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Closed-Loop Refinement and Execution for Learned Driving Planners

Huaijin Hu, Shanting Wang, Zhongyu Mo, Andreas A. Malikopoulos

arXiv:2610.00992v1eess.SYcs.ROcs.SY

TL;DR

Open-loop trajectory accuracy does not ensure reliable closed-loop driving, where planners may stall, move toward predicted conflicts, or require abrupt braking. CLRE refines and screens trajectories from a frozen planner, then adds execution-level intervention; on Bench2Drive, it improves route completion and driving score while reducing collision events.

  • Problem

    Open-loop trajectory performance can be misaligned with closed-loop execution, where learned planners may stall, move toward predicted conflicts, or require abrupt longitudinal intervention.

    Source: §I · p. 1

  • Method

    CLRE keeps the upstream planner frozen and combines trajectory refinement with prediction-conditioned OBB feasibility screening and execution-level intervention.

    Source: §I · pp. 1–2

  • Results

    On Bench2Drive, CLRE improved route completion and driving score while reducing collision events.

    Source: §V · p. 8

  • Takeaways & Limitations

    An eleven-route check with two additional planners provided preliminary evidence of cross-planner transfer without retuning.

    Source: §V · p. 8

    The study is limited to simulation, and its feasibility test depends on predicted agent motion.

    Source: §V · p. 8

Abstract

from arXiv · show

Learning-based driving planners are usually trained and evaluated in open loop against logged trajectories. In closed loop, a trajectory with small displacement error can still stall the vehicle, steer it into a conflict with surrounding agents, or be executed with abrupt braking. We introduce Closed-Loop Refinement and Execution (CLRE), a hierarchical receding-horizon control framework designed to mitigate these failure modes while leaving the upstream planner frozen and adding no new learned model. The upper layer treats the nominal trajectory as a reference and solves a finite-horizon optimal control problem that trades route progress against interaction with predicted agents. Solving it from several initializations gives a candidate set, and a prediction-conditioned oriented-bounding-box (OBB) feasibility test retains only candidates whose minimum predicted OBB clearance over the horizon meets a threshold. The lower layer executes the lowest-cost survivor, or a route-centerline backup when none remains, through the tracking controller supplied with the planner, augmented by a range-based speed bound and a saturated proportional braking law. In closed-loop simulation on 126 Bench2Drive routes with VAD as the upstream planner, CLRE raises the driving score from 43.41 to 56.42 and route completion from 57.27 to 72.23, and reduces collision events from 70 to 53.

I. INTRODUCTION

CLRE addresses the mismatch between open-loop trajectory accuracy and reliable closed-loop driving by refining a frozen planner’s trajectory, screening candidates, and intervening during execution. Its refinement uses a vehicle model and constraints in a local frame at the current ego pose.

I. INTRODUCTION

  • Open-loop accuracy does not ensure reliable closed-loop execution, where planners can stall, move toward predicted conflicts, or require abrupt longitudinal intervention.Prior runtime checks can leave an overly cautious plan unchanged, while optimization without a separate feasibility check can leave too little clearance.Source: §I · p. 1
  • CLRE refines the frozen planner’s nominal trajectory, screens the resulting candidates with a prediction-conditioned OBB test, and adds execution-layer intervention without a new learned model.The refinement balances nominal-path tracking, route progress, and predicted-agent interaction.Source: §I · pp. 1–2

A. Vehicle Model and Constraints

  • At each replanning instant, CLRE optimizes ego motion over a finite horizon using a local frame fixed at the current ego pose.The frame’s position axes point rightward and forward, and positive curvature denotes a right turn.Source: A. Vehicle Model and Constraints · p. 2, Figure 1 · p. 2
  • Ego motion is represented by position, heading, and longitudinal speed, with the initial speed set to the measured speed at replanning.The kinematic model uses acceleration and curvature to evolve speed, heading, and position.Source: A. Vehicle Model and Constraints · p. 2
  • The frozen planner’s nominal trajectory serves as a refinement reference but is not assumed to satisfy the vehicle dynamics or state and input constraints.The control input comprises longitudinal acceleration and path curvature, with a prescribed curvature limit.Source: A. Vehicle Model and Constraints · p. 2

B. Receding-Horizon Refinement Problem

The receding-horizon formulation optimizes a control sequence over a finite horizon, generating the corresponding ego-state sequence under the vehicle model and constraints. The selected refined path is downsampled for the planner’s downstream tracking controller.

B. Receding-Horizon Refinement Problem

A. Overview

CLRE combines trajectory refinement with candidate selection and an execution layer. Its trajectory-level objective retains the learned planner’s intent while allowing progress along the nominal path at a different rate.

A. Overview

  • The architecture constructs a refinement objective, candidate-selection test, and execution layer to solve and execute the receding-horizon problem.Source: A. Overview · p. 3

B. Trajectory-Level Terms

  • Point-wise nominal tracking anchors the refinement to the learned planner’s time-parameterized motion, while Huber loss permits larger deviations when other terms favor them.Small deviations are penalized quadratically and larger deviations linearly.Source: B. Trajectory-Level Terms · p. 3
  • Geometric path commitment penalizes spatial departure from the nominal path while allowing the refined trajectory to advance along it at a different rate.The path is extended along its terminal tangent to prevent the finite nominal endpoint from acting as an artificial stopping point.Source: B. Trajectory-Level Terms · p. 3
  • When the nominal trajectory shows little forward progress, the commitment gate decreases its influence, while minimizing the progress term rewards terminal displacement along the route direction.The remaining objective terms determine whether that progress is compatible with the nominal path, surrounding traffic, and road geometry.Source: B. Trajectory-Level Terms · p. 3

C. Interaction-Level Terms

CLRE shapes candidate trajectories with soft predicted-agent interaction costs and separately adds geometric boundary and route-aligned corridor penalties. The interaction terms use a simplified ego footprint and screen the highest-probability predicted motion mode.

C. Interaction-Level Terms

  • CLRE screens each retained agent using its highest-probability motion mode, while leaving uncertainty-aware multimodal screening to future work.An offline check found that screening all modes with probability at least 0.1 increased windows with no feasible candidate from 11.2% to 22.6%.Source: C. Interaction-Level Terms · p. 3
  • CLRE represents the ego footprint with three overlapping disks and uses outer and inner ellipses to shape interaction with each predicted agent.The outer ellipse penalizes approach in advance, while the tighter inner ellipse penalizes near-contact motion more strongly; these are soft penalties, not hard collision constraints.Source: C. Interaction-Level Terms · pp. 3–4
  • The interaction penalty rises smoothly as the ego footprint enters an interaction ellipse, remaining small outside it.The softplus approximation is applied to the penetration measure, with agent-mode confidence weighting the interaction cost.Source: C. Interaction-Level Terms · p. 4
  • A boundary-proximity cost increases as the refined trajectory enters a prescribed margin around solid boundaries and non-drivable regions.The margin sets where the penalty rises, and its sharpness parameter controls how quickly it increases.Source: C. Interaction-Level Terms · p. 4
  • A route-aligned corridor increasingly penalizes lateral departures beyond its half-width, complementing the explicit boundary cost.The corridor discourages large lateral excursions made in pursuit of route progress.Source: C. Interaction-Level Terms · p. 4

D. Ego-Motion-Level Terms

Ego-motion terms regularize speed relative to the nominal plan and discourage rapidly varying controls or unnecessarily sharp steering. The speed penalty is one-sided, allowing slower motion when interactions or road geometry require it.

D. Ego-Motion-Level Terms

  • The nominal speed reference comes from mean spacing between consecutive nominal waypoints divided by the waypoint interval, restricted to a prescribed speed range.This defines the reference used by the speed regularization term.Source: D. Ego-Motion-Level Terms · p. 4
  • The speed penalty discourages exceeding the nominal reference but does not penalize lower speeds, allowing slowing for predicted interactions or road geometry.This prevents route progress from being increased simply by driving substantially faster than the nominal plan.Source: D. Ego-Motion-Level Terms · p. 4
  • Input regularization discourages rapidly varying control profiles and unnecessarily sharp steering by penalizing control variation and curvature magnitude.The cited weights apply to curvature variation and curvature magnitude.Source: D. Ego-Motion-Level Terms · p. 5

E. Solution of the Refinement Problem

The refinement problem combines its objective terms with nonnegative weights fixed offline, then solves the nonconvex problem using Adam. Multiple behavior-structured initializations explore local solutions, but fixed-iteration solves are not claimed to find the global optimum.

E. Solution of the Refinement Problem

  • The objective weights are nonnegative and fixed offline.Their values are reported in Section IV-E.Source: E. Solution of the Refinement Problem · p. 5
  • Adam solves the nonconvex, piecewise-differentiable refinement problem with control variables clamped to acceleration and curvature bounds and rollout speed saturated to [0, vmax].The projected rollout therefore remains within the prescribed input and speed bounds.Source: E. Solution of the Refinement Problem · p. 5
  • Eight behavior-structured initializations explore local solutions, including nominal-following, braking, progress-oriented, accelerate-then-coast, and lateral-avoidance behaviors.When available, the shifted previous control sequence supplies an additional warm start, and its unoptimized shifted version is retained for candidate evaluation.Source: E. Solution of the Refinement Problem · p. 5
  • Each fixed-iteration Adam solve yields a locally refined candidate, not a claimed exact global solution.The paper evaluates sensitivity of candidate selection and feasibility to larger iteration budgets separately.Source: E. Solution of the Refinement Problem · p. 5

F. Prediction-Conditioned OBB Feasibility Test

The OBB test retains candidates whose minimum predicted clearance meets a prescribed threshold, then selects the feasible candidate with the lowest objective cost. The execution layer uses a route-centerline backup when none is feasible and augments waypoint tracking with a radar-based speed bound and braking intervention.

F. Prediction-Conditioned OBB Feasibility Test

  • The OBB clearance is positive for separated boxes and negative for overlapping boxes, with rectangle distance or penetration depth used accordingly.The signed clearance is computed from box-to-box geometry.Source: F. Prediction-Conditioned OBB… · p. 5
  • A candidate is feasible when its minimum predicted OBB clearance over the refinement horizon meets the prescribed positive threshold; CLRE selects the feasible candidate with the lowest objective cost.If no candidate passes, the backup policy is used; because the test relies on predicted agent motion, prediction errors can invalidate its clearance estimate.Source: F. Prediction-Conditioned OBB… · p. 5

G. Execution Layer

  • The tracking controller retains its baseline throttle PID, while the augmented execution logic can apply full braking after an emergency threshold persists or when commanded speed falls below a stopping threshold.The brake command is saturated to [0, 1].Source: G. Execution Layer · p. 5
  • When no refined candidate is feasible, the backup uses navigation-route centerline waypoints and holds the ego speed measured at backup entry until a feasible candidate returns.The reference waypoints are generated from the route centerline.Source: G. Execution Layer · p. 5
  • A forward-facing radar imposes a planner-independent speed ceiling for a closing return, based on stopping after reaction time while preserving a clearance margin.The stated stopping constraint uses reaction time, braking deceleration, range, and minimum clearance.Source: G. Execution Layer · p. 5

IV. SIMULATION RESULTS

The evaluation uses closed-loop Bench2Drive simulation with VAD as the main upstream planner. The setup fixes the refinement and execution parameters for reported evaluations, with objective weights and braking thresholds chosen heuristically.

A. Simulation Setup

  • 126 official validation routes: the main VAD evaluation covers 40 of 42 scenario types, while ablations use the first 36 routes.The reported metrics are Driving Score, Route Completion, Success Rate, and total collision events.Source: A. Simulation Setup · p. 6
  • The refinement horizon is 3 s with N = 12 intervals at ∆t = 0.25 s; the OBB clearance threshold is ϵ = 0.05 m.Other settings include vmax = 15 m/s and acceleration bounds of −6 to 2.5 m/s2.Source: A. Simulation Setup · p. 6
  • The execution layer uses forward radar only for its forward-speed bound and emergency-braking condition; lane boundaries and agent predictions come from the frozen planner.The route comes from the benchmark, and no ground-truth future states are used.Source: A. Simulation Setup · p. 6
  • Objective weights and braking thresholds were manually chosen heuristically, then frozen for all reported evaluations rather than optimized against benchmark performance.The sensitivity analysis assesses the nominal setting rather than tuning it.Source: A. Simulation Setup · p. 6

B. Closed-Loop Performance

On the 126-route VAD cohort, CLRE improves driving score and route completion while reducing collisions; ablations show distinct contributions from progress, OBB screening, and execution intervention. Sensitivity tests find no narrow operating range, though progress and commitment weights are more sensitive and systematic tuning remains future work.

B. Closed-Loop Performance

  • 13.01 and 14.96 points: CLRE improves Driving Score and Route Completion, respectively, on the 126-route VAD cohort.The paired bootstrap 95% confidence intervals are [8.3, 17.8] for Driving Score and [9.6, 20.5] for Route Completion; collision events fall from 70 to 53.Source: B. Closed-Loop Performance · p. 6
  • CLRE improves Driving Score and reduces collisions for UniAD and DriveTransformer, and improves Route Completion for UniAD, using shared CLRE parameters.The authors describe these results as preliminary evidence of transfer across upstream planners.Source: B. Closed-Loop Performance · p. 6

C. Planner Ablation

  • 17.6 points: removing the progress term lowers Route Completion; completed routes fall from 22 to 15 and mean speed from 2.43 to 1.64 m/s.The ablated variant also has fewer collisions, but stalled vehicles may not reach scenario conflict regions, so that reduction warrants caution.Source: C. Planner Ablation · p. 6
  • 70%: removing the OBB feasibility test raises collision rate per kilometer; collisions increase from 10 to 17 while distance driven stays within 0.2%.In the illustrated case, the lowest-cost candidate overlapping the predicted pedestrian box is rejected at −0.30 m, while a feasible candidate has a +0.96 m gap.Source: C. Planner Ablation · p. 6, Figure 5 · p. 7

D. Execution-Layer Ablation

  • 0.101 to 0.042: the execution layer reduces mean brake variation, while the jerk percentile falls from 38.3 to 28.7 and collisions per kilometer from 4.98 to 3.18.Driving Score, Route Completion, and completed-route count remain stable.Source: D. Execution-Layer Ablation · p. 6

B. Closed-Loop Performance

  • At t = 6.95 s, the nominal VAD vehicle has a zero-length plan and full brake applied, while CLRE travels at 6.1 m/s and completes the route.The baseline covers no distance and stalls until timeout.Source: Figure 4 · p. 7

E. Objective and Sensitivity Analysis

  • 94.0%: the progress term is the largest weighted contributor in moving solves; halving progress and commitment yields winner retention of 0.69 and 0.79.Other weights retain the selected candidate in over 0.9 of solves across the tested range; no weight has a narrow operating range, and systematic tuning remains future work.Source: E. Objective and Sensitivity Analysis · p. 7, Figure 6 · p. 7

F. Numerical Solver Analysis

Solver checks show that feasibility verdicts are stable across multi-restart search and clearance-threshold changes, although iteration budgets can alter selected candidates. The conclusion reports CLRE’s broader contributions and bounds the study to simulation and prediction-conditioned feasibility.

F. Numerical Solver Analysis

  • 99.54% agreement with the deployed feasibility verdict; raising the clearance threshold from 0.05 to 0.30 m changes the selected candidate in only 7.2% of windows.The table compares the deployed configuration with 2× and 5× iteration budgets.Source: F. Numerical Solver Analysis · pp. 7–8
  • Iteration-budget changes concentrate at the horizon end, while the first executed waypoint remains close across 2× and 5× solves.The largest deviation occurs at k = N in 74% and 76% of candidate-change cases; median first-waypoint difference is 14 mm under both budgets, with commanded-speed differences of 0.07 and 0.08 m/s.Source: F. Numerical Solver Analysis · pp. 7–8
  • CLRE’s current implementation is not real-time at 20 Hz, though vectorizing the feasibility test reduces its offline runtime from 28.9 ms to 0.64 ms with identical decisions.On the reported hardware, VAD averages 131.5 ms per control step and CLRE planning adds 100.4 ms; synchronous CARLA affects wallclock simulation speed, not simulated control timing.Source: F. Numerical Solver Analysis · p. 8

V. CONCLUSION

  • The conclusion presents CLRE as training-free closed-loop refinement and execution, with ablations supporting its components and an eleven-route check offering preliminary evidence of transfer to two additional planners without retuning.Source: §V · p. 8
  • The study is limited to simulation, and its feasibility test depends on predicted agent motion.Future work includes multimodal and uncertainty-aware screening, broader cross-backbone evaluation, and GPU-parallel real-time implementation.Source: §V · p. 8
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