Wikipedia / Navigation function
Wikipedia / 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.
Potential functions as navigation functions
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.
denote the goal region of the state space.
is called a (feasible) navigation function if
.
.
Probabilistic navigation function
, the Probabilistic navigation function is defined as:
is said to be a probabilistic navigation function if it satisfies the following conditions:
It is a navigation function.
.
Navigation Function in Optimal Control
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. Formalized as an optimal control problem, we can write
models the transition dynamics of the system.
Applying Bellman's principle of optimality the optimal cost-to-go function is defined as
Together with the above defined axioms we can define the optimal navigation function as
.
.
Even if a navigation function is an example for reactive control, it can be utilized for optimal control problems as well which includes planning capabilities.
Stochastic Navigation Function
If we assume the transition dynamics of the system or the cost function as subjected to noise, we obtain a stochastic optimal control. In the field of reinforcement learning.
See also
References
Sources
External links
NFsim: MATLAB Toolbox for motion planning using Navigation Functions.