The visibility graph (or V‐graph), formally described by Lozano‐Perez and Wesley in the 1970s [142], represents a complete and easy to implement algorithm. This algorithm is based on constructing an undirected graph where edges come as close as possible to obstacles, then resulting in minimum‐length paths. An important aspect is that obstacles can be inflated in order to avoid an incident where the robot could pass by too close to them, which could lead to collisions [58]. The main drawbacks of this algorithm are that it can demand a high computation time for getting a trajectory in environments with complicated obstacles, and some points of the path are too close to obstacles if inflation is not used. The fast dynamic visibility graph (DVG) approach proposed in [100] represents an efficient implementation of the traditional V‐graph. The V‐graph has been largely used by the robotics community from Shakey in the 1970s to recent publications like [39], where the V‐graph algorithm is used to find the obstacle‐free path after processing digital images acquired by a camera onboard a mobile robot.
The Voronoi diagram was proposed by G.F. Voronoi at the end of the nineteenth century and since then it has been widely used in many areas such as: mathematics, physics, astronomy, geographical information systems, computer graphics, image processing, and robotics. In contrast to the V‐graph approach, a Voronoi diagram tends to maximize the distance between the robot and obstacles in the map, that is, roads stay as far away as possible from obstacles. Therefore, there is no need to inflate the obstacles. As in the V‐graph after the set of roads are determined, a graph search algorithm is applied to estimate the best route between the desired points. An application of the Voronoi diagram in mobile robotics appears in [79]. Here, this algorithm is used by an agricultural robot to move in a greenhouse. There is also a broad field of research combining the advantages of the visibility graph (shortest routes) with those of the Voronoi diagram (maximum distance from the obstacles); see, for example, [151, 211].
Figure 1.4 Path planning methods. Main categories for path planning in mobile robotics and examples of some well‐known methods for each of them.
The second big category comprising exact planning algorithms is cell decomposition. This strategy is based on partitioning the free configuration space into a finite set of regions that can be safely traversed by a robot. Cell decompositions are often employed in high‐level planning approaches where the robot may visit some regions based on logic or temporal requirements (this feature is extended in Section 1.4 and in subsequent chapters).
It bears mentioning that the exact methods further need graph search algorithms in order to finally obtain the optimal route connecting a series of waypoints. For this aspect, three graph search algorithms that return a minimum cost path are usually employed, namely Dijkstra [53], A* [140, 160, 221], and D* [195]. Dijkstra and A* algorithms are great choices for static graphs with positive known arc weights, with A* additionally requiring a heuristic cost function. The D* algorithm, proposed by Anthony Stentz in 1995 [195], constitutes an extension of the classic A* algorithm for graphs, introduced by Nils J. Nilsson in the 1970s [160]. D* considers the 2D map as a cost map where each weight represents the cost of traversing each cell in the horizontal or vertical direction. For cells corresponding to obstacles we can give a huge cost, so D* finds the path minimizing the total cost of travel. This cost may deal with various aspects such as time to drive across a cell, roughness of the terrain, etc. The key feature of D* is that it supports incremental replanning, that is, the algorithm can replan the initial route while the robot discovers that the world is different from the initial plan. The incremental replanning has a lower computational cost than completely replanning, as would be required by A* [41]. The algorithm D* accounts for many successful applications in mobile robotics. For example, in [84], the D* algorithm is used for generating the optimal route for ground autonomous vehicles. More specifically, the cost function associated with the D* algorithm is configured in such a way that the route connecting two points minimizes the uncertainty associated with the elevation of the 3D points in the maps. This route also avoids points where the robot may experience a high risk of entrapment. After the successful introduction of the D* algorithm, there is a recent and broad body of research dealing with extending the features of this algorithm. Some of those recent algorithms are: D* Lite [125], PHI* [57], and E* [169].
Together with combinatorial or exact planning algorithms, another broad body of research in the field of path planning nowadays is related to sampling‐based planning methods. The first algorithm, called Probabilistic Roadmap (PRM), was proposed by Lydia E. Kavraki and Jean‐Claude Latombe in the 1990s [106]. The advantage of PRM is that relatively few points need to be tested to ascertain that the points and the paths between them are obstacle free [41]. The efficacy of several variations of the PRM algorithm is discussed in [75].
A major drawback of PRM is that it assumes that the robot is a point with omnidirectional capabilities. The Rapidly exploring Random Tree (RRT) algorithm takes into account the model of the robot, e.g. differential‐drive motion [134]. However, the main drawback of RRT is that it does not lead to an optimal path. This aspect is overcome by a variant of RRT called RRT*; this algorithm does guarantee the optimality and can find the optimal trajectory when applied to complex non‐holonomic systems [2, 104, 138]. In recent literature, there are numerous RRT‐based strategies trying to ensure optimality despite uncertainty in the motion of the robot [136, 143].
1.3 Motion control. Definition and historical background
The key goal of a mobile robot is to follow the route generated by the path planner, and this goal is responsible for the motion controller. More specifically, robot control deals with the problem of determining the forces (or velocities) that must be developed by the robotic actuators in order for the robot to go to a desired position, track a desired trajectory, and, in general, perform some tasks with desired performance requirements [202].
The control problem can be classified depending on how the reference path is defined: a single target point
Essentially the control problem must ensure
Figure 1.5 Motion control methods. Main categories for motion control in mobile robotics and an example of some well‐known methods for each of them.
where