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

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

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 MIT Computer Science and Artificial Intelligence Laboratory

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. MIT Computer Science and Artificial Intelligence Laboratory

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 MIT Computer Science and Artificial Intelligence Laboratory

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 MIT Computer Science and Artificial Intelligence Laboratory

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 MIT Computer Science and Artificial Intelligence Laboratory

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 MIT Computer Science and Artificial Intelligence Laboratory

Related Swarms Work Deploy a group of robots to optimize the observation of an unknown, possibly changing sensory function.

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

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

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

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

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 MIT Computer Science and Artificial Intelligence Laboratory

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 MIT Computer Science and Artificial Intelligence Laboratory

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 MIT Computer Science and Artificial Intelligence Laboratory 16

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 MIT Computer Science and Artificial Intelligence Laboratory 17

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

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: MIT Computer Science and Artificial Intelligence Laboratory 19

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 MIT Computer Science and Artificial Intelligence Laboratory 20

Decentralized Equal-mass Partitioning 4 6 10 MIT Computer Science and Artificial Intelligence Laboratory 21

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 MIT Computer Science and Artificial Intelligence Laboratory 22

Assembly Algorithm Truss: Select an edge with the maximum demanding mass Connector: Select a node with a positive demanding mass MIT Computer Science and Artificial Intelligence Laboratory

Implementation Delivery robot Assembly robot 4 assembling + 4 delivering robots 10 assembling + 10 delivering robots MIT Computer Science and Artificial Intelligence Laboratory 24

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 MIT Computer Science and Artificial Intelligence Laboratory 25

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 MIT Computer Science and Artificial Intelligence Laboratory

Distributed Assembly Example Truss Connector MIT Computer Science and Artificial Intelligence Laboratory

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. MIT Computer Science and Artificial Intelligence Laboratory

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. MIT Computer Science and Artificial Intelligence Laboratory

Experimental Results

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

Robustness to failure of robots Failure of delivering robots does not affect convergence but slow down the total construction time. MIT Computer Science and Artificial Intelligence Laboratory

Construction w/ dynamic constraints MIT Computer Science and Artificial Intelligence Laboratory

Reconfiguration Look for optimal change from one structure to the other MIT Computer Science and Artificial Intelligence Laboratory

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 MIT Computer Science and Artificial Intelligence Laboratory

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 MIT Computer Science and Artificial Intelligence Laboratory

Equal-mass partitioning Find graph Voronoi partitions that have equal amount of the node weights Must be distributed for large number of robots MIT Computer Science and Artificial Intelligence Laboratory

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 MIT Computer Science and Artificial Intelligence Laboratory

Implementation: Equal-mass partitioning 4 robots 144 nodes, 240 edges Must be distributed for large number of robots MIT Computer Science and Artificial Intelligence Laboratory

Implementation: Equal-mass partitioning 15 robot 384 nodes, 649 edges Must be distributed for large number of robots MIT Computer Science and Artificial Intelligence Laboratory

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. MIT Computer Science and Artificial Intelligence Laboratory 41