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Register a new userTractable Compositions of Discrete-Time Control Barrier Functions with Application to Lane Keeping and Obstacle Avoidance
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This paper introduces control barrier functions for discrete-time systems, which can be shown to be necessary and sufficient for controlled invariance of a given set. Moreover, we propose nonlinear discrete-time control barrier functions for partially control affine systems that lead to controlled invariance conditions that are affine in the control input, leading to a tractable formulation that enables us to handle the safety optimal control problem for a broader range of applications with more complicated safety conditions than existing approaches. In addition, we develop mixed-integer formulations for basic and secondary Boolean compositions of multiple control barrier functions and further provide mixed-integer constraints for piecewise control barrier functions. Finally, we apply these discrete-time control barrier function tools to automotive safety problems of lane keeping and obstacle avoidance, which are shown to be effective in simulation.
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Developing controllers for obstacle avoidance between polytopes is a challenging and necessary problem for navigation in a tight space. Traditional approaches can only formulate the obstacle avoidance problem as an offline optimization problem. To address these challenges, we propose a duality-based safety-critical optimal control using control barrier functions for obstacle avoidance between polytopes, which can be solved in real-time with a QP-based optimization problem. A dual optimization problem is introduced to represent the minimum distance between polytopes and the Lagrangian function for the dual form is applied to construct a control barrier function. We demonstrate the proposed controller on a moving sofa problem where non-conservative maneuvers can be achieved in a tight space.
Artificial potential fields (APFs) and their variants have been a staple for collision avoidance of mobile robots and manipulators for almost 40 years. Its model-independent nature, ease of implementation, and real-time performance have played a large role in its continued success over the years. Control barrier functions (CBFs), on the other hand, are a more recent development, commonly used to guarantee safety for nonlinear systems in real-time in the form of a filter on a nominal controller. In this paper, we address the connections between APFs and CBFs. At a theoretic level, we prove that APFs are a special case of CBFs: given a APF one obtains a CBFs, while the converse is not true. Additionally, we prove that CBFs obtained from APFs have additional beneficial properties and can be applied to nonlinear systems. Practically, we compare the performance of APFs and CBFs in the context of obstacle avoidance on simple illustrative examples and for a quadrotor, both in simulation and on hardware using onboard sensing. These comparisons demonstrate that CBFs outperform APFs.
Control barrier functions have shown great success in addressing control problems with safety guarantees. These methods usually find the next safe control input by solving an online quadratic programming problem. However, model uncertainty is a big challenge in synthesizing controllers. This may lead to the generation of unsafe control actions, resulting in severe consequences. In this paper, we develop a learning framework to deal with system uncertainty. Our method mainly focuses on learning the dynamics of the control barrier function, especially for high relative degree with respect to a system. We show that for each order, the time derivative of the control barrier function can be separated into the time derivative of the nominal control barrier function and a remainder. This implies that we can use a neural network to learn the remainder so that we can approximate the dynamics of the real control barrier function. We show by simulation that our method can generate safe trajectories under parametric uncertainty using a differential drive robot model.
In this paper, we introduce the notion of periodic safety, which requires that the system trajectories periodically visit a subset of a forward-invariant safe set, and utilize it in a multi-rate framework where a high-level planner generates a reference trajectory that is tracked by a low-level controller under input constraints. We introduce the notion of fixed-time barrier functions which is leveraged by the proposed low-level controller in a quadratic programming framework. Then, we design a model predictive control policy for high-level planning with a bound on the rate of change for the reference trajectory to guarantee that periodic safety is achieved. We demonstrate the effectiveness of the proposed strategy on a simulation example, where the proposed fixed-time stabilizing low-level controller shows successful satisfaction of control objectives, whereas an exponentially stabilizing low-level controller fails.
In this paper we present a multi-rate control architecture for safety critical systems. We consider a high level planner and a low level controller which operate at different frequencies. This multi-rate behavior is described by a piecewise nonlinear model which evolves on a continuous and a discrete level. First, we present sufficient conditions which guarantee recursive constraint satisfaction for the closed-loop system. Afterwards, we propose a control design methodology which leverages Control Barrier Functions (CBFs) for low level control and Model Predictive Control (MPC) policies for high level planning. The control barrier function is designed using the full nonlinear dynamical model and the MPC is based on a simplified planning model. When the nonlinear system is control affine and the high level planning model is linear, the control actions are computed by solving convex optimization problems at each level of the hierarchy. Finally, we show the effectiveness of the proposed strategy on a simulation example, where the low level control action is updated at a higher frequency than the high level command.
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