Legged navigation is typically examined within open-world, off-road, and challenging environments. In these scenarios, estimating external disturbances requires a complex synthesis of multi-modal information. This underlines a major limitation in existing works that primarily focus on avoiding obstacles. In this work, we propose TOP-Nav, a novel legged navigation framework that integrates a comprehensive path planner with Terrain awareness, Obstacle avoidance and close-loop Proprioception. TOP-Nav underscores the synergies between vision and proprioception in both path and motion planning. Within the path planner, we present and integrate a terrain estimator that enables the robot to select waypoints on terrains with higher traversability while effectively avoiding obstacles. In the motion planning level, we not only implement a locomotion controller to track the navigation commands, but also construct a proprioception advisor to provide motion evaluations for the path planner. Based on the close-loop motion feedback, we make online corrections for the vision-based terrain and obstacle estimations. Consequently, TOP-Nav achieves open-world navigation that the robot can handle terrains or disturbances beyond the distribution of prior knowledge and overcomes constraints imposed by visual conditions. Building upon extensive experiments conducted in both simulation and real-world environments, TOP-Nav demonstrates superior performance in open-world navigation compared to existing methods.

We extend and combine several tools of the literature to design fast, adaptive, anytime and scale-free online learning algorithms. Scale-free regret bounds must scale linearly with the maximum loss, both toward large losses and toward very small losses. Adaptive regret bounds demonstrate that an algorithm can take advantage of easy data and potentially have constant regret. We seek to develop fast algorithms that depend on as few parameters as possible, in particular they should be anytime and thus not depend on the time horizon. Our first and main tool, isotuning, is a generalization of the idea of balancing the trade-off of the regret. We develop a set of tools to design and analyze such learning rates easily and show that they adapts automatically to the rate of the regret (whether constant, $O(\log T)$, $O(\sqrt{T})$, etc.) within a factor 2 of the optimal learning rate in hindsight for the same observed quantities. The second tool is an online correction, which allows us to obtain centered bounds for many algorithms, to prevent the regret bounds from being vacuous when the domain is overly large or only partially constrained. The last tool, null updates, prevents the algorithm from performing overly large updates, which could result in unbounded regret, or even invalid updates. We develop a general theory using these tools and apply it to several standard algorithms. In particular, we (almost entirely) restore the adaptivity to small losses of FTRL for unbounded domains, design and prove scale-free adaptive guarantees for a variant of Mirror Descent (at least when the Bregman divergence is convex in its second argument), extend Adapt-ML-Prod to scale-free guarantees, and provide several other minor contributions about Prod, AdaHedge, BOA and Soft-Bayes.