Survey of Robot 3D Path Planning Algorithms <s> Artificial Potential Algorithms. <s> A method for planning smooth robot paths is presented. The method relies on the use of Laplace's equation to constrain the generation of a potential function over regions of the configuration space of an effector. Once the function is computed, paths may be found very quickly. These functions do not exhibit the local minima which plague the potential field method. Unlike decompositional and algebraic techniques. Laplace's equation is very well suited to computation on massively parallel architectures. > <s> BIB001 </s> Survey of Robot 3D Path Planning Algorithms <s> Artificial Potential Algorithms. <s> A methodology for exact robot motion planning and control that unifies the purely kinematic path planning problem with the lower level feedback controller design is presented. Complete information about a freespace and goal is encoded in the form of a special artificial potential function, called a navigation function, that connects the kinematic planning problem with the dynamic execution problem in a provably correct fashion. The navigation function automatically gives rise to a bounded-torque feedback controller for the robot's actuators that guarantees collision-free motion and convergence to the destination from almost all initial free configurations. A formula for navigation functions that guide a point-mass robot in a generalized sphere world is developed. The simplest member of this family is a space obtained by puncturing a disk by an arbitrary number of smaller disjoint disks representing obstacles. The other spaces are obtained from this model by a suitable coordinate transformation. Simulation results for planar scenarios are provided. > <s> BIB002 </s> Survey of Robot 3D Path Planning Algorithms <s> Artificial Potential Algorithms. <s> This paper presents a method for generating shortest paths in cluttered environments, based on the Hamilton-Jacobi-Bellman (HJB) equation. Formulating the shortest obstacle avoidance problem as a time optimal control problem, the shortest paths are generated by following the negative gradient of the return function, which satisfies the HJB equation. A method to generate near-optimal paths is also presented, based on a psuedo return function. Paths generated by this method are guaranteed to reach the goal, at which the psuedo return function is shown to have a unique minimum. The computation required to generate the near-optimal paths is substantially lower than those of traditional potential field methods, making it applicable to on-line obstacle avoidance. Examples with circular obstacles demonstrate close correlation between the near-optimal and optimal paths, and the advantages of the proposed approach over traditional potential field methods.<<ETX>> <s> BIB003 </s> Survey of Robot 3D Path Planning Algorithms <s> Artificial Potential Algorithms. <s> The paper first describes the problem of goals unreachable with obstacles nearby when using potential field methods for mobile robot path planning. Then, new repulsive potential functions are presented by taking the relative distance between the robot and the goal into consideration, which ensures that the goal position is the global minimum of the total potential. <s> BIB004 </s> Survey of Robot 3D Path Planning Algorithms <s> Artificial Potential Algorithms. <s> We address optimal path planning in three dimensional space for an unmanned aerial vehicle (UAV) in the stationary risk environment. We separate the task into two stage, in the first one we determine the risk optimal 2D path for fixed time problem. Then we solve the series of BVPs (Boundary Value Problems) with different UAV speeds and determine the admissible 2D path, which satisfies the time and risk constraints. In the last step one takes into account the relief along the chosen path and determine the approximated 3D path, which minimizes 2D threat along the path and satisfies other constraints. <s> BIB005
Since first proposed by Khatib  , potential field methods have been widely researched and implemented universally because of its low computational complexity. Potential field methods are based on the idea of assigning a potential function (relationship between obstacle free and obstacle space) to free space and simulating the vehicle as a particle reacting to force due to the potential field. It computes the goal attraction and obstacle repulsion simultaneously and guides the robot along the total force gradient. The potential field can be represented as where, in formula (5), ( ) is the total potential at state , ( ) is the attractive potential at state , and denotes the repulsive potential at state . Given ∇ ( ) = − ∑ all , ( ), that is, the virtual attractive force (from all neighbor obstacles, also please keep in mind the relation between the gradient of potential field and virtual force can be negative or positive for your consideration), then the force field at state can be represented as For the attractive potential, where is the goal node, − is the error vector with respect to the goal node, and ( ) is the attractive force at . Hence, attractive force increases with the distance between current node and goal node. For the repulsive potential, , ( ) = −∇ , ( ) where ( ) is the distance from to obstacle region which has been partitioned in convex components, the distance can be defined flexibly, Δ ( ) is a polar value, , is the th obstacle component repulsive force to , and 0, is the safe margin of th convex component which is defined by certain obstacle. For field implementation, APF tends to be more complicated as the bounder of the obstacles is not constant; thus the time efficiency is constructed based mainly on initial guess; thus we classify it as sampling based algorithm. However, such methods are incomplete because they are prone to drop into local minima area. Many studies have been done to help potential field algorithms to overcome the local minima by generating navigation functions or computing the potential with constraints. Reference BIB001 proposed a harmonic potential method to solve the local minima problem by using Laplace's Equation to constrain the generation of a potential function; Rimon and Koditschek BIB002 proposed a Morse function with a single minimum at the desired destination strategy to form a strong stable robot navigation method to jump out of the local minima, and this is the first formally proposed navigation method. Authors in BIB003 imported the idea of Hamilton-Jacobi-Bellman to form a HJB function: where is the control and ( ) is the potential factor. This function generates a unique global minimum value; thus it is able to yield a global feasible path that jumps out of local minima, and BIB005  put it to a higher stage. Ge and Cui BIB004 solved the goals nonreachable with obstacle nearby (GNRON) problem by introducing a new repulsive: where ( , ) is the minimal distance between the robot and the goal . ( , ) ensures the total potential ( ) arrives at its global minimum, that is, 0, if and only if = .