Recall the Dubins version of the simple car given in Section 13.1.2. The system was specified in (13.15). It is assumed here that the car moves at constant forward speed, . The other important constraint is the maximum steering angle , which results in a minimum turning radius . As the car travels, consider the length of the curve in traced out by a pencil attached to the center of the rear axle. This is the location of the body-frame origin in Figure 13.1. The task is to minimize the length of this curve as the car travels between any and . Due to , this can be considered as a bounded-curvature shortest-path problem. If , then there is no curvature bound, and the shortest path follows a straight line in . In terms of a cost functional of the form (8.39), the criterion to optimize is
Since the speed is constant, the system can be simplified to
It was shown in  that between any two configurations, the shortest path for the Dubins car can always be expressed as a combination of no more than three motion primitives. Each motion primitive applies a constant action over an interval of time. Furthermore, the only actions that are needed to traverse the shortest paths are . The primitives and their associated symbols are shown in Figure 15.3. The primitive drives the car straight ahead. The and primitives turn as sharply as possible to the left and right, respectively. Using these symbols, each possible kind of shortest path can be designated as a sequence of three symbols that corresponds to the order in which the primitives are applied. Let such a sequence be called a word . There is no need to have two consecutive primitives of the same kind because they can be merged into one. Under this observation, ten possible words of length three are possible. Dubins showed that only these six words are possibly optimal:
To be more precise, the duration of each primitive should also be specified. For or , let a subscript denote the total amount of rotation that accumulates during the application of the primitive. For , let a subscript denote the total distance traveled. Using such subscripts, the Dubins curves can be more precisely characterized as
It will be convenient to invent a compressed form of the words to group together paths that are qualitatively similar. This will be particularly valuable when Reeds-Shepp curves are introduced in Section 15.3.2 because there are 46 of them, as opposed to Dubins curves. Let denote a symbol that means ``curve,'' and represents either or . Using , the six words in (15.44) can be compressed to only two base words:
Powerful information has been provided so far for characterizing the shortest paths; however, for a given and , two problems remain:
In addition to use as an LPM, the resulting cost of the shortest path may be a useful distance function in many sampling-based planning algorithms. This is sometimes called the Dubins metric (it is not, however, a true metric because it violates the symmetry axiom). This can be considered as the optimal cost-to-go . It could have been computed approximately using the dynamic programming approach in Section 14.5; however, thanks to careful analysis, the exact values are known. One interesting property of the Dubins metric is that it is discontinuous; see Figure 15.6. Compare the cost of traveling using the primitive to the cost of traveling to a nearby point that would require a smaller turning radius than that achieved by the primitive. The required action does not exist in , and the point will have to be reached by a longer sequence of primitives. The discontinuity in is enabled by the fact that the Dubins car fails to possess the STLC property from Section 15.1.3. For STLC systems, is continuous.
Steven M LaValle 2020-08-14