Conformal Prediction (CP) is a powerful statistical machine learning tool to construct uncertainty sets with coverage guarantees, which has fueled its extensive adoption in generating prediction regions for decision-making tasks, e.g., Trajectory Optimization (TO) in uncertain environments. However, existing methods predominantly employ a sequential scheme, where decisions rely unidirectionally on the prediction regions, and consequently the information from decision-making fails to be fed back to instruct CP. In this paper, we propose a novel Feedback-Based CP (Fb-CP) framework for shrinking-horizon TO with a joint risk constraint over the entire mission time. Specifically, a CP-based posterior risk calculation method is developed by fully leveraging the realized trajectories to adjust the posterior allowable risk, which is then allocated to future times to update prediction regions. In this way, the information in the realized trajectories is continuously fed back to the CP, enabling attractive feedback-based adjustments of the prediction regions and a provable online improvement in trajectory performance. Furthermore, we theoretically prove that such adjustments consistently maintain the coverage guarantees of the prediction regions, thereby ensuring provable safety. Additionally, we develop a decision-focused iterative risk allocation algorithm with theoretical convergence analysis for allocating the posterior allowable risk which closely aligns with Fb-CP. Furthermore, we extend the proposed method to handle distribution shift. The effectiveness and superiority of the proposed method are demonstrated through benchmark experiments.

Collision-free navigation is a critical issue in robotic systems as the environment is often dynamic and uncertain. This paper investigates a data-stream-driven motion control problem for mobile robots to avoid randomly moving obstacles when the probability distribution of the obstacle’s movement is partially observable through data and can be even time-varying. A data-stream-driven ambiguity set is firstly constructed from movement data by leveraging a Dirichlet process mixture model and is updated online using real-time data. Then we propose an Online-Learning-based Distributionally Robust Nonlinear Model Predictive Control (OL-DR-NMPC) approach for limiting the risk of collision through considering the worst-case distribution within the ambiguity set. To facilitate solving the OL-DR-NMPC problem, we reformulate it as a finite-dimensional nonlinear optimization problem. To cope with the bilinear matrix inequality constraints in the nonlinear problem, we develop a parabolic relaxation and a sequential algorithm, by which the problem is further transformed into polynomial-time solvable surrogates. The simulations using a quadrotor model are employed to demonstrate the effectiveness and advantages of the proposed method.