# Mathematical Modelling and Methods for Load Balancing and Coordination of Multi-Robot Stations Doktorsavhandling, 2022

The automotive industry is moving from mass production towards an individualized production, individualizing parts aims to improve product quality and to reduce costs and material waste. This thesis concerns aspects of load balancing and coordination of multi-robot stations in the automotive manufacturing industry, considering efficient algorithms required by an individualized production. The goal of the load balancing problem is to improve the equipment utilization. Several approaches for solving the load balancing problem are suggested along with details on mathematical tools and subroutines employed.

Our contributions to the solution of the load balancing problem are fourfold. First, to circumvent robot coordination we construct disjoint robot programs, which require no coordination schemes, are flexible, admit competitive cycle times for several industrial instances, and may be preferred in an individualized production. Second, since solving the task assignment problem for generating the disjoint robot programs was found to be unreasonably time-consuming, we model it as a generalized unrelated parallel machine problem with set packing constraints and suggest a tailored Lagrangian-based branch-and-bound algorithm. Third, a continuous collision detection method needs to determine whether the sweeps of multiple moving robots are disjoint. We suggest using the maximum velocity of each robot along with distance computations at certain robot configurations to derive a function that provides lower bounds on the minimum distance between the sweeps. The lower bounding function is iteratively minimized and updated with new distance information; our method is substantially faster than previously developed methods. Fourth, to allow for load balancing of complex multi-robot stations we generalize the disjoint robot programs into sequences of such; for some instances this procedure provides a significant equipment utilization improvement in comparison with previous automated methods.

makespan minimization

Smart Assembly 4.0

Voronoi diagram

decomposition

automotive manufacturing

mathematical modelling

vehicle routing

motion planning

set packing

continuous collision detection

Pascal,Chalmers campus Johanneberg, Chalmers tvärgata 3
Opponent: Professor Martin Skutella, Institute of Mathematics, TU Berlin, Germany

## Författare

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Stiftelsen Fraunhofer-Chalmers Centrum för Industrimatematik

### Intersection-Free Geometrical Partitioning of Multirobot Stations for Cycle Time Optimization

IEEE Transactions on Automation Science and Engineering,;(2018)

Artikel i vetenskaplig tidskrift

### Exact makespan minimization of unrelated parallel machines

Open Journal of Mathematical Optimization,;Vol. 2(2021)

Artikel i vetenskaplig tidskrift

### Continuous collision detection of pairs of robot motions under velocity uncertainty

IEEE Transactions on Robotics,;Vol. 37(2021)p. 1780-1791

Artikel i vetenskaplig tidskrift

### Spatial–temporal load balancing and coordination of multi-robot stations

IEEE Transactions on Automation Science and Engineering,;Vol. 20(2023)p. 2203-2214

Artikel i vetenskaplig tidskrift

### E. Åblad, A.-B. Strömberg, D. Spensieri A Lagrangian-based method for makespan minimization of parallel machines with set packing constraints

Preventing collisions in multi-robot stations
In the automotive industry, it is desired to decrease the time from initial concept to final product. Moreover, the parts of a product have small deviations, hence tailoring each individual product, e.g., its welding program, can improve both quality and cost. To meet such demands with a high throughput, the decision process needs to be fully automated.

In multi-robot stations of manufacturing lines, robotics arms are positioned around the workpiece on which they carry out a set of (weld) tasks.
To maximize the throughput: the tasks are assigned to the robots, the order of the tasks is specified, and collision-free motions of the robots are computed. The robots may not collide with each other, hence the motion (and its duration) of a robot depend on the motions of the other robots, making the problem hard to solve optimally.

A common approach to solve the problem is to initially assume that the robots will not collide, and then let the robots wait on another to prevent collisions. This approach is computationally efficient since the robots’ motions can be computed independently, but it can lead to poor solutions.

In this thesis, the robots are instead constrained to work in separate volumes, thus preventing any robot-robot collisions. These volumes may change over time. This new approach allows for finding good solutions to some challenging scenarios, and thus, allowing for a higher degree of automation.

### Interlinked combinatorial and geometrical optimization problems in an autonomous automotive manufacturing industry

Stiftelsen för Strategisk forskning (SSF) (RIT15-0025), 2017-08-15 -- 2022-09-05.

Stiftelsen Fraunhofer-Chalmers Centrum för Industrimatematik, 2017-08-15 -- 2022-09-05.

### Smart Assembly 4.0

Stiftelsen för Strategisk forskning (SSF) (RIT15-0025), 2016-05-01 -- 2021-06-30.

### Drivkrafter

Hållbar utveckling

Produktion

### Ämneskategorier

Beräkningsmatematik

Robotteknik och automation

### ISBN

978-91-7905-661-2

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 5127

### Utgivare

Chalmers

Pascal,Chalmers campus Johanneberg, Chalmers tvärgata 3

Opponent: Professor Martin Skutella, Institute of Mathematics, TU Berlin, Germany

2023-11-12