This paper presents a novel approach for safe control synthesis using the dual formulation of the navigation problem. The main contribution of this paper is in the analytical construction of density functions for almost everywhere navigation with safety constraints. In contrast to the existing approaches, where density functions are used for the analysis of navigation problems, we use density functions for the synthesis of safe controllers. We provide convergence proof using the proposed density functions for navigation with safety. Further, we use these density functions to design feedback controllers capable of navigating in cluttered environments and high-dimensional configuration spaces. The proposed analytical construction of density functions overcomes the problem associated with navigation functions, which are known to exist but challenging to construct, and potential functions, which suffer from local minima. Application of the developed framework is demonstrated on simple integrator dynamics and fully actuated robotic systems.

翻译：本文提出了一种利用导航问题对偶形式进行安全控制综合的新方法。主要贡献在于通过解析构造密度函数，实现具有安全约束的几乎处处导航。与现有采用密度函数分析导航问题的方法不同，我们将密度函数用于安全控制器的综合。我们基于所提出的密度函数，给出了安全导航的收敛性证明。进一步地，利用这些密度函数设计了能够在杂乱环境及高维构型空间中导航的反馈控制器。所提出的密度函数解析构造方法克服了导航函数存在但难以构造的问题，以及势函数易陷入局部极小值的缺陷。将该框架应用于简单积分器动力学及全驱动机器人系统，验证了其有效性。