Hamilton-Jacobi (HJ) reachability analysis is an important formal verification method for guaranteeing performance and safety properties of dynamical systems; it has been applied to many small-scale systems in the past decade. Its advantages include compatibility with general nonlinear system dynamics, formal treatment of bounded disturbances, and the availability of well-developed numerical tools. The main challenge is addressing its exponential computational complexity with respect to the number of state variables. In this tutorial, we present an overview of basic HJ reachability theory and provide instructions for using the most recent numerical tools, including an efficient GPU-parallelized implementation of a Level Set Toolbox for computing reachable sets. In addition, we review some of the current work in high-dimensional HJ reachability to show how the dimensionality challenge can be alleviated via various general theoretical and applicationspecific insights.

Fast and safe navigation of dynamical systems through a priori unknown cluttered environments is vital to many applications of autonomous systems. However, trajectory planning for autonomous systems is computationally intensive, often requiring simplified dynamics that sacrifice safety and dynamic feasibility in order to plan efficiently. Conversely, safe trajectories can be computed using more sophisticated dynamic models, but this is typically too slow to be used for real-time planning. We propose a new algorithm FaSTrack: Fast and Safe Tracking for High Dimensional systems. A path or trajectory planner using simplified dynamics to plan quickly can be incorporated into the FaSTrack framework, which provides a safety controller for the vehicle along with a guaranteed tracking error bound. This bound captures all possible deviations due to high dimensional dynamics and external disturbances. Note that FaSTrack is modular and can be used with most current path or trajectory planners. We demonstrate this framework using a 10D nonlinear quadrotor model tracking a 3D path obtained from an RRT planner.

Reachability analysis provides formal guarantees for performance and safety properties of nonlinear control systems. Here, one aims to compute the backward reachable set (BRS) or tube (BRT) -the set of states from which the system can be driven into a target set at a particular time or within a time interval, respectively. The computational complexity of current approaches scales exponentially, making application to high-dimensional systems intractable. We propose a technique that decomposes the dynamics of a general class of nonlinear systems into subsystems which may be coupled through common states, controls, and disturbances. Despite this coupling, BRSs and BRTs can be computed efficiently using our technique without incurring additional approximation errors and without the need for linearizing dynamics or approximating sets as polytopes. Computations of BRSs and BRTs now become orders of magnitude faster, and for the first time BRSs and BRTs for many high-dimensional nonlinear control systems can be computed using the Hamilton-Jacobi (HJ) formulation. In situations involving bounded adversarial disturbances, our proposed method can obtain slightly conservative results. We demonstrate our theory by numerically computing BRSs and BRTs using the HJ formulation for several systems, including the 6D Acrobatic Quadrotor and the 10D Near-Hover Quadrotor.

Abstract-Traditional learning approaches proposed for controlling quadrotors or helicopters have focused on improving performance for specific trajectories by iteratively improving upon a nominal controller, for example learning from demonstrations, iterative learning, and reinforcement learning. In these schemes, however, it is not clear how the information gathered from the training trajectories can be used to synthesize controllers for more general trajectories. Recently, the efficacy of deep learning in inferring helicopter dynamics has been shown. Motivated by the generalization capability of deep learning, this paper investigates whether a neural network based dynamics model can be employed to synthesize control for trajectories different than those used for training. To test this, we learn a quadrotor dynamics model using only translational and only rotational training trajectories, each of which can be controlled independently, and then use it to simultaneously control the yaw and position of a quadrotor, which is non-trivial because of nonlinear couplings between the two motions. We validate our approach in experiments on a quadrotor testbed.

Real-world robots are becoming increasingly complex and commonly act in poorly understood environments where it is extremely challenging to model or learn their true dynamics. Therefore, it might be desirable to take a taskspecific approach, wherein the focus is on explicitly learning the dynamics model which achieves the best control performance for the task at hand, rather than learning the true dynamics. In this work, we use Bayesian optimization in an active learning framework where a locally linear dynamics model is learned with the intent of maximizing the control performance, and used in conjunction with optimal control schemes to efficiently design a controller for a given task. This model is updated directly based on the performance observed in experiments on the physical system in an iterative manner until a desired performance is achieved. We demonstrate the efficacy of the proposed approach through simulations and real experiments on a quadrotor testbed.All authors are with the

Real-world autonomous vehicles often operate in a priori unknown environments. Since most of these systems are safety-critical, it is important to ensure they operate safely in the face of environment uncertainty, such as unseen obstacles. Current safety analysis tools enable autonomous systems to reason about safety given full information about the state of the environment a priori. However, these tools do not scale well to scenarios where the environment is being sensed in real time, such as during navigation tasks. In this work, we propose a novel, real-time safety analysis method based on Hamilton-Jacobi reachability that provides strong safety guarantees despite environment uncertainty. Our safety method is planner-agnostic and provides guarantees for a variety of mapping sensors. We demonstrate our approach in simulation and in hardware to provide safety guarantees around a stateof-the-art vision-based, learning-based planner. Videos of our approach and experiments are available on the project website 1 .

Provably safe and scalable multi-vehicle path planning is an important and urgent problem due to the expected increase of automation in civilian airspace in the near future. Although this problem has been studied in the past, there has not been a method that guarantees both goal satisfaction and safety for vehicles with general nonlinear dynamics while taking into account disturbances and potential adversarial agents, to the best of our knowledge. Hamilton-Jacobi (HJ) reachability is the ideal tool for guaranteeing goal satisfaction and safety under such scenarios, and has been successfully applied to many small-scale problems. However, a direct application of HJ reachability in most cases becomes intractable when there are more than two vehicles due to the exponentially scaling computational complexity with respect to system dimension. In this paper, we take advantage of the guarantees HJ reachability provides, and eliminate the computation burden by assigning a strict priority ordering to the vehicles under consideration. Under this sequential path planning (SPP) scheme, vehicles reserve "space-time" portions in the airspace, and the space-time portions guarantee dynamic feasibility, collision avoidance, and optimality of the paths given the priority ordering. With a computation complexity that scales quadratically when accounting for both disturbances and an intruder, and linearly when accounting for only disturbances, SPP can tractably solve the multi-vehicle path planning problem for vehicles with general nonlinear dynamics in a practical setting. We demonstrate our theory in representative simulations.

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