, if the baseline model severely overshoots the designated slot, the PSS successfully prevents out-of-boundary run-off but lacks a proximity-based fall-back mechanism to command an immediate stop, causing the vehicle to eventually park in an incorrect non-target slot. Second, the system occasionally suffers from control chattering in obstacle-free paths, where conflicting neural outputs and conservative PSS regulations trigger alternating throttle and braking cycles. While the vehicle is still guided safely into the target slot, this efficiency degradation ultimately results in a terminal timeout failure. Fig. 16: Example of a closed-loop failure after target-slot overshoot. IV-E Ablation Study We ablate three guard components using the same 16×6×416× 6× 4 evaluation structure. Table V reports lower TSR after removing target alignment (93.49%), speed limiting (91.93%), or geofencing (89.84%) than with the complete PSS (97.66%). The respective APTs are 21.66, 18.07, 21.54, and 21.34 s. These descriptive results support the usefulness of the tested guards in this lot. They do not show that each guard is necessary in every parking environment, nor do they prove safety. TABLE V: PSS component ablations over 384 repeated attempts per configuration in one CARLA lot Configuration TSR (%) APT (s) (i) w/o Target Alignment 93.49 21.66 (i) w/o Speed Limit 91.93 18.07 (i) w/o Geofencing 89.84 21.54 (iv) Complete PSS 97.66 21.34 V Conclusion This study adds a hand-designed timed output supervisor to the existing E2E Parking network. In our single-lot CARLA evaluation, the retrained policy’s target successes increase from 327 to 375 out of 384 repeated attempts with the PSS. The ablations associate the observed improvement with several guards, but neither these results nor the state-machine specification establish a formal safety guarantee or generalization beyond this configuration. The rules were selected after manual inspection of failures and include fixed actuator values, durations, and map-specific coordinates. All 384 attempts per configuration arise from only 16 slots, six pose trials, and four rounds in the same CARLA lot, vehicle, and sensor setup. Four rounds do not characterize reliability across independently sampled environments; no different lot, vehicle, sensor suite, or real-world test was performed. The PSS uses simulator world coordinates that a real vehicle would need to estimate with uncertainty. Our earlier experiment with target coordinates in the vehicle frame was less stable, and no robust localization replacement is demonstrated here. Finally, a direct same-protocol experiment against ParkingE2E or HOPE remains outstanding. Future evaluation should report round- and scenario-level outcomes and test more independently sampled lots, obstacle layouts, vehicles, and sensors. A controlled comparison with recent parking methods requires a common scenario set and success definition. Before real-vehicle use, the guards also need a localization source, uncertainty handling, and safety analysis. To support reproducibility and further research, we release the full implementation of our framework. We hope this work encourages more exploration into the coordination between visual-based end-to-end models and rule-based traditional controllers in autonomous driving, especially in structured, low-speed and low-risk parking scenarios. References [1] K. Gao, L. Zhou, M. Liu, and A. Knoll (2025) E2E parking dataset: an open benchmark for end-to-end autonomous parking. External Links: 2504.10812, Link Cited by: §I, §I-A, §IV-B, TABLE IV, TABLE IV. [2] Y. Yang, D. Chen, T. Qin, X. Mu, C. Xu, and M. Yang (2024) E2e parking: autonomous parking by the end-to-end neural network on the carla simulator. In 2024 IEEE Intelligent Vehicles Symposium (IV), p. 2375–2382. Cited by: §I, §I-A, Fig. 1, Fig. 2, §I-C, §I, §IV-C, TABLE IV, TABLE IV, TABLE IV. [3] K. Chitta, A. Prakash, B. Jaeger, Z. Yu, K. Renz, and A. Geiger (2022) Transfuser: imitation with transformer-based sensor fusion for autonomous driving. IEEE transactions on pattern analysis and machine intelligence 45 (11), p. 12878–12895. Cited by: §I-A. [4] H. Shao, L. Wang, R. Chen, H. Li, and Y. Liu (2023) Safety-enhanced autonomous driving using interpretable sensor fusion transformer. In Conference on Robot Learning, p. 726–737. Cited by: §I-A. [5] Y. Hu, J. Yang, L. Chen, K. Li, C. Sima, X. Zhu, S. Chai, S. Du, T. Lin, W. Wang, et al. (2023) Planning-oriented autonomous driving. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, p. 17853–17862. Cited by: §I-A. [6] X. Shen, M. Lacayo, N. Guggilla, and F. Borrelli (2022) Parkpredict+: multimodal intent and motion prediction for vehicles in parking lots with cnn and transformer. In 2022 IEEE 25th International Conference on Intelligent Transportation Systems (ITSC), p. 3999–4004. Cited by: §I-A. [7] C. Li, Z. Ji, Z. Chen, T. Qin, and M. Yang (2024) ParkingE2E: camera-based end-to-end parking network, from images to planning. In 2024 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), p. 13206–13212. External Links: Document Cited by: §I-A, §IV-D4, TABLE I. [8] M. Jiang, Y. Li, S. Zhang, S. Chen, C. Wang, and M. Yang (2025) HOPE: a reinforcement learning-based hybrid policy path planner for diverse parking scenarios. IEEE Transactions on Intelligent Transportation Systems 26 (5), p. 6130–6141. External Links: Document Cited by: §I-A, §I-C, §IV-D4, TABLE I. [9] J. Philion and S. Fidler (2020) Lift, splat, shoot: encoding images from arbitrary camera rigs by implicitly unprojecting to 3d. In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XIV 16, p. 194–210. Cited by: §I-B, §I-C1. [10] Y. Li, Z. Ge, G. Yu, J. Yang, Z. Wang, Y. Shi, J. Sun, and Z. Li (2023) Bevdepth: acquisition of reliable depth for multi-view 3d object detection. In Proceedings of the AAAI conference on artificial intelligence, Vol. 37, p. 1477–1485. Cited by: §I-B, §I-D. [11] F. Bouchard, S. Sedwards, and K. Czarnecki (2022) A rule-based behaviour planner for autonomous driving. In International Joint Conference on Rules and Reasoning, p. 263–279. Cited by: §I-C. [12] A. Likmeta, A. M. Metelli, A. Tirinzoni, R. Giol, M. Restelli, and D. Romano (2020) Combining reinforcement learning with rule-based controllers for transparent and general decision-making in autonomous driving. Robotics and Autonomous Systems 131, p. 103568. Cited by: §I-C. [13] T. Chen, S. Saxena, L. Li, D. J. Fleet, and G. Hinton (2021) Pix2seq: a language modeling framework for object detection. arXiv preprint arXiv:2109.10852. Cited by: §I-B. Kejia Gao received a bachelor’s degree in Mechatronics from Tongji University in 2022 and a master’s degree in Robotics, Cognition, and Intelligence from the Technical University of Munich in 2025. He is currently working at the Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences. His research focuses on computer vision and AI-driven optics. Liguo Zhou received a bachelor’s degree in software engineering from Suzhou University, a master’s degree in pattern recognition and intelligent systems from Wuhan University, and a doctoral degree in computer science from the Technical University of Munich. He conducts research in artificial intelligence at Huaibei Normal University, with interests in computer vision, autonomous driving, and embodied intelligence. Lei Yu is a professor in the School of Computer Science and Technology at Huaibei Normal University. His research interests include artificial intelligence, information security, and security protocols. Alois Knoll (Fellow, IEEE) received a diploma in electrical engineering from the University of Stuttgart in 1985 and a doctorate in computer science from the Technical University of Berlin in 1988. He was a professor at Bielefeld University from 1993 to 2001 and has been a professor at the Technical University of Munich since 2001. His research focuses on autonomous systems, robotics, and artificial intelligence.

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