Tool routing problem based on the Hamiltonian cycle problem for wire arc additive manufacturing M. Murua
1
1 2
R. Santana
2
D. Galar
3
A. Suárez
1
Advanced Manufacturing Department, Tecnalia Research & Innovation.
Intelligent Systems Group, Computer Science and Artificial Intelligence Department, University of the Basque Country. 3
Smart Systems Department, Tecnalia Research & Innovation.
Problem definition
HCP formulation for WAAM
Why? Wire arc additive manufacturing, WAAM, is a popular wire-feed additive manufacturing, AM, technology [1] that creates components through the deposition of material layer-bylayer. AM can be classified into different categories depending on energy sources; laser, electron and arc and raw material. In the case of wire-feed AM, metallic wire is the additive material or feedstock [1]. Therefore, WAAM is by definition an arc-based process that uses wire as the additive material [2]. One of the major constraints of AM processes is the significant amount of time to fabricate parts. In view of this, researchers are proposing improvements to AM processes by applying better process planning algorithms. For further enhancement of the effectiveness and efficiency of the process, deposition path planning plays a major role. Consequently, finding the optimal tool-path becomes critical.
• It
is a stochastic process that fullfils the Markovian property (selection of the bead number in time t + 1 is only related to the bead selection in time t). • Some of discrete optimization problems can be considered as a particular cases of stochastic optimization problems [3]. • An MDP is a particular stochastic process that provides mathematical framework for modeling decision making.
Figure 2: Representation of a part with its vertices and edges.
Mathematical framework for WAAM
Figure 1: Stainless steel 316L part manufactured by WAAM technology.
Proposed approach The problem of finding an optimal toolpath can be stated as a Hamiltonian cycle problem, HCP, where the vertices in the graph correspond to the beads of a part, as given a part with N beads, the route of the torch has to be defined. The embedding of the HCP in a Markov Decision process, MDP, will be considered to model the problem and mathematical programming tools will be used to solve it.
A bead S is defined as comprising extreme two points, {(x1, y1), (x2, y2)}, a function g, and a layer number l: S = {(x1, y1), (x2, y2), g, l|x1, x2, (1) ∗ y1, y2 ∈ R, g ∈ F(R, R), l ∈ N } Where (x1, y1) and (x2, y2) are the cartesian coordinates of the extreme points in the bead and g returns y coordinate of a given x coordinate. A wall, W , is a set of beads, that share the same extreme points, and function g. Each bead has a different layer number l. ∗ W = {Sl | l = 1, ..., nl}, nl ∈ N and ∗ Sl = {(xa, ya), (xb, yb), g, l} nl ∈ N . (2) A part, P , is a set of n walls ∗
P = {Wi | i = 1, ..., n}, n ∈ N . (3) A part will be represented as a planar graph, G. We consider a region in the planar graph, the set of faces F (G), with the exception of the exterior face. A vertex v ∈ V (G) belongs to a region if it is bounding it. An adding option, Ii,j , is a (i, j) − walk where i is the origin and j is the terminus.
There are two different WAAM technologies considered in this investigation: Plasma arc welding (PAW) and metal inert gas (MIG). • PAW:
The torch has limitations in the movements. New graph will be built based on predefined adding options to solve HCP. • MIG: The torch has not limitations to turn. The original graph will be used to solve HCP.
Figure 4: Graph to find the Hamiltonian cycle with 2N states.
Work in progress • Solve
the proposed model using an iterative algorithm that prevents a policy from having two paths. • Extend the model for the scenario of 2N vertices. • Define different cost matrices depending on WAAM parameters such as time and temperature. Figure 3: A planar graph with its fixed adding options and regions.
A proximity notion has been developed between two adding options: • Take
the terminus vertex j from Ii,j . • Calculate to which region(s) vertex j belongs. • Take the vertex set V from the region(s). • Calculate the set of adding options 0 {Ik,l } that have origin in vertex k ∈V. Formulation of the problem for PAW. Consider two scenarios: 1) 2N vertices, the adding options represent the directionality. 2) N vertices, the adding options do not represent directionality. The embedding of the HCP in an MDP is considered and linear constraints of the mathematical problem are formulated. Linear constraints, where β ∈ [0, 1] is a discount factor, xi,j are the elements of the state-action occupation vector x, A is the set of following vertices and B is the set of preceding vertices:
References [1] D. Ding, Z.S. Pan, D. Cuiuri, and H. Li. Wire-feed additive manufacturing of metal components: technologies, developments and future interests. International Journal of Advanced Manufacturing Technology, 81:465–481, 2015. [2] D. Ding, Z. Pan, D. Cuiuri, and H. Li. A tool-path generation strategy for wire and arc additive manufacturing. International Journal of Advanced Manufacturing Technology, 73:173–183, 2014. [3] E.A. Feinberg. Constrained discounted Markov decision process and Hamiltonian cycles. Mathematics of Operations Research, 25:130–140, 2000.
Acknowledgements The authors acknowledge Basque Government for support from project EUSK-ADDI (Etorgai 2014) and Haritive (Hazitek 2017).
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