Recall the approach in Section 14.4.1 that enabled systems of the form to be expressed as for some suitable (this was illustrated in Figure 14.15). This enabled many systems to be imagined as multiple, independent double integrators with phase-dependent constraints on the action space. The same idea can be applied here to obtain a single integrator.
Let denote a 2D path-constrained phase space, in which each element is of the form and represents the position and velocity along . This parameterizes a 2D subset of the original phase space . Each original state vector is . Which accelerations are possible at points in ? At each , a subset of can be determined that satisfies the equations of the form (14.39). Each valid action yields an acceleration . Let denote the set of all values of that can be obtained from an action that satisfies (14.39) for each (except the ones for which ). Now the system can be expressed as , in which . After all of this work, we have arrived at the double integrator. The main complication is that can be challenging to determine for some systems. It could consist of a single interval, disjoint intervals, or may even be empty. Assuming that has been characterized, it is straightforward to solve the remaining planning problem using techniques already presented in this chapter. One double integrator is not very challenging; hence, efficient sampling-based algorithms exist.
An obstacle region will now be considered. This includes any states that belong to . Given and , the state can be computed to determine whether any constraints on are violated. Usually, is constructed to avoid obstacle collision; however, some phase constraints may also exist. The obstacle region also includes any points for which is empty. Let denote .
Before considering computation methods, we give some examples.
Suppose that , which means that the particle must move along a diagonal line through the origin of . This further simplifies (14.40) to and . Hence any may be chosen, but must then be chosen as . The constrained system can be written as one double integrator , in which . Both and are derived from as . Note that does not vary over ; this occurs because a linear path is degenerate.
Now consider constraining the motion to a general line:
Suppose that and . The path is
If independent, double integrators are constrained to a line, a similar result is obtained. There are equations of the form (14.40). The for which is largest determines the acceleration range as . The action is defined as , and the for are obtained from the remaining equations.
Now assume is nonlinear, in which case (14.39) becomes
The same ideas can be applied to systems that are much more complicated. This should not be surprising because in Section 14.4.1 systems of the form were interpreted as multiple, independent double integrators of the form , in which provided the possible accelerations. Under this interpretation, and in light of Example 14.6, constraining the motions of a general system to a path just further restricts . The resulting set of allowable accelerations may be at most one-dimensional.
The following example indicates the specialization of (14.39) for a robot arm.
Steven M LaValle 2020-08-14