text: When deploying autonomous agents in unstructured environments over sustained periods of time, adaptability and robustness oftentimes outweigh optimality as a primary consideration. In other words, safety and survivability constraints play a key role and in this paper, we present a novel, constraint-learning framework for control tasks built on the idea of constraints-driven control. However, since control policies that keep a dynamical agent within state constraints over infinite horizons are not always available, this work instead considers constraints that can be satisfied over a sufficiently long time horizon T > 0, which we refer to as limited-duration safety. Consequently, value function learning can be used as a tool to help us find limited-duration safe policies. We show that, in some applications, the existence of limited-duration safe policies is actually sufficient for long-duration autonomy. This idea is illustrated on a swarm of simulated robots that are tasked with covering a given area, but that sporadically need to abandon this task to charge batteries. We show how the battery-charging behavior naturally emerges as a result of the constraints. Additionally, using a cart-pole simulation environment, we show how a control policy can be efficiently transferred from the source task, balancing the pole, to the target task, moving the cart to one direction without letting the pole fall down.
  @cite_14: Motivated by the need to simultaneously guarantee safety and stability of safety-critical dynamical systems, we construct permissive barrier certificates in this paper that explicitly maximize the region where the system can be stabilized without violating safety constraints. An iterative search algorithm is developed to search for the maximum volume barrier certified region of safe stabilization. The barrier certified region, which is allowed to take any arbitrary shape, is proved to be strictly larger than safe regions generated with Lyapunov sublevel set based methods. The proposed approach effectively unites a Lyapunov function with multiple barrier functions that might not be compatible with each other. Simulation results of the iterative search algorithm demonstrate the effectiveness of the proposed method.
  @cite_7: 
  @cite_8: This paper presents safety barrier certificates that ensure scalable and provably collision-free behaviors in multirobot systems by modifying the nominal controllers to formally satisfy safety constraints. This is achieved by minimizing the difference between the actual and the nominal controllers subject to safety constraints. The resulting computation of the safety controllers is done through a quadratic programming problem that can be solved in real-time and in this paper, we describe a series of problems of increasing complexity. Starting with a centralized formulation, where the safety controller is computed across all agents simultaneously, we show how one can achieve a natural decentralization whereby individual robots only have to remain safe relative to nearby robots. Conservativeness and existence of solutions as well as deadlock-avoidance are then addressed using a mixture of relaxed control barrier functions, hybrid braking controllers, and consistent perturbations. The resulting control strategy is verified experimentally on a collection of wheeled mobile robots whose nominal controllers are explicitly designed to make the robots collide.
  @cite_15: In this paper we present a reformulation--framed as a constrained optimization problem--of multi-robot tasks which are encoded through a cost function that is to be minimized. The advantages of this approach are multiple. The constraint-based formulation provides a natural way of enabling long-term robot autonomy applications, where resilience and adaptability to changing environmental conditions are essential. Moreover, under certain assumptions on the cost function, the resulting controller is guaranteed to be decentralized. Furthermore, finite-time convergence can be achieved, while using local information only, and therefore preserving the decentralized nature of the algorithm. The developed control framework has been tested on a team of ground mobile robots implementing long-term environmental monitoring.
  @cite_41: We introduce Exponential Control Barrier Functions as means to enforce strict state-dependent high relative degree safety constraints for nonlinear systems. We also develop a systematic design method that enables creating the Exponential CBFs for nonlinear systems making use of tools from linear control theory. The proposed control design is numerically validated on a relative degree 6 linear system (the serial cart-spring system) and on a relative degree 4 nonlinear system (the two-link pendulum with elastic actuators.)
  @cite_29: An important tool for proving the safety of dynamical systems is the notion of a barrier certificate. In this paper, we prove that every robustly safe ordinary differential equation has a barrier certificate. Moreover, we show a construction of such a barrier certificate based on a set of states that is reachable in finite time.
  @cite_42: 
  @cite_1: As multi-agent systems become more wide-spread and versatile, the ability to satisfy multiple system-level constraints grows increasingly important. In applications ranging from automated cruise control to safety in robot swarms, barrier functions have emerged as a tool to provably meet such constraints by guaranteeing forward invariance of desirable sets. However, satisfying multiple constraints typically implies formulating multiple barrier functions, which would be ameliorated if the barrier functions could be composed together as Boolean logic formulas. The use of max and min operators, which yields nonsmooth functions, represents one path to accomplish Boolean compositions of barrier functions, and this letter extends previously established concepts for barrier functions to a class of nonsmooth barrier functions that operate on systems described by differential inclusions. We validate our results by deploying Boolean compositions of nonsmooth barrier functions onto a team of mobile robots.
  @cite_32: This paper presents a safe learning framework that employs an adaptive model learning method together with barrier certificates for systems with possibly nonstationary agent dynamics. To extract the dynamic structure of the model, we use a sparse optimization technique, and the resulting model will be used in combination with control barrier certificates which constrain policies (feedback controllers) in order to maintain safety, which refers to avoiding certain regions of the state space. Under certain conditions, recovery of safety in the sense of Lyapunov stability after violations of safety due to the nonstationarity is guaranteed. In addition, we reformulate action-value function approximation to make any kernel-based nonlinear function estimation method applicable to our adaptive learning framework. Lastly, solutions to the barrier-certified policy optimization are guaranteed to be globally optimal, ensuring greedy policy updates under mild conditions. The resulting framework is validated via simulations of a quadrotor, which has been used in the safe learnings literature under stationarity assumption, and then tested on a real robot called brushbot , whose dynamics is unknown, highly complex, and most probably nonstationary.
  @cite_21: This technical note shows that a barrier certificate exists for any safe dynamical system. Specifically, we prove converse barrier certificate theorems for a class of structurally stable dynamical systems. Other authors have developed a related result by assuming that the dynamical system has neither singular points nor closed orbits. In this technical note, we redefine the standard notion of safety to comply with dynamical systems with multiple singular elements. Hereafter, we prove the converse barrier certificate theorems and highlight the differences between our results and previous work by a number of illustrative examples.
  @cite_43: 
  @cite_45: Abstract Barrier functions (also called certificates) have been an important tool for the verification of hybrid systems, and have also played important roles in optimization and multi-objective control. The extension of a barrier function to a controlled system results in a control barrier function. This can be thought of as being analogous to how Sontag extended Lyapunov functions to control Lypaunov functions in order to enable controller synthesis for stabilization tasks. A control barrier function enables controller synthesis for safety requirements specified by forward invariance of a set using a Lyapunov-like condition. This paper develops several important extensions to the notion of a control barrier function. The first involves robustness under perturbations to the vector field defining the system. Input-to-State stability conditions are given that provide for forward invariance, when disturbances are present, of a “relaxation” of set rendered invariant without disturbances. A control barrier function can be combined with a control Lyapunov function in a quadratic program to achieve a control objective subject to safety guarantees. The second result of the paper gives conditions for the control law obtained by solving the quadratic program to be Lipschitz continuous and therefore to gives rise to well-defined solutions of the resulting closed-loop system.
  @cite_49: Safety Barrier Certificates that ensure collision-free maneuvers for teams of differential flatness-based quadrotors are presented in this paper. Synthesized with control barrier functions, the certificates are used to modify the nominal trajectory in a minimally invasive way to avoid collisions. The proposed collision avoidance strategy complements existing flight control and planning algorithms by providing trajectory modifications with provable safety guarantees. The effectiveness of this strategy is supported both by the theoretical results and experimental validation on a team of five quadrotors.
  @cite_46: Safety critical systems involve the tight coupling between potentially conflicting control objectives and safety constraints. As a means of creating a formal framework for controlling systems of this form, and with a view toward automotive applications, this paper develops a methodology that allows safety conditions—expressed as control barrier functions —to be unified with performance objectives—expressed as control Lyapunov functions—in the context of real-time optimization-based controllers. Safety conditions are specified in terms of forward invariance of a set, and are verified via two novel generalizations of barrier functions; in each case, the existence of a barrier function satisfying Lyapunov-like conditions implies forward invariance of the set, and the relationship between these two classes of barrier functions is characterized. In addition, each of these formulations yields a notion of control barrier function (CBF), providing inequality constraints in the control input that, when satisfied, again imply forward invariance of the set. Through these constructions, CBFs can naturally be unified with control Lyapunov functions (CLFs) in the context of a quadratic program (QP); this allows for the achievement of control objectives (represented by CLFs) subject to conditions on the admissible states of the system (represented by CBFs). The mediation of safety and performance through a QP is demonstrated on adaptive cruise control and lane keeping, two automotive control problems that present both safety and performance considerations coupled with actuator bounds.
  @cite_12: In this letter, we consider the problem of rendering robotic tasks persistent by ensuring that the robots' energy levels are never depleted, which means that the tasks can be executed over long time horizons. This process is referred to as the persistification of the task. In particular, the state of each robot is augmented with its battery level so that the desired persistent behavior can be encoded as the forward invariance of a set such that the robots never deplete their batteries. Control barrier functions are employed to synthesize controllers that ensure that this set is forward invariant and, therefore, that the robotic task is persistent. As an application, this letter considers the persistification of a robotic sensor coverage task in which a group of robots has to cover an area of interest. The successful persistification of the coverage task is shown in simulation and on a team of mobile robots.
  @cite_17: Abstract This paper presents a new safety feedback design for nonlinear systems based on barrier certificates and the idea of control Lyapunov functions. In contrast to existing methods, this approach ensures safety independently of abstract high-level tasks that might be unknown or change over time. Leaving as much freedom as possible to the safe system, the authors believe that the flexibility of this approach is very promising. The design is validated using an illustrative example.
summary: On the other hand, control barrier functions (CBFs) @cite_45 @cite_17 @cite_1 @cite_8 @cite_46 @cite_43 @cite_12 @cite_15 @cite_32 @cite_7 were proposed to guarantee that an agent remains in a certain region of the state space (i.e., forward invariance @cite_42 ) by using a locally accurate model of the agent dynamics (i.e., a model that accurately predicts a time derivative of the state at the current state and control input). When the system is linearizable and has a high relative degree, an exponential control barrier function @cite_41 was proposed and was applied to control of quadrotors @cite_49 . When a Lyapunov function is available, the work @cite_14 proposed a sum-of-squares approach to compute a valid barrier function. The idea of constraints-driven controls is in stark contrast to solving the task-specific problem that basically aims at singling out one optimal trajectory. However, although there exist converse theorems for safety and barrier functions which claim that a forward invariant set has a barrier function under certain conditions @cite_29 @cite_21 @cite_46 , finding such a set without assuming stability of the system is difficult in general (see @cite_45 for the conditions that a candidate barrier function can be a valid one).