Survey of Robot 3D Path Planning Algorithms <s> Optimal Control. <s> In this paper, we introduce the general purpose optimal control problem solver RIOTS_95, Matlab toolbox, as a solver for general linear and nonlinear model predictive control (MPC) problems. This optimization toolbox is designed to solve a wide variety of optimal control problems and therefore constitutes a good candidate as a optimization solver in the MPC framework. The illustrative example of a DC motor is introduced to demonstrate the feasibility of model predictive control using RIOTS_95. Nonlinearities are added to the model to exemplify that RIOTS_95 is also suitable for nonlinear model predictive control (NMPC). The effectiveness and advantages of our RIOTS_95 based MPC are exemplified by comparing its results with those obtained by two readily available MPC tools. <s> BIB001 </s> Survey of Robot 3D Path Planning Algorithms <s> Optimal Control. <s> This paper formulates the vehicle navigation task as a constrained optimal control problem with constraints bounding a traversable region of the environment. A model predictive controller iteratively plans an optimal vehicle trajectory through the constrained corridor and uses this trajectory to establish the minimum threat posed to the vehicle given its current state and driver inputs. Based on this threat assessment, the level of controller intervention required to prevent departure from the traversable corridor is calculated and driver/controller inputs are scaled accordingly. Simulated and experimental results are presented to demonstrate multiple threat metrics and configurable intervention laws. <s> BIB002
Path planning problem can be posed in an optimal control form, where optimal control can find the state and control oriented path based on a set of differential equations BIB002 . Optimal control can be interpreted as an extension of the linear algorithms to an infinite number of variables' condition, with the ability to model uncertainty as linear chance constraints much more easier. A basic optimization problem is depicted in a flowchart form in Figure 11 , where the initial state (or current state) and goal state are included in the constraints to ensure completeness. For optimal control case, a general system model is considered in time continuous form, where ( ) represent the evolving state and ( ) is the control parameters. The minimizing criterion is With the necessary constraints, Then cost function (an recommended performance index) can be derived by including all constraints above: Then we can achieve the Hamiltonian: where (22) is the initial condition and (23) is the final condition. Hamiltonian is used to solve the optimal problem based on the maximum principle and then follow the typical optimal solving procedure to generate an optimal path globally. This paper only explains in a general form; each equation can be extended to contain much more constraints. Tisdale et al.  implemented discrete receding-horizon control (RHC) in UAV, which is a variant of optimal control. Chen and Schwartz  described the optimal control problem in detail and proposed a free MATLAB toolbox RIOTS 95, and in BIB001 they further implemented it to solve model predictive problem.