### 8.4.4.1 An acceleration-based control model

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,

 (8.44)

The velocity is obtained by integration over time. The state trajectory, , is obtained by integrating (8.44) twice.

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

 (8.45)

in which denotes the unnormalized directional derivative in the direction of : . Suppose that an initial state is given, and that the initial velocity is . The feedback plan can now be defined as

 (8.46)

This is equivalent to the previous definition of a feedback plan from Section 8.4.1; the only difference is that now two integrations are needed (which requires both and as initial conditions) and a differentiability condition must be satisfied for the vector field.

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

 (8.47)

in which is a scalar gain constant. A larger value of will make the control system more aggressively attempt to reduce the error. If is too large, then acceleration or energy constraints may be violated. Note that if , then , which becomes equivalent to the earlier formulation.

Steven M LaValle 2012-04-20