INTERNATIONAL CONFERENCE ON , ICED21 16-20 AUGUST 2021, GOTHENBURG, SWEDEN

WHAT IS GENERATIVE IN TOOLS? UNCOVERING TOPOLOGICAL GENERATIVITY WITH A C-K MODEL OF EVOLUTIONARY ALGORITHMS.

Hatchuel, Armand; Le Masson, Pascal; Thomas, Maxime; Weil, Benoit

MINES ParisTech-PSL

ABSTRACT Generative design (GD) algorithms is a fast growing field. From the point of view of Design Science, this fast growth leads to wonder what exactly is 'generated' by GD algorithms and how? In the last decades, advances in enabled to establish conditions and operators that characterize design generativity. Thus, it is now possible to study GD algorithms with the lenses of Design Science in order to reach a deeper and unified understanding of their generative techniques, their differences and, if possible, find new paths for improving their generativity. In this paper, first, we rely on C-K ttheory to build a canonical model of GD, based independent of the field of application of the algorithm. This model shows that GD is generative if and only if it builds, not one single artefact, but a "topology of artefacts" that allows for design constructability, covering strategies, and functional comparability of . Second, we use the canonical model to compare four well documented and most advanced types of GD algorithms. From these cases, it appears that generating a topology enables the analyses of interdependences and the design of resilience.

Keywords: C-K theory, generative design algorithms, Design theory, Computational , Design informatics

Contact: Le Masson, Pascal MINES ParisTech-PSL Management Science France [email protected]

Cite this article: Hatchuel, A., Le Masson, P., Thomas, M., Weil, B. (2021) ‘What is Generative in Generative Design Tools? Uncovering Topological Generativity with a C-K Model of Evolutionary Algorithms.’, in Proceedings of the InternationalICED21 Conference on Engineering Design (ICED21), Gothenburg, Sweden, 16-20 August 2021. DOI:10.1017/ 1 pds.2021.603

ICED21 3419 1 INTRODUCTION: GENERATIVE DESIGN ALGORITHMS THROUGH THE LENSES OF DESIGN SCIENCE Generative design algorithms (GDA) is a fast growing field that develops “design approaches that use algorithms to generate designs” (Caetano et al., 2020). Advances in CAD software (in particular with the launch of Autodesk so-called generative design (Kazi et al., 2017)), computing power and new computer science algorithms have contributed to the emergence of various “generative design tools” (Buonamici et al., 2020) with multiple applications. This fast growth leads to critical questions about generative design algorithms from the point of view of design science. What is exactly ‘generated’ and how? What means progress in the field? And more fundamentally, what can design science say for the theoretical understandings and enhancement of generative algorithms? Design science have also achieved major steps. C-K theory has paved the way to new formalizations of a design process with a high of generality. Design science is now able to establish conditions and operators that define and enable design generativity and generative power. Such notions are now independent of “what” is designed and of classic technical models. They describe generativity as the transformation of known objects into new ones using a new abstract language which received wide validation in the literature: concepts, generic extensions, restrictive and expansive partitions, identity of objects, independence and splitting structures of knowledge (Hatchuel et al., 2011, Le Masson et al., 2016). Thus, it is now possible to study Generative design algorithms with the lenses of Design Science in order to reach a deeper and unified understanding of their generative techniques, their differences and, if possible, find new paths for improving their generativity. This paper develops such study through four steps that correspond to the sections of the article. In the literature review (part 2), we describe a brief state of the art of GDA and applications. In part 3, since any GDA is necessarily a constructive and iterative process, applying C-K theory to such constraints we can predict a general C-K logic for GDA. We define it as a canonical model of GDA that is not dependent of the field of application of the algorithm. This model mainly shows that GDA is generative if and only if it builds, not one single artefact, but a special “topology of artefacts” that allows for design constructability, covering strategies, and functional comparability of designs. In part 4, using the canonical model we compare the generative power of four well documented and most advanced types of GDA, which confirm their capacity to generate a ‘topology of artefacts’.

2 LITERATURE REVIEW: THE VARIETY OF GD ALGORITHMS INVENTIONS

2.1 GD definition. Recent reviews on GDA (Caetano et al., 2020, Mountstephens and Teo, 2020) share a definition of GDA: “a design approach that uses algorithms to generate designs” (Caetano et al., 2020); GDA is also seen “as the exploration of the principle of generating complex forms and patterns from a simple specification [with an algorithm]” (Shea et al., 2005, McCormack et al., 2005). Some authors insist on potential “creative outcomes” (Bernal et al., 2015) or “happy accidents” ie “unexpected results” born from “the number of design variations” and “the range of the variations”, that positively impact the design process” (Chaszar and Joyce, 2016). Hence GDA definition focuses on the presence of an algorithm but remains relatively fuzzy on what exactly generativity consists in. The notion of the variety of designs provided by the software is key in GD A, even if the link between design variations and GD generativity is not clearly explained and loosely related to how GDA provokes ‘surprises’.

2.2 A variety of Generative Design techniques. GDA is clearly related to algorithmic rule-based processes that mainly refer to evolutionary techniques but are not limited to them (Caetano et al., 2020) (p. 294) - More specifically, (Mountstephens and Teo, 2020) identifies four generation methods: genetic algorithms, Shape grammars, L-Systems, Swarm intelligence - the authors mention other useful techniques: parametric modeling and topology optimisation. One could also add the recent use of techniques coming from AI like Generative Adversarial Networks (GAN) Variational AutoEncoders (VAE) to enhance topology optimization (Oh et al., 2019) or to enable shape parameterization for further generation processes (Burnap et al., 2016, Umetani, 2017). In this paper, we will rely on well-known Multi-Objective Generative Algorithms (also known as Multi-Objective Evolutionary Algorithms - MOEA) - and we will pay a particular attention to a new family of algorithms called quality-diversity, which are particularly relevant for GDA: quality-

3420 ICED21 diversity algorithms (Pugh et al., 2016) “evolve an archive of solutions which is, according to a user- defined behaviour , as diverse as possible while obtaining for each solution a high performance” (Bossens et al., 2020). This family contains for the moment two prototypical algorithms: Novelty- Search with Local Competition (Lehman and Stanley, 2011a) and Multidimensional Archive of Phenotypic Elites (MAP-Elites) (Mouret and Clune, 2015).

2.3 Generativity in GD algorithms: a need for clarification and unification. The field of GDA presents multiple streams of works that develop original algorithms applied to ad hoc design cases. Each software has its applications and illustrates one form of generativity. For instance: - Byrne et al present a multi Objective Evolutionary Algorithm for design exploration and optimisation of a wing profile (Byrne et al., 2014); - Multi-Objective Genetic Algorithm (in Autodesk) has been used in architectural space planning (Nagy et al., 2017), the design of office table (Nagy et al., 2017) (Chen et al., 2018); - Novelty Search was used to evolve robot controllers into a deceptive maze (Pugh et al., 2016, Lehman and Stanley, 2011b) or to design images (Woolley and Stanley, 2011); - Map-Eliteswas used to design a self-repairing hexapod robot (Cully et al., 2015) or wing profile (Cully et al., 2015) as well as in video game (Fontaine et al., 2019), automated image generation (Nguyen et al., 2015), robot morphologies and controllers (Hart et al., 2018); - Autogenetic Design Theory was used for gearbox design (Wünsch et al., 2012). This variety of applications calls for a clarification of the design logic associated to each GD software.

2.4 The locus of generativity in GDA: an engine to generate a population of artefacts? Self-evidently, GDA operates on a parametrically defined object with constraints on the parameters, which restrict its use in design processes. (Mountstephens and Teo, 2020) proposed to distinguish between autonomous and interactive generative design: in a parametrically defined solution space, autonomous design might sound quite self-evident and generativity could appear only coming from interaction (capacity to use the GDA in a more or less creative way, at varied moments in the design process). Still, maybe counterintuitively, in this paper we first focus precisely on the specific generativity of the algorithm itself, its capacity to generate a collection of varied artefacts. Even if this generativity might sound limited (‘parametric’?), we investigate how a unique property of GDA software lies in its capacity to offer a structured set of alternatives, and, then, how the user might react to this set. In this paper, we focus on the generative logic of GDA such as: MOGA, quality-diversity algorithms, topologic optimisation, particle swarm optimisation, space filling techniques... Qualifying their generativity of these algorithms is not an easy task - and, to our knowledge, no systematic study was done until now. Hence, our research question: how can we characterize formally and systematically the generativity logic of GDA ‘engines’?

3 A C-K CANONICAL MODEL TO UNCOVER THE GENERATIVITY OF GDA

3.1 Method: casting GDA in C-K theory. C-K theory (Hatchuel and Weil, 2009), is one of the most advanced formulations of a design theory (Hatchuel et al., 2018). C-K theory presents the advantage to be independent of what is designed and can account for very strong forms of generativity (Hatchuel et al., 2011). Casting a design method in C-K theory has already been done in other papers (Reich et al., 2012, Kroll et al., 2014). We will follow the same method. We first codify in a canonical model what can make the generativity of a GDA. Based on this analytical model, we analyse the generativity of a sample of published GDA.

3.2 A canonical model of GDA: concepts, object model, expansions.

3.2.1 Knowledge base - object model and splitting condition

A GDA operates on an initial state of knowledge 퐾0 which contains an object model 푀0 with:

ICED21 3421 – Variables that can be assimilated to ‘design parameters’ or also called ‘genotypical parameters’ - parameters that are considered as directly actionable by the , noted

X in=1 .

– Variables that can be assimilated to functional (or behavioural or phenotypical) features, φ j .

They are not directly actionable; their value is computed in the object model: X in=1j→ φ .

– One artefact is a point value xin=1 that has the features φ j= 1 m . GDA apply in situations where:

– The object model M0 is not invertible: given specific φ j= 1 m , the object model M 0 doesn’t

enable to find even one X in=1 that meets φ j= 1 m in finite reasonable time (see H1 below). – The object model M0 is not derivable not continuous, which means that a small change in one

X i can provoke strong changes in φ j= 1 m and conversely a small change in φ j can

correspond to a strong change in X in=1 . In particular, this means that it is not possible, for a

known object (X in=1  , φ j = 1  m ), to know what is its neighbourhood in terms of genotype and (even less) in term of phenotype - so that in this kind of K0, there is no self-evident solution to a problem of optimisation, ie finding an object that, at least locally, phenotypically dominates the others. Design theory leads us to wonder whether this initial knowledge base K0 is splitting (Le Masson et al., 2016), ie non-modular and non-deterministic. Modularity would mean that some design parameters could be added without effect on phenomenology. As we just mentioned, initially, in K0, the design parameters are supposed to influence phenomenology. Determinism would mean that some design parameters would determine the phenomenological behaviour: again, as we just mentioned, initially, in K0, one can’t say that such deterministic law exist. So in usual contexts, we have: Property P1: usually GDA operates on a knowledge base that meet the splitting condition, hence GDA is compatible with a generative process.

3.2.2 Concepts as departures of a GDA: Following C-K theory, any GDA that aims to design some X begins necessarily with a concept of the form “X that fulfils P(X)”, P(X) being a series of properties of X such that: – P(X) are undecidable in K0 ie. there is no constructive rule that allows to design such X with K0 (of course, since we want to describe the mechanism of the GDA ‘engine’ following the C-K operators, the ‘engine’ itself is not in K; otherwise it would appear as a constructive rule and the design is finished) – P(X) will be constructible, true and compatible in some established Kn state of K. It has to be underlined that here is a specific feature of GDA: GDA actually work to generate a collection of artefacts, ie in the concept {X, P (X)}, X actually refers to a collection of artefacts; and P actually refers to a property of this collection: in the concept “the set of wing profiles that form a Pareto front”, we want to generate a collection (X) of wing profiles, and this collection has the property to form a Pareto front (this is a property of the population, and not of a lone artefact). Hence a second property: P2: usually GDA designs a collection of artefacts with specific property, this property can make that it is undecidable whether it is possible or impossible to get a population with property P.

P has to be interpretable (hence it is in K0) and just needs to make X undecidable in K0. : – "a collection of N artefacts": it can be generated by instantiating M0 N-times. It is not a concept. – "a collection of N artefacts generated by random variation of genotype": it can easily be generated as soon as one knows of random number generator. This is not a concept. – "a collection of N artefacts generated by variation of phenotype": if the object model is non- invertible, this is a concept.

3.2.3 Expansion in space K and concepts partition in space C C-K theory models Concept partitions and expansions through tree structured sequences of nested partitions. They describe a constructive refinement of C0 that should lead to the design of X; each of

3422 ICED21 these steps may activate space K, hence creating a knowledge expansion. At least, the last refinement produces an acceptable design that is integrated as a new true object in space K. In the case of GDA, the algorithm is parameterized to produce knowledge and concept expansions. In a genetic algorithm, this is done by variation-selection. But the partitions can’t be easily followed. When successive partitions can’t be easily followed, it is possible to evaluate the expansions: – If the initial proposition {X, P(X)} was a concept and has become knowledge (in C-K terms: there was an initial disjunction and there is a final conjunction), then there was C-expansion. – By comparing the knowledge base before (K0) and after the GD process (Kfinal), one can estimate K-expansion. P3: a criteria to evaluate the generativity of a GD is twofold: a- is there a conjunction after an initial disjunction? b- what is the knowledge expansion between K0 and Kfinal? P1, P2, P3 are the main properties of a canonical model of GDA in C-K framework (Fig. 2)

Iterations A population (� �= 1…� , φj= 1…m) … Variations on Capacity to compare C based on: K with property P genotype param • criteria P phenotypical � on the , … features Based on some parameters (initial collection data, computation param…) of artefacts Genotype param Phenotype param: • variations � , … = x , … � , … Topology other Topology with set d� , … Rules to compute: paramets, at time t, param, other time… object not-invertible, A topology on x�= 1…� →φj … object model � model � � splitting � Figure 1. A canonical model of GD algorithms in C-K. P1: object model is non-invertible, a priori non continuous (hence splitting); P2: Property P is undecidable; P3: generativity is measured on disjunction-conjunction and K-expansions (a topology on object model M0)

3.3 Avoiding the combinatorial trap in GDA: C-K conditions for generativity and the emergent topology of designs.

3.3.1 Variation and selection in GDA: the combinatorial trap GDA raise a critical question for design theory: in which way can an algorithm be generative in the sense of C-K theory? Usual applications of C-K theory consider that K-space contains propositions (that are true or false) as well as propositions that can be logically deducted one from the other - hence not every new proposition is a K-expansion. Hence the knowledge base contains an internal “knowledge production engine” and one considers that there is expansion only if one goes ‘beyond’ this internal knowledge production engine. In the case of a GDA, our hypothesis is: H1: we consider that classical computations techniques in finite time are available in K and their results are not considered as K-expansions. The algorithm that is under investigation is not in K.

3.3.2 Generativity in GD: the emergence of a topology of artefacts. Building on properties P1 to P3 and hypothesis H1, what does C-K predict on GDA generativity? – According to P1 and H1: even GDA-knowledge base is purely made of combinatorial knowledge, the knowledge base is splitting and enables generativity – According to P2 and H1: concepts in GDA are related to specific properties associated to a collection of artefacts - the concept doesn’t come from the number of entities (because of H1); hence the concepts comes from the structure and descriptors of this collection. – According to P3: expansions can be evaluated by analysing initial and final C and final K- expansion. In the end of a GDA process, one gets a collection of artefacts that meet the structural property P. So that the generativity is in this new structure of the collection of artefacts. At first sight, GDA appears trapped into a closed world of combinatorial designs. To avoid such trap, C-K theory calls for thorough examination of all knowledge produced by the algorithm. Clearly this knowledge is much more extended than the single artefacts that are designed. We have to recognize that the GDA not only explores single designs but compares them, positions them one to another, creating structures in the collection of artefacts. GDA provides new knowledge on the topology of Xis.

ICED21 3423 The expansion comes from the emergence of a ‘geometry’, a space in which artefacts can be relatively positioned. This new structure is a topology on the model of objects, in the sense that:

– GDA expresses each object in all its dimensions ( X in=1 , φ j = 1  m ) . Hence this space is multidimensional, linking genotypic dimensions and phenotypic dimensions. – GDA enables to distinguish certain objects - each object of the final collection is carefully separated (in singletons). – GDA also enables to not distinguish other objects: all the ‘dominated’ artefacts are considered in the same “neighbourhood” – In this topology, the object model can be inverted (almost) everywhere: for each point of the topology, one relevant artefact can be associated (with respect to the criteria P). It means that in the resulting topology the knowledge base is not splitting anymore.

3.3.3 How topological knowledge provides a source of generativity for the user For sure, there is a circular logic here: the topology that emerges is dependent of the iterative algorithm and another GDA technique would produce a different topology. Conversely, the information on the topology of designs can improve the GDA. However, what counts for the generativity of GDA is the type of new knowledge extracted from the topology that appeared. Information linked to this topology helps to explore dimensions of expansion predicted by canonical model Fig. 1:

– The topology can be extended by extending X in=1 , φ j = 1  m , and/or the model object M 0 :

adding or deleting some X i , changing range, or modifying φ j= 1 m .

– The topology of the X in=1 , φ j = 1  m revealed by the population of designs can help to compute some property P that will be introduced to change the iteration rules. E.g an algorithm can use the density of designs in some areas of the Xis to evolve the selection rules.

4 GDA: UNCOVERING TOPOLOGIES AND COMPARING GENERATIVITY We now have analytical tools (canonical model) and clear predictions (GDA tools generate topology of artefacts). We test them on a sample of most recent GDA. This sample was built on GDA recent reviews (Caetano et al., 2020, Mountstephens and Teo, 2020). We hence selected the following methods: – MOEA (with one particular illustrative use case: (Byrne et al., 2014)), – space-filling techniques (one particular illustrative use case: (Khan and Awan, 2018)), – topological optimization (illustrative use case: (Matejka et al., 2018)) – Quality-Diversity (QD) algorithms (illustrative use case: (Clune et al., 2013))

4.1 Analysis of four GDA tools with the C-K canonical model

4.1.1 Multi-Objective Evolutionary Algorithm (MOAE) Byrne et al. (Byrne et al., 2014) present a use case evolving parametric aircraft models. Coded with the canonical model (see Fig. 2 below): the designer knows a model of the aircraft, where given design parameters lead to two particular functional performances, Lift and Drag. The concept is: “A Pareto Front on the functions, max Lift, min Drag”. The designer makes variations on a subset of three parameters among the set of possible design parameters and run the GDA tool, powered by a multi- objective evolutionary algorithm (MOEA) non-sorting genetic algorithm-II (NSGA-II) (Deb et al., 2002). This leads to a first Pareto front (see blue dots in the figure). Then the designer selects a larger set of parameters and run the algorithm again, to get another Pareto front (see red dots in the figure).

3424 ICED21 J. Byrne et al. / Neurocomputing 142 (2014) 39–47 43

Fig. 9. Airfoil optimisation in the order of increasing lift (and increasing drag) from top left to bott om right. The overall shape of the design remains the same.

C A Pareto Front on (max Lift � ; min Drag � ), Genetic algorithm for Pareto fronts (NSG A II) • select 3 (or 5) parameters to be varied K • select functions for the Pareto front From a subset of 3 From a subset with more parameters {� } parameters {� … } … Variations on n=3 Fig. 10. The change in average lift/drag during the course of the run: (a) average lift maximisation and (b) average drag minimisation. Compare Pareto (or n=5) param domination

J. Byrne et al. / Neurocomputing 142 (2014) 39–47 43 Genotype param Selected functions F� � , … = x , …

Fig. 11. Wing optimisation in the order of increasing lift (and increasing drag) from the top left to the bott om right. The increased number of variables result ed in different w ing con gurations.

solutions are shown in red and green respectively with a line Rules to compute: connecting individual on the pareto front. Overall the pareto front object of the evolved solutions is equivalent to the randomly generated Fig. 9. Airfoil optimisation in the order of increasing lift (and increasing drag) from top left to bott om right. The overall shape of the design remains the same. x →F solutions, indicating that no bene t was provided by the genetic �= 1…� j , information. model � � That an evolutionary approach did not outperform a brute force approach could be the result of the constrained nature of the representation. Each of the three airfoil sections had two variables. Although each individual was encoded by 30 integers, the range of each variable was limited to viable designs. Such a representa- tion could generate good solutions purely by random variation, Figure 2. Pareto front GDindicatingAthat ittoolis too constra ined.(MOEAThis conclusion would be NSGA-II), from Byrne et al 2014. Parametric object supported by the fact that both approaches generat ed pareto optimal designs that outperformed the original model. A sample of individuals from the pareto front is shown in Fig. 9. Fig. 12. The Cessna 182 model. The optimised sections are highlighted in red. (For interpret ation of the referen ces to colour in this gure caption, the reader is model in K. GDALimiting the evolv generatesable representation to the airfoils produced the topology associated to a Pareto front. referr ed to the w eb version of this paper.) optimised solutions that maintained the same overall design as the BWB aircraft. evolved and brute force solutions are shown in red and green A scatter plot of wing and airfoil optimisation is shown in respectively. The graph sh ows how well the desi gn maximised Fig. 8(b). Again the original model is shown in black and the lift on the x-axis and how well it reduced drag on the y-axis.

Fig. 10. TheGenerativitychange in aver age lift/drag during the course of the run: (a) av eraganalysede lift maximisation and (b) average drag minimisation. within the canonical model: – the designer designs a Pareto front (not a single aircraft). We have a clear topology: single artefacts along the front, dominated artefacts below the front, no artefacts beyond.

Fig. 11. Wing optimisation in the order of increasing lift (and increasing drag) from the top left to the bottom right. The increased number of variables result ed in different w ing con gurations. – Modifying the design parameters to be varied, the designer gets several topologies (in case 1,

solutions ar e shown in red and green respectively with a line connecting individ ual on the pareto front. Over all the pareto front of the evolved solutions is equivalent to the randomlythegener ated designer only evolved the wing profile, in case 2, the designer also evolved the fuselage). solutions, indicating that no bene t was provided by the genetic information. That an evolutionary approach did not outperform a brute force approach could be the result of the constrained nature of the representation. Each of the three airfoil sections had two variables. Although each individual was encoded by 30 integers, the range of each variable was limited4.1.2to viable designs. SuchSpacea representa- -filling generative design tion could generat e good solutions purely by random variation, indicating that it is too constrained. This conclusion would be supported by the fact that both approaches generat ed pareto optimal designs that outperformed the original model. A sample of individuals from the pareto front is shown in Fig. 9. Fig. 12. The Cessna 182 model. The optimised sections are highlighted in red. (For interpret ation of the referen ces to colour in this gure caption, the reader is Limiting the evolvableKhanrepresentation to the airfoils&produced Awan (Khan and Awan, 2018) give (among others) an simple illustration of a “generative referr ed to the w eb version of this paper .) optim ised solutions that maintained the same overall design as the BWB aircraft. evolved and brute force solutions are shown in red and green A scatter plot of wing and airfoil optimisation is sh own in respectively. The graph sh ows how well the desi gn maximised Fig. 8(b). Again the ordesigniginal model is shown in blatechniqueck and the lift on the x-axis and how forwell it red uce exploringd drag on the y-axis. shape variation”, based on space-filling technique. In GD canonical model (see Fig. 3 below): in K, the designer disposes of a parameterized CAD-model (here a lamp, with two design parameters). The concept is: a map that represents the diversity of possible CAD shapes. To this end, the designer selects mapping criteria P: either space-filling (the criteria pushes to maximise the distance between shapes), or non-collapsing criteria (avoiding too different shapes (Draguljić et al., 2012)) or both. Powered also by MOES NSGA-II, the GD tool generates a map of CAD shapes. S. Khan, M.J. Awan A map that represents the variety of CAD models Genetic algorithm for Pareto fronts (NSG A II), Fig. 2. Design alternatives for a 3D CAD model C with two design parameters (a) are obtained in adapted to population constraints distribution K 2D spaces considering; (b) only space-filling Fr om selected design parameters � criterion, (c) only non-collapsing criterion, and • 2 parameters to be varied both space-filling and non-collapsing criteria • Different properties P (P1, P2…) using Sf-GDT (d). Design alternatives for the same CAD model with three parameters (e) are for the collection (mapping criteria) generated in 3D design space using Sf-GDT Based on P1: Based on P2: Based on P1 and P2: while considering both space-filling and non- space-filling non-collapsing space-filling and non- collapsing criteria (f). S. Khan, M.J.S. Khan,AwanM.J. AwancrS. Khan,iteriM.J.a onlAwany criteria only collapsing criteria Variations on n=2 param Fig. 2. DesignFig. 2.alternativesDesignFig.alternatives2.forDesigna 3D CADalternativesfor amodel3D CADformodela 3D CAD model No function, only with two designwith twoparametersdesignwithparameterstwo(a) designare obtained(a)parametersareinobtained(a) arein obtained in CAD file 2D spaces2Dconsidering;spaces considering;2D(b)spacesonly considering;space-filling(b) onlyGenspace-filling(b)otyponlye space-fillingparam criterion, criterion,(c) only non-collapsing(c)criterion,only non-collapsing(c)criterion,only non-collapsingandcriterion, andcriterion, and both space-fillingboth space-fillingandbothnon-collapsingspace-fillingand non-collapsingcriteria�and, non-collapsing…criteria= x , criteria… using Sf-GDTusing(d).Sf-GDTDesignusing(d).alternativesSf-GDTDesign (d).alternativesforDesignthe alternativesfor the for the same CADsamemodelCADwithmodelsamethreewithCADparametersthreemodelparameterswith(e) arethree (e)parametersare R(e)ulesare to compute: generatedgeneratedin 3D designingenerated3Dspacedesignusingin space3DoSf-GDTbdesignjectusing spaceSf-GDTusing Sf-GDT while consideringwhile consideringbothwhilespace-fillingconsideringboth space-fillingandbothnon-space-fillingand non- andx�=non-1…� →� � � file collapsingcollapsingcriteria (f).criteriacollapsing(f). criteriam(f).odel � �

Figure 3. Space-filling GD tool (MOEA-NSGA-II) from Khan et al. 2014 narrow down the design space and larger variation of designs may not stage and based on these specifi cations the designer limits the space. Generativity analysed within the canonical model: be achieved. Therefore, the decision on the selection of appropriate Autonomous Formulation: The autonomous formulation helps to design parameters should be carefully made. coarsely form the design space as a percentage of the initial parameter – the designer designs a map (not a single CAD shape) - hence a topology ofOne artefacts.strategy, which the designer can follow, is to fi rst detect the values of the design. This formulation happens when no primary un- important features of a given model and then these features can be derstanding of the design specifi cations are available in the conceptual – Formally speaking, the only difference with Pareto front case is thatparametrized the formerwith a relatively relieshigher number on of parameters and de- phase. The autonomous formulation gives a good initial guess of sui- signs can be generated with these parameters. Later, after some trials, table space limits. With this formulation, the designer can fi rst in- mapping criteria whereas Pareto case, the mapping criteria are the functionthe designer themselves.can detect quixotic parameters and eliminate them by adequately build up an initial map of promising regions of the design directly modifying the CAD model. Such capability of the generative space and then explore designs in that space. Afterward, the designer design system is recognized as ’designerly’ method, which allows de- can further reform the design space based on the previous exploration 4.1.3 Topological optimization algorithm signers to modify the model under consideration and use its generative results. There can be some infeasible designs in the autonomously narrow narrowdown thedowndesignnarrowthespacedesigndownandspacethelargerdesignandvariationlargerspacevariationandof designslargerofvariationmaydesignsnot mayof designsstagenot andmaystagebasednotandonbasedthesestageonspecificationsandthesebasedspecificationson thethesedesignerspecificationsthe designerlimits thethelimitsspace.designerthe space.limits the space. capabilities at any phase of the design process [ 12] . After exploring the formalized space, but this can be overridden by implementing geo- be achieved.be achieved.Therefore,beTherefore,achieved.the decisiontheTherefore,decisionon thetheselectionondecisionthe selectionof onappropriatetheofselectionappropriateof AutonomousappropriateAutonomousFormulation:AutonomousFormulation:The autonomousFormulation:The autonomousformulationThe autonomousformulationhelps toformulationhelps to helps to designs based on the important features, later, if required, design space metric constraints. design parametersdesignMatejkaparametersshoulddesignbeshouldparameters carefullyet beal.carefullymade.should prebemade.sentcarefully anmade. examplecoarsely ofcoarselyform topologicaltheformdesigncoarselythespacedesignformas spaceaoptimizationthepercentagedesignas a percentagespaceof theas initiala percentageof theGDparameterinitialAof parameterthe forinitial “parameterexplorationcan be explored andbased visualizationon its nominal features. Interactive Formulation: In the interactive formulation of the One strategy,One strategy,whichOnethewhichstrategy,designerthe designerwhichcan follow,thecandesignerisfollow,to firstcanisdetecttofollow,firstthedetectis to valuesfirstthe detectofvaluesthethedesign.of thevaluesThisdesign.formulationof theThisdesign.formulationhappensThis formulationhappenswhen nowhenprimaryhappensno un-primarywhen noun-primary un- design space, the designer creates multiple spaces and gradually pro- importantimportantfeaturesof featureslargeofimportanta givenof-scaleafeaturesmodelgivenandmodelof generativeathengivenandthesemodelthenfeaturestheseand featuresthencandesignbethesecanfeaturesderstanding bedatacanderstanding setofbethe”design derstandingof(Matejkathespecificationsdesignof specificationsthe design areetavailable al.,specificationsare availablein2018the conceptualarein)available.the Inconceptual inGDthe conceptual canonical model (see Fig. 4 ceeds to a fi nal design. First, the designer can autonomously form an 3.6. Formulation of design space parametrizedparametrizedwith aparametrizedrelativelywith a relativelyhigherwith numberahigherrelativelynumberof parametershigherof parametersnumberand de-of parametersand phase.de- Theandphase.autonomousde- The phase.autonomousformulationThe autonomousformulationgives a formulationgoodgivesinitiala goodgivesguessinitialaofgoodsui-guessinitialof sui-guess of sui- initial design space around the given CAD model and creates designs in signs cansignsbe generatedbelow)can besignsgeneratedwithcan: thesebe inwithgenerated parameters. K,these parameters.with theLater,these designerafterparameters.Later,someaftertrials,Later,some hastrials,aftertable some definedspacetabletrials,limits.space Withtablelimits.functionsthisspaceWithformulation,limits.this formulation, Withofthethis designer theformulation,the objectdesignercan firstthecanin-designer (anfirst in- can officefirst in- table) and the design this space. Afterward, the designer can select a design and then for- As stated before, the design space for any CAD model is formed by the designerthe designercan detectthecandesignerquixoticdetect quixoticcanparametersdetectparametersquixoticand eliminateparametersand eliminatethemandbythemeliminateadequatelyby themadequatelybuildby upadequatelybuildan initialup anmapbuildinitialofuppromisingmapan initialof promisingregionsmap ofofregionspromisingthe designof theregionsdesignof the design malize an autonomous space around that design. In this way, the de- parameters are only the presence or absence of matter. The concept can be formulatedthe number of the design as “parametersa largeand their bounds. The di- directly directlymodifyingmodifyingthedirectlyCADthemodifyingmodel.CAD Suchmodel.thecapabilityCADSuchmodel.capabilityof theSuchgenerativeofcapabilitythe generativeof spacethe generativeandspacethenandexplorethenspacedesignsexploreand thenindesignsthatexplorespace.in thatdesignsAfterward,space.in thatAfterward,thespace.designerAfterward,the designerthe designer signer can interactively proceed by selecting designs and forming the mensionality of the design space depends on the number of design design systemdesign issystemrecognizeddesignis recognizedsystemas ’designerly’isasrecognized’designerly’method,as ’designerly’method,which allowswhichmethod,de-allowswhichcande- furtherallowscanreformde-furtherthecanreformdesignfurtherthespacereformdesignbasedspacetheondesignbasedthe previousspaceon thebasedpreviousexplorationon theexplorationprevious exploration design spaces until he/ she achieves a fi nal desired design. For example, variety of (possibly surprising) office tables”. To design one office table (CADparameter shapeused )to, definethethe designerCAD model and the limits of the design signers tosignersmodifytothemodifysignersmodelthetoundermodelmodifyconsiderationunderthe modelconsiderationunderand useconsiderationitsandgenerativeuse its generativeand useresults.its generativeThereresults.canThereberesults.somecan Therebeinfeasiblesomecaninfeasiblebedesignssomeindesignsinfeasiblethe autonomouslyin designsthe autonomouslyin the autonomously Fig. 3 gives the illustration of the interactive formulation of the design space are set by defining the upper and lower bounds for each design capabilitiescapabilitiesat any phaseatcapabilitiesanyofphasethe designatofanytheprocessphasedesignof[process 12]the.designAfter[ 12]exploringprocess. After exploring[ 12]the. Afterformalizedtheexploringformalizedspace,the butformalizedspace,this butcan space,thisbe overriddencanbutbethisoverriddenbycanimplementingbe overriddenby implementinggeo-by implementinggeo- geo- space. In which initial space (design space 1) is formed around the can fix a level to each of the constraints and use a topological optimization algorithmparameter. However,, optimizformulationingof a suitablethe design space isa decisive designs designsbased onbasedthedesignsimportanton thebasedimportantfeatures,on thefeatures,later,importantif required,later,features,if required,designlater,spaceifdesignrequired,spacemetricdesignconstraints.metricspace constraints.metric constraints. initial design. A design (marked in green) is selected from this space task as the performance of a technique in term of creating better design can be exploredcan be exploredbasedcanonbebaseditsexplorednominalon itsbasednominalfeatures.on itsfeatures.nominal features. InteractiveInteractiveFormulation:InteractiveFormulation:In theFormulation:interactiveIn the interactiveformulationIn the interactiveformulationof the formulationof the of the and then a new space (design space 2) isformed around the previously alternatives mainly depends on it. Setting up the design space should be design space,designthespace,designerdesignthe designerspace,createsthemultiplecreatesdesignermultiplespacescreatesandspacesmultiplegraduallyandspacesgraduallypro- and pro-gradually pro- selected design. This process continues until the fi nal design is carefully done in order to achieve the maximum performance of the Sf- ceeds toceedsa finaltodesign.a finalceedsFirst,design.to athefinalFirst,designerdesign.the designercanFirst,autonomouslythecandesignerautonomouslyformcan autonomouslyan form an form an achieved. During selection, if the designer selects more than one design, GDT and should have sufficient high potential region. If design space is 3.6. Formulation3.6. Formulationof design3.6. Formulationofspacedesign spaceof design space initial designinitialspacedesignaroundinitialspacedesignthearoundgivenspacetheCADaroundgivenmodelCADtheandgivenmodelcreatesCADanddesignsmodelcreatesinanddesignscreatesin designs in then a new design space is created around the centroid of the selected too narrow then Sf-GDT will result in the creation of similar/ same de- this space.thisAfterward,space. thisAfterward,thespace.designerAfterward,the designercan selectthecandesignera designselect acananddesignselectthenandfor-a designthen for-and then for- designs. The designer can also refine the space after each interaction as signs. On the other hand, a vast design space can result in the waste of As statedAsbefore,ICEDstated thebefore,As21designstatedthe spacedesignbefore,forspacetheanydesignCADfor anymodelspaceCADisformodelformedany CADisbyformedmodelmalizebyis formedanmalizeautonomousby an autonomousmalizespaceanaroundautonomousspacethatarounddesign.spacethataroundIndesign.this way,thatIn thisdesign.the way,de-Inthethisde-way, the de- 3425 he/ she approaches the fi nal design. Once the fi nal design is selected computational effort in exploring undesirable regions of the design the numberthe numberof thetheofdesignnumberthe parametersdesignof theparametersdesignand theirparametersandbounds.their bounds.andThe theirdi- Thebounds.signerdi- canThesignerinteractivelydi- can interactivelysignerproceedcan interactivelybyproceedselectingbyproceedselectingdesignsbyanddesignsselectingforminganddesignstheformingandtheforming the then, if desired, it can be further modifi ed easily due to its parametric space. Typically, a design space is set up by defining the upper and mensionalitymensionalityof themensionalitydesignof the spacedesignofdependsspacethe designdependson thespacenumberondependsthe numberof designon theof numberdesigndesignofspacesdesignuntilspaceshe/design sheuntilachievesspaceshe/ sheuntilachievesa finalhe/desired shea finalachievesdesign.desireda finalFordesign.example,desiredFordesign.example,For example, lower bounds of the design parameters. Where each parameter re- parameterparameterused to useddefineparametertothedefineCADusedthemodeltoCADdefineandmodelthetheCADandlimitsthemodeloflimitstheanddesignofthethelimitsdesignFig.of 3thegivesFig.designthe3 givesillustrationFig.the 3illustrationgivesof thetheinteractiveillustrationof the interactiveformulationof the interactiveformulationof the designformulationof the designof the design presents a dimension in the design space. Defining the upper and lowers space arespaceset byaredefiningsetspaceby definingtheareupperset bytheanddefiningupperlowerandtheboundslowerupperforboundsandeachlowerfordesigneachboundsdesignforspace.eachInspace.designwhichIninitialwhichspace.spaceinitialIn which(designspaceinitialspace(designspace1) spaceis(designformed1) isspacearoundformed1)thearoundis formedthearound the bounds usually done based on the initial design specifi cations and de- parameter.parameter.However,However,parameter.formulationformulationHowever,of a suitableformulationof a suitabledesign spaceofdesigna suitableisaspacedecisivedesignisa decisivespaceinitialisadesign.decisiveinitial Adesign.designinitialA(markeddesigndesign.(markedinAgreen)designinis(markedgreen)selectedisinfromselectedgreen)thisisfromspaceselectedthis spacefrom this space signers’ understanding of the design. task as thetaskperformanceas the performancetaskofasathetechniqueperformanceof a techniquein termof ofaintechniquecreatingterm of creatingbetterin termdesignbetterof creatingdesignandbetterthenanddesigna newthenspacea newand(designthenspaceaspace(designnew 2)spaceisspaceformed(design2) isaroundformedspace 2)thearoundispreviouslyformedthearoundpreviouslythe previously In Sf-GDT, design space formulation can happen in three different alternativesalternativesmainly dependsmainlyalternativesdependson it.mainlySettingon it.dependsupSettingthe designonupit.theSettingspacedesignshouldupspacethe bedesignshouldspaceselectedbe shouldselecteddesign.be Thisdesign.selectedprocessThisdesign.processcontinuesThiscontinuesuntilprocessthecontinuesuntilfinalthedesignuntilfinal isdesignthe finalis design is way; explicit formulation, autonomous formulation, and interactive carefullycarefullydone in doneordercarefullyintoorderachievedoneto achievethein ordermaximumtheto achievemaximumperformancetheperformancemaximumof the Sf-performanceof theachieved.Sf- of achieved.theDuringSf- selection,Duringachieved.selection,if theDuringdesignerifselection,the selectsdesignerifmoretheselectsdesignerthanmoreoneselectsdesign,than onemoredesign,than one design, formulation. GDT andGDTshouldandhaveshouldGDTsufhavefandicientshouldsuhighfficienthavepotentialhighsuffpotentialiregion.cient highIfregion.designpotentialIfspacedesignregion.is spaceIfthendesignis a newspacethendesignisa newspacethendesignisa newcreatedspacedesignisaroundcreatedspacethearoundiscentroidcreatedthearoundofcentroidthe selectedtheofcentroidthe selectedof the selected Explicit Formulation: The explicit formulation of the design space too narrowtoo narrowthen Sf-GDTthentoo narrowwillSf-GDTresultthenwillinSf-GDTresultthe creationinwilltheresultcreationof similar/in theof samesimilar/creationde- sameof similar/designs.de- sameThedesigns.de-designerThedesigns.designercan alsoTherefinecandesigneralsotherefinespacecan thealsoafterspacerefineeachaftertheinteractionspaceeach interactionafteras each interactionas as happens when the design specifi cations are known at the conceptual Fig. 3. Interactive formulation of design space. signs. 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In Sf-GDT,In Sf-GDT,design spacedesignIn Sf-GDT,formulationspacedesignformulationcanspacehappenformulationcan inhappenthreecanindifthreefhappenerentdiffinerentthree different way; explicitway; explicitformulation,way;formulation,explicitautonomousformulation,autonomousformulation,autonomousformulation,and interactiveformulation,and interactiveand interactive formulation.formulation.formulation. ExplicitExplicitFormulation:Formulation:ExplicitThe explicitFormulation:The formulationexplicit Theformulationexplicitof the designformulationof thespacedesignofspacethe design space Fig. 3. InteractiveFig. 3.formulationInteractive formulationof design space.of design space. happenshappenswhen thewhendesignhappensthespecificationsdesignwhen specificationsthe designare knownspecificationsareatknownthe conceptualataretheknownconceptualat the conceptualFig. 3. Interactive formulation of design space. office table weight while meeting the functional requirements - depending on optimization parameters (eg voxel size). By varying the functional levels and the optimization parameters, the designer gets a variety of shapes.

C A large variety of (possibly surpisi ng) office Repeated topological optimization K tables, generated by topological optimization • With selected functions to be varied • Algo parameters (voxel size…)

Modifying initial constraints/functions Middle load Middle platform Edg e loads Edge platforms

Desk (obstacle) Modifying the parameters of the Variation step on topological optimisation algorithm functions � Fixed “ feet”

Problem Definition Dream Lens Genotype param Set functions, � , … = x , … optimize weight

middle load outer load Topological

middle outer loads optimization: object x Center of Mass→ � � � file (x, y, z), Weight, Overhang Percentag�e, =Su1rfa…ce �Area, Area/Volume Ratio model � � Maximum Displacement, Max. Strain, Total Strain, Max. Vonmises, Objective Value (for the simulation) Figure 4. Topological algorithm GD tool coded in C-K - based on Matejka et al. 2018 24”

Generativity analysed within the canonical model: 80mm Variation Creation

– the designer designs a variety of tables (Xi, Fj), on a functional mapconvergent . Hence a topology.

– the object model is different: in MOEA case, design parameters determinedivergent functions; here the model can be punctually inverted: for one functional definition, topological optimization determines the design parameters (matterwww.au todoreskrese arcnoth.com/publi catmatterions/dreamlens in each voxel). Still the model is only punctually invertible and so the topology is a concept. – This GDA tool seems different from MOEA - but formally it leads to quite similar results: a set of artefacts ordered according to functional dimensions, with design parameters for each artefact.

4.1.4 MAP-Elites Quality-Diversity (QD) algorithm MAP-Elites QD algorithm was used in a large variety of problems. One good illustration is the mapping of gaits of a hexapod robot (Cully et al., 2015, Koos et al., 2013). In GDA canonical model (see Fig. 5 below): the designer has a model of an hexapod described by 24 parameters, which create numerous gaits, from purely quadruped gait to classic tripod gait. The gait is a phenotype that can be described in several ways. For instance, one phenotype dimension can be 1=the speed in forward-axis - to be maximized. The concept can be formulated as: a map of gaits that present an optimal 1. This map is unknown and the designer can work on the space in which she will design these gaits. She has to propose a mapping criteria 2. She can map the 1-optimal gaits according to the y-axis speed - and this will probably result in a Pareto front. She can choose to map the gaits according to another phenotype dimension such as the fraction of time that each of the 6 legs touches ground (6-dimensional map). Based on 1 and 2, the MAP-Elites algorithm constructs an elite for each 2 niche. Generativity analysed within the canonical model: – The design process results in a topology of artefacts. Compared to Pareto front GDA, user relies on a phenotype feature that is not functional; compared to space-filling GDA, user defines his/her own phenomenological feature. Hence mapping criteria is a free parameter of the tool. – The map can be seen as a generalization of topological optimization GDA tools: map dimensions are made by a phenomenological feature 2 and there is 1-optimized genotype (in topological space: for each set of functional levels, there is a shape that minimizes the weight).

3426 ICED21 Illuminatio n algorithms (quality diversity algorithm) C A map of the � elites Genetic algorithm (MAP-El ites) • Genotype evolutions K • Fitness crit � , mapping crit � Fast Damage Recovery in Robotics with the Based on Based on mapping T-ResilienceVariations onAlgorithm Keep � -elite mapping dimension n=24 param for each � -niche dimension Sylvain Koos, A ntoine Cully and Jean-Baptiste M ouret ⇤ � � = {feature 1, Genotype param Phenotypical param � , � feature 2…} Damage� recovery =is criticalx for autonomous robots that need to operate, for…a long time,without… assistance. M ost current methods are complex and costly because they require antici- pating each potential damageRuinlorderes toto havecoma contingencypute: plan ready. As an alternative, we introduce the T-resilience al- gorithm,objecta new algorithm that allows robots to quickly and au- x�= 1…� →� , � tonomouslymodeldiscover � compensatory behaviors in unanticipated Goal: find the “fitness potential” for each combination features situations. This algorithm� equips the robot with a self-model and discovers new behaviors by learning to avoid those that (a) Normal state. (b) Two legs ripped out. The elites of the search space 3 perform differently in the self-model and in reality. Our algo-

1 rithm thus does not identify the damaged parts but it implicitly

Figure 5. Map-Elitesalgorithm GD tool, from (Cully0 searches et al.,for efficient 2015behaviors, thatKoosdo not use etthem. al.,We evalu- 2013)

2 ate the T-Resilience algorithm on a hexapod robot that needs to

adapt to leg removal, broken legs and motor failures; we com- Elite = best of the family b

e pare it to stochastic local search, policy gradient and the self-

Family = solution with similar features (niche) F modeling algorithm proposed by Bongard et al. The behavior of the robot isassessed on-board thanks to a RGB-D sensor and

4.2 Characterizing the generativity of GDA tools: the2 design of topologies

a SLAM algorithm. Using only 25 tests on the robot and an

(c) One broken leg. (d) Two unpowered motors.

] overall running time of 20 minutes, T-Resilience consistently leads to substantially better results than the other approaches. Figure 1: Examples of situations in which an autonomous robot We can synthesize the results just obtained above: O

needs to discover a qualitatively new behavior to pur- R

. sue its mission: in each case, classic hexapod gaits can-

s 1. Introduction

1. We confirm that GDA tools don’t generate one artefact but generate a topology ofnot artefactsbe used. The broken leg example (c) is a typical c

[ Autonomous robots are inherently complex machines that have damage that is hard to diagnose by direct sensing (be-

which is the locus of the generative power of the algorithm to cope with a .dynamic and often hostile environment. They cause no actuator or sensor isdamaged). 1 face an even more demanding context when they operate for a v long time without any assistance, whether when exploring re-

2. In this topology design process, GDA tools differ (at least!)6 mote places by(Bellingham theand degreeRajan, 2007) or, mor ofe pr osaicallyfreedom, in inefficiencies they(Goldber gofferand Chen, 2001; Caccavale and Villani, 8 a house without any robotics expert (Prassler and Kosuge, 2008). 2002; Qu et al., 2003; Lin and Chen, 2007). Such controllers usu- 3 ally do not require diagnosing the damage, but this advantage is

0 Asfamously pointed out by Corbato (2007), when designing such

for building the topology. Pareto front GDA tools maximize. correlated functions; space filling complex systems, “ [we should not] wonder if some mishap may tempered by the need to integrate the reaction to all faults in a 2 happen, but rather ask what one will do about it when it occurs” . single controller. Last, a robot can embed a few pre-designed be- techniques maximize phenotypic distances; topological0 Inoptimizationautonomous robotics, this remark pavesmeans that rfunctionalobots must be haviors tospacecope with antici withpated potential failures (Gorner¨ and

3 able to pursue their mission in situations that have not been an- Hirzinger, 2010; Jakimovski and Maehle, 2010; Mostafa et al., 1

: ticipated by their . Legged robots clearly illustrate this 2010; Schleyer and Russell, 2010). For instance, if a hexapod robot

weight optimal artefacts; Map-Elites has the interestingv need (generic)to handle the unexpected: propertyto be as versatile toas possible, be freedetects that inone of termits legs isnot rofeacting as expected, it can drop it i they involve many moving parts, many actuators and many sen- and adapt the position of the other legs accordingly (Jakimovski

X and Maehle, 2010; Mostafa et al., 2010).

phenomenological and genotypical features. r sors (Kajita and Espiau, 2008); but they may be damaged in nu- An alternative line of thought is to let the robot learn on its own a merous different ways. These robots would therefore greatly ben- 3. As predicted, the GDA tools also open diverse directionsefit from forbeing able designto autonomously expansionsfind a new behavior if some Etheachbest behavior GDforAthe currtoolent situation. If the learning pro- legs are ripped off, if a leg is broken or if one motor is inadver- cess is open enough, then the robot should be able to discover generates a specific topology which impacts DP’s or FRtently’s disconnectedgenerativity(Fig. 1). (see table belownew compensatory). Onebehaviors canin situations that have not been and resilience are classic topics in robotics and foreseen by its designers. Numerous learning systems have been engineering. The most classic approaches combine intensive test- experimented in robotics (for reviews, see Connell and Mahade- predict the emergence of complementary GDA algorithmsing with redundancy toof componentswork (Visinskyon et theal., 1994; relevantKo- van (1993); Ar gallXet al.in(2009);=1 Nguyen-T, uong and Peters (2011); ren and Krishna, 2007). These methods undoubtedly proved their Kober and Peters (2012)), with different levels of openness and usefulness in space, aeronautics and numerous complex systems, various a priori constraints. Many of them primarily aim at auto- explore the φ and evolve the resulting M . (Gaierbut they etalso al.,are expensive 2017,to operate andBossensto design. More imporet- al.,matically 2020)tuning controllers. Onefor complex robots (Kohl and Stone, j= 1 m 0 tantly, they require the identification of the faulty subsystems and 2004; Tedrake et al., 2005; Sproewitz et al., 2008; Hemker et al., a procedure to bypass them, whereas both operations are difficult 2009) whereas only a handful of these systems has been explicitly can also predict the emergence of algorithms with doublefor many generativitykinds of faults – for example (onmechanical DPsfailur es.andAn- testedFRs).in situations in which a robot needs to adapt itself to un- other classic approach to fault tolerance is to employ robust con- expected situations (M ahdavi and Bentley, 2003; Berenson et al., 4. Analysing GDA as tools for designing topologies hastr ollers severathat can workl in consequencesspite of damaged sensors or har dwar fore 2005;understandingBongard et al., 2006). Finding the behavior that maximizes performance in the cur- ⇤ Sylvain Koos, Antoine Cully and Jean-Baptiste Mouret are with the ISIR, Univer- site´ Pierre et Marie Curie-Paris 6, CNRS UMR 7222, F-75252, Paris Cedex 05, rent situation is a reinforcement learning problem Sutton and Barto their generativity, their value and their use: France. Contact: [email protected] (1998), but classic reinforcement learning algorithms (e.g. TD- – GDA tools help analyse interdependences between functions, not only correlation -as in Pareto front MOEA- but also independences - as inKoos QD, Cully and algorithmMouret . arXiv | 1 – The capacity to map independences can’t be underestimated in engineering design: this is precisely the capacity required to design resilient systems, capable of being independent of external events. Hence GDA tools might actually be useful for designing resilience. – Mapping phenotypes to genotypical elites, GDA tools also contribute to “invert” usual models that compute functional requirements from design parameters. The maps generated by GDA tools gives a genotype for each phenotype niche. Doing so, GD A tool actually contributes to uncover general laws and models linking phenotypes to their genotypical roots. Hence it contributes to establish design rules. Mac Cormack (McCormack et al., 2005) considered that GDA tools would lead designers to focus on the design of design rules and GDA tools would then generate artefacts based on these new design rules. Fifteen years later, one could rather say that GDA tools could be an essential tool to uncover design rules and hence improves the generativity of users. Table 1. Expansion directions opened by GD-generated topology

Pareto front Space filling Topological optimization MAP-Elites Topology Pareto front topology Identifies design neighbours Cover functional map with Find elite gait for each specific generativity (elites / dominated) (weight) optimal artefacts behavioral niche DP Identifies DP for None May help identify new None generativity improved Pareto front families of DPs (table legs…) FR None Paves the way to new FR, None Explore altearntive behavioral à generativity sensations, emotions niche types resilient design

ICED21 3427 5 CONCLUSION We analysed GDA in the light of design theory, to better characterize their generative power. Since these algorithms are based on a parametric of object, their generativity might sound limited to generate ‘varied artefacts’ from the same model, however we show that GDA design topologies of artefacts - ie large, ordered, structured, actionable sets of artefacts. GDA finally open the field of topological generativity. We show that GDA tools evolve today to enable more degree of freedom in the topology design, from the analysis of interdependences to the analysis of independences, leading possibly to designing resilience or to uncover design rules. This design-theory based analyses of GDA tools helps uncover some of their critical properties, identifies some development trends and even suggests ways for further improvement. Conversely, the analysis invites to deepen design theory: with the help of C-K theory applied to Topos (Hatchuel et al., 2019) it might be possible to give an enriched account of the design of topologies by GDA tools, hence better addressing the issues of resilient design and the design of design rules.

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