BINARY MORPHOLOGY

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BINARY MORPHOLOGY and APPLICATIONS IN ROBOTICS
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• Applications of Minkowski Sum
• Minkowski addition plays a central role in
  mathematical morphology
• It arises in the brush-and-stroke paradigm of 2D
  computer graphics (with various uses, notably by
  Donald E. Knuth in Metafont ) , and as the solid
  sweep operation of 3D computer graphics
• Motion planning
• Minkowski sums are used in motion planning of an
  object among obstacles.
• They are used for the computation of the
  configuration space which is the set of all
  admissible positions of the object.
• In the simple model of translational motion of an
  object in the plane, where the position of an
  object may be uniquely specified by the position
  of a fixed point of this object, the
  configuration space are the Minkowski sum of the
  set of obstacles and the movable object placed at
  the origin and rotated 180 degrees.
• NC machining
• In NC machining the programming of the NC tool
  exploits the fact that the Minkowski sum of the
  cutting piece with its trajectory gives the shape
  of the cut in the material.

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Application of Morphological Method to Robot
Motion Planning
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Mathematics of Motion Planning
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The Environment of Motion Planning and Obstacle
Avoidance
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The Concept of CONFIGURATION SPACE
Notation
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Configuration space of object A

1. Object is a point
2. Space is 2-Dimensional

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Configuration space of object A

1. Object is a point
2. Space is 3-Dimensional

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Possible Motions of the object in the
Configuration space

1. Object is NOT a point
2. Object has shape

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Some Other Examples of C-Space

• A rotating bar fixed at a point
• what is its C-space?
• what is its workspace
• A rotating bar that translates along the
  rotation axis
• what is its C-space?
• what is its workspace
• A two-link manipulator
• what is its C-space?
• what is its workspace?
• Suppose there are joint limits, does this change
  the C-space?
• The workspace?

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Configuration Space for a simple robot arm
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Configuration Space for a simple robot arm
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Configuration Space for a simple robot arm
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Disk (mobile robot with rounded base) in a
Configuration Space
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Example of a robot world with obstacles and a
robot
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Configuration space for the previous slide
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Motion Planning Revisited
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Configuration Space

• A key concept for motion planning is a
  configuration
• a complete specification of the position of
  every point in the system
• A simple example a robot that translates but
  does not rotate in the plane
• what is a sufficient representation of its
  configuration?
• The space of all configurations is the
  configuration space or Cspace.
• C-space formalism
• Lozano-Perez 79

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Formalization of Obstacles in Configuration Space
(C-Space)
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Obstacles Configuration Space
example

1. The robot A (a triangle) can translate freely in
   the plane at fixed orientation.
2. Its configuration is represented as q(x,y), the
   coordinates in FW of the vertex of A marked as a
   small circle (the origin of FA).
3. Hence, As configuration space is C R2.
4. The C-obstacle Cbi (shown dark) is obtained by
   growing the corresponding workspace obstacle Bi
   (a rectangle) by the shape of A.
5. Planning a motion of A relative to Bi is
   equivalent to planning a motion of the marked
   vertex of A relative to CBi

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Notation for Free Space in Motion Planning
This is expansion of one planar shape by another
planar shape
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Minkowski Sums in Robot Motion Planning
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C-obstacles in 3D Space take into account
possibility of rotations of the object (robot)
This is twisted
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C-obstacle in 3D
Can we stay in 2D?
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Assume now that the object can rotate while it
moves

• Any reference point configuration
• Taking the cross section of configuration space
• in which the robot is rotated 45 degrees...
• x
• y
• 45 degrees
• How many sides does P ?R have?

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Taking only one slice of the C-obstacle
This is only slice for 45 degrees
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• Why study the Topology
• Extend results from one space to another spheres
  to stars
• Impact the representation
• Know where you are
• Others?

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• The Topology of Configuration Space
• Topology is the intrinsic character of a
  space
• Two space have a different topology if cutting
  and pasting is required to make them the same
  (e.g. a sheet of paper vs. a mobius strip)
• think of rubber figures --- if we can stretch
  and reshape continuously without tearing, one
  into the other, they have the same topology
• A basic mathematical mechanism for talking about
  topology is the homeomorphism.

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More on dimension
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• What is the derivative of a rotation matrix?
• A tricky question --- what is the topology of
  that space

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• A Useful Observation
• The Jacobian maps configuration velocities to
  workspace
• velocities
• Suppose we wish to move from a point A to a
  point B in the
• workspace along a path p(t) (a mapping from some
  time index to
• a location in the workspace)
• dp/dt gives us a velocity profile --- how do we
  get the configuration
• profile?
• Are the paths the same if choose the shortest
  paths in workspace
• and configuration space?

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• Summary
• Configuration spaces, workspaces, and some
  basic ideas about
• topology
• Types of robots holonomic/nonholonomic,
  serial, parallel
• Kinematics and inverse kinematics
• Coordinate frames and coordinate
  transformations
• Jacobians and velocity relationships
• T. Lozano-Pérez.
• Spatial planning A configuration space approach.
• IEEE Transactions on Computing, C-32(2)108-120,
  1983.

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• A Few Final Definitions
• A manifold is path-connected if there is a path
  between any two
• points.
• A space is compact if it is closed and bounded
• configuration space might be either depending
  on how we model
• things
• compact and non-compact spaces cannot be
  diffeomorphic!
• With this, we see that for manifolds, we can
• live with global parameterizations that
  introduce odd singularities
• (e.g. angle/elevation on a sphere)
• use atlases
• embed in a higher-dimensional space using
  constraints
• Some prefer the later as it often avoids the
  complexities
• associated with singularities and/or multiple
  overlapping maps

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Use in path planning with large object in 2D
Projection
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Problems with this approach to Projection
The approach for the object shape from previous
slide is too conservative
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Sources

• Howie Choset
• G.D. Hager,
• Z. Dodds,
• Dinesh Mocha