This paper presents a method that addresses the conservatism, computational effort, and limited numerical accuracy of existing frameworks and methods that ensure safety in online model-based motion generation, commonly referred to as fast and safe tracking. Computational limitations restrict online motion planning to low-fidelity models. However, planning with low-fidelity models compromises safety, as the dynamic feasibility of resulting reference trajectories is not ensured. This potentially leads to unavoidable tracking errors that may cause safety-critical constraint violations. Existing frameworks mitigate this safety risk by augmenting safety-critical constraints in motion planning by a safety margin that prevents constraint violations under worst-case tracking errors. However, the methods employed in these frameworks determine the safety margin based on a heuristically selected performance of the planning model, which likely results in overly conservative reference trajectories. Furthermore, these methods are computationally intensive, and the state-of-the-art method is limited in numerical accuracy. We adopt a different perspective and address these limitations with a method that mitigates conservatism in existing frameworks by adapting the planning model performance to a given safety margin. Our method achieves numerical accuracy and requires significantly less computation time than existing methods by leveraging a captivity-escape game, which is a specific zero-sum differential game formulated in this paper. We demonstrate our method using a numerical example and compare it to the state of the art.

Fast and Safe Tracking (FaSTrack, Herbert* et al. (2017)) is a modular framework that provides safety guarantees while planning and executing trajectories in real time via value functions of Hamilton-Jacobi (HJ) reachability. These value functions are computed through dynamic programming, which is notorious for being computationally inefficient. Moreover, the resulting trajectory does not adapt online to the environment, such as sudden disturbances or obstacles. DeepReach (Bansal and Tomlin (2021)) is a scalable deep learning method to HJ reachability that allows parameterization of states, which opens up possibilities for online adaptation to various controls and disturbances. In this paper, we propose Parametric FaSTrack, which uses DeepReach to approximate a value function that parameterizes the control bounds of the planning model. The new framework can smoothly trade off between the navigation speed and the tracking error (therefore maneuverability) while guaranteeing obstacle avoidance in a priori unknown environments. We demonstrate our method through two examples and a benchmark comparison with existing methods, showing the safety, efficiency, and faster solution times of the framework.

Real-time navigation in a priori unknown environment remains a challenging task, especially when an unexpected (unmodeled) disturbance occurs. In this paper, we propose the framework Safe Returning Fast and Safe Tracking (SR-F) that merges concepts from 1) Robust Control Lyapunov-Value Functions (R-CLVF), and 2) the Fast and Safe Tracking (FaSTrack) framework. The SR-F computes an R-CLVF offline between a model of the true system and a simplified planning model. Online, a planning algorithm is used to generate a trajectory in the simplified planning space, and the R-CLVF is used to provide a tracking controller that exponentially stabilizes to the planning model. When an unexpected disturbance occurs, the proposed SR-F algorithm provides a means for the true system to recover to the planning model. We take advantage of this mechanism to induce an artificial disturbance by ``jumping'' the planning model in open environments, forcing faster navigation. Therefore, this algorithm can both reject unexpected true disturbances and accelerate navigation speed. We validate our framework using a 10D quadrotor system and show that SR-F is empirically 20\% faster than the original FaSTrack while maintaining safety.

Real time autonomous motion planning and navigation is hard, especially when we care about safety. This becomes even more difficult when we have systems with complicated dynamics, external disturbances (like wind), and a priori unknown environments. Our goal in this work is to "robustify" existing real-time motion planners to guarantee safety during navigation of dynamic systems. In control theory there are techniques like Hamilton-Jacobi Reachability Analysis that provide rigorous safety guarantees of system behavior, along with an optimal controller to reach a given goal (see Figure 1). However, in general the computational methods used in HJ Reachability Analysis are only tractable in decomposable and/or low-dimensional systems; this is due to the "curse of dimensionality."