A Temporal Barrier Framework for Collision Avoidance in Multi-Agent Autonomous Aerial Vehicles

Technical Summary: A Temporal Barrier Framework for Collision Avoidance in Multi-Agent Autonomous Aerial Vehicles

Problem Statement
Operating teams of autonomous aircraft in dynamic, uncertain, and potentially adversarial environments requires safety protocols that are both reliable and selective. Existing collision avoidance methods, such as Control Barrier Functions (CBFs) based on Euclidean distance or Velocity Obstacles (VO), often struggle in 3D aerial settings. Distance-based metrics fail to distinguish between safe close-proximity configurations (diverging trajectories) and dangerous ones (converging trajectories). Furthermore, standard time-to-collision (TTC) metrics, which assume constant velocity, are often ill-defined or infinite in 3D environments where agents possess multiple degrees of freedom and can maneuver to avoid collisions. The challenge is to develop a risk metric that is computable in real-time, accounts for agent dynamics and control limits, and remains effective when facing adversarial agents actively seeking collision.

Methodology
The authors propose a Temporal Barrier Framework centered on a new risk metric called Adversarial Time-to-Collision (aTTC).

1. Adversarial Time-to-Collision (aTTC):

   • Unlike standard TTC, which assumes constant velocity, aTTC defines the time-to-collision under a worst-case scenario: the ego agent moves at a constant velocity, while a surrounding "pursuer" agent maneuvers with maximum control effort (speed and acceleration limits) to intercept the ego.
   • The authors prove that under mild assumptions (specifically, if the pursuer's maximum speed exceeds the ego's current speed), the aTTC is guaranteed to be finite. This makes it a robust metric for 3D control.
   • The metric is inherently anticipatory, quantifying risk based on how quickly a collision could occur given dynamical constraints, rather than just current separation.

2. Integration into Control Barrier Functions (CBF):

   • The aTTC is embedded directly into the CBF framework. The barrier function $h$ is defined in the time domain ($h = \text{aTTC} - \tau_c$) rather than space or velocity.
   • This formulation results in a first-order relative degree with respect to the control input, simplifying the control synthesis compared to higher-order CBFs (HOCBFs) often required for distance-based barriers.
   • The framework incorporates robustness against process noise and adversarial inputs by treating the pursuer's control as an external forcing term and utilizing a perturbed Nagumo condition.

3. Real-Time Computation via Neural Surrogate:

   • Calculating the exact aTTC requires solving an optimal control problem and integrating dynamics forward, which is computationally prohibitive for real-time quadratic programming (QP).
   • To address this, the authors train a differentiable neural-network surrogate model ($G_\theta$) to approximate the aTTC.
   • The network takes relative position, velocities, and the pursuer's maximum speed as inputs. It utilizes Feature-wise Linear Modulation (FiLM) to condition the network on the pursuer's speed capability, allowing a single model to generalize across different agent types.
   • This surrogate allows the aTTC and its gradients to be computed efficiently within a standard CBF-QP solver.

Key Contributions

1. Derivation of aTTC: A worst-case extension of TTC that is always finite in 3D environments under realistic speed assumptions, serving as a natural dimension for collision risk.
2. Differentiable Surrogate Model: A lightweight neural network architecture that approximates the aTTC, enabling its direct integration into CBF-QP control laws without online trajectory integration.
3. Empirical Validation: Demonstration of the framework in long-horizon 3D simulations involving independent pursuit-evasion and formation-flight scenarios, comparing performance against a higher-order distance-based CBF (HOCBF) baseline.

Results
The paper presents results from simulations of 3D Dubins-type aircraft models over 50,000-second horizons.

• Independent Pursuit-Evasion:

  • In scenarios with fast pursuers (1.5x the evader's max speed), the aTTC-CBF achieved 58% of the nominal waypoint progress rate (compared to 30% for HOCBF) while maintaining half the collision rate of the HOCBF.
  • In slow pursuer scenarios, the aTTC-CBF achieved roughly 1.8x higher waypoint progress than the HOCBF with less than half the collision rate.
  • The aTTC-CBF demonstrated superior "near-miss" escape probabilities, successfully evading close encounters 50% of the time in fast-pursuer scenarios, whereas the HOCBF escaped only 5%.
  • The framework allowed agents to fly closer to the mission center (less conservative spreading) while maintaining safety.

• Formation Flight:

  • In formation scenarios, the aTTC-CBF maintained significantly better formation coherence. Under fast pursuers, the mean deviation from assigned formation slots was roughly half that of the HOCBF fleet.
  • The HOCBF tended to chase evaders far from the formation core when under pursuit, whereas the aTTC-CBF kept agents tighter to the formation while still evading.

• Neural Network Performance:

  • The surrogate model achieved a Mean Absolute Error (MAE) of less than 1.5 seconds for critical time horizons (<30s) and successfully generalized to unseen test data.
  • The model exhibited a conservative bias (under-predicting aTTC), which is favorable for safety-critical applications.

Significance and Claims
The authors claim that defining the barrier directly in time rather than distance or velocity provides a more intuitive and selective assessment of risk. By accounting for agent dynamics and adversarial intent, the aTTC-CBF allows agents to:

• Fly in closer proximity to other agents without unnecessary conservatism.
• Modulate velocity based on the potential for collision rather than just current trajectory alignment.
• Achieve greater mission progress and lower collision rates, particularly as the environment becomes more adversarial (i.e., as pursuer speed increases).

The paper modestly acknowledges limitations:

• The improvement over distance-based CBFs is marginal in non-adversarial (cooperative) settings, suggesting the current formulation may be too conservative for purely cooperative scenarios.
• The definition of aTTC as an infimum can lead to non-smooth gradients, requiring significant training data.
• The framework assumes knowledge of agent dynamics, which may not always be available in practice.
• Approximation errors in the neural surrogate could affect safety guarantees, though the authors note that training under one-sided constraints could recover strict safety.

The work positions the aTTC-CBF as a robust, data-driven approach to multi-agent collision avoidance that bridges the gap between theoretical worst-case guarantees and real-time computational feasibility.