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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.

**Subsections**
Steven M LaValle
2020-08-14