A vector field looks like a ``needle diagram'' over , as depicted in Figure 8.5. The idea is to specify a direction at each point . When used to represent a feedback plan, it indicates the direction that the robot needs to move if it finds itself at .
For every , associate an -dimensional vector space called the tangent space at , which is denoted as . Why not just call it a vector space at ? The use of the word ``tangent'' here might seem odd; it is motivated by the generalization to manifolds, for which the tangent spaces will be ``tangent'' to points on the manifold.
A vector field8.4 on is a function that assigns a vector to every . What is the range of this function? The vector at each actually belongs to a different tangent space. The range of the function is therefore the union
A vector field can therefore be expressed using real-valued functions on . Let for from to denote such functions. Using these, a vector field is specified as
Now consider some examples of vector fields over . Let a point in be represented as . In standard vector calculus, a vector field is often specified as , in which and are functions on
Due to obstacles that arise in planning problems, it will be convenient to sometimes restrict the domain of a vector field to an open subset of . Thus, for any open subset , a vector field can be defined.
Steven M LaValle 2012-04-20