Navigation and Motion Planning for Robots

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Navigation and Motion Planning for Robots

• Speaker Praveen Guddeti
• CSE 976, April 24, 2002

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Outline

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

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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.

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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 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.

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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.

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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.

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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.

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Configuration Spaces (6)

• Assumptions for planning in W
• Partition W into finitely many states.
• Plan object motion first and then plan for the
  robot.
• 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.

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Navigation and Motion Planning

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

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1. Cell Decomposition

• Divide F into simple, connected regions called
  cells. This is the cell decomposition.
• 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.
• Determine which cells the start and goal
  configurations lie in, and search for a path in
  the adjacency graph between those cells.
• 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.

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Cell Decomposition (2)

• Last step presupposes an easy method for
  navigating within cells.
• F typically has complex, curved boundaries.
• Two types of cell decomposition
• Approximate cell decomposition.
• Exact cell decomposition.

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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
• Conservative decomposition.
• Reckless decomposition.

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Approximate Cell Decomposition
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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.

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Approximate Cell Decomposition (3)Reckless
Decomposition

• Take all partially free cells as being free.
• Complete but not sound.

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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.

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Exact Cell Decomposition
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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.

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Skeletonization (2)

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

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Skeletonization1. Visibility Graphs

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

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Visibility Graphs
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Skeletonization2. 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.

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Voronoi Diagrams
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Skeletonization3. Roadmaps

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

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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 Xc changes abruptly as c varies.

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Roadmap of a Torus
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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.

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Voronoi-like Roadmap of a Polynomial Environment.
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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.
• Motion command.
• Termination condition.

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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.

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4. Landmark Based Navigation

• Assume the environment contains easily
  recognizable, unique landmarks.
• A landmark is surrounded with a circular field of
  influence.
• Robots control is assumed to be imperfect.
• The environment is know at planning time, but not
  the robots position.
• Plan backwards from the goal using
  backprojection.
• Polynomial complexity.

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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.

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Online Algorithms (2)

• A complete online strategy.
• Move towards the goal along the straight line L.
• 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.
• Walk around the obstacle from Q to P0. Now the
  shortest path to reach P0 is known. After
  reaching P0 repeat the above steps.

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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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CommentsQuestions