Path Planning. Jacky Baltes Dept. of Computer Science University of Manitoba 11/21/10
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1 Path Planning Jacky Baltes Autonomous Agents Lab Department of Computer Science University of Manitoba
2 Path Planning Jacky Baltes Dept. of Computer Science University of Manitoba
3 Path Planning Path planning problem Visibility Graphs Roadmaps Cell Decomposition Potential Fields Probabilistic Algorithms
4 Path Planning Problem Fundamental problem in robotics Navigation How do I get to where I want to go? Path planning Create a set of motions to moves the robot from an initial position to a goal position Without colliding with any obstacles
5 Path Planning Variations Motions 2D or 3D path for simple robots Trajectories (Position and time) Sensor-based motions Positions 2D or 3D position Higher level configuration space (e.g., humanoid robots)
6 Simple Path Planning Problem Goal Robot
7 Basic Formulation Compute a path from the initial position to the goal Collision-free Static obstacles Global knowledge of obstacle position Input Geometry of robot Kinematics of robot Geometry of obstacles Initial position Goal position Output Sequence of configurations of robot Reaches goal without collisions
8 Simple Domains Differential drive wheeled mobile robot Moving furniture
9 Complexity Results Theory proves that this is a hard problem in general PSPACE-hard Can be solved in polynomial space Assumed to be harder than NP-hard Completeness Correctness
10 Configuration Space (C-Space) Some robots have many degrees of freedom and complex configurations Humanoid robots Multi-dimensional space Number of degrees of freedom
11 Extensions Sensor noise Non-holonomic constraints Dynamic obstacles Partial knowledge of obstacles Movable obstacles or objects Deformable robots or obstacles Multiple robots
12 Theory and Practice Theory Complete algorithms Correct algorithms Low worst-case complexity Hide as much as you can in the O() notation Practice Heuristics, probabilistic approach Good performance on average case No performance guarantees
13 Simple Representation Robot is modeled as a point in the plane Obstacles are polygons Initial and goal positions are locations (x,y) Search the free space Very inefficient
14 Graph Search Algorithms Discretize space using fixed grid Fixed sized grids Dijkstra s graph search algorithm Single pair Single source Single destination Floyd Warshall Algorithm All pairs
15 Floyd Warshall Algorithm Calculate minimum distances between all nodes of a graph A dynamic programming approach Good runtime O(V^3) A recursive algorithm Assume that you know S(i,j,k), the shortest path between node i and j using nodes 0..k S(i,j,k+1) = Min(S(i,j,k),S(i,k,k)+S(k,j,k)) Base case S(i,j,0) = 0 if i=j, 1 if i adjacent j, and infinity else
16 Pseudo Code: Wikipedia Initialization /* Assume a function edgecost(i,j)= cost of the edge from i to j Also assume that n is the number of vertices and edgecost(i,i)=0 */ int path[n][n]; A 2-dimensional matrix. At each step in the algorithm, path[i][j] is the shortest path from i to j using intermediate values in (1..k-1). Each path[i][j] is initialized to edgecost(i,j).
17 Pseudocode procedure FloydWarshall () for k: = 1 to n for each (i,j) in (1..n) path[i][j] = min ( path[i][j], path[i][k]+path[k][j] );
18 Visibility Graph Any collision-free path can be converted into a set of line segments that only bend on corners Create the visibility graph of the environment Nilsson, 1969, Stanford Shakey Project Nodes: Init, Goal, Obstacle vertices Edges between u and v if and only if: u and v are vertices of an obstacle edge u and v do not intersect any obstacle
19 Visibility Graph
20 Search of Visibility Graph Visibility graph theorem: Any minimum length path between an initial position and a goal lies along the visibility graph Depth first search Breadth-first search A* Search, Iterative deepening A*
21 A* Search Using a priority queue to expand nodes based on the node cost The node cost f(n) is given as f(n) = g(n) + h(n) Where g(n) is the cost from the root to the node And h(n) is the estimated cost from the node to a goal If h(n) is an admissible heuristic function That is h(n) is guaranteed to be an underestimate Then A* is guaranteed to find the optimal (minimum cost) solution
22 A* Search Example Show the order of the nodes being expanded, their cost and heuristic value
23 Iterative Deepending A* If the error of the heuristic function is constant, then the time complexity of A* is O(n), where n is the number of nodes A linear error, exponential O(b n ) time complexity Space complexity is also exponential Iterative deepening increases cost cutoff value with each iteration
24 Visibility Graph Visibility graph is an example of a roadmap algorithm Visibility graph algorithm returns the optimal path in lower bound time Are we done? Robot drives as closely as possible to an obstacle Not a good idea
25 Voronoi Path Planning Find path with the maximum distance between obstacles Max. clearance for robot Problem is too afraid of obstacles
26 Cell Decomposition Represent free space as a set of cells Robot moves through set of adjacent cells Try to reduce the number of cells in the decomposition Shapes Trapezoids Triangles (Degauchy triangulation)
27 Trapezoidal Decomposition
28 Approximate Cell Decomposition Instead of exactly matching the free space, map a space that is guaranteed to be free Some free space is ignored Approximate cell decomposition planners are Correct If the planner returns a path, it is correct Not complete If the planner fails to find a path, there still may exist a path
29 Quadtree Decomposition (2D) Break space at center into four subsquares
30 Octtree Decomposition (3D)
31 Binary Space Partitioning BSP algorithms is a superset of the quadtree decomposition algorithm Break cell at centre either horizontally or vertically Used in the FPS Doom Used for processing of viewing rectangles, collisions in 3D rendering
32 Flexible Binary Space Partitioning (FBSP) Cut horizontally or vertically at arbitrary points Where to cut? At vertices of obstacles Bad case for quadtree decomposition
33 Partitioning Points Need to select partitioning dimension partitioning point Only check projection points Large jump in info. gain Linear complexity
34 Entropy Based on Shannon s idea of information entropy The expected value of information The minimum length needed to encode a message Minimum length to encode area
35 Entropy
36 Information Gain Heuristic Heurisitic used by ID3 and other machine learning algorithms Create a minimum depth decision tree Entropy Pick partition point p that maximizes products of entropy and cell size
37 Example: Quadtree vs FBSP
38 Evaluation 2.8m by 2.3m environment Number of obstacles 1 32 Size of obstacles randomized Repeated over a number of iterations Compared the number of cells generated Flexible BPS generated only 27% of the number of cells of standard Quadtree decomposition Even better results possible (11 vs. 47 cells)
39 Evaluation
40 Rapidly Expanding Random Trees A probabilistic algorithm for path planning
41 Naïve Random Algorithm Pick a vertex at random Move in a random direction
42 Rapidly Exploring Random Trees
43 Creation of RRT
44 Creation of RRT RRT(Tree T) Pick random sample S from search space Find the nearest neighbor N to this sample S in the tree T Select an action from N that heads in the direction of S If the outcome N is legal create a new node N in the tree T and connect N and N
45 RRT Extend Extend RRT until nearest vertex is close enough to the goal Local planner to reach goal Probabilistically complete, but slow convergence
46 Dual RRT Start two RRTs One from initial position One from goal position 737 nodes are used
47 RRT Connect Aggressively connect trees using a greedy heuristic Extend and connect alternatively 42 nodes are used
48 RRT Connect Start from initial and goal state Extend a tree Try to connect new vertex to other tree Keep trying to connect until you hit an obstacle Alternatively repeat until trees are connected q init q goal
49 Variations Extend Extend Less aggressive Works well for non-holonomic constraints Connect Connect More aggressive
50 RRT Variations Single RRT Goal biased sampling Connect heuristic Dual RRT Extend both trees towards samples Extend trees towards each other and samples
51 RRT Advantages Suitable for highly dimensional spaces Steering ability not required Exploration biased towards empty space Probabilistic completeness and uniform coverage Simplicity (few parameters) Connected structures using minimal edges
52 Other randomized approaches Randomized potential fields Move in a random direction biased by gradient Barraquand, Latombe, 1989 Probabilistic Roadmap Uniform sampling of free space Local planners try to connect Kavraki, Latombe, Overmars, Svestka 1996
53 Potential Fields Obstacle avoidance (Khatib 1986) Modeled on forces between electrically charged particles Robot drives in direction of the gradient of the potential field A navigation function is an ideal potential field Global minimum at goal No local minimas Infinity at border of obstacles Is smooth
54 Potential Fields 1/r 2 for obstacles Linear for goal 1-0.9*r
55 References Tae-Joon Kim, KAIST. J. F. Canny. The Complexity of Robot Motion Planning. MIT Press, Cambridge, MA, J. Barraquand and J.-C. Latombe. Nonholonomic multibody mobile robots: Controllability and motion planning in the presence of obstacles. In IEEE Int. Conf. Robot. & Autom., pages , L. E. Kavraki, P. Svestka, J.-C. Latombe, and M. H. Overmars. Probabilistic roadmaps for path planning in highdimensional configuration spaces. IEEE Trans. Robot. & Autom., 12(4): , June S. M. LaValle. Rapidly-exploring random trees: A new tool for path planning. TR 98-11, Computer Science Dept., Iowa State Univ. < Oct William Regli Steve LaValle James Bruce and Manuela Veloso ERRT Jacky Baltes and John Anderson. Flexible Binary Space Partitioning for Robotic Rescue. In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Las Vegas, pages , October 2003.
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