1. Coordinating Construction of Truss Structures using Distributed Equal-mass Partitioning
Affiliation
I am going to talk about decentralized control algorithms for coordinated construction.
, in which teams of robots divided up into specialized tasks such as part carrying robots and assembly robots to put the source part into a desired structure,
The robots have two algorithms, the first for assignment of work to assembling robots by distributed coverage, the second for balanced assembly.
The paper is on context of a truss structure, but our algorithms can be applied for general types of elements
Comment about distributed coverage
Daniela Rus
Seungkook Yun, Mac Schwager,
Computer Science and Artificial Intelligence Laboratory
MIT
MIT Computer Science and Artificial Intelligence Laboratory

2. Complex Mounds by Termites (Skelly talk)
outer walls
brood chambers
base with cooling vanes
royal chamber
fungus gardens
galleries
N-S orientation. Termites stay on cooler side during morning and afternoon. Experimental E-W orientation raises temp by 6 C in nest. May be related to propensity for flooding – termites have to stay in nest above ground through summer where it floods.
Large numbers of individuals
Limited information
Limited Capabilities
Stigmergy
Strongly condition Dependent

3. Robotic Construction: truss structure
Goal: build a truss structure w/
Robots with specialized tasks: assembly & delivery
Decentralized controller: parallelism
Adaptive algorithms for failure, amount of source material, etc.
Must be distributed for large number of robots
we want to make sure that if k robots work  in parallel the coordination algorithm ensures they all finish at the same time to maximize overall  parallelism
Delivery robot
Assembly robot
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4. Examples Construction Reconfiguration
The paper is about building a truss structure, but the algorithm can be used for any kind of source material.
Boeing uses several types of truss structures to support each subpart of an airplane. assembly costs a lot of money and labor. If we can automate building and reconfiguring, that would save a lot.
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5. Challenges Centralized Sequential Full travel distance for all
Uniform Size of construction grid (construction resolution)
Communication collision
We use a 2D problem to illustrate the technique
Tends to be centralized: order of assembly, pin-point delivery
Not adaptive: very inefficient when more sources
We have flexibility on density.
All the robots incrementally build a structure.
Voxel size could be Sparse
Naïve solution: Incremental construction
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6. Our Solution Decentralized Parallel Optimal travel distance
Construction resolution is adaptive to availability of source material
Less collision in comm.
Explain Distributed coverage as a way of dividing the area.
Now we want parallize
One dividing up solution is Voronoi partition.
Voronoi partition
Equal mass partitioning
Our solution
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7. Contribution Generalized 3D target structure Decentralized controller
Density function (given)
Decentralized controller
Decentralized method of assigning sub-assemblies
Delivery and assembly algorithms
No-dependency on
# of robots
amount of source material
Adaptable for
Failure of robots
Dynamic constraints
Reconfiguration
Provable convergence with guaranteed performance
Analysis based on the Balls into Bins problem
Emphasize it’s 3D algorithm
3D on the slide
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8. Related Work A couple of pictures from each side Distributed Coverage
J. Cortes, S. Martinez, T. Karatas, and F. Bullo. Coverage control for mobile sensing networks, 2004
L. C. A. Pimenta, M. Schwager, Q. Lindsey, V. Kumar, D. Rus, R. C. Mesquita, and G. A. S. Pereira. Simultaneous coverage and tracking (scat) of moving targets with robot networks, 2008
M. Pavone, E. Frazzoli, and F. Bullo. Distributed algorithms for equitable partitioning policies: Theory and applications, 2009
M. Schwager, D. Rus, and J.-J. E. Slotine. Decentralized, adaptive control for coverage with networked robots, 2009
Robotic Construction
S. Skaff, P. Staritz, and WL Whittaker. Skyworker: Robotics for space assembly, inspection and maintenance, 2001
J. Werfel and R. Nagpal. Three dimensional construction with mobile robots and modular blocks, 2008
S. Yun, D. Rus, Self Assembly of Modular Manipulators with Active and Passive Modules, ICRA 2008
S.Yun, D. Hjelle, E. Schweikardt, H. Lipson, D. Rus. Planning the Reconfiguration of Grounded Truss Structures with Truss Climbing Robots that Carry Truss Elements, ICRA 2009
J.Cortes, etc. 2004
A couple of pictures from each side
J. Werfel, etc. 2008
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9. Related Swarms Work
Deploy a group of robots to optimize the observation of an unknown, possibly changing sensory function.

10. Distributed Control with Optimization
: Sensory function
: Robot location
: Voronoi region
of robot
Robots move to optimize a cost function representing the collective sensing cost of the network
In high-dim parameter space each robot adapts a parameter vector to learn a distribution of sensory info in the environment
Centroid:
Mass:
Robots move to optimize a cost function representing the collective sensing cost of the network

11. Sensory Function Approximation
- basis functions
- parameters
Sensory function estimate:
Centroid estimate:
Learning ↔ Tuning

12. Decentralized Control Algorithms Extension Conclusion
Outline
Introduction
Problem Formulation
Decentralized Control Algorithms
Extension
Conclusion
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13. Decentralized Control Algorithms Extension Conclusion
Outline
Introduction
Problem Formulation
Decentralized Control Algorithms
Extension
Conclusion
MIT Computer Science and Artificial Intelligence Laboratory

14. Formulation Q Assumption: can communicate with neighbors
Target density function
Partition based on Voronoi Region
Partition based on Voronoi Region
Partition based on Voronoi Region
: Demanding mass
Assigned work: Mass in Voronoi region
Assigned work: Mass in Voronoi region
: Work done Q
Just an assumption not so big
encode geometry of a shape we wish to build in the target density function
Source cache:
Truss & Connector
Assumption: can communicate with neighbors
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15. Density function for a truss structure
Point-wise: distance between points is the length of a truss element
Generalizable to any part components
Density: a number of truss that can be connected to the point
Assembly decreases density
Interpolation for continuity
Target density function is like a blue-print.
Target density function is given!
Specified for a truss but can be generally used for any kinds of material
Φ=2
Φ=3
The algorithm can be applied for any kind of material
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16. Main Control Flow Deploy the assembly robot in the target area
Place the robots at optimal task location in Q
Delivery robot: carry parts
Assembly robot: assemble the parts
Here is the main flow in which how the system works.
Task.. Pronunciation
End
No more part
Sub structure
is done
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17. Balanced sub-structures
Why we want balance?
maximal parallelism, fair world.
Bring back the naïve case…
First assignment: Equal-mass partitioning
Balance during assembly: probabilistic selection & gradient of the demanding mass
No dependency on amount of source material
Parallelism
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18. Decentralized Control Algorithms Extension Conclusion
Outline
Introduction
Problem Formulation
Decentralized Control Algorithms
Extension
Conclusion
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19. Decentralized Equal-mass Partitioning
Target density function
Partition based on Voronoi Region
Assigned work: Mass in Voronoi region
Cost function: product of the masses
Minimize H not H dot _ H
GOAL:
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20. Decentralized Equal-mass Partitioning
Theorem: The proposed controller guarantees that H converges to either a local minimum or a global minimum
positive
function of @ H p i
We use Ji/Ji^2 form for future applications as adaptation.
Theorem is general, # of robots & # of types
Local min or saddle point
All go to zero (local min)
H goes to zero (global min)
Communication parameter: positions and masses of neighbors
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21. Decentralized Equal-mass Partitioning4 6 10
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22. Decentralized Delivery Algorithm
(1) p Á t
(2)
(2)
(1) Probabilistic deployment
- use the density function as a probability density function
- global balance
(2) Delivery by gradient of the demanding masses
- find the assembly robot with the maximum demanding mass
- local balance
Cartoon truss & connector
Drop the bar
Communication parameter: demanding masses of neighbors
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23. Assembly Algorithm
Truss: Select an edge with the maximum demanding mass
Connector: Select a node with a positive demanding mass
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24. Implementation Delivery robot Assembly robot
4 assembling + 4 delivering robots
10 assembling + 10 delivering robots
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25. Performance: balanced sub-structure
from naïve probabilistic choice
w/ movement by the gradient
10 robots
10 robots
# of truss
# of robots
Remind the algorithm
Also faster than probabilistic deployment
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26. Hardware Implementation
Truss element
IR Comm.
Assembly & delivery robot
iRobot platforms in a mesh network
Crawler arm w/ 5dof
Smart parts
Communication used for part location and grasping
No need for computer vision
Computation and communication beacon that leads to communication with robots
IR Comm.
Connector
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27. Distributed Assembly Example
Truss
Connector
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28. Distributed Assembly - platform
Sensor
Gripper
The gripper and sensor allow for communication with “fasteners”.
The sensor uses a single IR sensor and IR LED to send and receive data.
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29. Distributed Assembly User Interface
(other delivery robot waiting for the part source to become available)
Delivery robots start unpacking parts
Assembly robots distribute themselves across the job site.
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30. Experimental Results

31. Decentralized Control Algorithms Extension Conclusion
Outline
Introduction
Problem Formulation
Decentralized Control Algorithms
Extension
Conclusion
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32. Robustness to failure of robots
Failure of delivering robots does not affect convergence but slow down the total construction time.
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33. Construction w/ dynamic constraints
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34. Reconfiguration
Look for optimal change from one structure to the other
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35. Coverage in Discrete Environments
Coverage for environments
Explicitly modeled by a graph
Non-convex region
Must be distributed for large number of robots
Concept art by Jonathan Hiller, Cornell University
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36. Contributions for Graphs
Graph partitioning based on the Voronoi tessellation
Distributed control algorithms for coverage on a graph
Convergence proofs of the proposed controllers
Communication condition required for convergence
Must be distributed for large number of robots
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37. Equal-mass partitioning
Find graph Voronoi partitions that have equal amount of the node weights
Must be distributed for large number of robots
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38. Same strategy
Find the best node in Vi to minimize the local part of the cost function
Distributed vertex substitution is used with the cost function
Must be distributed for large number of robots
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39. Implementation: Equal-mass partitioning
4 robots
144 nodes, 240 edges
Must be distributed for large number of robots
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40. Implementation: Equal-mass partitioning
15 robot
384 nodes, 649 edges
Must be distributed for large number of robots
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41. Conclusion and Future work
We have proposed the decentralized controller for coordinated robotic construction
Work-assignment by distributed coverage
guaranteeing decay of the cost function
Consistently balanced sub-structures
probabilistic deployment & refinement by local gradient
Fully distributed algorithms
No dependency on
# of robots & amount of source material
Adaptive for
failure, reconfiguration, orderly construction
Next challenges
Multiple source components (dependency)
Constraints: Gravity, Strength, etc.
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