A fully autonomous inspection of a known, expansive structure will benefit from prin-
cipled, model-based path planning, combined with active perception [14] to modify the plan when unexpected events occur. We have advanced several steps closer to-
ward this capability by contributing to the former of these two areas, developing sampling-based methods that construct high-quality routes for covering large struc-
efficiently search a C-Space that is embedded not only with obstacles, but with a cov-
erage topology that maps robot configurations to sensor observations, and vice versa. Our proposed randomized planning techniques, implemented using high-performance
data structures and optimization methods, can be used to plan an inspection in whole or in part, over as long or as short a horizon as time allows. They will hopefully serve
Appendix A
Observation of a Continuous
Structure Boundary
In Section 3.4, we presented a probabilistic completeness analysis applicable to any
algorithm that employs random sampling of robot configurations to cover a finite set of discrete geometric primitives. As mentioned in that section, the discrete analysis
requires only two scalar parameters to describe the difficulty of a coverage problem: the total number of geometric primitives, and a ratio comparing the volumes of the
C-Space region being sampled and the smallest subset of views with a single primitive in common. To guarantee coverage of the full continuous boundary of a structure,
however, more problem-specific details are required in the analysis: the robot sensor’s field of view, the dimensionality of the workspace, and the degrees of freedom available
for positioning the sensor in the workspace.
Here we review the concepts that play a role in a continuous analysis of the
coverage sampling problem (CSP), as defined in Definition 1 of Chapter 3. The key parameter is the Vapnik- ˇCervonenkis dimension (VC-dimension) [161], a quantity
that captures the “hardness” of a problem’s geometry using a single scalar value. The derivation of this quantity for a specific robot, sensor, and workspace comprises
the main challenge of a continuous analysis. We will introduce the tools that can be used, in combination with the VC-dimension, to establish quantitative bounds on
We rely once again on the set system taxonomy introduced in Chapter 3, with
some modifications for continuous coverage. We refer to points on the continuous surface of the structure under inspection as pi ∈ P , and the sampled robot view configurations as qj ∈ Q. Si ∈ S refers to the set of all feasible configurations in Q that map to sightings of the point pi ∈ P . We once again invoke the primal set system (P, Q) and the dual set system (Q, S) to aid in the analysis.
A.1
Infinite P Preliminaries: VC-dimension and
-nets
If P is an infinite set, the limit in (3.5) no longer holds and a different approach is
required to show probabilistic completeness of a coverage sampling algorithm. Even if the number of sets Si ∈ S is infinite, we can still establish a bound on the number of samples needed to guarantee k-coverage of P , required by the redundant roadmap algorithm, with a specific probability of failure.
We first introduce the concept of shattering a set. Consider a finite subset of points B ⊆ P . If the intersection of B with the members qj ∈ Q yields every single one of the 2|B| combinatorially distinct subsets of B, then B is shattered by Q. Consequently, there must be at least 2|B| distinct sets in Q for B to be shattered. An important
property related to shattering is the VC-dimension, which we define below.
Definition 9 (Vapnik- ˇCervonenkis (VC) Dimension). The VC-dimension of a set
system (P, Q) is the cardinality of the largest subset of P that can be shattered by the family of ranges Q.
The VC-dimension figures critically in several theorems on set systems. It dictates
the approximation factor of a polynomial-time hitting set approximation algorithm [24], which has been used in planning and sensor placement problems [67], [81], [62] to
achieve a better worst-case approximation than the classical set cover approximation algorithms. The VC-dimension also appears in theorems on the sampling of random
governs the maximum number of samples required to achieve an -net with high
probability. An -net intersects all ranges whose intersection with B is greater than |B| in size.
Definition 10 (-net). Let (P, Q) be a set system, let B be a subset of P , let ∈ [0, 1]
be a real number, and let N ⊂ B be a set of samples drawn randomly from B. The subset N is an -net for B if every range qj ∈ Q of size |qj∩ B| > |B| contains at least one point from N .
In the dual set system (Q, S), Q is an infinite set of robot configurations, and S
is a family of infinite subsets of Q, each of which maps to a view of a specific point pi ∈ P . We can construct -nets for the dual system by sampling configurations from an infinite, continuous A ⊆ Q. The fact that A is infinite does not change the role of an -net; although most commonly presented over finite sets [70], [7], prior analyses
have considered -nets comprised of infinite sets as well, particularly with application to robotics and sensor placement [80], [81]. The sizes of sets A and Si∩ A can still be compared using a fraction , but the measure µ(A), which returns the volume of a set A in robot configuration space, will replace the cardinality |B|, and uniform random
sampling of continuous A will replace the drawing of samples from finite B.