8.4.4 A Step Toward Considering Dynamics

If dynamics is an important factor, then the discontinuous vector fields considered so far are undesirable. Due to momentum, a mechanical system cannot instantaneously change its velocity (see Section 13.3). In this context, vector fields should be required to satisfy additional constraints, such as smoothness or bounded acceleration. This represents only a step toward considering dynamics. Full consideration is given in Part IV, in which precise equations of motions of dynamical systems are expressed as part of the model. The approach in this section is to make vector fields that are ``dynamics-ready'' rather than carefully considering particular equations of motion.

A framework has been developed by defining a navigation function that satisfies some desired constraints over a simple region, such as a disc [829]. A set of transformations is then designed that are proved to preserve the constraints while adapting the navigation function to more complicated environments. For a given problem, a complete algorithm for constructing navigation functions is obtained by applying the appropriate series of transformations from some starting shape.

This section mostly focuses on constraints that are maintained under this transformation-based framework. Sections 8.4.2 and 8.4.3 worked with normalized vector fields. Under this constraint, virtually any vector field could be defined, provided that the resulting algorithm constructs fields for which integral curves exist in the sense of Filipov. In this section, we remove the constraint that vector fields must be normalized, and then consider other constraints. The velocity given by the vector field is now assumed to represent the true speed that must be executed when the vector field is applied as a feedback plan.

One implication of adding constraints to the vector field is that optimal solutions may not satisfy them. For example, the optimal navigation functions of Section 8.4.3 lead to discontinuous vector fields, which violate the constraints to be considered in this section. The required constraints restrict the set of allowable vector fields. Optimality must therefore be defined over the restricted set of vector fields. In some cases, an optimal solution may not even exist (see the discussion of open sets and optimality in Section 9.1.1). Therefore, this section focuses only on feasible solutions.