1. An Iterated Method to the Dubins Vehicle Travelling Salesman Problem OBJECTIVES Develop an algorithm to compute near optimal solutions to the Travelling Salesman Problem (TSP) with Dubins Vehicle constraints efficiently Provide a way to optimize vehicle travel time taking vehicle dynamics into consideration Student: J. Adam Nisbett, Electrical Engineering BACKGROUND A Dubins Vehicle ‘s motion is constrained by a minimum turning radius. Almost any vehicle in continuous forward motion must follow a Dubins Vehicle path when inertial effects are considered. The complexity of the TSP solution increases dramatically when the Dubins Vehicle path constraints are applied. Direction of motion while passing through each node in the TSP can significantly effect the length of the path to other nodes from that point. To reduce complexity many common algorithms use ordering of nodes based on a Euclidean TSP solution Problem is then somewhat simplified into finding the best angle to travel through each point. If several nodes lie close to each other (on the order of the minimum turning radius) this results in having to loop back around. An ideal path might save some points until it is necessary to pass near the area again. Most solutions also focus on path length and ignore any constraints on speed required to achieve the minimum turning radius. APPROACH Each node can be thought of as a bar magnet that is free to rotate Given the angle of a node, the direction of the ‘field’ at any other node is tangent to a circle containing both nodes. If the radius is less than the minimum turning radius the field strength is set to zero. The field strength is proportional to the arc distance between the points. The field strength can also be varied inversely with the radius to apply a penalty for requiring sharp turns if desired. Given the location of all nodes, the Euclidean distance and angles between them can be calculated once at the beginning of the algorithm The angle of the field produced by node A at B is described as The magnitude of the field produced by node A at B can be described by some ratio of arc length C and radius R With each iteration, every node tries to align itself with the fields produced by other nodes. The rate of change in each node’s alignment is based on the magnitude of the field at its location. When all nodes reach a stable position, the magnetic loop through all the nodes defines the solution path. Faculty Advisor: Donald C. Wunsch, Electrical & Computer Engineering RESULTS If all nodes are left free to interact with each other, multiple magnetic loops tend to form. After each iteration, every node should only adjust based on the effect of a limited number of other nodes which have the largest effect. By the end of the process each node’s orientation should only be affected by two other nodes. An added advantage of this process is that the ordering of the nodes can be recorded without tracing out the field lines. In some simple examples with few nodes this algorithm achieved shorter paths than is possible using the node ordering given by the Euclidean TSP When using a CPU, the time required to reach a stable solution becomes impractical for larger problem sizes If the algorithm can be shown to work well for small problem sizes, the parallel nature of the algorithm can be used by a GPU to enhance efficiency and make larger problem sizes practical. DISCUSSION Efficient solutions to the TSP for Dubins Vehicles have a wide range of practical applicability. The highly parallel nature of this method makes it a good candidate for use with a GPU Effective use of the GPU architecture could dramatically improve solution time efficiency and possibly make the algorithm practical for larger problem sizes. Direct solution of the Dubins Vehicle TSP rather than reliance on Euclidean TSP should result in improved performance in problems with higher node density FUTURE WORK Complete work on the algorithm that determines which nodes affect each other. Implement the algorithm using a GPU Perform analysis on system stability Acknowledgements Support from the National Science Foundation, the Intelligent Systems Center, the Chancellor’s Fellowship, and the M. K. Finley Missouri Endowment is gratefully acknowledged. A B C A.Euclidean TSP solution B.Alternating Algorithm solution Uses node ordering from Euclidean TSP Unnecessary loops are needed to follow node order C.Possible practical solution Node ordering and angle at each node are related Shorter path Turning radius can be relaxed without significantly lengthening path N S C R α θ β A B E ΔxΔx ΔyΔy