A lattice can even be obtained for the general case of a fully actuated mechanical system, which for example includes most robot arms. Recall from (13.4) that any system in the form can alternatively be expressed as , if is defined as the image of for a fixed . The main purpose of using is to make it easy to specify a fixed action space that maps differently into the tangent space for each .

A similar observation can be made regarding equations of the form , in which and is an open subset of . Recall that this form was obtained for general unconstrained mechanical systems in Sections 13.3 and 13.4. For example, (13.148) expresses the dynamics of open-chain robot arms. Such equations can be expressed as by directly specifying the set of allowable accelerations. Each will map to a new action in an action space given by

for each and .

Each directly expresses an acceleration vector in . Therefore, using , the original equation expressed using can be now written as . In its new form, this appears just like the multiple, independent double integrators. The main differences are

- The set may describe a complicated region in , whereas in the case of the true double integrators was a cube centered at the origin.
- The set varies with respect to and . Special concern must be given for this variation over the time sampling interval . In the case of the true double integrators, was fixed.

The first difference is handled by performing grid sampling over and making an edge in the reachability graph for every grid point that falls into ; see Figure 14.15a. The grid resolution can be improved along with to obtain resolution completeness. To address the second problem, think of as a shape in that moves over time. Choosing close to the boundary of is dangerous because as increases, may fall outside of the new action set. It is often possible to obtain bounds on how quickly the boundary of can vary over time (this can be determined, for example, by differentiating with respect to and ). Based on the bound, a thin layer near the boundary of can be removed from consideration to ensure that all attempted actions remain in during the whole interval . See Figure 14.15b.

These ideas were applied to extend the approximation algorithm framework to the case of open-chain robot arms, for which is given by (13.148). Suppose that is an axis-aligned rectangle, which is often the case for manipulators because the bounds for each correspond to torque limits for each motor. If and are fixed, then (13.140) applies a linear transformation to obtain from . The rectangle is generally sheared into a parallelepiped (a -dimensional extension of a parallelogram). Recall such transformations from Section 3.5 or linear algebra.

Steven M LaValle 2020-08-14