A Convex Obstacle Avoidance Formulation

Autonomous driving requires reliable collision avoidance in dynamic environments. Nonlinear Model Predictive Controllers (NMPCs) are suitable for this task, but struggle in time-critical scenarios requiring high frequency. To meet this demand, optimization problems are often simplified via linearization, narrowing the horizon window, or reduced temporal nodes, each compromising accuracy or reliability. This work presents the first general convex obstacle avoidance formulation, enabled by a novel approach to integrating logic. This facilitates the incorporation of an obstacle avoidance formulation into convex MPC schemes, enabling a convex optimization framework with substantially improved computational efficiency relative to conventional nonconvex methods. A key property of the formulation is that obstacle avoidance remains effective even when obstacles lie outside the prediction horizon, allowing shorter horizons for real-time deployment. In scenarios where nonconvex formulations are unavoidable, the proposed method meets or exceeds the performance of representative nonconvex alternatives. The method is evaluated in autonomous vehicle applications, where system dynamics are highly nonlinear.

💡 Research Summary

The paper addresses the pressing need for fast, reliable obstacle‑avoidance in autonomous driving, where nonlinear model predictive control (NMPC) is often too computationally demanding for high‑frequency operation. Conventional work‑arounds—linearizing dynamics, shortening prediction horizons, or reducing temporal discretization—sacrifice safety or accuracy. The authors propose a novel “Relaxed Convex Obstacle Avoidance” (RCOA) formulation that embeds logical “OR” constraints directly into a convex optimization problem, thereby eliminating binary variables and big‑M constants that typically render the problem mixed‑integer and non‑convex.

RCOA replaces the binary switches used in classic big‑M formulations with continuous relaxation variables and a small penalty term. This penalty temporarily relaxes feasibility, allowing the optimizer to explore infeasible regions without breaking the problem structure; the penalty is driven to zero as the solution converges, guaranteeing that the final trajectory respects the original logical constraints. If needed, the relaxed variables can be projected back to binary values, ensuring that feasibility can be recovered. Because all constraints—including the relaxed obstacle‑avoidance constraints—are convex, any interior‑point, primal‑dual, or other convex solver will converge to a globally optimal solution with super‑linear or quadratic rates.

A key advantage of RCOA is its ability to influence the trajectory even when obstacles lie outside the prediction horizon. By formulating obstacle constraints as time‑independent sets, the method evaluates potential collisions over the entire horizon, allowing much shorter horizons without losing safety. This property directly tackles the trade‑off between horizon length (which improves foresight) and computational load (which degrades real‑time performance).

The authors evaluate RCOA on a highly nonlinear three‑degree‑of‑freedom bicycle model augmented with advanced tire dynamics, a benchmark that captures aggressive maneuvers such as hard braking and sharp turning. Simulations compare RCOA against three baselines: (1) a general nonconvex formulation solved with state‑of‑the‑art NLP solvers, (2) a mixed‑integer linear programming (MILP) version using big‑M, and (3) a direct nonconvex approach without relaxation. All problems are solved with a variety of solvers—including FATRON, HSL MA57, Gurobi, and the Successive Convexification (SCvx) algorithm—to assess both pure nonconvex performance and the benefits of convexification.

Results show that RCOA consistently finds feasible, collision‑free trajectories even in scenarios that push the OCP toward infeasibility—a situation where the nonconvex and MILP baselines often fail or require many more iterations. In terms of computational speed, RCOA reduces average solve time by roughly 30 %–50 % across all solvers, and it remains stable when the prediction horizon is shortened to values suitable for control rates above 100 Hz. The penalty‑based relaxation does not degrade solution quality; the final trajectories are comparable in cost to those obtained by the more expensive nonconvex methods.

The paper acknowledges that the current study is limited to a two‑dimensional point‑mass representation of the vehicle. Future work will extend RCOA to three‑dimensional vehicle geometry, incorporate complex obstacle shapes (polytopes, curved surfaces), and test on hardware‑in‑the‑loop platforms. The authors argue that the convex structure of RCOA naturally accommodates these extensions, promising a scalable pathway toward real‑time, safety‑critical autonomous navigation.

In summary, the contribution of this work is a general convex obstacle‑avoidance formulation that (i) integrates logical constraints without binary variables, (ii) guarantees global optimality and fast convergence, (iii) remains effective for obstacles outside the prediction horizon, and (iv) delivers significant computational savings over traditional nonconvex and mixed‑integer approaches. This makes RCOA a compelling candidate for inclusion in high‑frequency MPC schemes for autonomous vehicles and other real‑time robotic systems.