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To derive solutions to path planning and navigation problem these approaches use geometric methods. “Piano- Movers” problem is a famous problem in this category which deals with the ‘maneuvers’ required to move a long piano out of a room through a narrow door.

Choset et. al. [90] have been concerned with new method, by using conventional (Polaroid) low-resolution ultrasonic sensors mounted in a circular array on a mobile robot for improving the azimuth accuracy of range information. Then to fuse sonar data to better approximate the actual obstacle location a new method was introduced. This new method is known as the arc transversal median method because the robot find out the location of an object 1) with intersecting other arcs by one arc as shown in Fig 2.2; 2) by considering only “transversal” intersections, those which exceed a threshold in angle as shown in Fig.2.3, and 3) by taking the median of the intersections. This method can improve the azimuth accuracy of the sonar sensor by a specified amount under well-defined conditions via some simple geometric relationships.

Fig.2.2. Object must be tangent to the arc, but there are an infinite number of orientation possibilities. Uncertainty is the beam width.

Fig.2.3 Object must be tangent to Arc1 and Arc2. Now there is only one possible orientation. Intersection points are the endpoints of the desired segment.

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Miah and Gueaieb [91] have been outlined two different aspects using received signal strength of a customized radio frequency identification system for mobile robot navigation problem in indoor environments. First, a trilateration method is applied where the localization problem is solved through a geometric approach based on Cayley-Menger determinants to estimate the robot’s current location. Second, a desired path along which a mobile robot is required to navigate can be found when the problem is explored by a set of points on the ground. Jan et. al. [92] have been introduced optimal path planning for navigating rectangular mobile robot based on a higher geometry maze routing algorithm among obstacles and weighted regions. To obtain an optimal collision free path with linear time and space complexities an 8-geomerty maze routing algorithm is applied. These methods cannot only search an optimal path among various terrains but also find the 2-D piano mover’s problem with 3 DOF optimal paths. In addition to that, the algorithm can be easily extended among multiple autonomous robots or path planning problem in the 3-D space to avoid the dynamic collision.

In cell decomposition the free space is decomposed into a large number of small regions called cells. These cells define configuration where the robot can trivially navigate between them. Path construction here can be realized by an algorithm for graph search. These methods are perhaps the most well studied and have been applied widely for robot navigation.

In the cell decomposition approach [93] the free space is represented as a union of cells and a sequence of cells comprises a solution path. For efficiency, hierarchical trees, e.g. octree, are often used. The path planning algorithm proposed by Zelinsky [94] is quite reliable and combines some advantages of the above algorithms. It makes use of quad tree data structure to model the environment and uses the distance transform methodology to generate paths. The obstacles are polygonal-shaped which yields a quad tree with minimum sized leaf quadrants along the edges of the polygon, but the quad tree is large and the number of leaves in the tree is proportional to the polygon’s perimeter and this makes it memory consuming. Acar et. al. [95] have been defined an exact cellular decomposition where critical points of Morse functions indicate the location of cell boundaries. They introduced a general framework for coverage of tasks while varying the Morse function. The variation in Morse function effects the recognized pattern change by the robot. To construct the decomposition they described a sensor-based algorithm. Acar and Choset [96] have been achieved an exact cellular decomposition whose cells are defined in terms of critical points of Morse functions. Generically, a cell is defined by two critical points. Using a graph they encoded the topology of the Morse decomposition. While incrementally constructing a graph, the nodes in the

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graph correspond to the critical points and the pairs of critical points represent the edges of the cells and simultaneously the robot covers its search space. To encounter all the critical points by the robot, they presented a complete algorithm thereby constructing the full graph, i.e., achieving complete coverage.

A planner presented in [97] that determines a robot path to reach its goal such that it does not have to heavily rely on odometry. To reach the goal the robot must follow sequence of obstacle boundaries determined by the planner. They used an exact cellular decomposition with simple structure to simplify the problem and introduced an algorithm to travel the robot between two points that does not heavily depend on dead-reckoning. Rekleitis et. al. [98] have been presented an algorithm on a team of mobile robots for the complete coverage of free space. Based on a single robot coverage algorithm their approach has been done. Single robot coverage is achieved by ensuring that the robot visits every cell. The new multi-robot coverage algorithm can be approached like a single robot planar cell-based decomposition. This method allows planning to occur for a team of N robots in a two-dimensional configuration space.