## 1 Introduction
Motion planning based on hydrodynamic potential applies (1) different fundamental elements such as a point sink (representing the goal), a point source (representing the robot location), or a uniform flow (defined as a flow with constant speed in a prespecified direction) plus a doublet (representing the obstacle), and their superposition, or (2) velocity potential solution to the Laplace's Equation with appropriate boundary conditions, to create a new HPF. The gradient of the HPF or the streamline that defines the vector field of the path at every point can be computed efficiently; analytically for a simple obstacle shape such as a circle or numerically. There are a few simulations showing that a vehicle modeled as a point (fluid) particle could smoothly navigate without collision with the (circular, elliptical, rectangle, or arbitrary-shaped) obstacles [6; 8; 11; 12; 13; 14; 15; 16; 30] by following streamlines from a variety of start points in an environment composed of multiple obstacles. To build an APF via hydrodynamics potential, [PERSON] [15] proposed a panel method by first approximating an arbitrarily shaped obstacle by an enclosing polygon (set of panels). Each obstacle panel is treated as a source/sink with the strength adjusted to make the obstacle a repelling potential function by summing the HPF of each panel, i.e., with nonzero outer normal velocity at the obstacle panel representative point. [PERSON] et al. [16] introduced a reactivity parameter to adjust the amplitude of the path's deflection around an obstacle and an optimal 3D path is obtained by a genetic algorithm. For path planning on an unstructured terrain consisting of meshes of different size and geometry of computational domain discretized by finite element, Reference [25] proposed to use a graph search to generate an initial path from the start to the goal, then used streamlines to smooth the initial path. Planning and optimization of state and input trajectories for stabilizing non-holonomic mobile robots studied in this paper or more general non-holonomic systems such as a car with trailers [33] in multiple obstacles environment subject to bounded state constraints (such as the environment constraint and path and its derivative constraints) and input constraints is challenging. The motion planners have to plan in the full state space (the space of position, heading, speed, curvature, and/or curvature derivative) and deal simultaneously with constraints of collision avoidance and non-holonomic constraint and admissible input (such as the computation of the invariant set or reachable set of the system [31]). Approaches that can either achieve the optimal or near-optimal position and heading and velocity or higher order derivatives simultaneously or sequentially are proposed for non-holonomic mobile robots. A dipolar potential field is combined with discontinuous state feedback for navigating a non-holonomic mobile robot in the presence of obstacles in Reference [32]. Pontryagin Maximum Principle in variational form was employed in Reference [33] to obtain a convergent sequence of controls for optimal motions of general non-holonomic systems with state and input constraints. New methods to non-holonomic path planning are given in Reference [34], which proposed to steer the non-holonomic mobile robot without additional constraints via rotation and linear translation using trigonometric switch inputs, a generic APF-based method via deforming a feasible path of non-holonomic systems (e.g., a mobile robot with trailers) without the violating non-holonomic constraint [35], and the ones based on partial differential equations other than the Laplace's Equation coined by References [7; 24], which applied a parabolic partial differential equation, and Reference [29], which used the Navier-Stokes equation of viscous flow for path planning. It is noted that HPF is interpreted in heat conduction, instead of hydrodynamics, in a recent work [28] with a demonstration of a real-time mobile robot navigation experiment.