Last lecture

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Last lecture

• Path planning for a moving
• Visibility graph
• Cell decomposition
• Potential field
• Geometric preliminariesImplementing geometric
  primitives correctly and efficiently is tricky
  and requires careful thought.

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Configuration Space
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What is a path?
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Rough idea

• Convert rigid robots, articulated robots, etc.
  into points
• Apply algorithms for moving points

5
Mapping from the workspace to the configuration
space
workspace
configuration space
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Configuration space

• Definitions and examples
• Obstacles
• Paths
• Metrics

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Configuration space

• The configuration of a moving object is a
  specification of the position of every point on
  the object.
• Usually a configuration is expressed as a vector
  of position orientation parameters q (q1,
  q2,,qn).
• The configuration space C is the set of all
  possible configurations.
• A configuration is a point in C.

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Topology of the configuration pace

• The topology of C is usually not that of a
  Cartesian space Rn.

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Dimension of configuration space

• The dimension of a configuration space is the
  minimum number of parameters needed to specify
  the configuration of the object completely.
• It is also called the number of degrees of
  freedom (dofs) of a moving object.

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Example rigid robot in 2-D workspace
workspace
robot

• 3-parameter specification q (x, y, q ) with q
  ?0, 2p).
• 3-D configuration space

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Example rigid robot in 2-D workspace

• 4-parameter specification q (x, y, u, v) with
  u2v2 1. Note u cosq and v sinq .
• dim of configuration space ???
• Does the dimension of the configuration space
  (number of dofs) depend on the parametrization?
• Topology a 3-D cylinder C R2 x S1
• Does the topology depend on the parametrization?

3
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Example rigid robot in 3-D workspace

• q (position, orientation) (x, y, z, ???)
• Parametrization of orientations by matrix q
  (r11, r12 ,, r33, r33) where r11, r12 ,, r33
  are the elements of rotation matrix
  with
• r1i2 r2i2 r3i2 1 for all i ,
• r1i r1j r2i r2j r3i r3j 0 for all i ? j,
• det(R) 1

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Example rigid robot in 3-D workspace

• Parametrization of orientations by Euler angles
  (f,q,y)

1 ? 2 ? 3 ? 4
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Example rigid robot in 3-D workspace

• Parametrization of orientations by unit
  quaternion u (u1, u2, u3, u4) with u12 u22
  u32 u42 1.
• Note (u1, u2, u3, u4) (cosq/2, nxsinq/2,
  nysinq/2, nzsinq/2) with nx2 ny2 nz2 1.
• Compare with representation of orientation in
  2-D(u1,u2) (cosq, sinq )

n (nx, ny, nz)
q
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Example rigid robot in 3-D workspace

• Advantage of unit quaternion representation
• Compact
• No singularity
• Naturally reflect the topology of the space of
  orientations
• Number of dofs 6
• Topology R3 x SO(3)

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Example articulated robot

• q (q1,q2,,q2n)
• Number of dofs 2n
• What is the topology?

An articulated object is a set of rigid bodies
connected at the joints.
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Example protein backbone

• What are the possible representations?
• What is the number of dofs?
• What is the topology?

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Configuration space

• Definitions and examples
• Obstacles
• Paths
• Metrics

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Obstacles in the configuration space

• A configuration q is collision-free, or free, if
  a moving object placed at q does not intersect
  any obstacles in the workspace.
• The free space F is the set of free
  configurations.
• A configuration space obstacle (C-obstacle) is
  the set of configurations where the moving object
  collides with workspace obstacles.

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Disc in 2-D workspace
workspace
configuration space
workspace
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Polygonal robot translating in 2-D workspace
configuration space
workspace
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Polygonal robot translating rotating in 2-D
workspace
configuration space
workspace
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Polygonal robot translating rotating in 2-D
workspace
q
y
x
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Articulated robot in 2-D workspace
workspace
configuration space
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Configuration space

• Definitions and examples
• Obstacles
• Paths
• Metrics

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Paths in the configuration space
configuration space
workspace

• A path in C is a continuous curve connecting two
  configurations q and q such that t(0) q
  and t(1)q.

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Constraints on paths

• A trajectory is a path parameterized by time
• Constraints
• Finite length
• Bounded curvature
• Smoothness
• Minimum length
• Minimum time
• Minimum energy

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Free space topology

• A free path lies entirely in the free space F.
• The moving object and the obstacles are modeled
  as closed subsets, meaning that they contain
  their boundaries.
• One can show that the C-obstacles are closed
  subsets of the configuration space C as well.
• Consequently, the free space F is an open subset
  of C. Hence, each free configuration is the
  center of a ball of non-zero radius entirely
  contained in F.

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Semi-free space

• A configuration q is semi-free if the moving
  object placed q touches the boundary, but not the
  interior of obstacles.
• Free, or
• In contact
• The semi-free space is a closed subset of C. Its
  boundary is a superset of the boundary of F.

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Example
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Example
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Homotopic paths

• Two paths t and t with the same endpoints are
  homotopic if one can be continuously deformed
  into the otherwith h(s,0) t(s) and h(s,1)
  t(s).
• A homotopic class of pathscontains all paths
  that arehomotopic to one another.

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Connectedness of C-Space

• C is connected if every two configurations can be
  connected by a path.
• C is simply-connected if any two paths connecting
  the same endpoints are homotopic.Examples R2 or
  R3
• Otherwise C is multiply-connected.Examples S1
  and SO(3) are multiply- connected
• In S1, infinite number of homotopy classes
• In SO(3), only two homotopy classes

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Configuration space

• Definitions and examples
• Obstacles
• Paths
• Metrics

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Metric in configuration space

• A metric or distance function d in a
  configuration space C is a function such that
• d(q, q) 0 if and only if q q,
• d(q, q) d(q, q),
• .

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Example

• Robot A and a point x on A
• x(q) position of x in the workspace when A is at
  configuration q
• A distance d in C is defined by d(q, q)
  maxx?A x(q) - x(q) where x - y
  denotes the Euclidean distance between points x
  and y in the workspace.

q
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Examples in R2 x S1

• Consider R2 x S1
• q (x, y,q), q (x, y, q) with q, q ?
  0,2p)
• a min q - q , 2p - q - q
• d(q, q) sqrt( (x-x)2 (y-y)2 a2 ) )
• d(q, q) sqrt( (x-x)2 (y-y)2 (ar)2 ),
  where r is the maximal distance between a point
  on the robot and the reference point

a
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Summary on configuration space

• Parametrization
• Dimension (dofs)
• Topology
• Metric

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Minkowski Sum
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Problem

• Input
• Polygonal moving object translating in 2-D
  workspace
• Polygonal obstacles
• Output configuration space obstacles represented
  as polygons

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Disc in 2-D workspace
workspace
configuration space
workspace
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Minkowski sum

• The Minkowski sum of two sets P and Q, denoted by
  P?Q, is defined as PQ pq p
  ?P, q?Q
• Similarly, the Minkowski difference is defined as
• P Q pq p?P, q?Q

q
p
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Minkowski sum of convex polygons

• The Minkowski sum of two convex polygons P and Q
  of m and n vertices respectively is a convex
  polygon P Q of m n vertices.
• The vertices of P Q are the sums of vertices
  of P and Q.

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Observation

• If P is an obstacle in the workspace and M is a
  moving object. Then the C-space obstacle
  corresponding to P is P M.

M
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Computing C-obstacles
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Computational efficiency

• Running time O(nm)
• Space O(nm)
• Non-convex obstacles
• Decompose into convex polygons (e.g., triangles
  or trapezoids), compute the Minkowski sums, and
  take the union
• Complexity of Minkowksi sum O(n2m2)
• 3-D workspace