Paradigms of A.I. Robotics

Generating Potential Field
Definition. Given S ⊂ R 2 , a vector field on S is a function F : S → R 2 . That is, F
associates a vector to every point in S.
FACT: Every potential field is a vector field, and most potential fields are
conservative vector fields, i.e. they are (more or less) the gradient of potential
energy functions.
Remark. Think of potential energy functions as a landscape with hills and valleys and the
potential field is the vector field generated by mapping the instantaneous change in each
direction at every point in the landscape.
FACT: Magnitude profiles define potential energy functions! I.e. Given a
potential energy function, U : R 2 → R + , the potential field associated to U is
F(⃗r) = −▽U(⃗r),⃗r ∈ R 2 .
Paradigms of A.I. Robotics – p.33/