We show how the concept of metamorphosis, together with a biologically inspired model of multicellular development, can be used to evolve soft-bodied robots that are adapted to two very different tasks, such as being able to move in an aquatic and in a terrestrial environment. Each evolved solution defines two pairs of morphologies and controllers, together with a process of transforming one pair into the other. Animats develop from a single cell and grow through cellular divisions and deaths until they reach an initial larval form adapted to a first environment. To obtain the adult form adapted to a second environment, the larva undergoes metamorphosis, during which new cells are added or removed and its controller is modified. Importantly, our approach assumes nothing about what morphologies or methods of locomotion are preferred. Instead, it successfully searches the vast space of possible designs and comes up with complex, surprising, lifelike solutions that are reminiscent of amphibian metamorphosis. We analyze obtained solutions and investigate whether the morphological changes during metamorphosis are indeed adaptive. We then compare the effectiveness of three different types of selective pressures used to evolve metamorphic individuals. Finally, we investigate potential advantages of using metamorphosis to automatically produce soft-bodied designs by comparing the performance of metamorphic individuals with their specialized counterparts and designs that are robust to both environments.

Metamorphosis is a process during which an organism that has already finished its embryonic growth undergoes a relatively fast and considerable modification of its body. It is usually accompanied by a change of the environment the organism lives in. While the processes through which these modifications occur are the very same as the ones through which multicellular development occurs, metamorphosis is considered a distinct phenomenon due to the scope of the changes and the fact that it effectively creates a complex life cycle with two or more phenotypes during the animal's lifetime. Familiar and striking examples include amphibians (such as frogs) and insects (such as butterflies). Frogs begin their lives in an aquatic environment and develop first into a fishlike larval stage (a tadpole). Matured tadpoles then undergo metamorphosis involving changes such as the loss (reabsorption) of gills, tail, and lateral-line system and the gradual growth of jaw and limbs. The modifications allow them to switch to a terrestrial habitat. A butterfly, on the other hand, develops after hatching into a caterpillar, a soft-bodied, elongated larva that typically feeds on leaves, to later transform into a winged imago1 that typically feeds on nectar.

Despite the additional developmental complexity created by metamorphosis, it is extremely common in the animal kingdom: Estimates suggest that just the metamorphosing insects amount to up to 65% of all animal species [11], while 80–90% of insect species themselves undergo metamorphosis [14]. The enormous success of this reproductive strategy suggests that there is a strong evolutionary advantage to having a complex life cycle. And while in the case of amphibians metamorphosis can be explained by their aquatic evolutionary ancestry, in the case of insects, both genetic and paleontological data suggest that early insects employed direct development (where juvenile stages resemble the adult form). It has thus been suggested that it was the very emergence of metamorphosis that sparked the extremely successful insect radiation [38, 28].

Why, however, is undergoing metamorphosis during one's life cycle beneficial? The main explanation of its evolutionary advantage is that it allows larvae and adults to occupy different ecological niches. By relying on different food resources or occupying different habitats, larvae avoid the cost of having to compete with adults of their own species [38]. And while there certainly must be a cost of having two distinct phenotypes encoded in a single genome, the genetic and chemical machinery required to assemble them is the same: What changes between the growth phases is the regulation that controls which genes are active and what the rate of their transcription is [41]. Thus, although having two phenotypes requires more complex control of development than having one, each metamorphosis stage does not emerge independently, but results from an evolutionary process specializing preexisting developmental stages. While the exact molecular mechanisms of metamorphosis are an active area of research, the emerging science of evo-devo has already provided us with a good glimpse of how plastic animal forms are when evolution acts on parts of the genomes that control development (see, e.g., [4]).

In this work, we aim to investigate whether some of the benefits that metamorphosis provides to evolving animal forms can also contribute to the domain of synthetic evolution of robotic morphologies. The very problem of evolving shape or, more typically, coevolving body and brain of an animat has been a staple of the artificial life field ever since Karl Sims' groundbreaking work in 1994 [34]. The classic approach is to evolve designs and evaluate each of them by measuring a robot's (animat's) performance on a task it is evolved for, such as ability to move in a given environment. Furthermore, a phenotype is typically indirectly encoded in a genotype, through some abstraction of development. Ultimately, however, a single genome encodes a single phenotype. What we thus decided to investigate in this article is to what extent, by applying appropriate evolutionary pressures, we can induce specialization of developmental stages. Therefore, we investigate here a scenario in which a single genome no longer encodes one phenotype but two, together with a method of transforming one into another.

It is easy to see how such robotic designs could prove themselves extremely useful: Whenever animats are required to operate in diverse environments during their lifetimes, they could switch their form when transferring from one environment to another. For example, a rescue robot launched for a mission from sea could swim to shore, metamorphose into its terrestrial form, and continue its mission on its newly grown legs. On the other hand, from the point of view of automatic design, the evolution of metamorphic solutions means obtaining two different designs with a single evolutionary run. Such an approach would likely be of limited use if the designs were to be entirely unrelated, as it would simply make the evolutionary search harder. However, in practical scenarios this would rarely be the case, as any two robot designs will share many features. And as the science of evo-devo reveals [4], evolution creates its “endless forms most beautiful” by actually reusing a limited number of body plans and only tweaking the genetic machinery that is largely shared by both vertebrates and invertebrates. Hence, if only we can make artificial evolution replicate this phenomenon, evolving multiple phenotypes in a single evolutionary run through the use of metamorphosis could be easier than attempting to evolve them separately. While this introduces heavy constraints on what kinds of morphologies can be obtained through evolutionary search, if the solutions are to share some of their structure, the constraints can work in favor of evolutionary search.

The idea of taking inspiration in metamorphosis to improve evolutionary search is not new. Bongard [1] has shown how a predefined morphological change during a legged robot's lifetime (progressive extension of its legs) occurring during early evolutionary stages facilitated evolution of higher-quality gaits. An evolutionary search of gaits for a long-legged robot is likely to be susceptible to the bootstrapping problem [22], where initial generations will produce robots that fall. Hence, starting with a short-legged form that is not prone to falling effectively guides evolutionary search towards the right regions of the search space where synchronized leg actuation emerges. As the author notes, the progressive change in morphology scaffolds behavior acquisition. While scaffolding techniques are well known as being helpful in guiding both biological and artificial learning processes [42, 1], our use of metamorphosis in this work has a very different motivation. Instead of using a predefined metamorphosis process as a scaffolding for the search process, we are interested in whether metamorphosis can evolve if appropriate evolutionary pressures are applied and whether it allows obtaining two, self-transforming phenotypes in a single evolutionary run.

In another metamorphosis-related work, Corucci et al. [7] took metamorphosis as an inspiration for a method of designing underwater robots. The authors explored a space of parametrized designs (e.g., having different length or stiffness of appendages) with novelty search [19] to find robots that display possibly different behaviors while being close to each other in the morphological space. The reasoning was that pairs of such designs make good candidates for being implemented as a robot that changes some of its morphological properties on the fly, in order to induce new types of behavior. While some of their motivation is shared with this work, namely, the goal of producing robots that change morphology depending on the problem, the approach of Corucci et al. is very different. It assumes a fixed, parametrized morphology and does not deal with the metamorphosis itself: It is implicit, understood as having two related robotic morphologies that produce very different motion patterns.

An approach much closer to ours can be found in the work of Tufte [39], who used metamorphosis to evolve simple digital circuits implemented in a cellular automaton (CA). The 2D CA would grow a circuit starting from a single cell, and the growth rules were evolved so that a circuit would maximize the number of flip-flops in the 1 state at the larval stage and make them perform as a counter in the adult stage. While Tufte's work seems to employ a similar setup, to our knowledge our work is the first attempt at applying the idea of metamorphosis to evolve actual self-transforming morphologies of robots.

As metamorphosis is an aspect of multicellular development, in our approach we employ artificial embryogenesis to construct animat forms from hundreds of cells through the processes of cellular division and death. In the next two sections we describe our developmental model. Then we show how the concept of metamorphosis can enhance artificial development by allowing an evolutionary algorithm to automatically produce solutions (here, soft robots) that can take two potentially very different forms, each adapted to its target environment. Importantly, one form can transform into another, offering the exciting potential of automatically designing robots that could efficiently operate in radically different environments and change their form on demand.

The field of artificial embryogeny, to which this work belongs, attempts to capture the seemingly endless capability of nature to generate forms by attempting to replicate key properties of development in silico. This typically involves bio-inspired construction processes in which a structure (such as a robot's body) is progressively built from smaller elements. Depending on the chosen level of abstraction, this may involve elements such as rods [18], primitives and joints [34, 24], or artificial cells, as in the case of the system employed by us and related ones [9, 10, 29, 2]. Self-assembly from higher-level components such as blocks and joints has been repeatedly demonstrated to be an effective way to generate interesting robotic designs. The reliance on high-level components makes it, however, only loosely reminiscent of biological development and its evolution. Hence, in our line of work, we aim to explore the potential and scalability of a much more biologically oriented fine-grained artificial development, where morphologies are assembled from large numbers (hundreds, thousands) of cells, each making independent decisions about their fate and interacting through simulated physics. So far we have demonstrated how this approach allows us to evolve a rich variety of complex soft-bodied animats and were surprised to observe how higher-level structures, such as simple appendages, emerge during evolution and function as legs or fins [15].

In the following sections, we describe the artificial embryogeny system that we employ. Its capability to evolve metamorphic individuals is a straightforward extension of the work we have introduced in [15]. Generally, the design of the system came from the desire shared among many artificial life approaches (and multi-agent systems in particular) of allowing complexity to emerge from a set of simple rules. However, finding simple sets of rules that allow complex phenomena to emerge is not an easy task and is a very counterintuitive one. After all, it is the very definition of emergence that behavior of lower-level entities leads to a higher-level behavior that would be difficult to predict. Hence, the evolutionary approach has been long thought to be a promising method of discovering simple rules that can lead to self-organization. Having this in mind, we designed our developmental system to rely on possibly simple laws that govern the behavior of cells during development, while providing a certain minimum level of richness of their interactions in the form of a simulated physical environment. While using artificial development to simulate growth of multicellular structures necessitates a level of unavoidable complexity, whenever we could, we attempted to keep our approach simple by avoiding features that are not needed to obtain desired results.

As is common among developmental systems, the growth of a virtual embryo in our system begins with a single cell and proceeds through subsequent cellular divisions and deaths (apoptosis). The fate of each cell is determined by the same control mechanism, a simple abstraction of the gene regulatory network (GRN) in the form of a feed-forward neural network. All cells are controlled by the same network, yet will act differently, as the external signals that are fed to the inputs of the network depend on their (potentially changing) position in the growing embryo, as well as signals output by other cells. Ultimately, the purpose of artificial development is to produce nontrivial morphologies that can then be used as templates for robots and evaluated for their performance in a given task.

In particular, in our approach we focus on generating morphologies for robots that are soft-bodied. Soft robotics is a very recent and rapidly developing branch of robotics focused on creating elastic artefacts that can reshape themselves in order to generate gaits or navigate tight spaces (e.g., in order to crawl through a small opening). Ideas for useful morphologies and methods of actuation are being continually explored, both in simulations (see, e.g., [5, 13, 23, 25, 33, 40]) and in physical implementations (e.g., [3, 12, 32, 36, 37]). Given that this area of engineering is still underexplored, we believe that an evolutionary approach, free from preconceived notions of a human designer, can provide us with original and inspiring designs. As the fine-grained developmental approach we employ does not assume the use of predefined building blocks and simply builds arbitrary morphologies from a mass of cells, it is, in principle, well aligned with the goal of automatic design of inventive soft robots.

In contrast with the version introduced in the original article [15], we have configured the system in a way that would prioritize simplicity over biological plausibility. This meant, among other things, allowing for only non-recurrent gene regulatory networks, sigmoidal-only gene activation functions, and a simplified mechanism of actuation. We would like to note, however, that we were able to obtain similar results with the original version of the model (recurrent GRNs); we simplified it out of our desire to show a minimal system. As this also meant a reduced search space, we were able to obtain high-quality solutions more consistently. Finally, a minimal design of the developmental system in use suggests that obtained results can be expected to be replicable with other developmental systems. Indeed, we think that the results in this article are indicative of metamorphosis being fairly easy to evolve in developmental systems in general.

The evolving networks that control cellular behavior during development are based on the MultiNEAT library [6], an implementation of the NEAT evolutionary algorithm [35]. We used a signed sigmoid as the default activation function, and kept most of the settings at their default values, with the link to the library parameter file provided in [6].

2.1 Physics of Development

We assume that development takes place in a continuous 2D space, where cells are represented as disks and undergo elastic collisions simulated with springs that connect them. A cell's physical state is defined by its position, its velocity, and an orientation vector that determines the direction of division. For the sake of simplicity, in the discussed experiments, all cells have uniform size and mass. Springs connect only the nearest neighbors and are determined dynamically, as the embryo grows, with the resting length set to the sum of two neighboring cells' radii. More precisely, we use Delaunay triangulation to determine the connectivity between cells and then remove links longer than 150% of a cell's diameter. As this may result in a disjoint structure, a long spring is removed only if some other path between the two nodes exists, thus preventing fragmentation of a growing embryo. To save some computational time, the neighborhood relation was recalculated every 10 steps of the physics simulation.

2.2 Morphogens

Morphogen gradients (both maternal and self-produced) have long been known to play a fundamental role in the developmental process and, in particular, in establishing the basic body plan [4]. Furthermore, given that the networks governing cell behavior in the presented experiments are stateless, an environmental signal is necessary to allow cells to differentiate their behavior. As a simple mechanism that substitutes for the maternal gradients present during development, we have provided the X and Y coordinates as inputs to every cell. Additionally, as a simple mechanism that substitutes for the morphogens produced by cells, the control network has two morphogen outputs and two associated morphogen inputs. For any given cell, the activation of the latter is set to be an average of the corresponding morphogen outputs of its neighbors (see summary in Table 1).

Table 1. 

Summary of inputs and outputs of the control network. Each cell has a copy of the same network, but receives different signals to its inputs.

TypeNameDescription
Input Time Linearly scales from 0 to 1 until the final developmental step is reached. 
X position Horizontal coordinate of physical position of the cell during development. 
Y position Vertical coordinate of physical position of the cell during development. 
Neighbors' morphogen 1 Average value of morphogen output 1 of neighboring cells. 
Neighbors' morphogen 2 Average value of morphogen output 2 of neighboring cells. 
Bias Fixed value 1. 
 
Output Inhibit division Prevents divisions of the cell if above 0. 
Relative division angle Determines the angle at which the cell will divide and its future orientation vector. Output value is multiplied by π and interpreted as a relative angle from −π to π. The final division angle is relative to the mean angle of neighboring cells' orientation. 
Die If above 0, the cell is removed from the embryo. 
Morphogen 1 The state of this output can be indirectly accessed by the cell's neighbors. 
Morphogen 2 Same as above. 
Oscillation period The value of this output at the end of development is used to determine the frequency of contractions of springs attached to the cell. 
Oscillation phase shift The value of this output at the end of development is scaled to the range (−π, π) and determines the phase shift of contractions of springs attached to this cell. 
TypeNameDescription
Input Time Linearly scales from 0 to 1 until the final developmental step is reached. 
X position Horizontal coordinate of physical position of the cell during development. 
Y position Vertical coordinate of physical position of the cell during development. 
Neighbors' morphogen 1 Average value of morphogen output 1 of neighboring cells. 
Neighbors' morphogen 2 Average value of morphogen output 2 of neighboring cells. 
Bias Fixed value 1. 
 
Output Inhibit division Prevents divisions of the cell if above 0. 
Relative division angle Determines the angle at which the cell will divide and its future orientation vector. Output value is multiplied by π and interpreted as a relative angle from −π to π. The final division angle is relative to the mean angle of neighboring cells' orientation. 
Die If above 0, the cell is removed from the embryo. 
Morphogen 1 The state of this output can be indirectly accessed by the cell's neighbors. 
Morphogen 2 Same as above. 
Oscillation period The value of this output at the end of development is used to determine the frequency of contractions of springs attached to the cell. 
Oscillation phase shift The value of this output at the end of development is scaled to the range (−π, π) and determines the phase shift of contractions of springs attached to this cell. 

2.3 Cell Division and Death

All cells are bound to divide with each subsequent update of the control network unless the output interpreted as an inhibitor of division has activation value above 0. Furthermore, the division is allowed to occur if and only if the space in the direction of the division is not occupied already by other cells. Although our elastic physics of development was designed to handle cells pushing others away as the embryo grows, we have found that constraining divisions produces fitter and more interesting animats (see [15] for comparison and discussion).

The newly created cell is placed next to the original cell in the direction determined by the division angle output. The angle is determined at the moment of division and is relative to the mean angle of the neighbors' orientation vectors (see Table 1). Unless the value of the division angle output in cells is different from zero, all will simply divide in the same direction.

Apoptosis (cell death) occurs whenever the state of associated network output is found to be above zero and leads to the cell being removed from the embryo.

We allowed for a maximum of 256 cells in the embryo. The limit was hard, that is, cells would stop dividing after the limit was reached. Only some of the individuals would, however, reach the maximum. Furthermore, in order to limit the occurrence of wasteful solutions in which cells continuously divide and die, we penalized individuals that had created more than 1024 cells during their development by multiplying their fitness value by 0.1.

2.4 Soft-Bodied Locomotion

Although the animat's representation during development is similar to its representation during the locomotion stage, the stages are separated. Both the growing embryo and the fully grown animat are represented as a spring-mass system and simulated using the same physics engine, but their physical properties and the rules that govern their reshaping are different: Development occurs without gravity, in an environment with high spring damping coefficient (simulating a viscous fluid), and involves continuous rearrangements of the body: Cells are created and removed and can change their neighbors. During the locomotion stage, the morphology that emerged through development is assumed to be final and is evaluated for its performance in the target environment (which may or may not include gravity). Only elastic changes are allowed (see example in Figure 1).

Figure 1. 

An example of a developmental process leading to the creation of a (non-metamorphic) animat. The last two frames illustrate the transition from the representation used during the developmental stage to the locomotion stage. Video for this figure can be accessed at https://goo.gl/PJbZTr.

Figure 1. 

An example of a developmental process leading to the creation of a (non-metamorphic) animat. The last two frames illustrate the transition from the representation used during the developmental stage to the locomotion stage. Video for this figure can be accessed at https://goo.gl/PJbZTr.

The locomoting animat is represented as a spring-mass system with point masses located at the final positions of the centers of the embryo's cells (Figure 1e), and springs forming a triangular mesh. The springs that connect outer cells in the embryo define the final body shape, while the internal structure of the body is fully triangulated with Delaunay triangulation. This works as a simple approximation of a body made from an elastic material. As this approach can and sometimes will produce a morphology with a protrusion that is connected by a single spring to the main body, such protrusions were removed during conversion of an embryo into the soft-body representation. This prevented the emergence of degenerate morphologies with infinitely thin body regions. Also, animats consisting of less than eight cells were considered not interesting and assigned a zero fitness.

The assigned resting length of springs is based on the distances between cell centers at the end of the developmental stage. Additionally, each triangular region has an equilibrium pressure S0 (represented by its surface area at the end of development), providing the animat with a hydrostatic skeleton and preventing excessive compression or stretching of body regions, and we have
formula
where Fp is the pressure force acting outward along the normal of the edge that is considered, L is the length of this edge, S and S0 represent the current and the original surface area of the triangle, and cp is the global pressure coefficient.
Springs are governed by Hooke's law with damping. All springs share the same Hooke's constant k. The force acting on a node is equal to
formula
where x is the difference between a spring's length at time t and its resting length, while c is the damping coefficient.
To avoid self-penetration of animat bodies, masses representing cells undergo elastic collisions with springs. Actuation is achieved by modifying the resting lengths of the springs attached to a given cell. This results in a body region contracting or expanding. The resting lengths are changed according to a sinusoidal oscillation pattern associated with every cell. The period of oscillation and the phase shift of every cell are determined by two corresponding outputs at the end of the developmental stage (see Table 1). During locomotion, the length of each spring is modified according to
formula
where t is the simulation time; A is the amplitude of the changes undergone by each of the two cells (A = 0.15); T1, T2 are the evolved periods of oscillation of the two cells at the endpoints of the spring; and ϕ1, ϕ2 are their evolved phase shifts (scaled to (−π, π)).

The ground-based environment was constructed by placing animats on top of a flat surface and introducing gravity and friction between their nodes and the surface. To prevent sudden changes in resting length for cells with nonzero phase shift at the start of the simulation, the amplitude of contractions was progressively increased during the first 200 steps of the locomotion stage. Furthermore, to prevent evolution from exploiting any initial motion that would come from relaxation of the animat body at the beginning of evaluation when gravity is present, before actuation was enabled, we waited for each animat to stabilize. This was implemented by making sure that the speeds of the nodes were sustained below a threshold over a period of 800 time steps.

For the aquatic environment, gravity was disabled and fluid drag was introduced. We used the fluid drag model used to simulate undulatory fish swimming by Sfakiotakis and Tsakiris [30], which assumes that the fluid is stationary and that the force acting on a single edge on the outline of the body is the sum of the tangential and normal drag components for the motion of this edge against the fluid:
formula
formula
where vT and vN are the normal and tangential motion components, and dT and dN are the drag coefficients for the tangential and normal drag on a unit-length edge (dN = 100dT). Since the animat structure is not rigid and the lengths of the springs change dynamically, the direction of motion of a given edge is understood as the direction of motion of its center. The calculated force is then divided by two and applied to the nodes associated with the edge. While this approach does not simulate turbulent flow, it provides enough environmental complexity to allow for fin-driven, undulatory, and jellyfish-like swimming patterns, observed in our earlier work [17].

2.5 Genetic Encoding and Genetic Algorithm

The neural network model and genetic representation are based on the MultiNEAT library [6], the implementation of the NEAT evolutionary algorithm [35]. In the NEAT method, networks are represented in the genomes as a list of nodes and their types (input, output, normal) and a list of connections. The NEAT algorithm keeps track of the innovation history and uses it to perform crossover between genomes. It also uses a fitness-sharing approach with the goal of preserving diversity and protecting new solutions before they have to compete with the rest of the population. We used a population size of 300 and runs of 2000 generations. The initial population was created as a fully connected feed-forward network with a hidden layer and random weights. During evolution, mutations that created cycles in the network were rejected.

2.6 Metamorphosis

The evolution of metamorphic individuals is based on a simple modification of the underlying developmental system. Rather than evaluate just once the morphology that has emerged during development in a virtual environment, we evaluate it twice, at different stages of its development (Figure 2). More precisely, we allow each genome to control embryonic development for 600 time steps. Then, the performance of the emerged (larval) morphology is evaluated on the first task (e.g., for its ability to swim). Next, the development is allowed to continue for another 600 time steps, and the resulting (adult) morphology is evaluated one more time, on the second task (e.g., for its capability to run in a terrestrial environment). Importantly, beyond defining the models of multicellular growth and locomotion, there are no prior assumptions on what kinds of morphologies are desired, their size, or their mode of locomotion. There is also no explicit assumption that morphological change needs to occur if development is continued beyond the larval stage.

Figure 2. 

A conceptual overview of a single genome's evaluation used to evolve metamorphic individuals.

Figure 2. 

A conceptual overview of a single genome's evaluation used to evolve metamorphic individuals.

The discontinuity between developmental and locomotion stages that is inherent in our approach means that after the performance of a larval morphology has been evaluated on its target task, the system needs to resume development. We do that simply by continuing development from the point before the larval morphology was converted into its representation as a soft-bodied animat.

To investigate the evolvability of metamorphic robots we considered a scenario in which soft-bodied robots had to be adapted to aquatic locomotion at one stage of life and terrestrial at another. This allowed for two different scenarios. The first one assumes that a robot first grows from a single cell into its aquatic larva. Then, after being evaluated for its performance in an aquatic environment, it is allowed to undergo metamorphosis that creates its terrestrial, adult form. The second scenario assumes reversal of the stages, that is, a terrestrial larva and an aquatic adult. The occurrence of the first scenario in the natural world is well known, as it is a typical developmental strategy of a whole class of vertebrates, namely the amphibians. Its occurrence is also well understood in evolutionary terms as being the result of amphibians' evolving from their aquatic ancestors. Natural examples of the opposite scenario, terrestrial larvae metamorphosing into aquatic adults, seem to be less common, but can be found, for example, among newts (the eastern newt has an aquatic larva, a terrestrial juvenile, and an aquatic adult form) and among some beetles (family Dryopidae [31]). What is, however, important from the point of view of artificial metamorphosis is that performing both of these types of experiments allows us to gain insight into what kind of morphological constraints are brought by the particular choice of developmental history.

As evolving robots optimize their performance for two different types of environments, the two scenarios used in the experiments constitute a multi-objective optimization problem. A considerable number of different approaches have been developed to simultaneously optimize for multiple objectives (see, e.g., [8]), typically employing the concept of Pareto dominance. To avoid having to modify the single-objective-only genetic algorithm in the MultiNEAT library that we were using, we first considered the simplest possible approach, namely scalarization, with a good result. Seeing a potential for improvement, we extended it and compared it with two other simple modifications of the search algorithm. We describe these three approaches below.

3.1 Simple Fitness Function

In the most basic approach, the fitness value of a single solution was scalarized using the geometric mean of the distances achieved in the developmental stages:
formula
where dl, da were the distances achieved by the larval and adult forms in their respective environments. The use of geometric rather than arithmetic mean prevents the emergence of individuals that overspecialize for one environment only and produce very little movement in the other.

3.2 Progressive Fitness Function

The above (Equation 6) simple method of scalarization assumes that the performance of the larval stage and that of the adult stage are of equal importance. This also means that viable solutions need to achieve a nonzero distance for each of their developmental stages. Generating such individuals randomly or by hand may not be difficult for tasks such as movement in aquatic and terrestrial environments; it may, however, be very difficult in more complex scenarios. Then, the lack of viable individuals in the initial generation will make it impossible for the evolutionary algorithm to perform selection, a situation known in the field of evolutionary search as the bootstrap problem (see, e.g., [22]). An example of such a more challenging task would be the necessity to produce a metamorphic individual adapted to moving down a slope in its larval form and up the slope as an adult. Finding random genomes that produce an adult form that moves up the slope even slightly is going to be very difficult. Even if successful, the types of solutions that can be found randomly are likely to be the types that lead evolutionary search to a low-quality local minimum (e.g., small, degenerate individuals). As a simple way of reducing the bootstrap problem, we took inspiration from the evolutionary emergence of metamorphosis in amphibians. Individuals were first evolved to move in one environment only. Next, the second developmental stage was evolved “on top” of an already found solution, by extending the length of the developmental process and evaluating the adult form in a new environment (but allowing the larval form to evolve as well). This creates a scenario in which, at least initially, “ontogeny recapitulates phylogeny.”2

More precisely, for the first third of an evolutionary run we evolved individuals to develop only for 600 time steps (i.e., the larval age) and move in the first environment. Then, we extended the development time to 1200 time steps and began evaluating the performance of individuals at both of their developmental stages. To avoid a potentially catastrophic reshuffling of individuals as soon as the performance of an adult form becomes part of the fitness function, we have made the fitness of the adult stage progressively increase in relevance until two-thirds of the evolutionary run length has elapsed. Finally, over the remaining part of the evolutionary run, individuals would be evaluated as if each phase had equal weight (using Equation 6). This approach allowed for a graceful and progressive improvement of the adult-stage performance while introducing changes to the morphology and controller of the larval stage. Formally, the scalarized, progressive fitness value fp was dependent on the current generation g and the total length L of the evolutionary run as follows:
formula

3.3 Novelty Search

The last approach we have evaluated as a method of evolving metamorphic agents was the novelty search algorithm [19]. In brief, the novelty search algorithm is based on the radical idea that abandoning an objective, goal-focused fitness function altogether can lead to discovering solutions of higher fitness. To do so, novelty search replaces the concept of objective fitness function with the notion of novelty, a scalar quantity corresponding to how much a given phenotype differs from phenotypes in the current population as well as from phenotypes that have been found to be novel in previous generations. This modification causes the evolutionary search to pursue phenotypes that are different from the already discovered ones instead of phenotypes that have higher fitness. While methods that increase genetic diversity have long been demonstrated to be useful in evolutionary algorithms (see, e.g., [21, 27]), novelty search differs from them in focusing solely on increasing phenotypic, not genotypic, diversity. Novelty search had been suggested to improve evolvability in problems that are deceptive in nature, that is, where greedily focusing the search on improving the fitness will likely lead the population into local minima in the fitness landscape that are difficult to escape from. It is suggested that the pressure to produce novel phenotypes will instead lead to the discovery of more and more complex solutions, among them the evolutionary stepping stones that open access to new regions of higher fitness in the solution space [19]. More precisely, Lehman and Stanley [19] define the novelty of a phenotype x as proportional to how sparsely the phenotype space surrounding x has been explored so far:
formula
where μi is the i th-nearest individual to x out of k according to the distance metric d.

The introduction of a novelty search algorithm into our system was limited to replacing the fitness function computation with calculation of the novelty value, and introducing of a novelty archive that stores past novel individuals and an algorithm for dynamic updating of the novelty threshold value (the novelty value at which an individual is added to the archive). We chose to increase the threshold by 10% if more than eight individuals were added to the archive one after another, and to decrease it by 10% if no individuals were added within 50 generations.

We have opted to use one of the simplest possible approaches to measuring similarity between phenotypes in our experiments: We characterized each individual's phenotype as a vector of size 4, consisting of the final X and Y coordinates of the center of mass of larval and adult forms at the end of their locomotion stages. We then used the Euclidean distance as the metric d, and k = 15.

All of the approaches discussed above were found to be successful in evolving individuals that undergo metamorphosis. More specifically, all of the 20 evolutionary runs performed for each approach would result in individuals with differing adult and larval forms that can move in their respective environments. The difference between the three evolutionary approaches was found to be mainly quantitative, with visually similar results, though with a different average quality of best individuals obtained in multiple evolutionary runs. We will thus first discuss features of evolved individuals and their metamorphosis, and will investigate quantitative differences between the several approaches to generating them in a following section.

The fact that morphological change evolved between two developmental stages is not surprising in itself. Unless individuals were specifically evolved to inhibit growth before the end of the first developmental stage (which can be done by introducing a penalty factor into the fitness function, as in [15]), some growth is almost certain to occur if development is extended and the hard limit of cells has not been reached. Thus, the type of metamorphosis that is of interest to us is not just any change occurring between the two developmental stages, but a change that can be considered adaptive.

4.1 From Water to Land: Aquatic Larvae and Terrestrial Adults

Figure 3 presents morphologies and motion patterns of two examples of metamorphic soft robots that are representative of high-fitness solutions. In both cases, a striking difference between the larval and the adult stage can be observed, with clear adaptations to their respective environments (likely best appreciated with the supplementary video materials; links are provided in figure captions). The aquatic larvae of both individuals have an elongated form. In the case of the individual in Figure 3a, the larva propels its body in a fishlike motion by bending it left and (mostly) right. The actuation pattern has the form of waves of contraction and expansion that travel through the body in the direction perpendicular to the direction of movement. The larva of the individual shown in Figure 3b has a snakelike, streamlined shape and displays undulatory locomotion resembling that of, for example, a nematode worm.

Figure 3. 

Motion snapshots of two solutions ((a),(b)) evolved to move first in the aquatic environment and then in the terrestrial environment (left column: larva; right column: adult). Arrows indicate the direction of movement; snapshots were selected to represent one motion cycle. Larval and adult stages are not to scale (larva is smaller). The individual in (a) was obtained with the default fitness function; (b) comes from the novelty search experiment. Colors represent whether the region of the body is currently expanded (red), contracted (blue), or at its resting size (white). Videos for (a) can be accessed at https://goo.gl/M9Ytym and https://goo.gl/M6K9gh; videos for (b) at https://goo.gl/GAvNtb and https://goo.gl/UNw7lf.

Figure 3. 

Motion snapshots of two solutions ((a),(b)) evolved to move first in the aquatic environment and then in the terrestrial environment (left column: larva; right column: adult). Arrows indicate the direction of movement; snapshots were selected to represent one motion cycle. Larval and adult stages are not to scale (larva is smaller). The individual in (a) was obtained with the default fitness function; (b) comes from the novelty search experiment. Colors represent whether the region of the body is currently expanded (red), contracted (blue), or at its resting size (white). Videos for (a) can be accessed at https://goo.gl/M9Ytym and https://goo.gl/M6K9gh; videos for (b) at https://goo.gl/GAvNtb and https://goo.gl/UNw7lf.

Perhaps the most interesting aspect of the evolved adult forms is that in both cases metamorphosis proceeds by restructuring the body in a way that involves growing two support appendages that are essential to the terrestrial gait and thus serve as primitive legs. The gaits rely on elasticity of the body and capability of the appendages to absorb and release elastic energy—once individuals get to speed, they cycle between their front and back appendages, expanding them on contact with the ground. The directions of motion of larval and adult stages (indicated by arrows in Figures 3 and 5) seem not to be correlated. Since, however, virtual animats have no sensors and therefore no concept of front or back, there is no direct reason for evolution to maintain the direction of movement unchanged between the developmental stages.

Importantly, the solution type where the larval stage is streamlined and the adult form grows appendages to support itself during terrestrial movement was commonly observed in the results of many other evolutionary runs, of which eight additional examples are presented in Figure 5. We find it remarkable that an attempt to evolve metamorphosis in silico produces solutions that immediately bring amphibians to mind, with streamlined, tadpole-like larvae and adults that often shed their tail and grow appendages to move in a terrestrial environment. Furthermore, we would like to note that the propensity to assemble cells into higher-level structures that can function as primitive legs or tails is nowhere explicitly encoded in the system. Their emergence is a testament to the creative potential of artificial evolution.

To gain qualitative insight into how the two individuals depicted in Figure 3 grow from a single cell and later undergo metamorphosis, we visualized developmental processes by coloring cells created during larval development in blue and cells created during metamorphosis in red (Figure 4). In these two cases, as well as in other inspected solutions (shown in Figure 5), metamorphosis proceeds by adding cells to the larval stage, resulting in a structure that is considerably larger and has the larval stage (or a part of it) embedded inside it. Apoptosis occurring during metamorphosis was not, however, uncommon and can be observed in the individuals seen in Figure 5 (e.g., the third from the right) or later in Figure 6b.

Figure 4. 

Developmental processes of the two individuals shown in Figure 3. The dashed line separates the larval stage from the beginning of metamorphosis; labels indicate the development time. Upper rows show the multicellular representation that is used during development; bottom rows show a preview of the soft-bodied representation. Note that protrusions with a thickness of one cell are removed, and the soft-bodied representation is used only for the evaluation of morphology, during locomotion stages. Blue color: cells that were created during the larval stage; red: cells created during metamorphosis. Videos can be accessed at: https://goo.gl/P7m287 and https://goo.gl/6OOosd.

Figure 4. 

Developmental processes of the two individuals shown in Figure 3. The dashed line separates the larval stage from the beginning of metamorphosis; labels indicate the development time. Upper rows show the multicellular representation that is used during development; bottom rows show a preview of the soft-bodied representation. Note that protrusions with a thickness of one cell are removed, and the soft-bodied representation is used only for the evaluation of morphology, during locomotion stages. Blue color: cells that were created during the larval stage; red: cells created during metamorphosis. Videos can be accessed at: https://goo.gl/P7m287 and https://goo.gl/6OOosd.

Figure 5. 

Animat zoo: additional examples of morphologies of metamorphic individuals with aquatic larval stage (upper row) and corresponding terrestrial adult stage (below). Examples come from all of the three different types of evolutionary search employed. Body parts grown during metamorphosis are colored red. Arrows indicate direction of movement.

Figure 5. 

Animat zoo: additional examples of morphologies of metamorphic individuals with aquatic larval stage (upper row) and corresponding terrestrial adult stage (below). Examples come from all of the three different types of evolutionary search employed. Body parts grown during metamorphosis are colored red. Arrows indicate direction of movement.

Figure 6. 

Motion snapshots of two solutions ((a),(b)) evolved to move first in the terrestrial environment and then in the aquatic environment (left column: larva; right column: adult). Arrows indicate direction of movement; snapshots were selected to represent one motion cycle. Larval and adult stages are not to scale (larva is smaller). The individual from (a) was obtained with the novelty search function, and (b) with the progressive fitness function. Colors represent whether the region of the body is currently expanded (red), contracted (blue), or at its resting size (white). Videos for (a) can be accessed at https://goo.gl/elTiW3 and https://goo.gl/8WROSq; videos for (b) at https://goo.gl/yE5d6U and https://goo.gl/z8CMzG.

Figure 6. 

Motion snapshots of two solutions ((a),(b)) evolved to move first in the terrestrial environment and then in the aquatic environment (left column: larva; right column: adult). Arrows indicate direction of movement; snapshots were selected to represent one motion cycle. Larval and adult stages are not to scale (larva is smaller). The individual from (a) was obtained with the novelty search function, and (b) with the progressive fitness function. Colors represent whether the region of the body is currently expanded (red), contracted (blue), or at its resting size (white). Videos for (a) can be accessed at https://goo.gl/elTiW3 and https://goo.gl/8WROSq; videos for (b) at https://goo.gl/yE5d6U and https://goo.gl/z8CMzG.

That the adult stage was larger than the larval one was universally observed, and we think it is most likely explained by developmental bias toward dividing cells by default unless inhibited by activation of a corresponding gene. Thus, given that evolution typically chooses simple solutions, a developmental controller is likely to lack self-termination of growth as long as the development is terminated by an external clock.

4.2 From Land to Water: Terrestrial Larvae and Aquatic Adults

Looking at the metamorphosis of amphibians, we can observe how the aquatic larvae (tadpoles) tend to display simpler morphologies, while terrestrial adults tend to have more complex shape, having grown appendages that allow them to move on the ground. Even ignoring their evolutionary history, this could be, in principle, explained simply by the fact that appendages are useful for moving in terrestrial habitats, while aquatic environments demand morphologies that have low drag and hence must be more streamlined. Furthermore, it is also the very nature of the developmental process that it constructs bodies by progressive complexification of an initially spherical embryo. Thus, the above-discussed scenario, in which soft robots have to move in an aquatic environment as larvae and a terrestrial one as adults, may be considered particularly well suited for evolving metamorphic robots: It gives artificial metamorphosis the opportunity to exploit morphological complexification that is inherent both in multicellular development and in the switch from aquatic to terrestrial environment.

To find out to what extent the robotic designs obtained in the above experiments are a result of evolution designing efficient morphologies and to what extent they are a fortunate result of complexification inherent in growth, we performed experiments with environments for the developmental stages inverted: Larvae had to move on land and adult forms had to move in water. To provide a fair comparison, all other parameters of the experiments remained unchanged.

Again, evolutionary runs were successful in producing individuals that undergo clearly adaptive metamorphosis, as seen in the selection of best individuals in Figure 7. Two examples of high-fitness individuals and their motion strategies are shown in more detail in Figure 6. What can be immediately observed is that evolution has again come up with legged locomotion for the terrestrial environment and fishlike morphologies for the aquatic one. Furthermore, the motion of the terrestrial larvae is largely similar to the gaits of the terrestrial adults in earlier experiments, with the exception that the legged individuals consist of smaller numbers of cells. Similarly, the adult forms propel themselves with undulating movements, just as in the previous experiments. In the case of the adult forms in Figure 6a they are the result of a wave of contractions of the body that moves through the body in the direction perpendicular to the direction of movement (just like the larva in Figure 3a). The adult form in Figure 6b, on the other hand, induces an undulating motion pattern by synchronously contracting all cells on its right side while expanding the cells on the left side and vice versa, with a division line separating the left and right sides of the body visible in the middle frame (t = 1854). The aquatic forms are largely symmetric along the direction of their motion, although hardly perfect. Since our developmental system does not explicitly facilitate symmetrical growth (at least as it was configured for the presented experiments), symmetric shapes emerge only as evolution's attempt to produce morphologies that swim consistently in one direction.

Figure 7. 

Animat zoo 2: further examples of metamorphic individuals evolved to first move in the terrestrial environment and then in the aquatic. Examples come from all of the three types of evolutionary runs employed. Body parts grown during metamorphosis are colored red. Arrows indicate direction of movement.

Figure 7. 

Animat zoo 2: further examples of metamorphic individuals evolved to first move in the terrestrial environment and then in the aquatic. Examples come from all of the three types of evolutionary runs employed. Body parts grown during metamorphosis are colored red. Arrows indicate direction of movement.

The fact that, in both discussed scenarios, artificial evolution converged to similar morphological solutions, regardless of the order of developmental stages, suggests high evolutionary plasticity of the employed approach. At the same time, the remaining differences, especially in the size of each stage, show how developmental constraints alter the evolutionary trajectory: The walkers are now smaller and grow into swimmers. Careful investigation of how the development proceeds (Figure 8) and comparison with that of the water-to-land scenario (Figure 4) allow us to observe what happens in detail: For both individuals, we can see how apoptosis removes some of the cells that formed appendages while at the same time growing cells between them to reduce drag and elongating the front and back of the creature.

Figure 8. 

Developmental processes of the two individuals shown in Figure 6. The dashed line separates the larval stage from the beginning of metamorphosis; labels indicate the development time. Upper rows show the multicellular representation that is used during development; lower rows show a preview of the soft-bodied representation. Note that protrusions with a thickness of one cell are removed, and the soft-bodied representation is used only for the evaluation of morphology, during the locomotion stages. Blue: cells that were created during the larval stage; red: cells created during metamorphosis. Videos can be accessed at https://goo.gl/nbg  JYp and https://goo.gl/HfjmJ3.

Figure 8. 

Developmental processes of the two individuals shown in Figure 6. The dashed line separates the larval stage from the beginning of metamorphosis; labels indicate the development time. Upper rows show the multicellular representation that is used during development; lower rows show a preview of the soft-bodied representation. Note that protrusions with a thickness of one cell are removed, and the soft-bodied representation is used only for the evaluation of morphology, during the locomotion stages. Blue: cells that were created during the larval stage; red: cells created during metamorphosis. Videos can be accessed at https://goo.gl/nbg  JYp and https://goo.gl/HfjmJ3.

4.3 Evolved Controllers

The movement of soft-bodied robots is a result of oscillating contractions, with each cell having an independent period and phase shift determined at the end of the developmental process (Equation 3). This simple method has been demonstrated to work well for a wide range of morphologies in experiments based on a similar approach in our earlier work [17]. The typical result of evolutionary runs, also compatible with our previous observations, is that the period of oscillations evolves to be uniform among the body and equal to the minimum allowed length. This is most likely explained by the fact that, given the lack of energetic costs of actuation, higher frequency of contractions delivers more energy to the system, which can then be converted into kinetic energy of an animat and used to overcome drag and friction. Hence, selection highly favors individuals that have the shortest allowed period of oscillation. Effective motion patterns are then produced by evolving a varied phase shift of contractions along the body.

Qualitative analysis of how controllers change during metamorphosis of the four individuals investigated earlier reveals that the oscillation patterns of cells are largely retained during metamorphosis (Figure 9), although small adjustments to it were common and larger changes were also observed. As an example of an evident change, we marked the same cell (id 67) in the larval and adult stages of the top individual in Figure 9b. This shows that evolved metamorphosis does not rely on morphological changes only, but fine-tunes the control mechanism as well.

Figure 9. 

Controller comparison for the larval and adult stages of individuals shown in Figures 3 and 6. Colors represent evolved phase shifts of cellular contractions and expansions.

Figure 9. 

Controller comparison for the larval and adult stages of individuals shown in Figures 3 and 6. Colors represent evolved phase shifts of cellular contractions and expansions.

4.4 Comparison of Evolutionary Search Methods

The best individuals obtained in the evolutionary experiments using the three different versions of the search algorithm (Section 3) were qualitatively similar. Yet, quantitative analysis of each algorithm's performance reveals that the success rate of producing high-quality individuals is very different for each of them.

Compared to the basal, simple fitness function, the progressive fitness approach was observed to produce on average higher quality of individuals (Figure 10), in both the cases of evolving aquatic and terrestrial larvae first. A likely explanation is that the progressive fitness function relaxes the requirement of both developmental phases having to produce moving individuals from the very beginning. At the same time, metamorphosis is easier to evolve if modifications that occur in the larvae have a limited scope and reuse its underlying structure to create the adult. The progressive approach supports metamorphosis by first solving an easier problem of finding a working larval morphology and then starting to evolve metamorphosis while already having discovered a promising region of search space: In this regard it can be considered a new type of scaffolding technique [1]. We expect that this method would perform especially well on problems in which bootstrapping evolutionary search would be difficult.

Figure 10. 

Comparison of the three different evolutionary search approaches to evolving metamorphic soft robots: scalarized fitness. Lines represent the average fitness of the best individual in a population from 20 evolutionary runs; the shaded area represents 95% bootstrapped confidence intervals. Fitness in the plot was calculated using the default fitness function (Equation 6). Note that it was the same as the fitness function used by evolutionary search only in the simple fitness method and after generation 1333 in the progressive fitness (Equation 7) method (solid blue line). Novelty search did not rely on objective fitnesses at all.

Figure 10. 

Comparison of the three different evolutionary search approaches to evolving metamorphic soft robots: scalarized fitness. Lines represent the average fitness of the best individual in a population from 20 evolutionary runs; the shaded area represents 95% bootstrapped confidence intervals. Fitness in the plot was calculated using the default fitness function (Equation 6). Note that it was the same as the fitness function used by evolutionary search only in the simple fitness method and after generation 1333 in the progressive fitness (Equation 7) method (solid blue line). Novelty search did not rely on objective fitnesses at all.

The most surprising finding was, however, that the novelty search algorithm outperformed our two objective fitness-based approaches by a large margin in both types of experiments (Figure 10). Novelty search succeeded even despite the fact that (by definition) it does not explicitly optimize the distances of each developmental stage and, instead, continuously searches for novel solutions. Furthermore, our application of the novelty search algorithm should be considered very basic, as we did not explore different phenotypic descriptors (e.g., based on overall motion patterns rather than the final position of the center of mass only) or search for optimal parameters. We also did not attempt to normalize final positions of individuals, and thus the novelty search may underperform by overexploiting one developmental stage. It has been suggested that novelty search excels in domains with highly deceptive fitness landscapes [20]. Our earlier experiments employing the same developmental model have also shown the advantage of using novelty search [16] on the gait evolution task. This suggests high deceptiveness of the artificial development problem domain and overall usefulness of novelty search for highly complex genotype-to-phenotype mappings.

Finally, Figures 11 and 12 present how the performance of developmental stages improves over evolutionary time when using each of the search algorithms. The larval stage evolved using the progressive fitness approach considerably outperforms the other two methods over the first third of the evolutionary run, that is, while this is the only developmental stage that is under selection. This is only, however, because the cost of balancing performance between the two developmental stages is not incurred yet. Nonetheless, adding a second developmental stage on top of an already evolved larva, even though it decreases its performance, does so only moderately (around 10–15% fitness drop: Figures 11a and 12a, the blue line after 667 generations). This approach, on the other hand, can be observed to bias the tradeoff in favor of the larval stage: The distance achieved by adults in both types of experiments was lowest for the progressive fitness approach (Figures 11b and 12b). The more detailed analysis also reveals that the novelty search approach did not universally outperform other approaches at the level of each developmental stage. The simple fitness approach produced on average better terrestrial adults, and so did the progressive fitness method for terrestrial larvae. The record holders for the best obtained larvae and adults in each type of experiment belong, however, to the novelty search.

Figure 11. 

Distance achieved by larvae and adults for best individuals in each type of experiment shown in Figure 10a. The very low performance of the adult stage in progressive fitness experiments before generation 667 stems from the lack of selection for their quality (see Equation 7).

Figure 11. 

Distance achieved by larvae and adults for best individuals in each type of experiment shown in Figure 10a. The very low performance of the adult stage in progressive fitness experiments before generation 667 stems from the lack of selection for their quality (see Equation 7).

Figure 12. 

Distance achieved by larvae and adults for best individuals in each type of experiment shown in Figure 10b. The very low performance of the adult stage in progressive fitness experiments before generation 667 stems from the lack of selection for their quality (see Equation 7).

Figure 12. 

Distance achieved by larvae and adults for best individuals in each type of experiment shown in Figure 10b. The very low performance of the adult stage in progressive fitness experiments before generation 667 stems from the lack of selection for their quality (see Equation 7).

4.5 Analysis of Evolutionary Runs

To understand how metamorphic individuals evolve over time, we have analyzed the most successful set of evolutionary runs, namely, the ones using the novelty search algorithm. Figure 13 directly compares the average fitnesses over time for two classes of experiments, namely, those with aquatic larvae and terrestrial adults and those with terrestrial larvae and aquatic adults. Other than a potentially small advantage of the first type of experiments (also observed for other search algorithms), the difference in average result is small and not statistically significant for the data shown. By comparing the performance in each developmental stage separately (Figure 14), we can see how the higher average fitness of the water-to-land individuals comes from the longer average distances achieved by their larvae than those achieved by aquatic adults in the land-to-water scenario, while the terrestrial stages in both types of experiments achieved almost the same average distances. Importantly, the variation between evolutionary runs is very large, indicating a very high effect of evolutionary contingency on the quality of a final solution. Furthermore, it was universally observed for adult forms to grow larger than larval forms (Figures 15 and 16).

Figure 13. 

Fitness of the best individual in the population over evolutionary time for the two types of experiments in evolving metamorphic individuals using the novelty search algorithm. Lines show average fitnesses of 20 independent evolutionary runs each; color stripes represent 95% bootstrapped confidence intervals for the averages. The plot is based on the same data as used in Figure 10a and 10b.

Figure 13. 

Fitness of the best individual in the population over evolutionary time for the two types of experiments in evolving metamorphic individuals using the novelty search algorithm. Lines show average fitnesses of 20 independent evolutionary runs each; color stripes represent 95% bootstrapped confidence intervals for the averages. The plot is based on the same data as used in Figure 10a and 10b.

Figure 14. 

Distances achieved by each developmental stage of the best individual in the population for both types of experiments while using the novelty search algorithm. Lines show average distances from 20 independent evolutionary runs of each experiment type; color stripes represent 95% bootstrapped confidence intervals for the averages. A dashed line represents the larval stage performance; the solid line represents the adult stage.

Figure 14. 

Distances achieved by each developmental stage of the best individual in the population for both types of experiments while using the novelty search algorithm. Lines show average distances from 20 independent evolutionary runs of each experiment type; color stripes represent 95% bootstrapped confidence intervals for the averages. A dashed line represents the larval stage performance; the solid line represents the adult stage.

Figure 15. 

Number of cells in larvae over evolutionary time for the two types of experiments in evolving metamorphic individuals using the novelty search algorithm. Lines show averages of 20 independent evolutionary runs each; color stripes represent 95% bootstrapped confidence intervals for the averages.

Figure 15. 

Number of cells in larvae over evolutionary time for the two types of experiments in evolving metamorphic individuals using the novelty search algorithm. Lines show averages of 20 independent evolutionary runs each; color stripes represent 95% bootstrapped confidence intervals for the averages.

Figure 16. 

Number of cells in adults over evolutionary time for the two types of experiments in evolving metamorphic individuals using the novelty search algorithm. Lines show averages of 20 independent evolutionary runs each; color stripes represent 95% bootstrapped confidence intervals for the averages.

Figure 16. 

Number of cells in adults over evolutionary time for the two types of experiments in evolving metamorphic individuals using the novelty search algorithm. Lines show averages of 20 independent evolutionary runs each; color stripes represent 95% bootstrapped confidence intervals for the averages.

There are, however, large differences between the two classes of experiments. More precisely, Figure 15 shows how terrestrial larvae of the land-to-water scenario grow to be more than twice the size of aquatic larvae in the alternative scenario. The growth happens very quickly on the evolutionary timeline (within the first 100 generations), suggesting very high selective pressure to do so. The adults of this experiment also grew to be larger (Figure 15). Analysis of their process of metamorphosis further reveals that in the land-to-water scenario the changes that occur in the body are more pronounced. The number of cells that undergo apoptosis is 2–3 times higher than in the water-to-land scenario and averages to around 25% of the number of cells present in the larvae (Figure 17). When it comes to a number of cells added during metamorphosis (Figure 18), the first 200 generations in both types of experiments show a steady increase of the parameter, likely indicating a period in which the first functional metamorphic individuals emerge. Just as with apoptosis, larger growth during metamorphosis occurred in the land-to-water scenario.

Figure 17. 

Number of cells that undergo apoptosis during metamorphosis over evolutionary time for the two types of experiments in evolving metamorphic individuals while using the novelty search algorithm. Lines show averages of 20 independent evolutionary runs each; color stripes represent 95% bootstrapped confidence intervals for the averages.

Figure 17. 

Number of cells that undergo apoptosis during metamorphosis over evolutionary time for the two types of experiments in evolving metamorphic individuals while using the novelty search algorithm. Lines show averages of 20 independent evolutionary runs each; color stripes represent 95% bootstrapped confidence intervals for the averages.

Figure 18. 

Number of cells that are added during metamorphosis over evolutionary time for the two types of experiments in evolving metamorphic individuals while using the novelty search algorithm. Lines show averages of 20 independent evolutionary runs each; color stripes represent 95% bootstrapped confidence intervals for the averages.

Figure 18. 

Number of cells that are added during metamorphosis over evolutionary time for the two types of experiments in evolving metamorphic individuals while using the novelty search algorithm. Lines show averages of 20 independent evolutionary runs each; color stripes represent 95% bootstrapped confidence intervals for the averages.

Why, however, would terrestrial larvae grow larger than aquatic larvae? We think this can be explained by two effects. Firstly, the cells grow by default, unless the growth process is inhibited. This has the side effect of biasing the second-stage morphologies towards being bigger: From the control point of view, it is easier to grow cells than prevent growth, as the latter requires an explicit control signal to do so. In the case of aquatic larvae it means that evolution produces highly elongated, very slim shapes that later increase in size and produce appendages. The larvae, however, remain relatively small, often displaying simple, snakelike morphology. This is not, however, possible in the case of terrestrial larvae, which will move best only if they have appendages, and growing these requires additional cells. Therefore, a second effect at play is that a minimal functional morphology for terrestrial locomotion is of larger size and complexity than its aquatic counterpart. During the transformation, simpler aquatic larvae need only to grow appendages, whereas terrestrial larvae have to somehow suppress the appendages to produce a streamlined shape. Suppression is achieved most easily by growing additional cells that will fill the space between appendages, though as mentioned previously, partial apoptosis of appendages would also occur (Figure 7). Finally, to see what kinds of morphological changes the evolving animats undergo during their evolution, we have selected the best individuals from each generation from the evolutionary run that produced the individual in Figure 3a. Although the selected historical individuals shown in subsequent frames (Figure 19) do not represent direct descendants, it is clear that throughout the later part of the evolutionary history subsequent best individuals are closely related to each other. It can also be observed how during the first 100 generations of the evolutionary run, morphological changes occurred very frequently. At generation 125, the overall morphology of the best individual already resembles the best final individual found in generation 1386 (although its fitness is only 5.07, compared to 11.15 for the latter). The following generations bring multiple small changes to the morphology of the adult form and almost no changes to the larval form. Both stages, however, undergo continuous modifications of their controllers, and it is these alterations that contribute the most to the improvements in fitness. This pattern was also observed in other evolutionary runs: The final morphology would emerge in the first few hundred generations, and the remainder of the run would be spent on small tweaks to the bodies and optimization of controllers.

Figure 19. 

Evolution of a metamorphic individual: morphologies of larval (top) and adult stages (below) of the best individual in a given generation for the evolutionary run that produced the individual shown in Figure 3a. Colors correspond to the evolved phase shift of muscular contractions.

Figure 19. 

Evolution of a metamorphic individual: morphologies of larval (top) and adult stages (below) of the best individual in a given generation for the evolutionary run that produced the individual shown in Figure 3a. Colors correspond to the evolved phase shift of muscular contractions.

4.6 Comparison with Robust Robots

An alternative to creating a self-reconfiguring robotic design that can function in two different environments would be either to create a single, robust solution that can function in both environments or to create two different designs, each specialized to its target environment. Each of these approaches has its design compromises and advantages. In this and the following subsection we attempt to find how the performance of evolved metamorphic individuals compares in the two alternatives: a single, environmentally robust (here, amphibious) animat, and a pair of specialized individuals evolved separately. A robust robot has its morphology and controller designed in a way that allows it to function in any type of environment. If the environments are as different as an aquatic and a terrestrial one, this imposes considerable tradeoffs on its design, and it is likely to perform much worse in each environment than two specialized robots would. On the other hand, an evolved metamorphic robot also faces tradeoffs coming from the fact that a single genome encodes two phenotypes, as well as the process of converting one into another. This makes evolutionary design of a metamorphic robot much more challenging. Naturally, a physically implemented shape-changing robot would be immensely useful owing to this unique property. What we thus focus on and compare here is solely the evolvability of the metamorphic approach in the form presented in this article, understood as the quality of designs obtained at the end of evolutionary runs. Confirming that the specialization observed in the experiments indeed outperforms evolved, robust designs sets the baseline of the usefulness of metamorphosis as a technique of automated design.

We have evolved robotic designs using the same setting and algorithm as the metamorphic ones, but with the goal of producing amphibious animats that can move both in aquatic and in terrestrial environments. In this scenario, there was no metamorphosis, but adult forms were evaluated in both environments. Hence, evolution had to strike a compromise between robots' performance in each of the environments.

A direct comparison poses challenges, however. In our experiments with metamorphosis, we assumed that the growth of the larval stage takes 600 time steps, whereas the adult stage grows over 1200 time steps. As we have shown in the previous sections, allowing for a longer growth time leads to the emergence of larger individuals. At the same time, larger individuals were observed to achieve larger distances. Therefore, attempting to evolve robust individuals that grow for 1200 time steps will lead to the emergence of larger swimming individuals than if the growth was allowed for only 600 time steps, giving them an unfair advantage over the smaller larvae of metamorphic individuals. One way of ensuring a fairer comparison would thus be to employ some method of scaling the achieved distance with body size. As this has challenges on its own (the choice of fair scaling function), in this experiment we instead decided to compare metamorphic individuals with two types of robust individuals: animats that were allowed to grow for 600 time steps and animats that were grown for 1200 time steps. We expected that robust individuals would thus have a size advantage in the latter case. Having already observed the superior performance of the novelty search algorithm, we evolved amphibious designs using novelty search as well.

Figure 20a shows the aggregate results of repeated evolutionary runs. The metamorphic individuals clearly outperform robust individuals that were allowed to grow for 600 time steps. The parsimonious explanation here is however that the longer development time of adults biased evolution towards producing larger individuals, hence the larger distances achieved. The benefit of increased development time becomes obvious when we look at the results achieved by the robust robots that were allowed to grow for 1200 time steps: They clearly outperform those grown for a shorter time. Although the average fitness of the robust individuals is not significantly different from that of metamorphic ones in the presented experiment, the variance of the data in the former case is much lower, revealing that robust solutions are all of mediocre quality. On the other hand, despite novelty search discovering some poor-quality metamorphic individuals, in both types of metamorphic experiments the best individuals found were much better than the best ones from the robust experiments.

Figure 20. 

(a) Comparison of the performance of metamorphic individuals with amphibious designs. (b) Comparison of the distances achieved by each developmental stage for the most favorable metamorphic scenario (aquatic larva and terrestrial adult) and the most favorable amphibious scenario (development time of 1200 steps).

Figure 20. 

(a) Comparison of the performance of metamorphic individuals with amphibious designs. (b) Comparison of the distances achieved by each developmental stage for the most favorable metamorphic scenario (aquatic larva and terrestrial adult) and the most favorable amphibious scenario (development time of 1200 steps).

Figure 20b further compares the actual distances achieved in each type of environment in the experiments that produced the best metamorphic individuals (viz., the ones with aquatic larvae and terrestrial adults) and the best robust ones (viz., the ones grown for 1200 time steps). The two types of experiments lead to comparable distances achieved in the terrestrial environment, and thus the overall higher fitnesses of metamorphic individuals come from evolution being able to specialize the aquatic stage and achieve longer distances when swimming.

The above results show that evolutionary search indeed finds metamorphic solutions that have better performance than their robust alternatives. What may be surprising, however, is that robust individuals grown for 1200 steps were able to reach performance almost at the level of metamorphic individuals. We think it is largely a result of the configuration of our simulated system, in which it so happens that a morphology and controller that can move in the simulated fluid can also exert enough forces to move the same body on the ground. Changing physical constants related to body strength would likely tip the balance much more in favor of metamorphic individuals.

4.7 Comparison with Specialized Robots

Each developmental stage of a metamorphic individual shares the same genetic machinery. Thus, it should be expected that it introduces certain tradeoffs and limitations on the designs that could be achieved using this approach—after all, the morphology of a larva not only has to be adapted to its specific environment, but also needs to be malleable to the environment it has to move in during the adult stage. The same applies to the genetic program that creates it: A single network has to be able not only to control development in a way that leads to the emergence of the first morphology, but also to control its metamorphosis. Specialized robots, each optimized independently for its target environment, are free from these additional costs and thus should perform better in their respective environments, much as a sports car will outperform an all-terrain vehicle on a highway. Indeed, attempting to evolve individuals specialized for either aquatic or terrestrial environments would result in individuals that can move faster than corresponding life stages of metamorphic individuals. We wondered, however, if scenarios exist in which the very necessity to come up with a malleable design could make it easier to evolve a single metamorphic robot rather than to evolve two specialized ones independently. We hypothesized that one scenario in which this could happen would be if evolving a design specialized for a particular problem de novo were difficult, but at the same time, evolving a solution for a related problem could be much easier. In such a case, the need to perform metamorphosis would guide an evolving population towards a region of search space that would otherwise be difficult to discover.

To see if this could be the case, we investigated a scenario in which individuals were evolved for two types of terrestrial gaits: Larvae had to move on a flat, horizontal surface, whereas adults had to climb a sloped (17 degrees) surface and avoid sliding or rolling down. As both types of environments are terrestrial, it can be expected that morphologies for both would share the overall plan and hence metamorphosis would need only relatively small adjustments to the body structure. At the same time, an attempt to evolve specialized individuals to climb a slope is likely to experience the bootstrapping problem, as finding initial solutions that can move uphill rather than roll or slide down the slope may be difficult. To shorten the computation time, we reduced the maximum allowed number of cells in the individuals to 128, which also allowed us to shorten the total development time, from 1200 to 800 time steps (400 for the larval stage). Anticipating the bootstrapping problem, we have used the progressive fitness approach (Section 3.2). Specialized individuals were evolved using normal, NEAT search (the fitness was a single objective, viz., the distance achieved by a specialized robot). In each type of experiment, the fitness function rewarded only moving right.

We performed three types of evolutionary runs, each repeated 20 times using different random seed values. The first involved evolving a metamorphic individual, and the remaining two were individuals specialized for each of the environments. To make the comparison fair and avoid our system's bias towards producing smaller larvae and larger adults in the metamorphic scenario, we allowed specialized individuals for the horizontal surface to develop for exactly the same number of time steps as larvae of metamorphic individuals, namely, for 400 time steps.

The number of generations was the same in each type of experiment. Therefore, the amount of computational time spent on obtaining a single metamorphic solution was not larger than that spent for two separate individuals. In fact, it was smaller, as growing two specialized individuals would mean 400 and 800 separate developmental steps, but only 800 in the case of a single metamorphic individual.

The amount of observed change during metamorphosis in evolved animats was smaller than in the case of aquatic and terrestrial individuals, though that should not be surprising: The two target environments are closely related. We present two examples of metamorphic individuals obtained with this method in Figure 21. Both individuals show only a minor morphological change. The individual in Figure 21a increases the surface of contact of its tail, while the one in Figure 21b shifts its center of mass towards the front with additional cells and an appendage. The best individuals that we observed would, however, only change their controllers, by adjusting the oscillation frequency of their cells.

Figure 21. 

Two examples of metamorphic individuals that move on a horizontal surface in their larval stage and climb a slope in their adult form.

Figure 21. 

Two examples of metamorphic individuals that move on a horizontal surface in their larval stage and climb a slope in their adult form.

Figure 22 compares the average performance of individuals obtained using each evolutionary search method, showing the performance of each stage of metamorphic individuals separately. As can be seen in Figure 22a, after evolutionary pressure for slope climbing starts to increase after generation 333, the performance of the larval stage of metamorphic individuals progressively decreases, being traded for the performance of the adult form. After 1000 generations, the individuals specialized in moving on the horizontal surface achieve much longer distances than the corresponding larvae of metamorphic solutions. The situation is reversed, however, if we look at the performance of metamorphic adults and animats specialized in climbing the slope. Here, for the first 333 generations, the fitness of the metamorphic adults remains at 0, which means that before the selection pressure was applied, all of these adults failed to move up the slope. Their improvement over evolutionary time proceeds rapidly, and after 1000 generations they outperform specialized solutions, though at the cost of lower performance on a flat surface. Additionally, if we considered only the slope-climbing phenotypes and a matching number of evaluations in the virtual environment for the two methods (i.e., metamorphic individuals evolved for 1000 generations and specialized ones evolved for 2000 generations), the average performance of the specialized ones was still lower (mean 271 versus 340, although the difference between the final fitnesses was not statistically significant, with large standard deviations of 204 and 319, respectively).

Figure 22. 

Performance of metamorphic individuals and specialized ones over evolutionary time: (a) larvae moving on a horizontal surface, and specialized individuals evolved for the same environment; (b) adults climbing uphill, and specialized individuals evolved for climbing. Lines show averages of 20 independent evolutionary runs each; color stripes represent 95% bootstrapped confidence intervals for the averages.

Figure 22. 

Performance of metamorphic individuals and specialized ones over evolutionary time: (a) larvae moving on a horizontal surface, and specialized individuals evolved for the same environment; (b) adults climbing uphill, and specialized individuals evolved for climbing. Lines show averages of 20 independent evolutionary runs each; color stripes represent 95% bootstrapped confidence intervals for the averages.

Ultimately, we think that the much faster improvement over evolutionary time of metamorphic individuals can be best explained by the fact that the adult form does not have to be discovered de novo, but instead emerges as a modification on top of the already functional larva. Naturally, other methods exist to address the bootstrap problem that the specialized individuals faced. For example, it is likely that increasing the angle of the slope progressively (or in stages) would help evolutionary search. Furthermore, based on our preliminary runs, evolving metamorphic individuals is advantageous only in a certain range of slope angles. This is to be expected: A very low angle will pose no challenge when attempting to evolve a specialized hill climber, whereas for high angles, both approaches will fail to find working solutions. Nonetheless, we see this experiment as a confirmation that in certain scenarios, evolving metamorphic robots could allow us to discover useful solutions faster than simply trying to evolve a set of specialized ones. In such cases, the necessity to evolve a metamorphic design works as a scaffolding technique, helping the evolutionary search discover stepping stones necessary to produce the desired solution.

In this work, we have investigated how the concept of metamorphosis can enhance artificial developmental processes used to evolve designs of robotic morphologies and controllers. Metamorphosis allowed us to evolve solutions in which a single genome encodes two related phenotypes, each crafted for a different problem and/or environment, together with a method of self-restructuring one phenotype into the other. We found that, given a biologically inspired model of development (one that progressively constructs a body), this required simply employing a fitness function that rewards each developmental stage for a different objective. We observed how evolution, searching for morphologies de novo, without any prior assumptions about what they should be, was able to discover designs that are well adapted to different tasks at each stage of development. Observed rearrangements were highly reminiscent of amphibian metamorphosis in that we commonly observed the growth of appendages for walking in terrestrial environments and tails for swimming, as well as their removal after they were no longer needed. Reversing the order of environments in which growing individuals were evaluated allowed us to confirm that observed morphological changes were clearly a result of evolution shaping growing morphologies in a convergent way: legged locomotion for a terrestrial environment and undulatory swimming for an aquatic one, regardless of the order of evaluation. A further comparison with the evolution of a single amphibious solution allowed us to find that metamorphic individuals indeed produced useful specialization, outperforming environmentally robust solutions.

Finally, we were able to show that in scenarios where the two tasks in which each developmental stage is evaluated are closely related, evolving a metamorphic, dual design can actually be faster than evolving two specialized solutions independently. This means that even if we do not consider the enormous advantages metamorphosis would bring to robots by allowing them to switch between different environments, simply using metamorphosis as a method of evolving two designs at the same time and disregarding the element of self-transformation can still be computationally advantageous in some situations.

The results presented in this work should be considered far from being a limit of what could be achieved by exploiting the concept of metamorphosis. One of our design goals for the developmental system was to reduce constraints put on the evolutionary process, and we did so in order to investigate what level of complexity of designs can be produced through evolutionary search alone and how different they can be from what a human engineer might come up with. In our line of research we are interested in finding ways of harnessing the creative potential of the evolutionary process, with human bias removed, and we see in mimicking the way shape and form evolve in nature a promising method to achieve that. We can however easily imagine methods that improve the quality of obtainable designs by including more knowledge about problem domain in the search algorithm and constraining the search space. For example, the use of multi-objective optimization is likely to produce better results than the approach used in this article. Also, the way the development is organized currently is such that mutations are likely to interfere with both developmental stages. However, biological metamorphosis is well studied, and it is known that often a single hormonal trigger induces metamorphosis. It does so by activating whole ensembles of physically closely located genes associated with the new developmental stage. This was observed as early as in the 1950s in developing Drosophila larvae and is visible owing to what is known as chromosomal puffing: an enlargement of chromosomal regions that are undergoing transcription [26]. Hence, we expect that using a mechanism that either fully or partially separates developmental stages on a genetic level (e.g., by having genes explicitly assigned to each stage) is likely to enhance the evolvability of metamorphosis. We plan to investigate this in the future.

Finally, we believe that, given its relatively simple and generic design, the important result of this article is that it shows how metamorphosis and fine-grained self-assembly, such as multicellular development, are a natural combination, where the former enhances the latter. It is also easy to see why: Metamorphosis and multicellular development are two sides of the same coin. We thus hope that metamorphosis can find its use whenever artificial development is used as a method of automated design.

This work was supported by the Japan Society for the Promotion of Science through the JSPS Fellowship for Foreign Researchers and JSPS KAKENHI grant 26-04349. High-performance computing resources were provided by the Interdisciplinary Center for Molecular and Mathematical Modeling (University of Warsaw) and the Tri-City Academic Computer Center (TASK). The CGAL library was used.

1 

Imago is the general name of the adult stage of insect development.

2 

Note, however, that Ernst Haeckel's theory of recapitulation to which we refer here has only historical significance.

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Author notes

*

Contact author.

**

Department of Complex Systems Science, Graduate School of Information Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan. E-mail: mjoach@alife.cs.is.nagoya-u.ac.jp (M.J.); reiji@nagoya-u.jp (R.S.); arita@nagoya-u.jp (T.A.)