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的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建密度函数，在满足安全约束的前提下实现几乎处处导航。与现有方法中密度函数仅用于分析导航问题不同，本文将密度函数用于安全控制器的综合。我们给出了利用所提密度函数实现安全导航的收敛性证明。进一步，利用这些密度函数设计了能够在杂乱环境和高维构型空间中导航的反馈控制器。该解析构建方法克服了导航函数（虽已知存在但构建困难）和势函数（存在局部极小值问题）的固有问题。所提框架的应用通过简单积分器动力学和全驱动机器人系统进行了验证。