CS 326 A: Configuration Space

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CS 326 A Configuration Space
qn
q(q1,,qn)
q3
q1
q2
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What is a Path?
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Definition of a Robot Configuration

• A robot configuration is a specification of the
  positions of all robot points relative to a fixed
  coordinate system
• Usually a configuration is expressed as a
  vector of position/orientation parameters

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Rigid Robot Example
Robots configuration is q (x,y,q)
In a 3-D workspace q would be of the form
(x,y,z,a,b,g)
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Articulated Robot Example
q (q1,q2,,q10)
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Configuration Space of a Robot

• Space of all its possible configurations
• But the topology of this space is usually not
  that of a Cartesian space

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Structure of the Configuration Space C

• C is a manifold, which means that, at each point
  q, there is a local 1-to-1 map between the
  neighborhood of q and a Cartesian space Rn, where
  n is the dimension of C
• This local map is a local coordinate system
  called a chart. C can be covered by a finite
  number of charts. Such a set of charts is called
  an atlas

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Case of a Planar Rigid Robot

• q (x,y,q) with q limited to 0,2p)
• Other representation q (x,y,cosq,sinq)
  configuration space is a 3-D cylinder embedded
  in a 4-D space, denoted by R2
  x S1
• Two charts are needed to build an atlas over R2 x
  S1

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Case of a Rigid Robot in 3-D Workspace

• q (x,y,z,a,b,g)
• Other representation q (x,y,z,r11,r12,,r33)
  where r11, r12, , r33 are the nine components of
  a 3x3 rotation matrix r11 r12 r13
  r21 r22 r23 r31 r32
  r33with
• ri12ri22ri32 1
• ri1rj1 ri2r2j ri3rj3 0
• det 1 the configuration space is a 6-D
  space (manifold) embedded in a 12-D
  Cartesian space. It is denoted by R3xSO(3)

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Parameterization of SO(3)

• Euler angles (f,q,y)
• Unit quaternion (cos q/2, n1 sin
  q/2, n2 sin q/2, n3 sin q/2)

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Metric in Configuration Space

• A metric or distance function d in C is a map
  d (q1,q2) d(q1,q2) gt 0such that
• d(q1,q2) 0 if and only if q1 q2
• d(q1,q2) d (q2,q1)
• d(q1,q2) lt d(q1,q3) d(q3,q2)
• Example Given a robot A and a point x of A, let
  x(q) be the point of the workspace occupied by x
  when the robot is at configuration q. A distance
  d is defined by d(q,q) max
  x(q)-x(q) over all points x of Awhere a -
  b denotes the distance between points a and b

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A Specific Example Metric in R2 x S1

• Let q (x,y,q) and q (x,y,q) with q and q
  in 0,2p)
• d(q,q) sqrt(x-x)2 (y-y)2
  a2where a minq-q , 2p-q-q
• d(q,q) sqrt(x-x)2 (y-y)2
  (a/r)2where r is the maximal distance between
  the reference point and a robot point

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Notion of a Path

• A path in C is a piece of continuous curve
  connecting two configurations q and q t s
  in 0,1 t(s) in C
• Other possible constraints finite length,
  smoothness,
• A trajectory is a path parameterized by time

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Homotopic Path

• Two paths with the same endpoints are homotopic
  if one can be continuously deformed into the
  other
• R x S1 examplePaths t1 and t2 are
  homotopicPaths t1 and t3 are not
  homotopicInfinity of homotopic classes

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Configuration Space Connectedness

• C is connected, meaning that every two
  configurations can be connected by a path
• C is simply-connected if any two paths connecting
  the same endpoints are homotopicExample R2 or
  R3
• Otherwise C is multiply-connectedExamples S1
  and SO(3) are multiply connectedIn S1, infinity
  of homotopic classesIn SO(3), only two homotopic
  classes

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Obstacles in Configuration Space

• A configuration is collision-free, or free, if
  the robot placed at this configuration has no
  intersection with the obstacles in the workspace
• The free space is the set of all free
  configurations
• A C-obstacle is the set of all configurations
  where the robot collides with a given workspace
  obstacle
• A configuration is semi-free if the robot at this
  configuration touches obstacles without overlap

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Disc Robot in 2-D Workspace
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Rigid Robot Translating in 2-D
CB B A b - a a in A, b in B
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Linear-Time Computation of C-Obstacle in 2-D
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Rigid Robot Translating and Rotating in 2-D
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C-Obstacle for Articulated Robot
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A Remark on the Topology of the Free Space

• The robot and the obstacles in the workspace are
  modeled as closed subsets. This means that they
  contain their boundaries
• One can show that the C-obstacles are closed
  subsets of the configuration space C
• 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
• The semi-free space is a closed subset of C. Its
  boundary is a superset of the boundary of F

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Free and Semi-Free Paths

• A free path is one that lies entirely in the free
  space
• A semi-free path is one that lies entirely in the
  semi-free space

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Classes of Homotopic Free Paths