## 1 Introduction  
Due to advances in sensing, actuation, communication, computing, storage, and AI technologies with affordable costs, increasing deployment and applications of intelligent autonomous mobile robots or ground vehicles for long-term operation are prevalent. The integrated platforms at our disposal are intended for missions such as SAR (search and rescue), autonomous driver-less driving, manufacturing, and surveillance in cluttered environments [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40]. Along with an increasingly heavy interaction between human and robots, update-to-date but cost-effective implementation or prototyping of intelligent mobile robot systems to accomplish missions autonomously employing recent advances in localization, mapping, and navigation have been the focus of some endeavors put into the robotic systems, as shown, for instance, in References [39; 41; 42; 43]. Obstacle avoidance and motion planning [2; 3; 4; 31] are essential for the completion of missions in a smooth and optimal manner. A variety of obstacle avoidance algorithms are designed and implemented for navigating a mobile robot to avoid static and dynamic obstacles in open space or in narrow passages. The artificial potential field (APF) approach is one of the most well-known reactive online obstacle avoidance methods applicable in known or unknown environments [5]. The goal position of the robot is assigned as an artificial attractive potential and obstacles are applied as artificial repulsive forces. Then, the collision avoidance path is derived by using the gradient of the linear superposition of each potential. However, by following the gradient path of APF, navigation routes generated by APFs suffer from the local minima due to the presence of obstacles, i.e., the set of trap positions where the gradient of APF vanishes so that the robot gets stuck there, thus preventing the robot from reaching the target or lower the navigation efficiency. The number of obstacles affects the number of local minima significantly [44]. In general, it is desired that no other local minima except the goal exist in APF so that the navigation is efficient. For this purpose, hydrodynamic or harmonic (or velocity) potential functions (HPFs) (see References [7; 9; 15; 17]) derived from the velocity potential of the solution to the boundary value problem of Laplace's Equation with appropriate boundary conditions in a computational domain are proposed as an appealing class of APFs for navigation. The properties of HPFs such as min-max principle and superposition in the context of path planning were proved in the foundation work [7; 15; 17]. To ensure the repulsion of the flow from the obstacles, one effective way is to impose two types of boundary conditions on the obstacle and domain boundaries for the solution to the Laplace's Equation in a computational domain: Dirichlet type (the potentials on the boundaries of obstacles and domain are assigned a constant high value, i.e., the fluid motion on the obstacle boundary is along the normal direction of the obstacle boundary/wall) and Neumann type (i.e., the flow cannot pass through the boundary, or the fluid motion on the obstacle boundary and wall is parallel to the tangential direction of the obstacle boundary since the normal component is null). For path planning, the HPF has the property of the min-max principle, so that no local minima other than the goal in the interior of cluttered or bounded environments with state constraints of a point robot are defined by Dirichlet boundary conditions or Neumann boundary conditions on the borders of the obstacles and computational domain.