Thinking about space

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Thinking about space

• Robotics
• Bridgewater State College
• Spring 2005

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Thinking about space

• Brooks once claimed unnecessary
• But for complex tasks found necessary.
• Can't go from here to your car without thinking
  about the space and obstacles in between.
• No planning done without thinking about space

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Representing Space

• Easiest way of internal representation
• Use a map.
• Needed for
• Knowing where to go/navigation
• free space
• Recognize regions or locations in environment
• Recognize objects in environment

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How does robot represent environment?

• Let environment represent itself.
• All planning and manipulation of environment take
  place on real environment at given time.
• Plans only based on instantaneous immediate
  sensor input.
• If internal representation
• What are the primitives?
• Objects? Features? Symbolic entities? Spacial
  occupancy? etc?

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Decomposing space.

• Simple method
• Divide the space into equal sized squares
• Decide for every square, is this square free or
  filled.
• Could implement with 2d array.
• bitmap/occupancy grid
• Space considerations (how much?)
• Plus side- General
• Can also add other information like confidence,
  terrain info etc.

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Occupancy grid storage

• Cells that are all the same still represented
  separately
• Wastes lots of space
• Can't similar pixels be compressed?

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Quad Tree

• Recursive data structure.
• Represents square 2-D region
• Hierarchical
• May reduce storage
• Large spaces of occupied or unoccupied space
• Scattered spaces storage no better than
  occupancy grid.
• Worst case worse than occupancy since extra
  overhead.
• Base-2 nice for decisions.

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BSP-tree

• Binary Space Partitioning trees.
• Usually in computer graphics.
• Useful for 2d spaces
• Free space divided into regions by lines parallel
  to outside edge.
• Each boundary divide cell into two but not
  necessarily evenly.
• Arbitrary division point
• No unique BSP-tree for a given space.

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Exact division

• Subdivide space into several non-overlapping
  regions.
• Union of parts is exactly the whole space.
• Non-overlapping not necessary, but convenient
• End spatial decomposition methods.

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Geometric modelling

• Spatial decomposition can use a lot of memory
• Geometric maps one answer
• Describe world in terms of simple geometric
  shapes, points, line, polygons and circles.
• Will assume 2d representation since we don't have
  manipulators
• Key components
• Set of primitives for describing objects
• Set of composition and deformation operators to
  manipulate objects

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Shortcomings

• Lack of stability
• Representation changes a lot from small input
  change.
• Lack of expressive power
• Different environments map to same representation
• Difficult (impossible) to represent salient
  features in modelling system.

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Lack of uniqueness

• Models may approximate input
• Express single set of observations more than one
  way.
• Minimal descriptive length
• Associate cost with each model and attempt to
  find single model that minimizes cost.

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Stabilizer

• Minimize problems by adding stabilizer
• Discard outlying data points
• Prefer longer line segments.
• And parallel to other directions.
• Prefer models that are more compact possible for
  set of data points.
• Prefer models with uniform curvature.

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Topological Representations

• Noisy data -gt hard to make good geometry
• Humans seem to use more topological approach,
• So how about for robots
• Cognitive science approach
• Key to topological
• Some explicit representation of connectivity
  between two places
• May have no metric data at all.

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Topological Reps II

• Abstract the environment
• Descrete spaces connected by edges
• Represent as graph
• G (V, E)
• V Set of nodes E set of edges that connect
  them.
• Book example of ceiling mounted landmarks
• Airplane travel.

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Part IIRepresenting the robot in space

• Up to now,
• just concerned with representing the world
• But need to represent the robot in the world.
• Now let's represent the robot.

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Configuration Space

• Configuration space formalism
• Specification for physical state of robot
  relating to fixed environmental frame
• For simple rigid robot that can move and turn on
  flat surface but has no limbs
• Configuration qx,y,??
• What does this remind you of from mobility?

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C-Space constraints

• Contraints prohibit robot from entering some
  areas
• Circular robot with radius r can't go closer to
  any object than distance r
• Written as G(q)0 where q is the pose
• Holonomic constraint
• Any constraint that does not involve velocity
• For any real world obstical can expand to set
  of points and solve to keep the robot's pose
  correct.

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Non-holinomic Contraints

• Non-holinomic Contraints
• Contraints on what velocities are allowed
• Much more difficult to calculate.
• Getting from point A to point B without complete
  pose control
• Parallel parking
• Backing a tractor trailor truck.
• Free flying robots (conservation of angular
  momentum.
• C-space Search path can get large.

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Simplification

• Point-robot assumption
• Assume robot is a point capable of
  omni-directional movement
• Physics jokes?
• But robotics so have to make up for this
• Do so with object dilation
• Show on board.

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Path Planning

• Now we've covered
• representing space
• Representing robot in space
• Now plan motion
• Big field entire book written just on path
  planning for robots
• Skim surface.

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Path Planning Summary

• Want to get from point A to point B
• Often want minimum length path.
• Sometimes minimum cost path
• Not always the same
• Rochester NY example.
• Path curvature
• Obsicles
• Anytime algorithms
• safe paths

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Standard Path Planning

• standard planning basic assumptions
• Rigid robot A
• Static, known environment W
• Robot domain is euclidean (we'll use 2-d space)
• Known set of obsticles in W B1 B2 ... Bq
• Robot travels in straight line segments perfectly

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Evaluating Planning Techniques

• Environment and robot
• Different techniques are needed for different
  robot capabilities
• Soundness is the planned path guaranteed to be
  collision free?
• Complete if a path exists is is guaranteed to be
  found?
• Optomality cost of path found vs. best path
• Space and time complexity.

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Graph Search

• Basic graph search learned in most CS classes
• If you have a topological graph
• Start from initial node, expand that node
• Add adjacent nodes to OPEN list
• Found goal? If not, choose one of the nodes on
  OPEN and put start node on CLOSED list.
• Depth-first / Breadth-first / A search

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Optimizing Search

• Evaluation function f(n)
• n is a node on the graph
• Smaller values of f(n) means n more likely to be
  on optimal path.
• Choose node with smallest value of f(n)
• Assume no 0 or negative values for f(n) (every
  move costs something)
• F(n) g(n)h(n)
• G(n) is estimated cost from start to n
• H(n) is estimated cost from n to goal

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GraphSearch Procedure

• Procedure GraphSearch(s, goal)
• OPEN s.
• CLOSED .
• Found false.
• While (OPEN ? Ø) and (not found) do
• OPEN OPEN n.
• CLOSED CLOSED ? n
• If n ? goal then
• found true
• else
• OPEN OPEN ? M. (where M is set of all nodes
  directly accessible from n)

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A search

• Use G(n) as upper bound on cost thus far
• maximize g(n)
• Use h(n) as lower bound on cost of time to go
• Underestimate cost remaining
• Will find optimal path to goal.

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Dynamic Programming

• Solution must adhere to Bellman's principle
• Given points A, B, C
• Optimal path from A to B through C found by
  optimal path from A to C and optimal path from C
  to B
• Create cost table for moving from place to
  place till have a good solution to end point.
• Use heuristics to not create entire cost table.

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Visibility Graph planning.

• V-Graph
• Technique produces minimum length path
• Through graph traversal.
• Defined
• G(V, E) such that V is made up of union of all
  verticies of polygonal obsticles as well as start
  and end points of path.
• Edges of graph connect all vertices that are
  visible to one another.

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V-Graph planning II

• Once V-Graph setup
• Have graph whose edges can be traversed w/out
  hitting anything
• Start and goal nodes are included in verticies
• So can travel from start to goal w/out hitting
  anything
• Can search graph in time O(N2) on the number of
  verticies.

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To Be continued

• More Navigation next time.