Navigation function

Navigation function usually refers to a function of position, velocity, acceleration and time which is used to plan robot trajectories through the environment. Generally, the goal of a navigation function is to create feasible, safe paths that avoid obstacles while allowing a robot to move from its starting configuration to its goal configuration. Navigation function usually refers to a function of position, velocity, acceleration and time which is used to plan robot trajectories through the environment. Generally, the goal of a navigation function is to create feasible, safe paths that avoid obstacles while allowing a robot to move from its starting configuration to its goal configuration. Potential functions assume that the environment or work space is known. Obstacles are assigned a high potential value, and the goal position is assigned a low potential. To reach the goal position, a robot only needs to follow the negative gradient of the surface. We can formalize this concept mathematically as following: Let X {displaystyle X} be the state space of all possible configurations of a robot. Let X g ⊂ X {displaystyle X_{g}subset X} denote the goal region of the state space. Then a potential function ϕ ( x ) {displaystyle phi (x)} is called a (feasible) navigation function if While for certain applications, it suffices to have a feasible navigation function, in many cases it is desirable to have an optimal navigation function with respect to a given cost functional J {displaystyle J} . Formalized as an optimal control problem, we can write whereby x {displaystyle x} is the state, u {displaystyle u} is the control to apply, L {displaystyle L} is a cost at a certain state x {displaystyle x} if we apply a control u {displaystyle u} , and f {displaystyle f} models the transition dynamics of the system. Applying Bellman's principle of optimality the optimal cost-to-go function is defined as ϕ ( x t ) = min u t ∈ U ( x t ) { L ( x t , u t ) + ϕ ( f ( x t , u t ) ) } {displaystyle displaystyle phi (x_{t})=min _{u_{t}in U(x_{t})}{Big {}L(x_{t},u_{t})+phi (f(x_{t},u_{t})){Big }}}