Self-Reconfigurable Robots

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Self-Reconfigurable Robots

• Reconfiguration Algorithms
• Lei Wei
• lwei_at_cs.unc.edu
• 12/05/2006

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Reconfigurable Robot A Brief Review

• A self-reconfigurable modular robotics system
  comprises of a collection of homogeneous modules
  that can connect, disconnect, and move around
  adjacent modules.
• What is a reconfiguration plan?
• A reconfiguration plan is a sequence of module
  motions that changes the shape of the system from
  a start configuration to a goal configuration
  while enforcing constraints such as avoiding
  collision and maintaining connectivity.

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When we need reconfiguration?

• Obstacle avoidance in highly constrained and
  unstructured environments
• Growing structures composed of modules to form
  bridges, buttresses, and other civil structures
  in times of emergency

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Primary design goal

• To allow the robot to assume any geometric shape

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Applications

• Robot could match its geometric structure to the
  shape of the surrounding terrain for versatile
  locomotion.
• To realize self-repair

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Two basic types of self-reconfiguring system

• Heterogeneous
• the modules may be different
• Homogeneous
• all the modules are identical

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Crystalline Atoms

• Identical robot modules
• Actuated by expansion and contraction
• Each module is an atom, each connector a bond,
  group of atoms are crystals

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Physical Prototype
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Physical Prototype (Contd)
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Physical Prototype (Contd)

• When fully contracted, the Atom occupies a square
  with a 2 inch side.
• When fully expanded, the Atom occupies a square
  with a 4 inch side.
• By manipulating the size of the Atom, it is
  possible to approximate any finite sold shape to
  an arbitrary precision.

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Physical Prototype (Contd)

• Crystalline robot systems are dynamic structures
• They can move using sequences of reconfigurations
  to implement locomotion gaits
• They can undergo shape metamorphosis

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Primitive Operations for Crystal Modules

• Expand ltatomgt ltdimensiongt
• Expand a compressed atom
• Contract ltatomgt ltdimensiongt
• Compress an expanded atom
• Bond ltatomgt ltdimensiongt
• Activate one of the atoms connectors to bond
  with a neighboring atom
• Free ltatomgt ltdimensiongt
• Deactivate one of the atoms connectors to break
  a bond with a neighboring atom

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Revisit inchworm gait

• Use of these primitives for generating a linear
  locomotion algorithm

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Another view of the primitives
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Basis for Self-Reconfiguration

• As mentioned before, the primary design goal for
  a self-reconfiguring robot is to allow the robot
  to assume any geometric shape in a dynamic
  fashion.
• How can we guarantee a unit modular robotic
  system can reconfigure itself?

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Requirements for self-reconfiguration

• Structure Formation
• groups of modules can be assembled into
  arbitrarily shaped rigid structures.
• Ensures any geometric structure can be
  aggregated from some collection of modules.
• Module relocation
• in every structure composed of unit modules,
  some unit module can be relocated to each
  location on the surface of the structure without
  human intervention.
• Provides for shape metamorphosis in a general
  way From starting structure S to goal G
  incrementally.

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Why Crystalline Modules?

• Crystalline Atoms can be packed tightly
  approximate any three dimensional structure. By
  manipulating the size of the Atom, we can use
  this to represent any solid geometric shape to an
  arbitrary precision.
• How to implement relocating a Module on a
  Crystal?

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Module Relocation on Convex Structure

• Unlike other proposed unit modules(Yim,1993Murata
  et al.,1994,1998Pamecha et al.,1996Kotay and
  Rus,1998,1999)which can relocate only by
  traveling on the surface of a structure,
  Crystalline Atoms can be relocated by traveling
  through the volume of a Crystal.
• An Atom can be relocated in constant time on any
  convex structure.

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Module Relocation on non-convex Structure

• When the Crystalline robot structure is
  non-convex, a similar algorithm effects the
  module relocation operation in O(k)-time, where k
  is the number of concave angles in the structure.

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A planner for Shape Metamorphosis

• Given a pair of Crystals (S,G), each composed of
  n Atoms, a planner finds a feasible
  reconfiguration plan P that transform S into G.
• A reconfiguration plan P is a partially ordered
  sequence of Atom primitive operations.

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The Melt-Grow Algorithm---At a Glance

• Notation S is the starting Crystal
• G is the goal Crystal
• I is the intermediate Crystal
• Input S, G
• Melt-Grow Melt S into I
• Grow G out of I
  

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Melt-Grow Planner (Continued)

• Centralized planning algorithm
• Run in O(n2) time, where n is the number of
  Atoms in the Crystal
• Trades optimality for simplicity
• Works on a useful subset Grain(4) of Crystals

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Grain(n) Class of Crystals

• Contains all crystals that can be tiled by cubic
  (or square, in 2D) blocks of Atoms of side-length
  n
• The set of planes( or edges, in 2D) that coincide
  with all sides of all blocks intersect only at
  block edges and corners
• Each block of Atoms is a Grain

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Grain(4) example
A 2D Crystal in Grain(4)
A 2D Crystal not in Grain(4) because the bottom
grain is not aligned with the top ones
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Why use Grains?

• Moving atoms requires other Atoms to act as
  helpers
• Using Grains ensures that helper Atoms are always
  available
• Expansion and contraction operation now act on a
  whole face of the Grain at one time

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Grain Motion Primitives

• Scrunch
• Create a planar compression in a mobile Grain at
  one of its faces
• Relax
• Expand a compression at one face of a Grain
• Transfer
• Move a compression at one face of a Grain into
  the adjacent neighbor Grain
• Propagate
• Move a compression at one face of a Grain to the
  opposing face of that Grain
• Convert
• Relocate a compression at one face of a Grain to
  one of the orthogonal faces of that Grain

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Two Goals of these primitives

• They can assembled into linear sequences to
  effect Grain relocation

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Two Goals of these primitives (Contd)
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The second Goal

• All the 5 primitives all always feasible
• Each primitive maintains 3 invariants
• The moving Grain remains internally connected
• The Atoms in the moving Grain never crash
• There is some connected path from some Atom in
  every neighboring Grain, through the moving
  Grain, to some Atom in every other neighboring
  Grain

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The Melt-Grow Planner

• Melt algorithm
• Melt works by finding a mobile Grain g in S,
  transporting g to a place in I, and repeating
  until all Grains are in I.
• Grow algorithm
• Grow works by selecting mobile Grains from I and
  transporting them to locations in G until all
  Grains are in G

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Why use an intermediate Crystal?

• To help maintain stability during reconfiguration
  in situations where gravity is present
• To simplify the selection of mobile Grains
• A Grain is mobile iff it can be removed without
  disconnecting the Crystal

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Revisit example of Melt-Grow algorithm
S
I
G
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An Example Simulated Preplanned Reconfiguration
(Dog to Couch)
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An Example Simulated Preplanned Reconfiguration
(Dog to Couch)
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Potential Applications and Drawbacks

• Ability to adapt would allow such a robot system
  to maneuver around, through, or over a wide
  degree of obstacles
• Could navigate through narrow or awkward passage
  ways
• Seems suitable only for tasks that do not require
  rapid movement

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Potential Applications and Drawbacks (Contd)

• The reconfiguration planning proposed is a
  centralized algorithm requiring a controller to
  know status of the entire system
• The reconfiguration algorithm trades optimality
  for simplicity

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How to achieve optimal plan?

• We need to define what the optimal means.
• Optimality for reconfiguration strategies can be
  measured in different ways
• Minimizing the number of module moves
• Minimizing time for reconfiguration
• Minimizing energy consumption during
  reconfiguration

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Lattice Kinematics

• See the white board
• Lattice metric denoted by
• where a and b are lattice points.
• If the robot system is composed of square
  modules, distance between modules would be given
  by the Taxicab/Manhattan metric.

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General Formulation of Reconfiguration Problem

• The Kinematics constraints governing the motion
• Modules can only move into spaces which are not
  already occupied.
• Every module must remain connected to at least
  one other module.
• A single module may only move one lattice space
  per timestep.

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General Formulation of Reconfiguration Problem
(Contd)

• Under those constraints, the reconfiguration
  problem becomes
• Determination of the sequence of module
  motions from any given initial configuration to
  any given final configuration in a preferably
  minimal number of moves.
• In order to solve this problem, a concept of
  distance between configurations is needed.

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Defining Distance Between Configurations

• Let the present configuration of the robot be
  described by the set of modules A. And Let the
  new configuration be defined by the set B.
• One possible configuration metric
• Another is the Overlap metric

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Optimal Assignment Metric

• The optimal assignment metric between
  two configurations A and B is given by an optimal
  assignment of each element in A to an element in
  B, such that the sum of the distances between
  modules for the assignment is minimized.
• Equivalently, this can be represented as finding
  a minimum weight matching in a bipartite graph.

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Optimal Assignment Metric (Contd)

• Lets formalize the problem
• is the lattice distance between
  module and

otherwise
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Optimal Assignment Metric (Contd)

• An arbitrary assignment will have an associated
  cost function
• With the constraints

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Optimal Assignment Metric (Contd)

• Finally, we can define
• is the set of all possible assignments and
  is equivalent to the set of permutations of
  module labels.

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An Example

• Fig (a) shows the present configuration of a six
  module robot. (b) shows the new configuration and
  (c) shows a labeling of the modules in the two
  configurations

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Complexity of the Problem

• For any number of modules n, the number of
  connected configurations possible appears to be
  exponential in n.
• We have to look for heuristics which can give a
  near optimal solution.

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References

• 1. Daniela Rus and Marsette Vona, Crystalline
  Robots Self-Reconfiguration with Compressible
  Unit Modules, Autonomous Robots,2001
• 2. Kotay,K., Rus,D.,M., and McGray,C., The
  Self-reconfigurable robotics molecule. In
  Proceedings of the 1998 International Conference
  on Robotics and Automation
• 3. Kotay,K., Rus,D.,M., and McGray,C., The
  Self-Reconfiguring robotic molecule Design and
  Control algorithms. In the 1998 Workshop on
  Algorithmic Foundations of Robotics.
• 4. Kotay,K., Rus,D.,1998. Motion synthesis for
  the self-reconfiguring robotic module. In
  proceedings of the 1998 International Conference
  on Intelligent Robots and Systems.

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References

1. Kotay,K. and Rus, D. 1999. Locomotion
   versatility through self-reconfiguration.
   Robotics and Autonomous Systems.
2. Yim, M.1993. A reconfigurable modular robot with
   multiple modes of locomotion. In Proceedings of
   the 1993 JSME Conference on Advanced
   Mechatronics, Tokyo, Japan.
3. Murata,S., Kurokawa,H.,and Kokaji,S. 1994.
   Self-assembling machine. In Proceedings of the
   1994 IEEE International Conference on Robotics
   and Automation, San Diego.
4. Murata,S.,Kurokawa,H.,Yoshida,E.,Tomita,K., and
   Kokaji, S. 1998. A 3-D self-reconfigurable
   structure. In proceedings of the 1998 IEEE
   International Conference on Robotcs and
   Automation, Leuven.

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References

• 9. Pamecha,A.,Chiang,C.-J.,Stein,D., and
  Chirikjian,G.1996. Design and implementation of
  metamorphic robots. In proceedings of the 1996
  ASME Design Engineering Technical Conference and
  Computers in Engineering Conference, Irvine,CA
• 10. A.Pamecha, I. Ebert-Uphoff, and G.
  Chirikjian, Useful metrics for modular robot
  motion planning, in IEEE Transactions on Robotics
  and Automation. Vol.13, 1997.