Reconfiguration Planning Among Obstacles for
Heterogeneous Self-Reconfiguring Robots
Robert Fitch* (NICTA)
Zack Butler (RIT)
Daniela Rus (MIT)
* Research performed at Dept. of Computer Science, Dartmouth
Heterogeneous Lattice-Based SR Robots
• Composed of many modules, homogeneous and
heterogeneous
• Match structure to task (modularity)
• Match capability to task (heterogeneity)
• Complexity of heterogeneous reconfig. planning same as
homogeneous planning (in relevant cases)!
Heterogeneous Systems
• Vision: SR robots that match capability to task
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Specialized sensors
Communication with human
Dedicated battery modules
Diverse module shapes (not addressed here)
• Research challenges: planning and control
– Reconfiguration
– Locomotion
• So what is complexity (time and moves) of heterogeneous
reconfiguration?
Reconfiguration with Heterogeneity
Coordinated Motion Planning Problems
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Warehouse Problem
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(n2-1)-Puzzle
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Rectangles (not squares)
Multiple sizes
Rectangular bounding region
No connectivity constraints
PSPACE-hard
Polynomial-time if enough free space or
1x1 squares
“Sliding-Block” puzzles
8-puzzle, 15-puzzle
Not all instances solvable
NP-complete for optimal solution
NP-hard additive constant approx.
Polynomial-time constant-factor approx.
Heterogeneous Reconfiguration Problem
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1x1 modules
Connectivity constraints
Polynomial-time solvable with sufficient
free-space
Quadratic-time lower-bound
1 13 14
5 10 9 15
7 6 2 12
3 11 4 8
Approach
• Heterogeneous reconfiguration among obstacles
– Available free space influences problem complexity
• Hierarchy of motion primitives
– Discrete motions
– Module trajectories
– Reconfiguration plans
• Decentralized control
– Centralized version first
– Decentralized with message passing
Complexity Results
MeltSortGrow
TunnelSort
Constrained-TunnelSort
Free space
Unlimited
Crust
Bounding region
Planning (shape forming)
O(4n2)
centralized,
O(3n2 + n3)
decentralized
n/a
O(n2)
Plan length (shape forming)
O(3n2)
n/a
O(np)
Planning (sorting by type)
n/a
O(25mn + min(4mt2, 4n2))
O(mn + m2 + 25m2n +
4m2t2)
Plan length (sorting by type)
n/a
O(22mp + min(4mt2, 4n2))
O(22m2p + 4m2t2)
Assumptions: SlidingCube module abstraction, configurations with or without holes..
Number of modules is n, m is number of type errors in goal configuration, t is bound on tunnel length, p is
bound on surface path length. All analysis is worst-case.
Related Work
• Computational complexity
– Reconfiguration problem [Chirikjian]
– Warehouse problem [Hopcroft, Sharma]
– Sliding block (n2-1) puzzle [Hearn, Demaine]
• Reconfiguration planning
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Unit-compressible systems [Rus, Vona, Butler, Yim, …]
Scaffolding [Kotay, Stoy]
Chain-based [Yim, Shen, …]
Self-assembly, self-repair [Murata et al]
Outline
• Introduction
• Reconfiguration, no obstacles
– Motion over surface (IROS’03)
– Motion through volume (DARS’04)
• Algorithm: ConstrainedTunnelSort (ICRA’05)
– Motion both over surface and through volume
– Planned swap sequence
– Complexity analysis
• Discussion
Reconfiguration Planning Problem
• Given two shapes, morph between them
– Configurations (shapes) specify module position, type
– Find sequence of primitive motions
• Obstacles
– Constrain space available during reconfiguration
• Sliding Cube Model
Sliding Cube Model
• Instantiated by various hardware
prototypes
• Motion primitives
– Sliding across
– Convex transition
• Other Properties
– Square lattice
– Connection at faces
– Neighbor-to-neighbor
communication
– Onboard computation
– Onboard power
Trajectory Primitive: Motion Over Surface
Reconfiguration with Surface Motion
Trajectory Primitive: Motion through Volume
Reconfiguration with Tunneling
• TunnelSort
• Uses limited freespace
• O(n2) in worst-case
(optimal)
Motion Over Surfaces and Through Volume
Mobile if
neighbors are
connected
Can do both virtual and actual module relocation
Reconfiguration Algorithm
• ConstrainedTunnelSort
– Form goal shape
homogeneously
– While not done
• Greedily choose modules to
swap
• Swap using trajectory
primitives
Only one
move
possible –
disconnec
tion!
No moves
possible!
Need to plan swap sequence!
Maybe no
solution!
1 13 14
5 10 9 15
7 6 2 12
3 11 4 8
Choosing Swap Order
• Build connectivity graph
– For each module to be swapped, find
all other modules it can swap with
• Find MST (minimum diameter ST)
– BFS from each node
– Choose tree with minimum diameter
• Find correct graph coloring
– Permute colors by swapping
parent/child nodes
– Iterate over nodes in depth-first order
• Approximation to optimal
Algorithm: Constrained TunnelSort
Homogeneous phase
• While not done
– Choose module m and
position p, where m
needs to move and p
needs to be filled
– Find tunnel path from
m to p
– Use virtual module
relocation to move m
along path
Heterogeneous phase
O(n)
O(n)
O(n)
O(n)
O(n2)
• Build connectivity graph O(n2) O(n2)
of swappable modules
• Search for feasible swap O(n2) O(n)
sequence using MSTbased algorithm
2)
O(n
O(n)
• Execute swaps using
tunneling
O(n4) O(n2)
Example
Example
Outline
• Introduction
• Reconfiguration, no obstacles
– Motion over surface
– Motion through volume
• Algorithm: ConstrainedTunnelSort
– Motion both over surface and through volume
– Planned swap sequence
– Complexity analysis
• Discussion
Discussion
• Algorithmic results
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Solves heterogeneous reconfiguration among obstacles
Worst-case is uncommon in practice (m = t = p = n)
Average-case quadratic with more realistic estimates of m,t,p.
Both centralized and decentralized versions
• Compliant locomotion
– Series of goal configurations specified as overlapping bounding
boxes
• Position constraints
Position Constraints
• Objective
– Maintain relative position of
single module during
reconfiguration
• Assumptions
– Non-exact goal configuration
representation
• Results
– Initial solution
Next Steps
• Decrease number of moves, increase computation
– Approximation of optimal path length
• Goal configuration determination
– Alternative goal specifications (bounding box, etc.)
– Use learning
Acknowledgements
• This talk describes work performed in the Dartmouth Robotics
Laboratory. Support for this work was provided through NSF CAREER
award IRI-9624286 and NSF awards IRI-9714332, EIA-9901589, IIS9818299, and IIS-9912193, and a NASA SpaceGrant award. We are
very grateful.
• Portions of this work were performed at National ICT Australia (NICTA).
NICTA is funded by the Australian Government's Department of
Communications, Information Technology and the Arts and the
Australian Research Council through Backing Australia's Ability and the
ICT Centre of Excellence program.