Navigation and Motion Planning for Robots Speaker: Praveen Guddeti CSE 976, April 24, 2002

Outline 1. Configuration spaces. 2. Navigation and motion planning. 1. Cell decomposition. 2. Skeletonization. 3. Bounded-error planning. 4. Landmark based navigation. 5. Online algorithms. 3. Conclusions.

Configuration Spaces Framework for designing and analyzing motion-planning algorithms. Why? – State space is the all-possible configurations of the environment. – In robotics, the environment includes the body of the robot itself. – Robotics usually involves continuous state space. – Impossible to apply standard search algorithms in any straightforward way because the numbers of states and actions are infinite.

Configuration Spaces (2) If the robot has k degrees of freedom, then the state or configuration of the robot can be described with k real values q 1,…,q k. K values can be considered as a point p in a k- dimensional space called the configuration space, C of the robot. Set of points in C for which any part of the robot bumps into something is called the configuration space obstacle, O. C – O is the free space, F.

Configuration Spaces (3) Given an initial point c1 and a destination point c2 in configuration space, the robot can safely move between the corresponding points in physical space if and only if there is a continuous path between c1 and c2 that lies entirely in F. Generalized configuration space: systems where the state of other objects is included as part of the configuration. The other objects may be movable and their shapes may vary.

Configuration Spaces (4) E : space of all possible configurations of all possible objects in the world, other than the robot. If a given configuration can be defined by a finite set of parameters  1,…  m, then E will be an m-dimensional space. W = C  E, that is W is the space of all possible configurations of the world, both robot and obstacles. If no variation in the object shapes, then E is a single point and W and C are equivalent.

Configuration Spaces (5) Generalized W has (k + m) degrees of freedom, but only k of these are actually controllable. Transit paths: the paths where the robot moves freely. Transfer paths: the paths where the robot moves an object. Navigation in W is called a foliation. Transit motion within any page of the book. Transfer motion allows motion between pages.

Configuration Spaces (6) Assumptions for planning in W: 1. Partition W into finitely many states. 2. Plan object motion first and then plan for the robot. 3. Restrict object motions. Rather than a point in configuration space, if the robot starts with a probability cloud, or an envelope of possible configurations, then such an envelope is called a recognizable set.

Navigation and Motion Planning 1. Cell decomposition. 2. Skeletonization. 3. Bounded-error planning. 4. Landmark based navigation. 5. Online algorithms.

1. Cell Decomposition 1. Divide F into simple, connected regions called “cells”. This is the cell decomposition. 2. Determine which cells are adjacent to which others, and construct an “adjacency graph”. The vertices of this graph are cells, and edges join cells that abut each other. 3. Determine which cells the start and goal configurations lie in, and search for a path in the adjacency graph between those cells. 4. From the sequence of cells found at the last step, compute a path within each cell from a point of the boundary with the previous cell to a boundary point meeting the next cell.

Cell Decomposition (2) Last step presupposes an easy method for navigating within cells. F typically has complex, curved boundaries. Two types of cell decomposition: 1. Approximate cell decomposition. 2. Exact cell decomposition.

Approximate Cell Decomposition Approximate subdivisions using either boxes or rectangular strips. Explicit path from start to goal is constructed by joining the midpoints of each strip with the midpoints of the boundaries with neighboring cells. Two types of strip decomposition: 1. Conservative decomposition. 2. Reckless decomposition.

Approximate Cell Decomposition

Approximate Cell Decomposition (2) Conservative Decomposition Strips must be entirely in free space. “Wasted” wedges of free space at the ends of strip. What resolution of decomposition to choose? Sound but not complete.

Approximate Cell Decomposition (3) Reckless Decomposition Take all partially free cells as being free. Complete but not sound.

Exact Cell Decomposition Divide free space into cells that exactly fill it. Complex shaped cells. Cells cylinders: – Curved top and bottom ends. – Width of cylinders not fixed. – Left and right boundaries are straight lines. Critical points: points where the boundary curve is vertical.

Exact Cell Decomposition

2. Skeletonization Collapse the configuration space into a one- dimensional subset, or skeleton. Paths lie along the skeleton. Skeleton: A web with a finite number of vertices, and paths within the skeleton can be computed using graph search methods. Generally simpler than cell decomposition, because they provide a “minimal” description of free space.

Skeletonization (2) To be complete for motion planning, skeletonization methods must satisfy two properties: 1. If S is a skeleton of free space F, then S should have a connected piece within each connected region of F. 2. For any point p in F, it should be “easy” to compute a path from p to the skeleton. Skeletonization methods: 1. Visibility graphs. 2. Voronoi diagrams. 3. Roadmaps.

Skeletonization 1. Visibility Graphs Visibility graph for a polygonal configuration space C consists of edges joining all pairs of vertices that can see each other.

Visibility Graphs

Skeletonization 2. Voronoi Diagrams For each point in free space, compute its distance to the nearest obstacle. Plot that distance as a height coming out of the diagram. Height of the terrain is zero at the boundary with the obstacles and increases with increasing distance from them. Sharp ridges at points that are equidistant from two or more obstacles. Voronoi diagrams consists of these sharp ridge points. Complete algorithms. Generally not the shortest path.

Voronoi Diagrams

Skeletonization 3. Roadmaps Two curves: 1. Silhouette curves ( freeways). 2. Linking curves (bridges). Travel on a few freeways and connecting bridges rather than an infinite space of points. Two versions of roadways: 1. Silhouette method. 2. Extension of voronoi diagrams.

Silhouette Method Silhouette curves are local extrema in Y of slices in X. Linking curves join critical points to silhouette curves. Critical points are points where the cross-section X=c changes abruptly as c varies.

Roadmap of a Torus

Extension of Voronoi Diagrams. Silhouette curves: extremals of distance from obstacles in slices X = c. Linking curves: start from a critical point and hill-climb in configuration space to a local maxima of the distance function.

Voronoi-like Roadmap of a Polynomial Environment.

3. Bounded-error Planning (Fine-motion Planning) Planning small,precise motions for assembly. Sensor and actuator uncertainly. Plan consists of a series of guarded motions. 1. Motion command. 2. Termination condition.

Bounded-error Planning (2) Fine-motion planner takes as input the configuration space description, the angle of velocity uncertainty cone, and a specification of what sensing is possible for termination. Should produce a multi-step conditional plan or policy that is guaranteed to succeed, if such a plan exists. Plans are designed for the worst case outcome. Extremely high complexity.

4. Landmark Based Navigation Assume the environment contains easily recognizable, unique landmarks. A landmark is surrounded with a circular field of influence. Robot’s control is assumed to be imperfect. The environment is know at planning time, but not the robot’s position. Plan backwards from the goal using backprojection. Polynomial complexity.

5. Online Algorithms Environment is poorly known. Produce conditional plan. Need to be simple. Very fast and complete, but almost always give up any guarantee of finding the shortest path. Competitive ratio.

Online Algorithms (2) A complete online strategy. 1. Move towards the goal along the straight line L. 2. On encountering an obstacle stop and record the current position Q. Walk around the obstacle clockwise back to Q. Record points where the line L is crossed and the distance taken to reach them. Let P 0 be the closest such point to the goal. 3. Walk around the obstacle from Q to P 0. Now the shortest path to reach P 0 is known. After reaching P 0 repeat the above steps.

Conclusions Five major classes of algorithms. Algorithms differ in the amount of uncertainty and knowledge of the environment they require during planning time and execution time.

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