Robotic Swarms for Assembly Spring Berman and Vijay Kumar GRASP Laboratory, University of Pennsylvania Loïc Matthey University College London, formerly EPFL February 22, 2010

Motivation Design a reconfigurable manufacturing system that quickly assembles target amounts of products from a supply of heterogeneous parts

Design Criteria (1) Strategy should be scalable in the number of parts Decentralized approach: - Parts scattered randomly inside an arena Randomly moving autonomous robots assemble products Local sensing, local communication (2) Minimal adjustments when product demand changes - Probabilities of assembly and disassembly are robot control policies - Can be updated via a broadcast (3) System can be optimized for fast production Spatial homogeneity  Chemical Reaction Network model

Related Work Collision-based self-assembly modeled as Chemical Reaction Networks Hosokawa et al., Artificial Life, 1994 - Predict yield of complete assemblies from passive modules Baskaran et al., Micro Electro Mech. Syst., 2008 - Model micro scale batch assembly Klavins et al., Proc. Robotics Sci. Syst., 2007 - Optimize assembly detachment probabilities to maximize yield of a particular assembly type - Optimization requires enumeration of all reachable states

Outline Abstraction: continuous model of robots executing an assembly task Synthesis: top-down design of decentralized robot controllers Optimization: controller parameters are independent of the number of robots and parts Implementation: provable guarantees on performance

System Abstraction Ordinary differential equations M states: continuous populations of parts Reduced continuous model Robots find parts quickly, Ordinary differential equations States: continuous populations of robots and free/carried parts Complete continuous model Large Spatial homogeneity [D. Gillespie, Annu. Rev. Phys. Chem., 2007] 3D physics simulation N robots, Pi parts; i = 1,…,M types Physical system

Reduced continuous model Problem Statement ODE’s are functions of probabilities of assembly and disassembly: Optimize for fast assembly of target amounts of products Reduced continuous model Robots start assemblies and perform disassemblies according to optimized probabilities Physical system

2 types of final assemblies Example Implemented in the robot simulator Webots (www.cyberbotics.com) - Uses Open Dynamics Engine to simulate physics Predefined assembly plan: 2 types of final assemblies 4 types of basic parts

Magnets that bond to other parts Example Magnets that bond to other parts Khepera III + bar (www.k-team.com) Rotational servo Magnet Bonds to bar Magnets can be turned on or off Servo rotates bonded part to orientation for assembly Infra-red distance sensors for collision avoidance Emitter/receiver on each robot and basic part for local communication, computing relative bearing

Complete Continuous Model pe = prob. that a robot encounters a part or another robot ≈ [Correll and Martinoli, Coll. Beh. Workshop, ICRA 2007] A = arena area

Complete Continuous Model prob. of two robots successfully completing assembly process j (measured from simulations)

Complete Continuous Model Tunable: prob. of two robots starting assembly process j prob. per unit time of a robot performing disassembly process j

(Illustrated later for reduced model) Complete Continuous Model ODE Model (Illustrated later for reduced model)

Validation of Complete Continuous Model 15 robots, 15 basic parts measured Error bars show standard deviations Part populations averaged over 100 Webots simulations F2 F2 F1 F1 Time (sec)

Validation of Complete Continuous Model Continuous model is fairly accurate Discrepancies are due to: Relatively low populations; ODE most accurate for large ones Assembly disruption in simulation (not modeled) Final product populations Webots, average of 100 simulations F2 Complete cont. model (numerically integrated) F1 Time (sec)

Reduced Continuous Model Lower-dimensional model (abstract away robots): Vector of complexes: Conservation constraints:

Reduced Continuous Model The system has a unique, positive, globally asymptotically stable equilibrium. Proof: Reaction network is deficiency zero and weakly reversible, does not admit equilibria with some xi = 0  We can design K such that the system always converges to a target equilibrium, xd > 0

Design of Optimal Recall that K is a function of p, the vector of Select xd that satisfies conservation constraints Compute p that minimizes the system convergence time to xd subject to constraints:

Optimization Problems I. Linear Program Objective: Maximize the average inverse relaxation time τj τj = time for system mode to return to equilibrium after perturbation Estimated by linearizing the ODE model around xd [Heinrich and Schuster, The Regulation of Cellular Systems, 1996] For reaction : II. Monte Carlo Method Objective: Minimize time for system to reach

Mapping onto Physical System = simulation timestep (32 ms) = random number uniformly distributed over [0,1] Robot computes R at each Δt, disassembles the part if Robot computes R, executes assembly if

Optimization Improves Convergence Rate 15 robots, 15 basic parts Simulations averaged over 30 runs Simulation Model Target α = 0.5 (1– α) = 0.5 Randomly selected from from Linear Program Monte Carlo Time (sec)

Linearization is most effective for α ≈ 0.2 – 0.5 For all α, linear program only changes rates of disassembling F1, F2 Monte Carlo yield fastest convergence but take ~10 hrs to compute, vs. <1 s using the linear program (2 GHz laptop) 300 basic parts Time for reduced model to reach from: Random selection (Average of 100 values) Linear Program Monte Carlo Method

Linearization is most effective for α ≈ 0.2 – 0.5 15 robots, 15 basic parts Simulations averaged over 30 runs Simulation Model Target α = 0.9 (1– α) = 0.1 Randomly selected from from Linear Program Monte Carlo Time (sec)

Conclusions and Future Work Abstracted multi-robot assembly system to an accurate continuous model Optimization of stochastic control policies results in faster production of target quantities Linearization of model for optimization is most effective for a certain range of target product distributions Extensions: Optimize assembly plans Use inter-robot communication to improve the yield rate Ex. Recruitment of robots carrying needed parts