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Probability-Based Collision Risk Evaluation of Trajectories for Optimal Control Problems with Moving Obstacles

Florian Steppich, Matthias Gerdts

arXiv:2609.38534v1eess.SYcs.SY

TL;DR

Potential-field planning can omit global information and use non-differentiable functions that hinder derivative-based optimization. This paper approximates occupancy distributions with smooth functions interpreted as Poisson random-field intensities, and reports up to 50% lower solver computational load than an error-only approximation configuration. The method’s uniform knot placement may be suboptimal for fine spatial structures.

  • Problem

    Potential-field methods may neglect global information and use non-differentiable functions that hinder optimization methods relying on derivative information.

    Source: §1 · p. 3

  • Method

    The paper approximates occupancy distributions with smooth, non-negative functions and interprets them as Poisson random-field intensity functions to aggregate multiple obstacles.

    Source: §1 · p. 3, §5 · p. 25

  • Results

    Up to 50% lower solver computational load was reported for a configuration balancing approximation fidelity and surface variability, compared with one minimizing approximation error alone.

    Source: §5 · p. 25

  • Takeaways & Limitations

    The intensity interpretation retains probabilistic semantics while enabling straightforward aggregation of multiple obstacles.

    Source: §5 · p. 25

    Uniform knot placement may be suboptimal for fine spatial structures, and adaptive placement raises temporal-consistency challenges when occupancy distributions are unavailable at every OCP shooting node.

    Source: §6 · p. 25

Abstract

from arXiv · show

This paper presents a method for approximating occupancy distributions, a common probabilistic representation used in trajectory planning for autonomous vehicles operating in uncertain environments. The proposed method employs B-spline surfaces in conjunction with regularization techniques to ensure smoothness and differentiability. Interpreting the resulting approximation as an intensity function of a Poisson random field enables seamless integration of multiple occupancy distributions into a coherent risk representation, while retaining compatibility with the semantics of probability distributions. The method is tailored to optimal control problems (OCPs), where solvers benefit from gradient and higher-order derivative information. It also preserves information about the spatial and temporal structure of occupancy distributions, which is critical for handling dynamic obstacles. We demonstrate the feasibility of the approach in a trajectory planning scenario involving autonomous vehicles and moving obstacles with time-varying uncertainty. By enabling controlled risk acceptance, our formulation extends beyond conservative "no-collision" strategies and allows for admissible trajectories that would otherwise be ruled out. In our application scenario, we report a reduction in solver computational load of up to 50\% through proper tuning of the regularization.

1. Introduction

Planning around moving obstacles requires accounting for their changing uncertainty while producing trajectories that remain tractable for optimization. The paper proposes a time-dependent collision-risk map designed for optimal control and differentiable solvers.

1. Introduction

  • Existing approaches use simplified or static obstacle distributions, and typically do not update obstacle motion during each planning iteration.The introduction also surveys approaches using potential fields, occupancy grids, Gaussian collision probabilities, and CVaR risk measures.Source: §1 · pp. 2–3
  • Potential-field methods can neglect global information and use non-differentiable functions, while strict no-collision strategies may leave the planner without a feasible trajectory.Smoothness and curvature constraints can also be cumbersome in potential fields, and occupancy grids need additional processing to produce smooth, physically feasible trajectories.Source: §1 · p. 3
  • The proposed framework quantifies time-dependent collision risk from multiple moving obstacles’ occupancy distributions using smooth, non-negative functions suited to gradient-based OCP solvers.An occupancy distribution describes the probability distribution over an obstacle’s occupiable future locations; regularization is used to support smoothness.Source: §1 · p. 3
  • The framework assumes updated obstacle-state estimates are available at the start of each planning iteration; the guiding MPC scheme and uncertainty propagation are outside its scope.The cited examples of uncertainty propagation include.Source: §1 · p. 3
  • Although developed for OCPs, the continuous probabilistic collision-risk measure may also benefit graph-based and sampling-based planning algorithms.The related work includes collision-severity cost fields, with human collisions penalized more than vehicle collisions.Source: §1 · pp. 1–3

2. Prerequisites

The section models moving obstacles through time-varying occupancy distributions and compares ways to combine them into collision risk. It favors a Poisson-intensity formulation and uses B-spline approximations to provide the smoothness required by derivative-based optimization.

2. Prerequisites

  • The ego vehicle uses a single-track kinematic model, while each obstacle is approximated as a point mass; vehicle dimensions are handled in the OCP objective.Obstacle states include position and heading in the global frame.Source: §2 · p. 4

2.1. Interpretation

  • Occupancy distributions are assumed to be supplied and can be moved and rotated from a nominal state to match an obstacle’s initial state.This relies on obstacle dynamics being invariant to transformations of the initial state.Source: §2.1 · pp. 5–6
  • A disk around the ego trajectory represents its physical footprint, with its size derived from ego and obstacle dimensions plus any safety buffer.This frames risk around whether the vehicle is in a risky region, rather than at a risky point.Source: §2.1 · p. 6
  • The collision-free-probability view multiplies the complements of obstacle collision probabilities, yielding a value in [0, 1] but potentially very small for high risks or long trajectories.The passage notes that log-likelihoods could address the numerical issue.Source: §2.1 · p. 6
  • CVaR measures expected risk in the worst (1 − α) · 100% of collision scenarios, but its sampling or approximation requirements may make real-time computation infeasible.The paper says this tail-focused perspective is not relevant to its scenario, where risk acceptance is considered only when external circumstances require it.Source: §2.1 · pp. 6–8
  • The Poisson-field interpretation treats occupancy distributions as spatial occurrence rates, so their integral over a trajectory’s spatio-temporal region gives the expected number of collisions.The formulation assumes finite total intensity, measurable trajectory disks, and independent point occurrences.Source: §2.1 · pp. 7–8

2.2. B-Spline Surface

  • Nonnegative B-spline surfaces approximate occupancy distributions in Wd,∞, providing bounded derivatives for OCP solvers; normalization preserves their probability interpretation.The intensity, as a linear combination of these approximations, also lies in Wd,∞.Source: §2.2 · pp. 8–10
  • Uniform knots keep the approximation structure consistent for temporal interpolation, but can be inefficient for distributions with fine or strongly varying structure.Adaptive knot placement could improve approximation accuracy but conflicts with that consistency.Source: §2.2 · pp. 9–10

3. Approximation Problem

The approximation problem is to preserve occupancy-distribution features while producing a smooth representation suitable for risk-minimizing trajectory generation. The paper uses fidelity and derivative-based regularization metrics to balance approximation quality against smoothness.

3. Approximation Problem

  • Smooth approximations must contend with plateaus, sharp transitions, and narrow corridors, which can respectively induce spikes, oscillations, or over-smoothing.Source: §3 · pp. 10–11
  • Total variation penalizes large gradients, favoring piecewise-constant approximations while suppressing spikes and preserving sharp transitions.Source: §3 · p. 11
  • The squared-gradient formulation differs from classical total variation and provides a smooth, differentiable regularization term for the minimization problem.Source: §3 · pp. 11–12
  • Tikhonov regularization penalizes curvature, discouraging large second derivatives and suppressing oscillatory behavior.Source: §3 · p. 12
  • The objective combines MSE fidelity with total-variation and curvature penalties to compute B-spline coefficients.The weights ρMSE, ρTV, and ρT control the trade-off between fidelity, variation, and smoothness.Source: §3 · pp. 11–12
  • The authors omit a conservative pointwise lower-bound constraint because it may produce overly cautious behavior and suboptimal trajectories; normalization is instead applied after optimization.Source: §3 · p. 12

4. Evaluation

The evaluation first benchmarks regularization-weight combinations on occupancy-distribution test cases, then tests selected configurations in a trajectory-planning OCP. Repeated solves with varied initial conditions assess solver performance.

4. Evaluation

  • The evaluation benchmarks combinations of the MSE, total-variation, and Tikhonov weights across test cases with varying stochastic structures and complexity.Source: §4 · p. 13
  • Three configurations are selected: one minimizing MSE, one minimizing TV, and one balancing MSE and TV.Source: §4 · p. 13
  • The selected configurations are applied to a vehicle-motion occupancy time series and integrated into a representative path-planning OCP.Source: §4 · p. 13
  • Repeated OCP solves with slight changes to initial conditions provide statistical insights into how regularization influences solver performance.Source: §4 · p. 13

4.1. Metrics

The evaluation uses one metric set to assess approximation fidelity, variation, and smoothness, and another to assess OCP solver performance.

4.1. Metrics

  • Approximation metrics include MSE, gradient and second-derivative suprema, and total variation to assess fidelity, edge strength, curvature, and surface variability.Source: §4.1 · p. 13
  • OCP solver performance is assessed by iteration count, convergence time, and the final objective value.Source: §4.1 · p. 13
  • Table 1 summarizes metrics for approximation quality and OCP solver performance.Source: §4.1 · p. 13

4.2. Test Cases for the Approximation of Occupancy Distributions

The approximation tests use three distributions spanning different stochastic structures and complexity, including a dynamic occupancy distribution derived from vehicle uncertainty propagation. Knot configurations are selected without regularization to establish fixed baselines.

4.2. Test Cases for the Approximation of Occupancy Distributions

  • Three probability distributions with varying stochastic structure and complexity serve as test cases for evaluating regularization configurations.Source: §4.2 · p. 13
  • TC1 is a bivariate normal distribution with diagonal covariance, so the two dimensions have no correlation.Source: §4.2 · p. 13
  • TC2 combines a Beta marginal for X with a truncated-normal conditional distribution for Y whose spread varies with the Beta likelihood.Source: §4.2 · p. 15
  • TC3 uses occupancy distributions from vehicle uncertainty propagation and reflects time-evolving occupancy; the described method derives uncertainty from OCP-based representative trajectories and motion primitives.Source: §4.2 · p. 15
  • The study focuses on regularization rather than knot configuration, selecting unregularized B-spline baselines using support expansion and MSE, with quadratic and cubic bases considered.Source: Table 4 · p. 16

4.3. Results for the Test Cases

Across the test cases, regularization reduces surface variability, but the best fidelity–variability balance depends on the occupancy distribution. TV tends to preserve sharp features better than Tikhonov, though this distinction diminishes at higher weights.

4.3. Results for the Test Cases

  • The experiments fixed knot configurations and compared MSE with TV, monitoring Tikhonov regularization; weighted-sum sweeps may miss non-convex Pareto-front regions.The evaluation focused on regularization effects rather than knot count or basis degree.Source: §4.3 · pp. 16–17

4.3.1. Test Case 1

  • For TC1, no regularization achieved the lowest MSE, while the minimum-TV configuration used ρMSE = 0.5, ρTV = 9.5, and ρT = 0.The best fidelity–variability trade-off used TV weight 8.5, no Tikhonov regularization, and MSE weight 1.5; approximation error stayed low across configurations.Source: §4.3.1 · p. 17

4.3.2. Test Case 2

  • TC2's average MSE was 2.1 · 10^-3; its best MSE–TV compromise used ρMSE = 2, ρTV = 8, and ρT = 0.The approximation struggled to capture the occupancy support's concavity, with the effect varying by regularization weights.Source: §4.3.2 · p. 17

4.3.3. Test Case 3

  • TC3 had average MSE of 2.8 · 10^-8; its best fidelity–variability trade-off used ρMSE = 2.5, ρTV = 7.5, and ρT = 0.The approximation smoothed rugged PMF contours while preserving step-like transitions between plateaus.Source: §4.3.3 · p. 19

4.3.4. Conclusion for Test Cases

  • Across cases, regularization reduces variability; TV primarily helps TC3, while TC1 and TC2 benefit from combining TV and Tikhonov, and no general weight-selection pattern emerges.Greater smoothness also lowers gradients and curvature; TV's edge-preservation advantage diminishes when ρTV and ρT exceed approximately 2 in the tested cases.Source: §4.3.4 · pp. 19, 21

4.4. Optimal Control Problem

The OCP combines collision risk with control effort, allowing low risk to be exchanged for more effective controls. Compared with the MSE-minimizing unregularized configuration, regularized configurations reduced solver iterations and runtime.

4.4. Optimal Control Problem

  • The OCP's risk integral is weighted about three times as heavily as the control-norm penalty, whose inclusion supports accepting low risk for more effective controls.The destination constraint permits flexibility in final heading.Source: §4.4 · p. 21

4.4.1. Results for the Optimal Control Problem

  • Compared with no regularization, solver iterations fell 40% for TV-minimizing and 28% for balanced regularization; runtime fell 50% and 33%, respectively.The reported objective values are difficult to interpret absolutely, but regularized approximations had lower variance and narrower min–max ranges across initial conditions.Source: Table 5 · p. 22, §4.4.1 · p. 22, Table 6 · p. 23
  • The balanced strategy's smoother risk field encouraged steering through uncertain regions rather than strict avoidance.Figure 6 depicts ego-vehicle positions alongside historical and future obstacle locations.Source: §4.4.1 · p. 22, Figure 6 · p. 24

5. Conclusion

The paper approximates occupancy distributions with B-spline intensity maps, interpreted as Poisson random-field intensities. In the tested configurations, balancing fidelity and variability reduced solver computational load by up to 50%.

5. Conclusion

  • The proposed method approximates occupancy distributions with smooth functions interpreted as Poisson random-field intensity functions, retaining probabilistic semantics and enabling aggregation of multiple obstacles.B-spline intensity maps lie in a Sobolev function-space subspace and are suited to OCP objective functions.Source: §5 · p. 25
  • A tested configuration balancing MSE fidelity and TV surface variability reduced solver computational load by up to 50% versus minimizing approximation error alone.The paper identifies this gain as especially relevant to constrained autonomous-vehicle computing and real-time high-level MPC planning.Source: §5 · p. 25
  • The method and results also apply to domains benefiting from derivative access or Poisson-field aggregation of occupancy distributions.These properties support sensitivity analysis, gradient-based planning, and coherent risk modeling across multiple occupancy distributions.Source: §5 · p. 25

6. Outlook

The outlook identifies uniform knot placement, time-varying regularization needs, and the current two-dimensional formulation as opportunities for further development.

6. Outlook

  • Uniform knot placement may miss fine spatial structure, while adaptive knots would require interpolation between intensity maps when occupancy distributions are unavailable at OCP shooting times.Mesh morphing is proposed as one possible interpolation approach.Source: §6 · p. 25
  • Regularization weights may need to vary over time, motivating future work on automated, adaptive weighting.The authors note that adaptive weighting could yield robust, structure-aware approximations.Source: §6 · p. 25
  • Extending the framework to three-dimensional systems would require generalizing its spline-based intensity formulation and regularization to volumetric domains.The authors cite aerial vehicles and drone swarms as potential applications.Source: §6 · p. 25

Funding

The research was conducted within dtec.bw’s project MORE, funded by the European Union’s NextGenerationEU.

Funding

  • The research was conducted within dtec.bw’s project MORE, funded by the European Union’s NextGenerationEU.Source: Funding · p. 26
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