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

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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 2 types of final assemblies
Example
Implemented in the robot simulator Webots (
- Uses Open Dynamics Engine to simulate physics
Predefined assembly plan:
2 types of final assemblies
4 types of basic parts

9. Magnets that bond to other parts
Example
Magnets that bond to other parts
Khepera III + bar (
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

10. 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

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

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

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

14. 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)

15. 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)

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

17. 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

18. 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:

19. 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

20. 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

21. 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)

22. 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

23. 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)

24. 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