Introduction to Robot Motion Planning

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Introduction to Robot Motion Planning
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Example

• A robot arm is to build an assembly from a set of
  parts.
• Tasks for the robot
• Grasping position gripper on object
• design a path to this position
• Trasferring determine geometry path for arm
• avoide obstacles clearance
• Positioning

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Information required

• Knowledge of spatial arrangement of wkspace.
  E.g., location of obstacles
• Full knowledge full motion planning
• Partial knowledge combine planning and
  execution
• motion planning collection of problems

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Basic Problem

• A simplified version of the problem assumes
• Robot is the only moving object in the wkspace
• No dynamics, no temporal issues
• Only non-contact motions
• MP pure geometrical problem

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Components of BMPP

• A single rigid object - the robot - moving in
  Euclidean space W (the wkspace).
• W RN, N2,3
• Bi , I1,,q. Rigid objects in W. The obstacles
• Assume
• Geometry of A and Bi is perfectly known
• Location of Bi is known
• No kinematic constraints on A a free flying
  object

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Components of BMPP (cont.)

• The Problem
• - Given an initial position and orientation
• a goal position and orientation
• - Generate continuous path t from initial
  postion to goal
• t is a continuous sequence
  of position and orientation

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Configuration Space

• Idea represent robot as point in space
• map obstacles into this space
• transform problem from planning object
  motion to planning point motion
• Notion of CS
• A at a given position is a compact in W.
• attached frame FA
• Bi closed subset of W .
• Fw is a frame fixed in W

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Notion of CS (cont)

• Def configuration of an object
• Is the position of every point of the object
  relative to FW
• Configuration q of A is the postion T and
  orientation O of FA w.r.t. FW
• Def configuration space of A
• Is the space T of all configurations of A
• A(q) subset of W occupied by A at q
• a(q) is a point in A(q)

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Information Required

• Example T N-dimensional vector
• O NxN rotation matrix
• In this case, q (T,O), a subset of RN(N1)
• Note that C is locally like R3 or R6.
• Notice no global correspondence

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Notion of Path

• Need a notion of continuity
• Define a distance function d C x C -gt R
• Example
• d(q,q) maxa in A a(q) - a(q)
• Def A path of A from qinit to qgoal
• Is a continuous map t 0,1 -gt C
• s.t. t(0) qinit and t (1) qgoal
• Prop. t is continuous if for each so in (0,1),
  lim d(s,so) 0 when s -gt so

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Obstacles in Configuration Space

• Bi maps in C to a region
• CBi q in C, s.t. A(q) ? Bi ? ?
• Obstacles in C are called C-obstacles.
• C-obstacle region
• Free space Cfree C -
• q is a free configuration if q belongs to Cfree
• Def Free Path.
• Is a path between qinit and qgoal , t 0,1 -gt
  Cfree

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Example of a C-Obstacle
CBi
Bi

• The robot is a triange that translates but not
  rotates

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Obstacles in C (cont.)

• Def Connected Component
• q1, q2 belong to the same connected component
  of Cfree iff they are connected by a free path
• Objective of Motion Planning generate a free
  path between 2 configurations if one exists or
  report that no free path exists.

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Examples of C-Obstacles

• Translational Case
• A is a single point -gt no orientation
• W RN C
• A is a disk or dimensioned object allow to
  translate freely but without rotation.
• C-Obstacles Are the obstacles grow by the
  shape of A

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Planning Approaches

• 3 approaches road maps, cell decomposition and
  potential field
• 1- Roadmap
• Captures connectivity of Cfree in a network of
  1-D curves called the roadmaps.
• Once a roadmap is constructed use a standard
  path.
• Roadmap Construction Methods 1) Visibility
  Graph, 2) Voronoi Diagram, 3) Freeway Net and 4)
  Silhouette.

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a- Visibility Graphs

• Mainly 2-D C-space and polygonal C-obstacles
• qinit , qgoal
• C-obstacle vertices
• b- Retractions Voronoi Diagram
• V.D. set of all configurations whose minimal
  distance to C-obstacle region is achieved with at
  least 2 points in the boundary of CB

• nodes

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c. Cell decomposition

• Decompose the robots free space into simple
  regions called cells
• A path within a cell -gt easy to generate
• Connectivity graph non-directed graph
  representing adjacency relation between cells
• Nodes cells extracted from free space
• 2 nodes connected by a link iff cells are
  adjacent
• channel resulting sequence of links
• Cell Decomp Exact or Approximate

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d- Potential Fields

• In principle, we can discretize the C - space by
  using a greed. Then search for path.
• Computational expensive gt need heuristics.
  Example of heuristics potential fields
• Idea robot attracted by qgoal and repulsed
  by CBis
• Potential methods can be very efficient, but
• Can be trapped in local minima!

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Interaction with Sensing

• Robot with no prior knowledge about environment
  cannot plan
• Instead, rely on sensory information
• E.g., robot with proximity sensor can attempt to
  create repulsive potential. This can result on
  falling into local minima.
• Absence of info reactive scheme due to Lumelsky.
  Algorithm guaranteed to reach qgoal whenever C
  -obstacles are bounded by a simple closed curve
  of finite length
• Works in 2-D