In a rectilinear route, a moving sink is restricted to travel either horizontally or vertically along the connecting edges. We present a new algorithm that finds the shortest round trip rectilinear route covering the specified nodes in a grid based Wireless Sensor Network. The proposed algorithm determines the shortest round trip travelling salesman path in a two-dimensional grid graph. A special additional feature of the new path discovery technique is that it selects that path which has the least number of corners (bends) when more than one equal length shortest round trip paths are available. This feature makes the path more suitable for moving objects like Robots, drones and other types of vehicles which carry the moving sink. In the prosed scheme, the grid points are the vertices of the graph and the lines joining the grid points are the edges of the graph. The optimal edge set that forms the target path is determined using the binary integer programming.