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Chance-Constrained Trajectory Planning with Multimodal Environmental Uncertainty

Kai Ren, Heejin Ahn, Maryam Kamgarpour

TL;DR

The paper addresses safe trajectory planning under multimodal environmental uncertainty by modeling obstacle futures with Gaussian Mixture Models (GMM). It develops deterministic reformulations and CVaR-based approximations of chance constraints, plus tight moment-concentration bounds to handle sampling-based moments, resulting in tractable mixed-integer second-order cone programs for trajectory planning. The approach enables per-mode safety guarantees with finite-sample confidence and demonstrates reduced conservatism compared to unimodal Gaussian models on autonomous driving data, while CVaR-based methods control the magnitude of potential constraint violations. This yields practical, provably safe planning for autonomous vehicles in complex, interactive traffic environments.

Abstract

We tackle safe trajectory planning under Gaussian mixture model (GMM) uncertainty. Specifically, we use a GMM to model the multimodal behaviors of obstacles' uncertain states. Then, we develop a mixed-integer conic approximation to the chance-constrained trajectory planning problem with deterministic linear systems and polyhedral obstacles. When the GMM moments are estimated via finite samples, we develop a tight concentration bound to ensure the chance constraint with a desired confidence. Moreover, to limit the amount of constraint violation, we develop a Conditional Value-at-Risk (CVaR) approach corresponding to the chance constraints and derive a tractable approximation for known and estimated GMM moments. We verify our methods with state-of-the-art trajectory prediction algorithms and autonomous driving datasets.

Chance-Constrained Trajectory Planning with Multimodal Environmental Uncertainty

TL;DR

The paper addresses safe trajectory planning under multimodal environmental uncertainty by modeling obstacle futures with Gaussian Mixture Models (GMM). It develops deterministic reformulations and CVaR-based approximations of chance constraints, plus tight moment-concentration bounds to handle sampling-based moments, resulting in tractable mixed-integer second-order cone programs for trajectory planning. The approach enables per-mode safety guarantees with finite-sample confidence and demonstrates reduced conservatism compared to unimodal Gaussian models on autonomous driving data, while CVaR-based methods control the magnitude of potential constraint violations. This yields practical, provably safe planning for autonomous vehicles in complex, interactive traffic environments.

Abstract

We tackle safe trajectory planning under Gaussian mixture model (GMM) uncertainty. Specifically, we use a GMM to model the multimodal behaviors of obstacles' uncertain states. Then, we develop a mixed-integer conic approximation to the chance-constrained trajectory planning problem with deterministic linear systems and polyhedral obstacles. When the GMM moments are estimated via finite samples, we develop a tight concentration bound to ensure the chance constraint with a desired confidence. Moreover, to limit the amount of constraint violation, we develop a Conditional Value-at-Risk (CVaR) approach corresponding to the chance constraints and derive a tractable approximation for known and estimated GMM moments. We verify our methods with state-of-the-art trajectory prediction algorithms and autonomous driving datasets.

Paper Structure

This paper contains 9 sections, 3 theorems, 16 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Chance Constraint with GMM Uncertainty
  3. Linear Constraint
  4. Chance and CVaR Constraints under GMM
  5. Deterministic reformulation & Moment robustification
  6. Chance-Constrained Trajectory Planning
  7. Deterministic reformulation & Moment robustification
  8. Simulations and Results
  9. Conclusion

Key Result

Lemma 1

When the GMM moments $\mu_k$ and $\Sigma_k$ are known for all $k \in \mathbb{Z}_{1:K}$, the chance constraints constraint:CCsingleGMMmode and the CVaR approximation constraint:CVaRsingleGMMmode can be equivalently reformulated as the following second-order cone constraint: where $\Gamma_k=\text{\textPsi}^{-1}(1-\epsilon_k)$ for chance constraint Calafiore2006CC and $\Gamma_k=\Phi\left(\text{\text

Figures (5)

  • Figure 1: Linear constraint in 2-dimensional space
  • Figure 2: Comparison of $C_1$, $C_2$ for fixed $\beta = 10^{-3}$ and varying $N_k$ (left), fixed $N_k = 5000$ and varying $\beta$ (right).
  • Figure 3: EV Trajectories in scene 1 with MTA, MRA, CVaR and CVaRR trajectory planners.
  • Figure 4: EV Trajectories in scene 2 with MTA, MRA, CVaR and CVaRR trajectory planners.
  • Figure 5: Worst-case collisions for MTA, MRA, CVaR and CVaRR trajectory planners.

Theorems & Definitions (7)

  • Lemma 1
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Theorem 2
  • proof