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

A potential function. Imagine dropping a marble on the surface. It will avoid the three obstacles and eventually reach the goal position in the center.

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 $X$ be the state space of all possible configurations of a robot. Let $X_{g}\subset X$ denote the goal region of the state space.

Then a potential function $\phi (x)$ is called a (feasible) navigation function if

1. $\phi (x)=0\ \forall x\in X_{g}$
2. $\phi (x)=\infty$ if and only if no point in ${X_{G}}$ is reachable from $x$.
3. For every reachable state, $x\in X\setminus {X_{G}}$, the local operator produces a state $x'$ for which $\phi (x')<\phi (x)$.

See also

• control theory
• robot control
• motion planning

References

• LaValle, Steven M. (2006), Planning Algorithms (First ed.), Cambridge University Press, ISBN 978-0-521-86205-9
• Laumond, Jean-Paul (1998), Robot Motion Planning and Control (First ed.), Springer, ISBN 3-540-76219-1

External links

• NFsim: MATLAB Toolbox for motion planning using Navigation Functions.