Navigation and Motion Planning Navigation and Motion Planning for Robotsfor Robots

Speaker: Praveen GuddetiCSE 976, April 24, 2002

OutlineOutline1. 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 SpacesConfiguration 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 Configuration Spaces (2)(2) If the robot has k degrees of freedom, then the

state or configuration of the robot can be described with k real values q1,…,qk.

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 Configuration Spaces (3)(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 Configuration Spaces (4)(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 Configuration Spaces (5)(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 Configuration Spaces (6)(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 PlanningNavigation and Motion Planning

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

1. Cell Decomposition1. Cell Decomposition1. 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 Cell Decomposition (2)(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 DecompositionApproximate 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 DecompositionApproximate Cell Decomposition

Approximate Cell Decomposition (2)Approximate Cell Decomposition (2)

Conservative DecompositionConservative DecompositionStrips 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)Approximate Cell Decomposition (3)

Reckless DecompositionReckless DecompositionTake all partially free cells as being free.Complete but not sound.

ExactExact Cell DecompositionCell 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.

ExactExact Cell DecompositionCell Decomposition

2. Skeletonization2. SkeletonizationCollapse 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 Skeletonization (2)(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.

SkeletonizationSkeletonization1. Visibility Graphs1. Visibility Graphs

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

Visibility GraphsVisibility Graphs

SkeletonizationSkeletonization2. Voronoi Diagrams2. 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 DiagramsVoronoi Diagrams

SkeletonizationSkeletonization3. Roadmaps3. 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 MethodSilhouette MethodSilhouette 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 TorusRoadmap of a Torus

Extension of Voronoi Diagrams.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 Voronoi-like Roadmap of a Polynomial Environment.Polynomial Environment.

3. Bounded-error Planning 3. Bounded-error Planning (Fine-motion 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 Bounded-error Planning (2)(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 Navigation4. Landmark Based NavigationAssume 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 Algorithms5. Online AlgorithmsEnvironment 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 Online Algorithms (2)(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 P0 be the closest such point to the goal.

3. Walk around the obstacle from Q to P0. Now the shortest path to reach P0 is known. After reaching P0 repeat the above steps.

ConclusionsConclusionsFive 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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