For statistical learning in high dimension, sparse regularizations have proven useful to boost both computational and statistical efficiency.

For time-series modeling we employ a Gaussian process with a nonlinear exogenous input structure.

For time-series modeling we employ a Gaussian process with a nonlinear exogenous input structure.

We believe these changes, and in particular the additional results in continuous MDPs, address all reviewer concerns.

Abstract--Legged robots can sense terrain through force interactions during locomotion, offering more reliable traversability estimates than remote sensing and serving as scouts for guiding wheeled rovers in challenging environments. However, even legged scouts face challenges when traversing highly deformable or unstable terrain. We present Safe Active Exploration for Granular T errain (SAEGT), a navigation framework that enables legged robots to safely explore unknown granular environments using proprioceptive sensing, particularly where visual input fails to capture terrain deformability. SAEGT estimates the safe region and frontier region online from leg-terrain interactions using Gaussian Process regression for traversability assessment, with a reactive controller for real-time safe exploration and navigation. SAEGT demonstrated its ability to safely explore and navigate toward a specified goal using only proprioceptively estimated traversability in simulation.

As will be shown in Section 4.1,

--This paper proposes a data-driven motion-planning framework for nonlinear systems that constructs a sequence of overlapping invariant polytopes. Around each randomly sampled waypoint, the algorithm identifies a convex admissible region and solves data-driven linear-matrix-inequality problems to learn several ellipsoidal invariant sets together with their local state-feedback gains. The convex hull of these ellipsoids--still invariant under a piece-wise-affine controller obtained by interpolating the gains--is then approximated by a polytope. Safe transitions between nodes are ensured by verifying the intersection of consecutive convex-hull polytopes and introducing an intermediate node for a smooth transition. Control gains are interpolated in real time via simplex-based interpolation, keeping the state inside the invariant polytopes throughout the motion. Unlike traditional approaches that rely on system dynamics models, our method requires only data to compute safe regions and design state-feedback controllers. The approach is validated through simulations, demonstrating the effectiveness of the proposed method in achieving safe, dynamically feasible paths for complex nonlinear systems. Over the years, several motion-planning approaches have been proposed, including graph search-based methods [2], sampling-based methods like rapidly exploring random trees (RRT) [3], behavior-based approaches [4], machine learning-based approaches [5], potential fields [6], and optimization-based techniques such as differential dynamic programming [7]. Among them, RRT, as a sampling-based approach, has received a surge of interest due to its success in robotic applications. However, most of these successful strategies are under assumptions that cannot be certified in many applications [8], [9]. For instance, the planning is typically performed assuring that the waypoints are kinematically feasible.