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A Time-to-Collision Barrier Function Approach to Collision Avoidance for Stochastic Systems

Benedikt Barthel Sorensen, Mitchell Black, Erfaun Noorani, Themistoklis P. Sapsis

arXiv:2609.17347v1eess.SY

TL;DR

Collision avoidance methods often react to geometric proximity without directly representing how quickly an adversary can force a collision. This paper builds a stochastic CBF around adversarial time-to-collision, uses a differentiable neural surrogate for real-time QP control, and reports stronger avoidance against pursuers with a speed advantage.

  • Problem

    Existing adversarial collision-avoidance studies considered 2D fixed-speed motion and short missions, without statistical analysis of long-term collision-avoidance capability.

  • Method

    The paper defines a stochastic CBF using adversarial TTC, approximates aTTC with a differentiable neural network, and embeds it in a stochastic CBF-QP control law.

  • Results

    The aTTC-CBF maintains low barrier values and collision-free progress in planar trials, while outperforming the higher-order distance-based baseline in the pursuit-evasion study.

  • Takeaways & Limitations

    Time-based barrier control uses adversarial dynamics to support anticipatory collision avoidance across planar and multi-agent pursuit-evasion scenarios.

Abstract

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Collision avoidance constraints for autonomous systems are typically formulated in position or velocity space, implicitly reacting to geometric proximity. We propose an alternative paradigm based on the adversarial time-to-collision (aTTC): the minimum time in which an adversary could achieve a collision given its dynamical constraints. By defining a control barrier function (CBF) directly in the time domain, the resulting controller is inherently anticipatory. The evading agent responds not only to whether a pursuer is on a collision course, but to how quickly it could reach one. This formulation enables velocity modulation that exploits the pursuers dynamic limits as an evasive strategy, a behavior not captured by standard distance-based CBFs. Since exact aTTC computation requires integrating the full system dynamics, we employ a lightweight neural network surrogate that admits a real-time quadratic program-based control law. We validate the approach in a 2D comparative study and a 3D multi-agent pursuit-evasion scenario, where the aTTC-based CBF outperforms a higher-order distance-based baseline by more effectively buying time against superior pursuers with a significant speed advantage.

I. INTRODUCTION

Collision avoidance in stochastic multi-agent systems requires real-time control under dynamic, uncertain interactions. The paper replaces conventional spatial or velocity-based safety descriptions with a temporal formulation based on collision time and establishes probabilistic safety conditions.

  • Motivation: Adversarial collision avoidance must account for agents that actively seek collision, whereas prior studies were limited to 2D fixed-speed motion and short missions.The paper identifies the absence of statistical analysis of long-term collision avoidance capability in that setting.
  • Motivation: The proposed temporal CBF defines safety directly in time rather than Euclidean position or velocity space, using TTC to combine distance and speed into collision risk.A critical time threshold represents the minimum response time available before collision.
  • Contributions: Adversarial TTC assumes a pursuing agent drives toward collision under maximum control effort, producing a finite worst-case metric under controllable dynamics.This metric is intended for real-time control of adversarial multi-agent interactions.
  • Contributions: The stochastic CBF formulation provides a lower bound on the probability that the multi-agent system remains collision-free.The safety condition is enforced through barrier-function evolution and yields a joint probabilistic guarantee.
  • Contributions: The paper trains a differentiable aTTC surrogate, embeds it in a stochastic CBF-QP controller, and evaluates it in planar and 3D pursuit-evasion studies.The joint dynamics retain a block-diagonal structure, allowing local dynamics to evaluate the condition while preserving a joint-level guarantee.

III. TIME-TO-COLLISION CBFS

The method uses TTC to formulate a stochastic barrier function, then strengthens it with adversarial TTC and a differentiable surrogate for predictive control. This produces a real-time controller with probabilistic collision-avoidance guarantees.

  • III. TIME-TO-COLLISION CBFS: The section introduces TTC as the basis for a stochastic TTC-CBF and defines aTTC as the earliest collision time achievable by a pursuing agent.The pursuit policy maximizes closing rate subject to the pursuer’s velocity and acceleration bounds.
  • III. TIME-TO-COLLISION CBFS: A fast, fully differentiable surrogate model approximates aTTC so it can be incorporated into a predictive control law with probabilistic collision-avoidance guarantees.The surrogate avoids repeatedly integrating full system dynamics during optimization.

A. Time-to-Collision

The Stochastic Time-to-Collision CBF defines collision-free states through temporal distance and provides probabilistic safety guarantees under suitable control and backup-policy assumptions.

  • A. Time-to-Collision: The S-TTC-CBF uses predicted time until collision rather than spatial separation to account for the time needed to execute evasive or backup maneuvers.This temporal formulation incorporates system dynamics and is intended to preserve sufficient reaction time.
  • A. Time-to-Collision: The framework activates backup policies when TTC falls below a critical threshold, assuming recovery and non-interference conditions for the affected agent pair.The backup policy must keep the system collision-free over the critical interval with probability at least pB and return the state to the safe set.
  • A. Time-to-Collision: A collision requires both a CBF-margin breach and subsequent backup failure, while the system-level guarantee follows by applying a union bound across pairwise collisions.This decomposition connects pairwise SCBF bounds with the multi-agent collision probability.

B. Adversarial Time-to-Collision

Adversarial time-to-collision measures the earliest collision time against a maximally pursuing agent under dynamical limits, yielding a finite worst-case risk metric when the pursuer is faster.

  • B. Adversarial Time-to-Collision: aTTC is the earliest time a maximally adversarial pursuer can bring the agents within the collision threshold while the other agent applies zero control.The pursuer follows a policy that maximizes closing rate subject to its velocity and acceleration bounds.
  • B. Adversarial Time-to-Collision: The aTTC ignores stochasticity to obtain a deterministic metric with well-defined gradients suitable for integration into a QP-based control law.It represents worst-case collision time rather than the pursuer’s actual intent.
  • B. Adversarial Time-to-Collision: When the pursuer’s maximum speed exceeds the nonreactive agent’s speed and its dynamics are controllable, aTTC is finite for every initial configuration.A finite reorientation time is followed by closing at a rate at least equal to the pursuer’s surplus speed.

C. Surrogate Modeling

Because exact aTTC evaluation requires forward integration and differentiability for optimization, the paper uses a neural network surrogate based primarily on relative states and system limits.

  • C. Surrogate Modeling: Exact aTTC computation is computationally demanding because it requires integrating system dynamics and differentiating the state-to-aTTC mapping for control optimization.The surrogate is introduced to provide a fast, differentiable representation.
  • C. Surrogate Modeling: The surrogate models aTTC from relative position, both agents’ velocities, maximum velocity, collision radius, and maximum acceleration.The exposition assumes constant collision radius and maximum acceleration.
  • C. Surrogate Modeling: A 3-layer fully connected neural network with layer sizes 128-128-64 maps relative states and maximum velocity to the predicted aTTC.Because aTTC is deterministic from the instantaneous state, the model does not require a recurrent architecture.
  • C. Surrogate Modeling: Training uses a weighted Huber loss with weight 1/τ* to emphasize low-aTTC, high-risk configurations while avoiding excessive emphasis on large aTTC values.The Huber loss is quadratic for small outputs and linear for larger ones, supporting smooth behavior near zero.

D. Control Strategy

The controller embeds the differentiable aTTC surrogate in a stochastic CBF quadratic program that stays close to a nominal controller while enforcing the safety constraint.

  • D. Control Strategy: The CBF-QP minimizes deviation from the nominal controller while enforcing the stochastic CBF constraint.The weighting matrix M determines the penalty on deviation from the nominal control.
  • D. Control Strategy: The barrier function is defined from the learned aTTC surrogate, allowing its generator to be computed at each time step because the neural network is differentiable.The safety threshold τc is user defined.

A. Comparative Study: Planar Collision Avoidance

The planar study evaluates the aTTC-CBF against three alternative CBF controllers across 1000 stochastic trials. It achieves high safety and goal progress while avoiding the excessive conservatism or collision rates observed in other methods.

  • A. Comparative Study: Planar Collision Avoidance: The experiment compares four controllers over 1000 stochastic planar collision-avoidance trials using safety, goal progress, and trajectory measures.The study uses a 2D double-integrator agent perturbed by Brownian process noise and compares aTTC-CBF with C3BF, VO-CBF, and RFF-CBF.
  • A. Comparative Study: Planar Collision Avoidance: The aTTC-CBF records the second-longest average time to first collision, behind the highly conservative VO-CBF.VO-CBF crosses the origin in only 4% of trials.
  • A. Comparative Study: Planar Collision Avoidance: The aTTC-CBF makes steady, collision-free progress toward its goal, unlike methods that are overly conservative or exceed the theoretical collision bound.The comparison includes C3BF, VO-CBF, and RFF-CBF behavior.

1) Neural Network Training:

The neural-network surrogate is trained on simulated 3D Dubins-agent trajectories spanning varied relative states and maximum speeds. Its training pairs map relative position and velocities, together with vmax, to adversarial time-to-collision.

  • 1) Neural Network Training:: The surrogate training data comprise 50,000 seconds of 3D Dubins simulation sampled every 0.1 seconds across varied relative positions and velocities.Separate paired datasets are generated for distinct vmax values.
  • 1) Neural Network Training:: Each training pair maps [∆x, vi, vj, vmax] to the corresponding aTTC τ*(vmax).The scenarios use two agents navigating 3D space through randomly spaced waypoints.

2) Numerical Experiment:

The 3D stochastic pursuit-evasion experiment tests the aTTC-CBF against a higher-order distance-based CBF under pursuers with a substantial speed advantage. The aTTC-CBF reduces collisions by modulating velocity in ways that account for the agents’ dynamical limits.

  • 2) Numerical Experiment:: The stochastic 3D Dubins experiment uses one evader and five pursuers, with pursuers’ maximum speed set to 0.75km/s versus 0.5km/s for the evader.The simulation runs for 50,000s, with process noise modeled as a Wiener process.
  • 2) Numerical Experiment:: The aTTC-CBF has 3.7 collisions per 100s versus 16.2 for the HOCBF, representing more than 75% fewer collisions.Table II reports total collision rate and close-encounter outcomes for the multi-agent pursuit-evasion scenario.
  • 2) Numerical Experiment:: The aTTC-CBF’s velocity modulation reflects learned control limits, whereas the geometric HOCBF favors horizontal maneuvers without encoding those limits.In this experiment, maximum yaw rate is four times the maximum pitch rate.
  • 2) Numerical Experiment:: The aTTC-CBF more effectively navigates between pursuers, spending less time in the collision zone and more time at intermediate distances of 0.25-1.25km.This behavior is associated with improved escape from close encounters.
  • 2) Numerical Experiment:: For close encounters, 69% result in collisions with HOCBF agents, whereas aTTC-CBF agents escape 50% of them.Close encounters are defined using the collision-distance thresholds reported for each evader-pursuer pair.
  • 2) Numerical Experiment:: The study reports that computational cost and total collision rate scale linearly with the number of pursuing agents.The repeated experiments hold other parameters constant, although tabulated data are omitted.
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