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MIT16_410F10_lec18

# MIT16_410F10_lec18 - 16.410/413 Principles of Autonomy and...

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Unformatted text preview: 16.410/413 Principles of Autonomy and Decision Making Lecture 18: (Mixed-Integer) Linear Programming for Vehicle Routing and Motion Planning Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 15, 2010 Frazzoli (MIT) Lecture 18: (MI)LP for Motion Planning November 15, 2010 1 / 31 Assignments Readings Lecture notes [IOR] Chapter 11. Frazzoli (MIT) Lecture 18: (MI)LP for Motion Planning November 15, 2010 2 / 31 (Mixed) Integer Linear Programming Many problems of interest can be formulated as mathematical programs in which some of the decision variables are constrained to take one of a finite set of values Typically, these represent logical decisions: visit a location or not, do a task before another, pass to the left or to the right of an obstacle, etc. These can often be modeled as “LP’s”, in which some of the variables must take discrete values (or binary 0/1 values.) Let us look at some examples. Frazzoli (MIT) Lecture 18: (MI)LP for Motion Planning November 15, 2010 3 / 31 Vehicle Routing Problems We have already studied a basic problem in robotics and automation, i.e., the computation of a shortest path between a start and a goal location. In many applications, e.g., UAV mission planning problems, it is of interest to compute paths for one or more vehicles to reach a number of locations, while optimizing some performance criterion. Vehicle Routing Problems are essentially shortest path problems for multiple vehicles and/or multiple destinations, subject to a variety of constraints or performance objectives. VRPs come in a large number of varieties, we will look at some examples. Frazzoli (MIT) Lecture 18: (MI)LP for Motion Planning November 15, 2010 4 / 31 The Traveling Salesman Problem The Traveling Salesman Problem (TSP) is an example of a VRP, in which a single vehicle must visit N locations (cities) following a minimum-cost closed path, starting and ending at a depot. Formal definition, as a graph problem: Let G = ( V , E , w ) be a complete undirected weighted graph, whose vertices include the depot V , and the locations to be visited V 1 ,..., V N . Compute a minimum-weight Hamiltonian cycle for G (i.e., a closed path through all vertices). Prototypical hard combinatorial problem (NP-hard). (But polynomial-time approximations exist for metric TSPs, i.e., TSPs in which the weights satisfy the triangle inequality!) It is possible to write the TSP in a LP form... Frazzoli (MIT) Lecture 18: (MI)LP for Motion Planning November 15, 2010 5 / 31 A na¨ ıve LP formulation Binary decision variables x e , e ∈ E : x e = 1 if the path includes edge e , and x e = 0 otherwise. Let S be a proper subset of V , i.e., ∅ ⊂ S ⊂ V , and indicate with δ ( S ) ⊂ E the set of edges that have exactly one endpoint in V ....
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MIT16_410F10_lec18 - 16.410/413 Principles of Autonomy and...

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