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the empirical parameters which govern the performance of the search. To recap, these parameters are:

1. The maximum distance from the current node in the A* search to bound the number of nodes that are expanded at each iteration: we define this as the Walkable set of nodes available for expansion at each iteration.

2. The Nielson-Foley parametrization, which we use as a distance metric.

3. The definition of a discrete radius measurement, which we use to penalize segment costs which likely would require a full loop to reach a goal (in addition to cost calcu- lated by basic geometry).

4. The definition of a penalty associated with segments with inflections which would violate curvature constraints when smooth paths are calculated.

5. A Greedy Algorithm, which is used to compute the A* _{heuristic cost-to-go.}

Walkable Nodes: There are multiple ways a traveling salesman problem can be con- figured within an A*search framework. Rather than using the entire set of points of interest as the search graph, which would permit expansion from a current node to all other unvis- ited nodes, we limit the horizon the A* search can use when selecting a new set of nodes to expand. Without this consideration, the complexity factor of the search would be N!, since all remaining nodes would be available for expansion at each time step. Nodes that are distant from the one under consideration are unlikely to be part of an optimal path. We only consider them when no others are available for expansion. When the area of interest is convex, a search horizon of 3 times the nodal point spacing appears to be an appropriate balance of sufficient exploration versus minimization of computation time. However, when the area of interest is non-convex or non-contiguous, a wider horizon for the search is war- ranted, since otherwise, the search may be forced to jump wide distances when there are no relatively adjacent nodes remaining. The intersection-shaped pattern explored in the results section required a horizon of at least four times the nodal point spacing to avoid costly jumps across non-convex portions. Expansion over a horizon of more than four times the point spacing resulted in an excessive number of nodes being expanded in each time step. Moreover, results with a horizon of four do not include many inefficient crossovers from one branch of the intersection to another. Future applications should assess this parameter at least cursorily if areas of interest have shapes not considered in this thesis.

straints can be abstracted into the graph search. We augment their parametrization to permit proportional representation of the cost of turning segments up to the maximum turning angle of π radians. Moreover, when the final segment turning angle has a zero magnitude, we assign it a small positive value so that in the event of multiple points along a straight line, the node with the smallest Euclidean offset distance will be selected.

Discrete Curvature requirements are estimated for each segment of the graph search to embed the cost of a loop when the proximity of endpoints implies a directC2_{path would}

violate feasibility (curvature) constraints. We use an inscribed circle to define the turning radius implies by a candidate path segment. When it is less than the actual turning radius, we add the cost of a loop to a candidate path segment. Consequently, even in cases when a feasible path between points may require looping when a smooth C2 path is generated, the augmented Nielson-Foley metric provides an appropriate balance when ranking distance versus turning angle requirements with minimal complexity to determine a tour sequence.

Inflections that would be required in the smooth path generation (second) stage of the planner are detected in the search phase by sign reversals in the required turning an- gles over each four node set as path candidates are defined. Inflection costs are accounted as factor on the angles used in the Nielson-Foley parametrization, since the angles in the parametrization already have distance normalization factors. The magnitude of this factor was empirically found to be 15 for the types of areas of interest studied to date. These include:

• Low and high aspect ratio contiguous rectangular areas • Non-convex (cross-shaped) areas

• Non-contiguous areas with randomly located nodes

Future applications should assess this parameter at least cursorily if areas of interest have shapes not considered in this thesis.

A Greedy Search is used as the A* heuristic to complete partial tour paths so that each candidate path is full cycle TSP tour. As noted earlier, the consequence is that paths cannot be considered to be optimal. However, since the A* search is computed for a user

defined number of iterations, this heuristic does not cause early termination of the search as could be the case in a simple start-to-goal type search.

Note, the aforementioned heuristicsprevent selection of candidate paths that would violate curvature constraints when smoothC2paths are calculated but do not guarantee the smooth paths are fully free of such instances. In these cases, there are options available during the smooth path generation stage to fully guarantee paths are feasible.

Results for two examples of our A* _{search are illustrated on Figures 5.5 and 5.6. The}

search is initialized with a three point starting sequence to seed the Nielson-Foley metric, as identified by red symbols in the figure. The cycle is then closed to the initial point. Both the Greedy (Nearest Neighbor) and A* path sequences are shown on the right side of these figures for reference. In the first example case, the point spacing set by data resolution requirements (≈218m) is significantly larger than the minimum turning radius ( 58m), so adjacent points can readily be traversed through direct turns when curvature constraints are observed. Hence, a relatively simple cycle tour is determined. In the second example case, the point spacing set by data resolution requirements (≈121m) is only slightly larger than the minimum turning radius ( 85m), so adjacent points cannot typically be traversed through direct turns when curvature constraints are observed. Consequently, feasible paths through adjacent points may require looping or other indirect routing.

The next step is to determine a C2 path that satisfies turning constraints, based on the results of the A* search.

Figure 5.5: A* and Nearest Neighbor sequences for an example case where turn radius is small relative to the sensor/imaging footprint (starting node sequence is 1-2-3).

Figure 5.6: A* and Nearest Neighbor sequences for an example case where turn radius is large relative to the sensor/imaging footprint (starting node is sequence 2-3-4.

5.2

Continuous Feasible Trajectories

Parametric polynomial curves are a natural framework for this application. Complete tra- jectories can be encoded with a small set of control points. Parametric curves can be directly integrated with autopilot commands so that navigation commands can be rapidly computed based on an arbitrary vehicle location and an associated point on the trajectory curve [67]. Computation requirements to determine any point along a path as well as other parameters such as tangent, normal, and curvature are minimal. The approach we take is to first find paths solely based on boundary conditions using the fewest number of control points. These paths are then augmented to guarantee feasibility. If required, additional control points can be added to provide additional degrees of freedom so that boundary conditions are preserved whilst the interior portion of internal segments can be tailored to satisfy feasibility. The fewest number of control points that provides C2 continuity at both ends with fixed boundary conditions is six10. We used this approach to develop a simultaneous arrival coordination schema for a team [71].