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Unformatted text preview: CS221 Problem Set #1 1 CS 221, Autumn 2007 Problem Set #1: Search, Motion Planning, CSPs Due by 9:30am on Tuesday, October 16. Please see the course information page on the class website for late homework submission instructions. SCPD students can also fax their solutions to (650) 725-1449. We will not accept solutions by email or courier. Written part (100 points) NOTE: These questions require thought, but do not require long answers. Please try to be as concise as possible. 1. [9 points] Configuration Spaces d robot arm floor ceiling base gripper ρ α Consider the robot arm pictured above with two degrees of freedom, operating in a two dimensional workspace. The robot arm has a revolute joint and a prismatic joint. The revolute joint has a range of 0 ≤ α ≤ π , where α is the angle of the arm relative to the floor. The prismatic joint has a range of ρ min ≤ ρ ≤ ρ max , where ρ is the length of the arm from the base to the gripper. The ceiling is a distance d from the floor, with ρ min < d < ρ max . The width of the arm and gripper may be considered negligible. CS221 Problem Set #1 2 θ a o h =o/h θ sin cos θ =a/h Draw the configuration space of the robot arm. Please specify the coordinates of the important points of the obstacles in configuration space (e.g. leftmost point, rightmost point, etc.). Also, while a freehand drawing is ac- ceptable, please make sure that the shape of the obstacle is clear from your drawing. 2. [21 points] Optimization Search / Configuration Spaces For this problem, we consider the problem of moving a robot in various configuration spaces with obstacles. We start with a simple configuration space, with only convex obstacles, and then make it more complicated by allowing non-convex obstacles as well. convex obstacle nonconvex obstacle A discrete search space for this planning problem can be defined using the visibility graph method . In this method, the search space is restricted to consist of only the following states in configuration space: the initial position of the robot, the goal position of the robot, and the vertices of the polygonal obstacles. Further, the search operators only allow the robot to walk in a straight line between two of these points: a state s in the search space is connected to any other state s ′ which can be reached from s by walking along a straight line either completely in free space or along the boundary of an obstacle (i.e. s is connected to s ′ if s ′ is “visible” from s ). Assume that our robot navigates this space using a simple greedy hill-climbing search with a straight-line distance heuristic function. (Note that the robot climbs “down” in that the heuristic function decreases as the robot approaches the goal. Nevertheless, we’ll continue to use the “climbing” metaphor, as we did in class.) We assume that the robot is allowed to move in any direction, but is not allowed rotation.We assume that the robot is allowed to move in any direction, but is not allowed rotation....
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