To motivate the introduction of constraints, consider a control model proposed in [235,830]. The action space, defined as in Formulation 8.2, produces a velocity for each action . Therefore, . Suppose instead that each action produces an acceleration. This can be expressed as , in which is an acceleration vector,
Suppose that a vector field is given in the form . How can a feedback plan be derived? Consider how the velocity vectors specified by change as varies. Assume that is smooth (or at least ), and let
Now the relationship between and will be redefined. Suppose that is the true measured velocity during execution and that is the prescribed velocity, obtained from the vector field . During execution, it is assumed that and are not necessarily the same, but the task is to keep them as close to each other as possible. A discrepancy between them may occur due to dynamics that have not been modeled. For example, if the field requests that the velocity must suddenly change, a mobile robot may not be able to make a sharp turn due to its momentum.
Using the new interpretation, the difference, , can be considered as a discrepancy or error. Suppose that a vector field has been computed. A feedback plan becomes the acceleration-based control model
Steven M LaValle 2012-04-20