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## Robot Lab: Robot Path Planning

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**Robot Lab:Robot Path Planning**William RegliDepartment of Computer Science(and Departments of ECE and MEM)Drexel University Slide 1**Introduction to Motion Planning**• Applications • Overview of the Problem • Basics – Planning for Point Robot • Visibility Graphs • Roadmap • Cell Decomposition • Potential Field Slide 2**Goals**• Compute motion strategies, e.g., • Geometric paths • Time-parameterized trajectories • Sequence of sensor-based motion commands • Achieve high-level goals, e.g., • Go to the door and do not collide with obstacles • Assemble/disassemble the engine • Build a map of the hallway • Find and track the target (an intruder, a missing pet, etc.) Slide 3**Fundamental Question**Are two given points connected by a path? Slide 4**Basic Problem**• Problem statement: Compute a collision-free path for a rigid or articulated moving object among static obstacles. • Input • Geometry of a moving object (a robot, a digital actor, or a molecule) and obstacles • How does the robot move? • Kinematics of the robot (degrees of freedom) • Initial and goal robot configurations (positions & orientations) • Output Continuous sequence of collision-free robot configurations connecting the initial and goal configurations Slide 5**Example: Rigid Objects**Slide 6**Example: Articulated Robot**Slide 7**Is it easy?**Slide 8**Hardness Results**• Several variants of the path planning problem have been proven to be PSPACE-hard. • A complete algorithm may take exponential time. • A complete algorithm finds a path if one exists and reports no path exists otherwise. • Examples • Planar linkages [Hopcroft et al., 1984] • Multiple rectangles [Hopcroft et al., 1984] Slide 9**Tool: Configuration Space**Difficulty • Number of degrees of freedom (dimension of configuration space) • Geometric complexity Slide 10**Extensions of the Basic Problem**• More complex robots • Multiple robots • Movable objects • Nonholonomic & dynamic constraints • Physical models and deformable objects • Sensorless motions (exploiting task mechanics) • Uncertainty in control Slide 11**Extensions of the Basic Problem**• More complex environments • Moving obstacles • Uncertainty in sensing • More complex objectives • Optimal motion planning • Integration of planning and control • Assembly planning • Sensing the environment • Model building • Target finding, tracking Slide 12**Practical Algorithms**• A complete motion planner always returns a solution when one exists and indicates that no such solution exists otherwise. • Most motion planning problems are hard, meaning that complete planners take exponential time in the number of degrees of freedom, moving objects, etc. Slide 13**Practical Algorithms**• Theoretical algorithms strive for completeness and low worst-case complexity • Difficult to implement • Not robust • Heuristic algorithms strive for efficiency in commonly encountered situations. • No performance guarantee • Practical algorithms with performance guarantees • Weaker forms of completeness • Simplifying assumptions on the space: “exponential time” algorithms that work in practice Slide 14**Problem Formulation for Point Robot**• Input • Robot represented as a point in the plane • Obstacles represented as polygons • Initial and goal positions • Output • A collision-free path between the initial and goal positions Slide 15**Framework**Slide 16**Visibility Graph Method**• Observation: If there is a collision-free path between two points, then there is a polygonal path that bends only at the obstacles vertices. • Why? • Any collision-free path can be transformed into a polygonal path that bends only at the obstacle vertices. • A polygonal path is a piecewise linear curve. Slide 17**Visibility Graph**• A visibility graphis a graph such that • Nodes: qinit, qgoal, or an obstacle vertex. • Edges: An edge exists between nodes u and v if the line segment between u and v is an obstacle edge or it does not intersect the obstacles. Slide 18**Computational Efficiency**• Simple algorithm O(n3) time • More efficient algorithms • Rotational sweep O(n2log n) time • Optimal algorithm O(n2) time • Output sensitive algorithms • O(n2) space Slide 20**Framework**Slide 21**Breadth-First Search**Slide 22**Breadth-First Search**Slide 23**Breadth-First Search**Slide 24**Breadth-First Search**Slide 25**Breadth-First Search**Slide 26**Breadth-First Search**Slide 27**Breadth-First Search**Slide 28**Breadth-First Search**Slide 29**Breadth-First Search**Slide 30**Breadth-First Search**Slide 31**Other Search Algorithms**• Depth-First Search • Best-First Search, A* Slide 32**Framework**Slide 33**Summary**• Discretize the space by constructing visibility graph • Search the visibility graph with breadth-first search Q: How to perform the intersection test? Slide 34**Summary**• Represent the connectivity of the configuration space in the visibility graph • Running time O(n3) • Compute the visibility graph • Search the graph • An optimal O(n2) time algorithm exists. • Space O(n2) Can we do better? Slide 35**Classic Path Planning Approaches**• Roadmap – Represent the connectivity of the free space by a network of 1-D curves • Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells • Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function Slide 36**Classic Path Planning Approaches**• Roadmap– Represent the connectivity of the free space by a network of 1-D curves • Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells • Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function Slide 37**Roadmap**• Visibility graph Shakey Project, SRI [Nilsson, 1969] • Voronoi Diagram Introduced by computational geometry researchers. Generate paths that maximizes clearance. Applicable mostly to 2-D configuration spaces. Slide 38**Voronoi Diagram**• Space O(n) • Run time O(n log n) Slide 39**Other Roadmap Methods**• Silhouette First complete general method that applies to spaces of any dimensions and is singly exponential in the number of dimensions [Canny 1987] • Probabilistic roadmaps Slide 40**Classic Path Planning Approaches**• Roadmap – Represent the connectivity of the free space by a network of 1-D curves • Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells • Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function Slide 41**Cell-decomposition Methods**• Exact cell decomposition The free space F is represented by a collection of non-overlapping simple cells whose union is exactly F • Examples of cells: trapezoids, triangles Slide 42**Trapezoidal Decomposition**Slide 43**Computational Efficiency**• Running time O(n log n) by planar sweep • Space O(n) • Mostly for 2-D configuration spaces Slide 44**Adjacency Graph**• Nodes: cells • Edges: There is an edge between every pair of nodes whose corresponding cells are adjacent. Slide 45**Summary**• Discretize the space by constructing an adjacency graph of the cells • Search the adjacency graph Slide 46**Cell-decomposition Methods**• Exact cell decomposition • Approximate cell decomposition • F is represented by a collection of non-overlapping cells whose union is contained in F. • Cells usually have simple, regular shapes, e.g., rectangles, squares. • Facilitate hierarchical space decomposition Slide 47**Quadtree Decomposition**Slide 48**Octree Decomposition**Slide 49**Algorithm Outline**Slide 50