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Probabilistically Robust Trajectory Planning of Multiple Aerial Agents

Christian Vitale, Savvas Papaioannou, Panayiotis Kolios, Georgios Ellinas

TL;DR

This work tackles robust multi-UAV trajectory planning under non-Gaussian disturbances by introducing a distributed Det-MPC NG framework that uses exact mixed-trigonometric-polynomial moment propagation to convert non-Gaussian chance constraints into deterministic moment-based bounds. The approach enables decentralized optimization where each UAV forecasts its own state moments and uses deterministic safety guarantees derived from a VP inequality, avoiding overly conservative or linear-Gaussian assumptions. Key contributions include a closed-form moment propagation method, a deterministic bound for probabilistic safety via the Vysochanskij–Petunin inequality, and a distributed MPC formulation that maintains safe inter-agent distances while pursuing destination objectives. Simulations with non-Gaussian noise demonstrate that the method achieves safe, near-optimal trajectories with modest violations, indicating practical applicability and scalability for complex aerial teams.

Abstract

Current research on robust trajectory planning for autonomous agents aims to mitigate uncertainties arising from disturbances and modeling errors while ensuring guaranteed safety. Existing methods primarily utilize stochastic optimal control techniques with chance constraints to maintain a minimum distance among agents with a guaranteed probability. However, these approaches face challenges, such as the use of simplifying assumptions that result in linear system models or Gaussian disturbances, which limit their practicality in complex realistic scenarios. To address these limitations, this work introduces a novel probabilistically robust distributed controller enabling autonomous agents to plan safe trajectories, even under non-Gaussian uncertainty and nonlinear systems. Leveraging exact uncertainty propagation techniques based on mixed-trigonometric-polynomial moment propagation, this method transforms non-Gaussian chance constraints into deterministic ones, seamlessly integrating them into a distributed model predictive control framework solvable with standard optimization tools. Simulation results demonstrate the effectiveness of this technique, highlighting its ability to consistently handle various types of uncertainty, ensuring robust and accurate path planning in complex scenarios.

Probabilistically Robust Trajectory Planning of Multiple Aerial Agents

TL;DR

This work tackles robust multi-UAV trajectory planning under non-Gaussian disturbances by introducing a distributed Det-MPC NG framework that uses exact mixed-trigonometric-polynomial moment propagation to convert non-Gaussian chance constraints into deterministic moment-based bounds. The approach enables decentralized optimization where each UAV forecasts its own state moments and uses deterministic safety guarantees derived from a VP inequality, avoiding overly conservative or linear-Gaussian assumptions. Key contributions include a closed-form moment propagation method, a deterministic bound for probabilistic safety via the Vysochanskij–Petunin inequality, and a distributed MPC formulation that maintains safe inter-agent distances while pursuing destination objectives. Simulations with non-Gaussian noise demonstrate that the method achieves safe, near-optimal trajectories with modest violations, indicating practical applicability and scalability for complex aerial teams.

Abstract

Current research on robust trajectory planning for autonomous agents aims to mitigate uncertainties arising from disturbances and modeling errors while ensuring guaranteed safety. Existing methods primarily utilize stochastic optimal control techniques with chance constraints to maintain a minimum distance among agents with a guaranteed probability. However, these approaches face challenges, such as the use of simplifying assumptions that result in linear system models or Gaussian disturbances, which limit their practicality in complex realistic scenarios. To address these limitations, this work introduces a novel probabilistically robust distributed controller enabling autonomous agents to plan safe trajectories, even under non-Gaussian uncertainty and nonlinear systems. Leveraging exact uncertainty propagation techniques based on mixed-trigonometric-polynomial moment propagation, this method transforms non-Gaussian chance constraints into deterministic ones, seamlessly integrating them into a distributed model predictive control framework solvable with standard optimization tools. Simulation results demonstrate the effectiveness of this technique, highlighting its ability to consistently handle various types of uncertainty, ensuring robust and accurate path planning in complex scenarios.
Paper Structure (11 sections, 2 theorems, 22 equations, 4 figures)

This paper contains 11 sections, 2 theorems, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Problem Formulation
  4. A Distributed Multi-UAV Trajectory Planning Under Non-Gaussian Noise
  5. UAV's System State Moments Computation
  6. Deterministic Safety Distance Under Non-Gaussian Noise
  7. The Distributed Det-MPC NG Framework
  8. PERFORMANCE EVALUATION
  9. Simulation Setup
  10. Distributed Det-MPC NG Simulation Results
  11. CONCLUSIONS

Key Result

Lemma 1

Let $\theta$ be a random variable with characteristic function $\Phi_\theta(t)$. Given $(\alpha_1,\alpha_2,\alpha_3) \in \mathbb{R}^3$ and a constant scaling factor $\delta$, the expectation of the mixed-trigonometric polynomial function $m_{\delta,\theta^{\alpha_1} c^{\alpha_2}_\theta s^{\alpha_3}_

Figures (4)

  • Figure 1: (a, b) shows executed UAV trajectories respecting safety distance $\text{d}_\text{MIN}=10$ m with probability $1-\epsilon=0.9$ for non-Gaussian noise. Time marks along the trajectories indicate progression. (c) shows pairwise distances among UAVs that, in all time steps, respect safety distance $\text{d}_\text{MIN}=10$ m.
  • Figure 2: Trajectories planned at time $2.6$s respecting safety distance $\text{d}_\text{MIN}=10$ m with probability $1-\epsilon=0.9$ for non-Gaussian noise. Subplots (a$-$d) correspond to UAV $2$, (e$-$h) to UAV $3$, and (i$-$l) to UAV $4$. The second column shows a sampling of the safety distances, representing the volume in the environment that is closer than $\text{d}_\text{MIN}=10$ to another UAV, for the second time step of the planning horizon. The third and fourth columns depict the same for the fourth and ninth time steps of the planning horizon.
  • Figure 3: Trajectories planned by all UAVs at time $2.6$s. Green denotes locations reachable by UAV $4$ at the second, fourth, and ninth time steps, following safety-compliant control sequences. Pink indicates locations reachable using controls that do not adhere to safety constraints. UAV positions at selected time steps, planned using Distributed Det-MPC NG, are marked with black dots.
  • Figure 4: Pairwise distances between UAVs $1$ and $4$, and UAVs $3$ and $4$, calculated at 2.6 seconds using Monte Carlo simulations within the planning horizon. Boxplots for $\epsilon=0.1$ (a) and $\epsilon=0.01$ (b) show distances from the $25^{th}$ to the $75^{th}$ percentiles, with whiskers indicating the $1^{st}$ to the $99^{th}$ percentiles.

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • proof