However, flying the robot at a constant speed and turning it as it approaches a given point, so it continues on to the next point without stopping, consumes significantly less fuel. This scenario requires a path with more gentle curves than the sudden turns that occur if the vehicle is allowed to stop. Consider a certain minimum curvature to the path and then seek the shortest path between two successive positions. This problem does not concern itself with the shortest or most efficient path for a spacecraft to follow between Earth and Mars. A position implies more than a location.
Position also implies the orientation of the robot at a given location.
When a position is considered, we must recognize that it does no good for the robot to arrive at an inspection point if its camera is pointing in the wrong direction. The problem is to determine the most efficient path from one position to another in order to plan an inspection tour for the robot. In 1957, L. E. Dubins 1 solved this problem for a robot moving in two-dimensional space, such as on a factory floor. These shortest paths always consist of a circular segment, a linear segment, and another circular segment if the initial and final positions are sufficiently far apart. In other cases, the shortest path may consist of three circular segments.